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+[    {        "title": "0705.0406v1.Planar_spin_transfer_device_with_a_dynamic_polarizer.pdf",        "content": "arXiv:0705.0406v1  [cond-mat.mtrl-sci]  3 May 2007Planar spin-transfer device with a dynamic polarizer.\nYa. B. Bazaliy,1D. Olaosebikan,2and B. A Jones1\n1IBM Almaden Research Center, 650 Harry Road, San Jose, CA 951 20\n2Department of Physics, Cornell University, Ithaca, NY 1485 3\n(Dated: July, 2006)\nIn planar nano-magnetic devices magnetization direction i s kept close to a given plane by the large\neasy-plane magnetic anisotropy, for example by the shape an isotropy in a thin film. In this case\nmagnetization shows effectively in-plane dynamics with onl y one angle required for its description.\nMoreover, the motion can become overdamped even for small va lues of Gilbert damping. We\nderive the equations of effective in-plane dynamics in the pr esence of spin-transfer torques. The\nsimplifications achieved in the overdamped regime allow to s tudy systems with several dynamic\nmagnetic pieces (“free layers”). A transition from a spin-t ransfer device with a static polarizer to\na device with two equivalent magnets is observed. When the si ze difference between the magnets\nis less than critical, the device does not exhibit switching , but goes directly into the “windmill”\nprecession state.\nPACS numbers: 72.25.Pn, 72.25.Mk, 85.75.-d\nI. INTRODUCTION\nThe prediction1,2and first experimental\nobservations3,4,5,6,7,8of spin-transfer torques opened a\nnew field in magnetism which studies non-equilibrium\nmagnetic interactions induced by electric current. Since\nsuch interactions are relatively significant only in very\nsmall structures, the topic is a part of nano-magnetism.\nThe current-induced switching of magnetic devices\nachieved through spin-transfer torques is a candidate\nfor being used as a writing process in magnetic random\naccess memory (MRAM) devices. The MRAM memory\ncell is a typical example of a spintronic device in which\nthe electron spin is used to achieve useful logic, memory\nor other operations normally performed by electronic\ncircuits.\nTo produce the spin-transfer torques, electric currents\nhave to flow through the spatially non-uniform mag-\nnetic configurations in which the variation of magneti-\nzation can be either continuous or abrupt. The first case\nis usually experimentally realized in magnetic domain\nwalls.3,9,10,11Here we will be focusing on the second case\nrealized in the artificially grown nano-structures. Such\nspin-transfer devices contain severalmagnetic pieces sep-\narated by non-magnetic metal spacers allowing for arbi-\ntraryanglesbetweenthemagneticmomentsofthepieces.\nMagnetizationmyvarywithineachpieceaswell, butthat\nvariation is usually much smaller and vanishes as the size\nof piece is reduced, or for larger values of spin-stiffness\nof magnetic material. The typical examples of a system\nwith discrete variation of magnetization are the “nano-\npillar” devices8(Fig.1A). Their behavior can be reason-\nablywellapproximatedbyassumingthatmagneticpieces\nare mono-domain, each described by a single magnetiza-\ntion vector /vectorM(t) =Ms/vector n(t) where /vector nis the unit vector\nandMsis the saturation magnetization. The evolution\nof/vector n(t) is governedby the Landau-Lifshitz-Gilbert (LLG)\nequation with spin-transfer terms.2,12\nIt is often the case that magnetic pieces in a spin-transfer device have a strong easy-plane anisotropy. For\nexample, in nano-pillars both the polarizer and the free\nmagnetic layer are disks with the diameter much larger\nthan the thickness. Consequently, the shape anisotropy\nmakes the plane of the disk an easy magnetic plane. In\nthe planar devices13built from thin film layers (Fig. 1B)\nthe shape anisotropy produces the same effect. When\nthe easy-plane anisotropy energy is much larger then all\notherenergies,thedeviationsof /vector n(t)fromthein-planedi-\nrection are very small. An approximation based on such\nsmallness is possible and providesan effective description\nof the magnetic dynamics in terms of the direction of the\nprojection of /vector n(t) on the easy plane, i.e. in terms of one\nazimuthalangle. Inthispaperwederivetheequationsfor\neffective in-plane motion in the presence of spin-transfer\neffect and discuss their use by considering several exam-\nples.\nIn the absence of spin-transfer effects the large easy-\nplane anisotropy creates a regime of overdamped mo-\ntion even for the small values of Gilbert damping con-\nstantα≪1.14In that regime the equations simplify\nfurther. Here the overdamped regime is discussed in\nthe presence of electric current. The reduction of the\nnumber of equations allows for a simple consideration\nof a spin-transfer device with two dynamic magnetic\npieces. We show how an asymmetry in the sizes of these\npieces createsa transition between the polarizer-analyzer\n(“fixed layer - free layer”) operation regime2,8,12,15and\nthe regime of nearly identical pieces where current leads\nns\n\rjj j\nsnA B\nFIG. 1: Planar spin-transfer devices2\nnot to switching, but directly to the Slonczewski “wind-\nmill” dynamic state.2Finally, we point out the limita-\ntions of the overdamped approximation in the presence\nof the spin-transfer torques.\nII. DYNAMIC EQUATIONS IN THE LIMIT OF\nA LARGE EASY-PLANE ANISOTROPY\nMagnetizationdynamicsin thepresenceofelectriccur-\nrent is governed by the LLG equation with the spin-\ntransfer term.2,12For each of the magnets in the device\nshown on Fig. 1A\n˙/vector n=γ\nMs/bracketleftbigg\n−δE\nδ/vector n×/vector n/bracketrightbigg\n+u[/vector n×[/vector s×/vector n]]+α[/vector n×˙/vector n] (1)\nwhere/vector s(t) is the unit vector along the instantaneous\nmagnetization of the other magnet and the spin-transfer\nmagnitude\nu=g(P)γ(¯h/2)\nVMsI\ne(2)\nis proportional to the electric current I. Hereeis the\n(negative) electroncharge, so uis positivewhen electrons\nflow into the magnet. Due to the inverse proportionality\nto the volume V, the larger magnets become less sensi-\ntive to the current and can serve as spin-polarizers with\na fixed magnetization direction. As for the other pa-\nrameters, γis the gyromagnetic ratio, g(P,(/vector n·/vector s)) is the\nSlonczewski spin polarization factor2which depends on\nmanysystemparameters,16,17andαis the Gilbert damp-\ning which also depends on /vector nand/vector swhen spin pumping18\nis taken into account. We will restrict our treatment to\nthe constant gandαto focus on the effects specific to\nthe strong easy plane anisotropy.\nIn terms of the polar angles ( θ,φ) the LLG equation\n(1) has the form\n˙θ+α˙φsinθ=−γ\nMsinθ∂E\n∂φ+u(/vector s·/vector eθ)\n˙φsinθ−α˙θ=γ\nM∂E\n∂θ+u(/vector s·/vector eφ) (3)\nwhere the tangent unit vectors /vector eθand/vector eφare defined in\nAppendix A.\nWe will consider a model for which the energy of a\nmagnet is given by\nE=K⊥cos2θ\n2+Er(φ) (4)\nwithK⊥being the easy-plane constant, Erbeing the\n“residual”in-plane anisotropy energy and z-axis directed\nperpendicular to the easy plane. The limit of a strong\neasy-planeanisotropyisachievedwhenthe maximalvari-\nation of the residual energy is small compared to the\neasy-plane energy, ∆ Er≪K⊥. In this case θ=π/2+δθ\nwithδθ≪1.To estimate δθ, consider the motion of magnetization\ninitially lying in-plane offthe minimum of Erand neglect\nfor the moment the spin-transfer terms in Eq.(3). Mag-\nnetization starts movingand a certain deviation from the\neasy plane is developed. For the estimate, assume that\nthe energy is conserved during this motion (the presence\nof damping will only decrease δθ). Then\n|δθ| ∼/radicalbigg\n∆Er\nK⊥≪1 (5)\nWecannowlinearizetherighthandsidesofequations(3)\nin smallδθ. On top of that, some terms on the left hand\nsides of (3) turn out to be small and can be discarded.\nIndeed, taking into account the smallness of αone gets\nthe estimates\n˙θ∼ −γ\nMs∂Er\n∂φ∼ −γ\nMs∆Er\n˙φ∼γ\nMsK⊥δθ∼γ\nMs/radicalbig\nK⊥∆Er\nConsequently ˙θ∼˙φ/radicalbig\n∆Er/K⊥≪˙φand˙φ≫α˙θ, there-\nfore the second term on the left hand side of the second\nequation of the system (3) can be discarded. No simpli-\nfication happens on the left hand side of the first equa-\ntion, where ˙θandα˙φcan be of the same order when\nα<∼/radicalbig\n∆Er/K⊥.\nPutting the spin-transfer terms back we get the form\nof equations in the limit of large easy-plane anisotropy:\n˙δθ+α˙φ=−γ\nMs∂E\n∂φ+u(/vector s·/vector eθ)\n˙φ=γK⊥\nMsδθ+u(/vector s·/vector eφ) (6)\nExpressions for the scalar products in (6) in terms of\npolar angles are given in Appendix A.\nThe second equation shows that δθcan be expressed\nthrough ( φ,˙φ). Small out-of-plane deviation becomes a\n“slave” of the in-plane motion.14We get\nMs\nγK⊥/parenleftbigg\n¨φ−ud(/vector s·/vector eφ)\ndt/parenrightbigg\n+αi˙φ=−γ\nMs∂Er\n∂φ+u(/vector s·/vector eθ) (7)\nThe term with the second time derivative ¨φdecreases\nwith increasing K⊥. As pointed out in Ref. 14, in the\nabsence of spin-transfer this term can be neglected when\nK⊥>∆Er/α2. Mathematically this corresponds to a\ntransition from an underdamped to an overdamped be-\nhavior of an oscillator as the oscillator mass decreases.\nWith spin-transfer terms the overdamped approxima-\ntion gives an equation\nα˙φ−ξd\ndt(/vector s·/vector eφ) =−γ\nMs∂Er\n∂φ+u(/vector s·/vector eθ) (8)\nwhereξ=uMs/(γK⊥). The range of this equation’s\nvalidity will be discussed in Sec. IV. The scalar products\nin Eq. (8) have to be expressed through the polar angles3\n(θs(t),φs(t)) ofvector /vector s, and linearizedwith respectto δθ\n(see Appendix, Eq. A4), which is then substituted from\nEq. (6). Finally, the equation is linearized with respect\nto small spin-transfer magnitude u. We get:\nα˙φ−ξ/parenleftbiggd\ndt/bracketleftbig\nsinθssin(φs−φ)/bracketrightbig\n−sinθscos(φs−φ)˙φ/parenrightbigg\n=−γ\nMs∂Er\n∂φ−ucosθs, (9)\ndescribing the in-plane overdamped motion of an ana-\nlyzer with a polarizer pointed in the arbitrary direction.\nNext, we show how some known results on spin-transfer\nsystems are recovered in the approximation (9).\nConsider the device shown on Fig. 1A and assume that\nthe first magnet is very large. As explained above, this\nmagnetisnotaffectedbythecurrentandservesasafixed\nsource of spin-polarized electrons for the second magnet\ncalled the analyzer, or the “free” layer. The magneti-\nzation dynamics of the analyzer is described by Eq. (3).\nThe case of static polarizer is extensively studied in the\nliterature.\nFirst, consider the case of collinear switching , exper-\nimentally realized in a nano-pillar device with the ana-\nlyzer’s and polarizer’s easy axes along the ˆ xdirection:\nEr= (1/2)K||sin2φ,/vector s= (1,0,0).7Using Eq. (9) with\nθs=π/2,φs= 0 we get\n(α+2ξcosφ)˙φ=−γK||\n2Mssin2φ (10)\nWithout the current, there are four possible equilibria\nof the analyzer. Two stable equilibria are the parallel\n(φ= 0) and anti-parallel ( φ=π) states. Two perpen-\ndicular equilibria ( φ=±π/2) are unstable. Lineariz-\ning Eq. (10) near equilibria one finds solutions the form\nδφ(t)∼exp(ωt) with eigenfrequencies\nω=−γK||\nMs(α+2ξ),(φ≈0)\nω=−γK||\nMs(α−2ξ),(φ≈π)\nω=γK||\nMsα,(φ≈ ±π/2)\nThe equilibria are stable for ω <0 and unstable other-\nwise. Thus the parallel state is stable for ξ >−α/2, the\nantiparallel state is stable for ξ < α/2, and the perpen-\ndicular states cannot be stabilized by the current. These\nconclusions agree with the results of Refs. 2,7,12. The\nstability regions are shown in Fig. 2A.\nNote how Eq. (10) emphasizes the fact that spin-\ntransfer torque destabilizes the equilibria by making the\neffective damping constant αeff=α+2ξcosφnegative,\nwhiletheequilibriumpointsremainaminimumoftheen-\nergyEr. Any appreciable influence of the current on the\nposition and nature (minimum or maximum) of the equi-\nlibrium can only be observed at the current magnitudes\n1/αtimes larger than the actual switching current.12ξ\nα/2−α/2(A) static polarizer\n−α/[2(1−ε)](B) dynamic polarizer\nα/2 −α/(2ε)\"windmill\"\nprecession\"windmill\"\nprecession ξ\nFIG. 2: Stability regions for systems with static (A) and\ndynamic (B) polarizers as a function of applied current,\nξ=g(P)(¯h/2VK⊥)I/e∝I.\nSecond, consider the case of magnetic fan .19Here the\neasy axis of the polarizer is again directed along ˆ x, but\nthe polarizer is perpendicular to the easy plane: /vector s=\n(0,0,1),θs= 0. This arrangement is known to produce\na constant precession of vector /vector n. Eq. (9) gets a form:\nα˙φ=−γK||\n2Mssin2φ−u (11)\nfor|u|< γK ||/(2Ms) the current deflects the analyzer\ndirection from the easy axis direction. For larger values\nofuthere is no time-independent solution. The angles\nφgrows with time which corresponds to /vector nmaking full\nrotations. At |u| ≫γK||/(2Ms) the rotation frequency\nof the magnetic fan is given by ω∼u/α.\nIII. DEVICE WITH TWO DYNAMIC\nMAGNETS (TWO “FREE LAYERS”)\nNo let us assume that both magnets in Fig. 1A have\nfinitesize. Eachmagnetservesasapolarizerfortheother\none. Without approximations, the evolution of two sets\nof polar angles ( θi,φi),i= 1,2 is described by two LLG\nsystems of equations\n˙θ(i)+αi˙φ(i)sinθ(i)=−γ\nMsisinθ(i)∂E(i)\n∂φ(i)+\n+uji(/vector n(j)·/vector e(i)\nθ) (12)\n˙φ(i)sinθ(i)−αi˙θ(i)=γ\nMsi∂E(i)\n∂θ(i)+uji(/vector n(j)·/vector e(i)\nφ)\nwherejmeanstheindexnotequalto iandnosummation\nis implied.\nWe now apply the overdamped, large easy-plane\nanisotropyapproximationtobothmagnets. Equation(9)4\nfor each magnet is further simplified since for the magnet\nithe angle θs=θj=π/2 +δθj,δθj≪1. Expanding\n(9) in small δθjand using the slave condition (6) for δθj\nwith (/vector s·/vector eφ) = (/vector n(j)·/vector e(i)\nφ) expanded in both small angles\n(see Eq. (A5)) we get the system:\n(αi+2ξjicos(φj−φi))˙φi− (13)\n−ξji(cos(φj−φi)+1)˙φj=−∂E(i)\n∂φi,\nwithξji=ujiMsi/(γK⊥). It was assumed that K⊥is\nthe same for both magnets.\nThe spin-transfer torque parameters u21andu12have\nopposite signs and their absolute values are different due\nto different volumes of the magnets, accordingto Eq. (2).\nWe assume V1≥V2and denote u12=u,u21=−ǫu.\nThe larger magnet experiences a relatively smaller spin\ntransfer effect, and the asymmetry parameter satisfies\n0≤ǫ≤1. In general, material parameters α1,2,Ms1,2\nand magnetic anisotropy energies E(1,2)of the two mag-\nnets are also different, but here we focus solely on the\nasymmetry in spin-transfer parameters. Both E(1)and\nE(2)are assumed to be given by formula (4) with the\nsame direction of in-plane easy axis. The situation can\nbe viewed as a collinear switching setup with dynamic\npolarizer. Equations (13) specialize to\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleα−2ǫξC ǫξ(C+1)\n−ξ(C+1)α+2ξC/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg˙φ1\n˙φ2/bracketrightbigg\n=−ω0\n2/bracketleftbigg\nsin2φ1\nsin2φ2/bracketrightbigg\nC= cos(φ1−φ2), ω0=γK||\nMs(14)\nNext, we study the stability of all equilibrium configu-\nrations ( φ1,φ2) of two magnets. There are four equilib-\nrium states that are stable without the current: two par-\nallel states along the easy axis (0 ,0) and (π,π), two an-\ntiparallelstatesalongthe easyaxis(0 ,π)and(π,0). Four\nmore equilibrium states have magnetization perpendicu-\nlar to the easy axis and are unstable without the current:\n(±π/2,±π/2). Once again, since spin-transfer does not\ndepend on the relative direction of current and magneti-\nzation, the configurations which can be transformed into\neach other by a rotation of the magnetic space as a whole\nbehaveidentically. Thusitisenoughtoconsiderfourcon-\nfigurations: (0 ,0), (0,π), (π/2,π/2), and ( π/2,−π/2).\nWe linearize equations (14) near each equilibrium and\nsearch for the solution in the form δφi∼exp(ωt). The\neigenfrequencies are found to be:\n(0,0) :ω1=−ω0\nα, ω2=−−ω0\nα+2ξ(1−ǫ)\n(0,π) :ω1=−ω0\nα+2ǫξ, ω2=−ω0\nα−2ξ\n(π\n2,π\n2) :ω1=ω0\nα, ω2=ω0\nα−2ξ(1−ǫ)\n(π\n2,−π\n2) :ω1=ω0\nα+2ǫξ, ω2=ω0\nα−2ξThe state is stable when both eigenfrequencies are\nnegative. We conclude that initially unstable states\n(π/2,±π/2) are never stabilized by the current, while\nthe (0,0) and (0 ,π) state remain stable for\n(0,0) :ξ >−α\n2(1−ǫ)\n(0,π) :−α\n2ǫ< ξ <α\n2\nThese regions of stability are shown schematically in\nFig. 2B in comparison with the case of static magnetic\npolarizer (Fig. 2A) which is recovered at ǫ→0.\nAs the size of the polarizer is reduced, the asymme-\ntry parameter ǫgrows. The stability region of the an-\ntiparallel state acquires a lower boundary ξ=−α/(2ǫ).\nUp toǫ= 1/2, this boundary is still below the lower\nboundary of the parallel configuration stability region.\nConsequently, the parallel configuration is switched to\nthe antiparallel at a negative current ξ=−α/(2(1−ǫ)).\nThe system then remains in the antiparallel state down\ntoξ=−α/(2ǫ). Below that threshold no stable configu-\nrations exist, and the system goes into some type of pre-\ncession state. This dynamic state is related to the “wind-\nmill” state predicted in Ref. 2 for two identical magnets\nin the absence of anisotropies. Obviously, here it is mod-\nified by the strong easy-plane anisotropy.\nTheǫ= 1/2 value represents a transition point in the\nbehavior of the system. For 1 /2< ǫ <1, the stability\nregion of the parallel configuration completely covers the\none of the antiparallel state. A transition without hys-\nteresis now happens at ξ=−α/(2(1−ǫ)) between the\nparallel state and the precession state. If the system is\ninitially in the antiparallel state, it switches to the par-\nallel state either at a negative current ξ=−α/(2(1−ǫ))\nor at a positive current ξ=α/2, and never returns to\nthe antiparallel state after that.\nIV. CONCLUDING REMARKS\nWe studied thebehaviorofplanarspin-transferdevices\nwith magnetic energy dominated by the large easy-plane\nanisotropy. The overdamped approximation in the pres-\nence of current-induced torque was derived and checked\nagainst the cases already discussed in the literature. In\nthe new “dynamic polarizer” case, we found a transition\nbetween two regimes with different switching sequences.\nThe large asymmetry regime is similar to the case of\nstatic polarizer and shows hysteretic switching between\nthe parallel and antiparallel configurations, while in the\nsmall asymmetry regime the magnets do not switch, but\ngo directly into the “windmill” precession state.\nWe saw that the current-induced switching occurs\nwhen the effective damping constant vanishes near a par-\nticularequilibrium. Thismakestheoverdampedapproxi-\nmationinapplicableinthe immediatevicinityofthetran-\nsition and renders Eqs. (14) ill-defined at some points.\nHowever, the overall conclusions about the switching5\nevents will remain the same as long as the interval of\ninapplicability is small enough.\nWe also find that the overdamped planar approxima-\ntion does not work well when a saddle point of magnetic\nenergy is stabilized by spin-transfer torque, e.g. during\nthe operation of a spin-flip transistor.20Description of\nsuch cases in terms of effective planar equations requires\nadditional investigations.\nV. ACKNOWLEDGEMENTS\nWe wish to thank Tom Silva, Oleg Tchernyshyov,Oleg\nTretiakov, and G. E. W. Bauer for illuminating discus-\nsions. This work was supported in part by DMEA con-\ntract No. H94003-04-2-0404, Ya. B. is grateful to KITP\nSanta Barbara for hospitality and support under NSF\ngrant No. PHY99-07949. D. O. was supported in part\nby the IBM undergraduate student internship program.\nAPPENDIX A: VECTOR DEFINITIONS\nz\neφ\nθen\nφθ\r\nxFIG. 3: Definitions of the tangent vectors and polar angles.\nWe use the standard definitions of polar coordinates\nand tangent vectors (see Fig. 3):\n/vector n= (sinθcosφ,sinθsinφ,cosθ)\n/vector eθ= (cosθcosφ,cosθsinφ,−sinθ) (A1)\n/vector eφ= (−sinφ,cosφ,0)\nWhenθ=π/2+δθa linearization in δθgives\n/vector n≈(cosφ,sinφ,−δθ)\n/vector eθ≈ −(δθcosφ,δθsinφ,1) (A2)\n/vector eφ≈(−sinφ,cosφ,0)\nFor two unit vectors /vector n(i),i= 1,2 with polar angles\n(θi,φi) the scalar product expressions are\n(/vector n(j)·/vector e(i)\nθ) = sin θjcosθicos(φj−φi)−cosθjsinθi\n(/vector n(j)·/vector e(i)\nφ) = sin θjsin(φj−φi) (A3)\nLinearizing (A3) with respect to small δθifor arbitrary\nvalues of θjone gets:\n(/vector n(j)·/vector e(i)\nθ)≈ −sinθjδθicos(φj−φi)−cosθj\n(/vector n(j)·/vector e(i)\nφ)≈sinθjsin(φj−φi) (A4)\nLinearization of (A3) with respect to both δθiandδθj\ngives\n(/vector n(j)·/vector e(i)\nθ)≈ −δθicos(φj−φi)+δθj\n(/vector n(j)·/vector e(i)\nφ)≈sin(φj−φi) (A5)\n1L. 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Phys., 45, 3863 (2006)."    },    {        "title": "0910.0163v1.Spin_motive_forces_and_current_fluctuations_due_to_Brownian_motion_of_domain_walls.pdf",        "content": "arXiv:0910.0163v1  [cond-mat.mes-hall]  1 Oct 2009Spin motive forces and current fluctuations due to\nBrownian motion of domain walls\nM.E. Lucassen, R.A. Duine\nInstitute for Theoretical Physics, Utrecht University, Le uvenlaan 4, 3584 CE Utrecht, The\nNetherlands\nAbstract\nWe compute the power spectrum of the noise in the current due to s pin mo-\ntive forces by a fluctuating domain wall. We find that the power spect rum of\nthe noise in the current is colored, and depends on the Gilbert dampin g, the\nspin transfer torque parameter β, and the domain-wall pinning potential and\nmagnetic anisotropy. We also determine the average current induc ed by the\nthermally-assisted motion of a domain wall that is driven by an extern al mag-\nnetic field. Our results suggest that measuring the power spectru m of the noise\nin the current in the presence of a domain wall may provide a new meth od for\ncharacterizing the current-to-domain-wall coupling in the system .\nKeywords: A. Magnetically ordered materials; A. Metals; A. Semicon ductors;\nD. Noise\nPacs numbers: 72.15 Gd, 72.25 Pn, 72.70 +m\n1. Introduction\nVoltage noise has long been considered a problem. Engineers have be en con-\ncerned with bringing down noise in electric circuits for more than a cen tury.\nThe seminal work by Johnson[1] and Nyquist[2] on noise caused by t hermal ag-\nitation of electric charge carriers (nowadays called Johnson-Nyqu ist noise) was\nlargely inspired by the problem caused by noise in telephone wires. The experi-\nmental work by Johnson tested the earlier observations by engine ers that noise\nincreases with increasing resistance in the circuit and increasing tem perature.\nHe was able to show that there would always be a minimal amount of nois e, be-\nyond which reduction of the noise is not possible, thus providinga ver ypractical\ntool for people working in the field. At the same time, the theoretica l support\nfor these predictions was given by Nyquist. It is probably not a coinc idence\nEmail address: m.e.lucassen@uu.nl (M.E. Lucassen)\nPreprint submitted to Elsevier December 5, 2018that, at the time of his research, Nyquist worked for the American Telephone\nand Telegraph Company.\nAs long as noise is frequency-independent, i.e., white like Johnson-Ny quist\nnoise, it is indeed often little more than a nuisance (a notable exceptio n to\nthis is shot noise[3] at large bias voltage). However, frequency-de pendent, i.e.,\ncolored noise can contain interesting information on the system at h and. For\nexample, in a recent paper Xiao et al.[4] show that, via the mechanism of spin\npumping[5], a thermally agitated spin valve emits noisy currents with a c olored\npower spectrum. They show that the peaks in the spectrum coincid e with the\nprecession frequency of the free ferromagnet of the spin valve. This opens up\nthe possibility of an alternative measurement of the ferromagnetic resonance\nfrequencies and damping, where one does not need to excite the sy stem, but\nonly needs to measure the voltage noise power spectrum. Here, we see that\nproperties of the noise contain information on the system. Clearly, this proposal\nonly worksif the Johnson-Nyquistnoise is not too large comparedto the colored\nnoise.\nNot only precessing magnets in layered structures induce current s: Re-\ncent theoretical work has increased interest in the inverse effect of current-\ndriven domain-wall motion, whereby a moving domain wall induces an ele ctric\ncurrent[6, 7, 8, 9]. Experimentally, this effect has been seen recen tly with field-\ndriven domain walls in permalloy wires[10]. These so-called spin motive for ces\nultimately arise from the same mechanism as spin pumping induced by th e pre-\ncessing magnet in a spin valve, i.e., both involve dynamic magnetization t hat\ninduces spin currents that are subsequently converted into a cha rge current.\nIn this paper, we study the currents induced by domain walls at nonz ero\ntemperature. In particular, we determine the (colored) power sp ectrum of the\nemitted currents due to a fluctuating domain wall, both in the case of an un-\npinned domain wall (Sec. 2.2), and in the case of a domain wall that is ex -\ntrinsically pinned (Sec. 2.3). We also compute the average current in duced by\na field-driven domain wall at nonzero temperature. We end in Sec. 4 w ith a\nshort discussion and, in particular, compare the magnitude of the c olored noise\nobtained by us with the magnitude of the Johnson-Nyquist noise.\n2. Spin motive forces due to fluctuating domain walls\nIn this section, we compute the power spectrum of current fluctu ations due\nto spin motive forces that arise when a domain wall is thermally fluctua ting.\nWe consider separately the case of intrinsic and extrinsic pinning.\n22.1. Model and approach\nThe equations of motion for the position Xand the chirality φof a rigid\ndomain wall at nonzero temperature are given by[11, 12, 13]\n˙X\nλ=α˙φ+K⊥\n/planckover2pi1sin2φ+/radicalbigg\nD\n2η1, (1)\n˙φ=−α˙X\nλ+Fpin+/radicalbigg\nD\n2η2, (2)\nwhereαis Gilbert damping, K⊥is the hard-axis anisotropy, and λ=/radicalbig\nK/Jis\nthe domain-wall width, with Jthe spin stiffness and Kthe easy-axisanisotropy.\nWe introduce a pinning force, denoted by Fpin, to account for irregularities in\nthe material. We have assumed that the pinning potential only depen ds on the\nposition of the domain wall. Pinning sites turn out to be well-described b y a po-\ntentialthat isquadraticin X, suchthat wecantake Fpin=−2ωpinX/λ[11]. The\nGaussian stochastic forces ηidescribe thermal fluctuations and are determined\nby\n/angbracketleftηi(t)/angbracketright= 0 ; /angbracketleftηi(t)ηj(t′)/angbracketright=δijδ(t−t′). (3)\nThey obey the fluctuation-dissipation theorem[12]\nD=2αkBT\n/planckover2pi1NDW. (4)\nNote that in this expression, the temperature Tis effectively reduced by the\nnumber of magnetic moments in the domain wall NDW= 2λA/a3, withAthe\ncross-sectional area of the sample, and athe lattice spacing. Up to linear order\ninthe coordinate φ, validwhen K⊥> kBT, wecanwritethe equationsofmotion\nin Eqs. (1) and (2) as\n∂t/vector x=M/vector x+N/vector η , (5)\nwhere\nM=2\n1+α2\n−αωpinK⊥\n/planckover2pi1\n−ωpin−αK⊥\n/planckover2pi1\n;/vector x=/parenleftbiggX\nλ\nφ/parenrightbigg\n, (6)\nand\nN=1\n1+α2/radicalbigg\nαkBT\nNDW/planckover2pi1/parenleftbigg1α\n−α1/parenrightbigg\n;/vector η=/parenleftbiggη1\nη2/parenrightbigg\n. (7)\nWe readily find that the eigenfrequencies of the system, determine d by the\neigenvalues Λ ±of the matrix M, are\nΛ±≡iω±−Γ∓=−α\n1+α2/parenleftbigg\nωpin+K⊥\n/planckover2pi1/parenrightbigg\n±α\n1+α2/radicalBigg/parenleftbigg\nωpin−K⊥\n/planckover2pi1/parenrightbigg2\n−4\nα2ωpinK⊥\n/planckover2pi1, (8)\n30.0001 0.0002 0.0003 0.0004/HBarΩpin/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtK/UpTee\n-1.5-1-0.50.511.52xΑ/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt1/Plus Α2\n/HBarΩ/PlusMinus/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtK/UpTee/HBar/CΑpGΑmmΑ/PlusMinus/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtK/UpTee\nFigure 1: Values of Γ ±(red curves) and ω±(blue curves) as a function of the pinning for\nα= 0.02 .\nwith both the eigenfrequencies ω±and their damping rates Γ ±real numbers.\nTheir behavior as a function of /planckover2pi1ωpin/K⊥is shown in Fig. 1. Note that this\nexpression has an imaginary part for pinning potentials that obey /planckover2pi1ωpin/K⊥≥\n(α/2)2, and, because for typical materials the damping assumes values α∼\n0.01−0.1, the eigenfrequency assumes nonzero values already for very s mall\npinning potentials. Without pinning potential ( ωpin= 0) the eigenvalues are\npurely real-valued and the motion of the domain wall is overdamped sin ce Γ±≥\n0 andω±= 0.\nIf we include temperature, we find from the solution of Eq. (5) [witho ut loss\nof generality we choose X(t= 0) =φ(t= 0) = 0] that the time derivatives of\nthe collective coordinates are given by\n∂t/vector x(t) =MeMt/integraldisplayt\n0dt′e−Mt′N/vector η(t′)+N/vector η(t), (9)\nfor one realization of the noise. By averaging this solution over realiz ations of\nthe noise, we compute the power spectrum of the currentinduced by the domain\nwall under the influence of thermal fluctuations as follows.\nIt was shown by one of us[8] that up to linear order in time derivatives , the\ncurrent induced by a moving domain wall is given by\nI(t) =−A/planckover2pi1\n|e|L(σ↑−σ↓)/bracketleftBigg\n˙φ(t)−β˙X(t)\nλ/bracketrightBigg\n, (10)\nwithLthe length of the sample, and βthe sum of the phenomenological dissi-\npative spin transfer torque parameter[14] and non-adiabatic con tributions. The\npower spectrum is defined as\nP(ω) = 2/integraldisplay+∞\n−∞d(t−t′)e−iω(t−t′)/angbracketleftI(t)I(t′)/angbracketright. (11)\n4Note that in this definition the power spectrum has units [ P] = A2/Hz, not to\nbe mistaken with the power spectrum of a voltage-voltage correlat ion, which\nhas units [ P] = V2/Hz. In both cases, however, the power spectrum can be\nseen as a measure of the energy output per frequency interval. W e introduce\nnow the matrix\nO=/parenleftbiggA/planckover2pi1\n|e|L/parenrightbigg2\n(σ↑−σ↓)2/parenleftbiggβ2−β\n−β1/parenrightbigg\n, (12)\nso that we can write the correlations of the current as\n/angbracketleftI(t)I(t′)/angbracketright=/angbracketleftBig\n[∂t/vector x(t)]TO∂t′/vector x(t′)/angbracketrightBig\n=\n/integraldisplayt\n0/integraldisplayt′\n0dt′′dt′′′/angbracketleftBig\n/vector η(t′′)TNTeMT(t−t′′)MTOMeM(t′−t′′′)N/vector η(t′′′)/angbracketrightBig\n+/integraldisplayt\n0dt′′/angbracketleftBig\n/vector η(t′′)TNTeMT(t−t′′)MTON/vector η(t′)/angbracketrightBig\n+/integraldisplayt′\n0dt′′/angbracketleftBig\n/vector η(t)TNTOMeM(t′−t′′)N/vector η(t′′)/angbracketrightBig\n+/angbracketleftBig\n/vector η(t)TNTON/vector η(t′)/angbracketrightBig\n=\nθ(t−t′)/braceleftBigg/integraldisplayt′\n0dt′′Tr/bracketleftBig\nNTeMT(t−t′′)MTOMeM(t′−t′′)N/bracketrightBig\n+Tr/bracketleftBig\nNTeMT(t−t′)MTON/bracketrightBig/bracerightBigg\n+θ(t′−t)/braceleftBigg/integraldisplayt\n0dt′′Tr/bracketleftBig\nNTeMT(t−t′′)MTOMeM(t′−t′′)N/bracketrightBig\n+Tr/bracketleftBig\nNTOMeM(t′−t)N/bracketrightBig/bracerightBigg\n+δ(t−t′)Tr/bracketleftBig\nNTON/bracketrightBig\n.(13)\nWe evaluate the traces that appear in this expression to find that t he power\nspectrum is given by\nP(ω) =\n2/parenleftbiggA/planckover2pi1\n|e|L/parenrightbigg2(σ↑−σ↓)2\n1+α2αkBT\n/planckover2pi1NDW×/bracketleftBigg\n(1+β)2−/braceleftBigg\n(1+β2)(1+α2)2/parenleftBigg\n/planckover2pi1ωpin\nK⊥/parenrightBigg2\n−/bracketleftBigg\nβ2−α2+2(1+β2)/planckover2pi1ωpin\nK⊥+(1−α2β2)/parenleftBigg\n/planckover2pi1ωpin\nK⊥/parenrightBigg2/bracketrightBigg/parenleftBigg\n/planckover2pi1ω\nK⊥1+α2\n2/parenrightBigg2/bracerightBigg/slashBig\n/braceleftBigg\n(1+α2)2/parenleftBigg\n/planckover2pi1ωpin\nK⊥/parenrightBigg2\n+/bracketleftBigg\nα2−2/planckover2pi1ωpin\nK⊥+α2/parenleftBig/planckover2pi1ωpin\nK⊥/parenrightBigg2/bracketrightBigg/parenleftBigg\n/planckover2pi1ω\nK⊥1+α2\n2/parenrightBigg2\n+/parenleftBigg\n/planckover2pi1ω\nK⊥1+α2\n2/parenrightBigg4/bracerightBigg/bracketrightBigg\n. (14)\n52.2. Domain wall without extrinsic pinning\nWe first consider a domain wall with Fpin= 0. In this case, only the chirality\nφdetermines the energy, a situation referred to as intrinsic pinning[1 1]. From\nthe result in Eq. (14) we find that the power spectrum is given by\nP(ω) = 2/parenleftbiggA/planckover2pi1\n|e|L/parenrightbigg2(σ↑−σ↓)2\n1+α2αkBT\n/planckover2pi1NDW×\n/braceleftBigg\n(1+β)2+β2−α2\nα2/bracketleftBig\n1+/parenleftBig/planckover2pi1ω\nK⊥1+α2\n2α/parenrightBig2/bracketrightBig−1/bracerightBigg\n. (15)\nIndeed, we find that next to a constant contribution there is also a frequency-\ndependent contribution for β/negationslash=α, i.e., the power spectrum is colored. The fact\nthatβ=αis a special caseis understood from the fact that in that case we ha ve\nmacroscopic Galilean invariance. This translates to white noise in the c urrent\ncorrelations. The power spectrum is a Lorentzian, centered arou ndω= 0\nbecause the domain wall is overdamped in this case, with a width deter mined\nby the damping rate in Eq. (8) as /planckover2pi1Γ+/K⊥= 2α/(1 +α2). Relative to the\nwhite-noise contribution\nPW= 2(1+β)2/parenleftbiggA/planckover2pi1\n|e|L/parenrightbigg2(σ↑−σ↓)2\n1+α2αkBT\n/planckover2pi1NDW, (16)\nthe height of the peak is given by ∆ P=PW(β2−α2)/α2. The behavior of the\npower spectrum is illustrated in Fig. 2 for several values of β/α.\n-0.3 -0.2 -0.1 0.1 0.2 0.3/HBarΩ/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExtK/UpTee1234/LParen11/PlusΒ/RParen12P/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtPW\nΒ/Slash1Α/EquΑl2Β/Slash1Α/EquΑl1.5Β/Slash1Α/EquΑl1Β/Slash1Α/EquΑl0.6Β/Slash1Α/EquΑl0\nFigure 2: The power spectrum for α= 0.02 and several values of β.\n2.3. Extrinsically pinned domain wall\nFor extrinsically pinned domain walls the behavior of the power spectr um\ngivenbyEq.(14) isdepicted in Fig.3. We seethat for /planckover2pi1ωpin/K⊥/greaterorsimilarα2the peaks\n60.05 0.1 0.15 0.2/HBarΩ/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExtK/UpTee12345/LParen11/PlusΒ/RParen12P/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtPW\n/HBarΩpin/EquΑl0.002 K/UpTee/HBarΩpin/EquΑl0.001 K/UpTee/HBarΩpin/EquΑl0.0005 K/UpTee/HBarΩpin/EquΑl0\n(a)\n0.05 0.1 0.15 0.2/HBarΩ/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExtK/UpTee2468/LParen11/PlusΒ/RParen12P/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtPW\n/HBarΩpin/EquΑl0.002 K/UpTee/HBarΩpin/EquΑl0.001 K/UpTee/HBarΩpin/EquΑl0.0005 K/UpTee/HBarΩpin/EquΑl0\n(b)\n0.05 0.1 0.15 0.2/HBarΩ/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExtK/UpTee5001000150020002500/LParen11/PlusΒ/RParen12P/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtPW\n/HBarΩpin/EquΑl0.002 K/UpTee/HBarΩpin/EquΑl0.001 K/UpTee/HBarΩpin/EquΑl0.0005 K/UpTee/HBarΩpin/EquΑl0\n(c)\nFigure 3: The power spectrum as a function of the frequency ωand the pinning potential ωpin\nforβ=α/2 (a),β= 2α(b) and β= 50α(c), all for α= 0.02.\n7in the power spectrum are approximately centered around the eige nfrequencies\n/planckover2pi1ω/K⊥≃ ±2/radicalbig\n/planckover2pi1ωpin/K⊥, consistent with Eq. (8). We can discern between\ntwo regimes, one where β∼αin figs. 3 (a-d), and one where β≫αin figs. 3\n(e-f). In the former regime, the height of the peaks in the power s pectrum\ndepend strongly on the pinning. For small ωpinwe see a clear dependence on\nthe value of β, whereas for large ωpinthis dependence is less significant. In the\nregime of large β, the height of the peaks hardly depends on the pinning and\nis approximately given by P≃PWβ2/α2. Note that the width of the peaks is\nindependent of β. For pinning potentials α2</planckover2pi1ωpin/K⊥the width is given by\n/planckover2pi1Γ/K⊥=α(1+/planckover2pi1ωpin/K⊥)/(1+α2), so for/planckover2pi1ωpin≪K⊥the dependence of the\nwidth on the pinning is negligible.\n3. Spin motive forces due to thermally-assisted field-drive n domain\nwalls\nIn this section we compute the average current that is induced by a moving\ndomain wall. The domain wall is moved by applying an external magnetic fi eld\nparallel to the easy axis of the sample. In this section we take into ac count\ntemperature but ignore extrinsic pinning (see Ref. [15] for a calcula tion of spin\nmotive forces in a weakly extrinsically pinned system for β= 0). The equations\nof motion for a field-driven domain wall are then given by[11, 12, 13]\n˙X\nλ=α˙φ+K⊥\n/planckover2pi1sin2φ+/radicalbigg\nD\n2η1, (17)\n˙φ=−α˙X\nλ+gµBBz\n/planckover2pi1+/radicalbigg\nD\n2η2, (18)\nwheregBzis the magnetic field in the z direction, and the stochastic forces are\ndetermined by Eq.(3) with the strength givenbyEq. (4). In earlier work[13], we\ncomputed from these coupled equations the average drift velocity of a domain\nwall (here, we set the applied spin current zero)\nα/angbracketleft˙X/angbracketright\nλ=−/angbracketleft˙φ/angbracketright+gµBBz\n/planckover2pi1, (19)\nwith (we omit factors 1+ α2≃1)\n/angbracketleft˙φ/angbracketright=−2παkBT\n/planckover2pi1NDW/parenleftbig\ne−2πgµBBzNDW\nαkBT−1/parenrightbig/slashBig\n/braceleftBigg/integraldisplay2π\n0dφeNDW\nkBT/parenleftbig\ngµBBz\nαφ+K⊥\n2cos2φ/parenrightbig/bracketleftBigg/integraldisplay2π\n0dφ′e−NDW\nkBT/parenleftbig\ngµBBz\nαφ′+K⊥\n2cos2φ′/parenrightbig\n+/parenleftbig\ne−2πgµBBzNDW\nαkBT−1/parenrightbig/integraldisplayφ\n0dφ′e−NDW\nkBT/parenleftbig\ngµBBz\nαφ′+K⊥\n2cos2φ′/parenrightbig/bracketrightBigg/bracerightBigg\n.(20)\n8Combining Eqs. (10) and (19) gives us the average current straigh tforwardly\n/angbracketleftI/angbracketright=−A/planckover2pi1\nα|e|L(σ↑−σ↓)/bracketleftbigg\n(α+β)/angbracketleft˙φ/angbracketright−βgµBBz\n/planckover2pi1/bracketrightbigg\n. (21)\nWe evaluate this expression for several temperatures in Fig. 4. Th e black curve\n0.5 1 1.5 2 2.5 3Bz/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExtBW\n-3-2-112/LAngleBracket1I/RAngleBracket1/VertBar1e/VertBar1L/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtA/Spaceg/SpaceΜB/SpaceBW/Space/LParen1Σ/UpArrow/MinuΣ Σ /DownArrow/RParen1\nkBT/EquΑl10K/UpTeeNkBT/EquΑl0.5K/UpTeeNkBT/EquΑl0.1K/UpTeeNkBT/EquΑl0\nFigure 4: The average current induced by a domain wall as a fun ction of the magnetic field\napplied to this domain wall. We show curves for several tempe ratures, with α= 0.02 and\nβ= 2α. The normalization of the axes BWis the Walker-breakdown field which is given by\nBW=αK⊥/gµB.\nin Fig. 4 is computed at zero temperature. It increases linearly with t he field\nup to the Walker-breakdown field BW=αK⊥/gµB, where it reaches a max-\nimum current of /angbracketleftI/angbracketright|e|L/Agµ BBW(σ↑−σ↓) =β/α. Then, it drops and even\nchanges sign. This curve is consistent with the curves obtained by o ne of us[8].\nThere, an open circuit is treated where there is no current but a vo ltage. The\nvoltage is related to the current as ∆ V=/angbracketleftI/angbracketrightL/A(σ↑+σ↓), such that indeed\n∆V/V0=/angbracketleftI/angbracketright|e|L/Agµ BBW(σ↑−σ↓), where the normalization is defined as\nV0=PgµBBW/|e|and the polarization is given by P ≡(σ↑−σ↓)/(σ↑+σ↓). In-\ncreasing temperatures smoothen the peak around the Walker-br eakdown field,\nand for high temperatures, the peak vanishes altogether. There fore, for fields\nsmaller than the Walker-breakdown field and for small temperature s, the ther-\nmal fluctuations tend to decrease the average induced current. However, for\nhigher temperatures, the current reverses and increases again . For fields suffi-\nciently larger than the Walker-breakdown field, we always find a slight increase\nof the current as a function of temperature.\n4. Discussion and conclusions\nIn Sec. 2 we computed power spectra for domain walls, both with and with-\nout extrinsic pinning. In ferromagnetic metals, the spin transfer t orque param-\neter has values β∼α, for which we see that for large pinning /planckover2pi1ωpin/K⊥/greaterorsimilarα2\n9the power spectra only weakly depend on the spin transfer torque parameter β.\nTherefore, determination of βis only possible for very small pinning potentials.\nIn an ideally clean sample without extrinsic pinning the height and sign of the\npeak in the power spectrum can give a clear indication whether βis smaller,\n(approximately) equal or larger than α.\nIn order to perform these experiments, the contributions due to the domain\nwallshouldnotbecompletelyoverwhelmedbyJohnson-Nyquistnoise . Theratio\nof the peaks in our power spectrum and the Johnson-Nyquist noise determine\nthe resolution of the experiment, and we want it to be at least of the order\nof a percent. We use that the resistance of the domain wall is small, a nd\nthe Johnson-Nyquist noise is governed by the resistance of the wir e. For a\nwire of length Land cross-section A, the power spectrum due to Johnson-\nNyquist noise is given by PJN= 4kBT(σ↑+σ↓)A/L. From section (2.2) we can\nestimate the ratio of the height of the peak and the Johnson-Nyqu ist noise as\n∆P/PJN≃[(β/α)2−1]αA/planckover2pi1(σ↑+σ↓)/(4LNDWP2e2). To make a rough estimate\nof this ratio, we use λ≃20nm such that A/NDW≃2·10−2˚A2fora≃2˚A, a\npolarization P ≃0.7 and a conductivity σ↑+σ↓≃107A/Vm. We then find\n∆P/PJN≃10−2˚A/L, which shows that the wire length must satisfy L <1˚A\nin order to have a resolution of 1%. We therefore conclude that this effect at\nzero pinning is impossible to see in experiment, where wires are typically at\nleast of the order of L≃10µm, i.e., five orders of magnitude larger. We can\ntry to increase the signal by turning on a pinning potential. Insertio n of the\neigenfrequencies ω±from Eq. (8) in Eq. (14) shows us that we can maximally\ngain a factor α−2≃104forα= 0.01, as illustrated in Fig. 5. We can also still\n20 40 60 80 100/Radical3/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/Radical3Extens/HBarΩpin/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtK/UpTee0.20.40.60.81/LParen11/PlusΒ/RParen12Α2 P/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExt/FrΑctionBΑrExt /FrΑctionBΑrExtPW\nΒ/EquΑl1Β/EquΑl0.02\nFigure 5: The height of the peaks in the power spectrum as a fun ction of the pinning. We\nusedα= 0.02 and two values for β. We checked that the curves do not differ significantly for\ndifferent values for αand therefore, the maximal value of the peaks in the power spe ctrum is\n∆P(1+β2)/PJN≃α−2. Note that the dependence on βis negligible for large pinning. This\ncurve was obtained by inserting the eigenfrequencies ω±from Eq. (8) in the expression for\nthe power spectrum in Eq. (14).\ngain some resolution by considering a domain wall in a nanoconstriction , where\n10the width can be as small as λ= 1nm[16]. All this adds up to a resolution\nof ∆P/PJN∼1%. Therefore, with all parameter values set to ideal values\nit appears to be possible to observe our predictions in ferromagnet ic metals,\nalthough the experimental challenges are big.\nIn a magnetic semiconductorlike GaMnAs the number ofmagnetic mom ents\nin the domain wall is two orders of magnitude smaller than in a ferromag netic\nconductor. This increases the ratio ∆ P/PJNwith a factor 100. However, the\nconductivity is also considerably smaller than that in a ferromagnetic metal.\nThe conductivity of GaMnAs depends on many parameters, but a re asonable\nestimate is that it is at least about three orders of magnitude smaller than\nthat of a metal, although usually even smaller. Therefore, the adva ntage of a\nsmall number of magnetic moments is cancelled. Another property, however, of\nGaMnAs is that the parameter βcan assume considerably higher values[17], up\ntoβ= 1. This does not influence the maximal value of the power spectrum ,\nbut it does dramatically increase the value for small pinning.\nNote that there are contributions to the noise that we did not discu ss in this\narticle. For example, the time-dependent magnetic field caused by a moving\ndomain wall will induce electric currents that contribute to the color ed power\nspectrum. Distinguishing such contributions from the spin motive fo rces was\nessential for the experimental results by Yang et al.[10], and would also be\nimportant here.\nIn Sec. 3, we calculated the current induced by a field-driven domain wall\nunder the influence of temperature. We estimate that in ferromag netic metals\nat room temperature that kBT/K⊥N≃10−3, which is indistinguishable from\nthe zero-temperature curve in Fig. 4. However, for magnetic sem iconductors,\nlike GaMnAs, we find kBT/K⊥N≃10−1atT= 100K, which corresponds to\nthe red curve in Fig. 4. We therefore expect finite-temperature e ffects to be\nimportant in magnetic semiconductors like GaMnAs.\n5. acknowledgement\nThis work was supported by the Netherlands Organization for Scien tific Re-\nsearch(NWO) and by the EuropeanResearchCouncil (ERC) under the Seventh\nFramework Program. We would like to thank Yaroslav Tserkovnyak f or useful\ndiscussions.\nReferences\n[1] J.B. Johnson, Nature 119, 50 (1927); Phys. Rev. 29, 367 (1927); Phys.\nRev.32, 97 (1928).\n[2] H. Nyquist, Phys. Rev. 29, 614 (1927); Phys. Rev. 32, 110 (1928).\n[3] M.J.M. de Jong and C.W.J. Beenakker, in Mesoscopic Electron Transport ,\nedited by L.L. Sohn, L.P. Kouwenhoven and G. Schoen, NATO ASI Ser ies\n(Kluwer Academic, Dordrecht, 1997), Vol. 345, pp. 225-258.\n11[4] J. Xiao, G.E.W. Bauer, S. Maekawa and A. Brataas, Phys. Rev. B 79,\n174415 (2009).\n[5] Y. Tserkovnyak, A. Brataas and G.E.W. Bauer, Phys. Rev. Lett .88,\n117601 (2002).\n[6] S.E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601 (2007).\n[7] W.M. Saslow, Phys. Rev. B 76, 184434 (2007).\n[8] R.A. Duine, Phys. Rev. B 77, 014409 (2008); R.A. Duine, Phys. Rev. B\n79, 014407 (2009).\n[9] Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77, 134407 (2008).\n[10] S.A. Yang, G.S.D. Beach, C. Knutson, D. Xiao, Q. Niu, M. Tsoi, and J.L.\nErskine, Phys. Rev. Lett. 102, 067201 (2009).\n[11] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004); 96, 189702\n(2006).\n[12] R.A. Duine, A.S. N´ u˜ nez and A.H. MacDonald, Phys. Rev. Lett. 98, 056605\n(2007).\n[13] M.E. Lucassen, H.J. van Driel, C. Morais Smith and R.A. Duine, Phys .\nRev. B79, 224411 (2009).\n[14] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[15] V. Lecomte, S.E. Barnes, J.-P. Eckmann and T. Giamarchi, Phys . Rev. B\n80, 054413 (2009).\n[16] P. Bruno, Phys. Rev. Lett. 83, 2425 (1999).\n[17] K.M.D. Hals, A.K. Nguyen and A. Brataas, Phys. Rev. Lett. 102, 256601\n(2009).\n12"    },    {        "title": "1704.05718v1.Refractive_index_of_dense_materials.pdf",        "content": "Refractive index of dense materials  \nGilbert Zalczer  \nSPEC, CEA, CNRS, Université Paris Sac lay, CEA Saclay,  91191 Gif sur Yvette, France  \nWe show that applying the Lorentz -Lorenz transformation to the refractive index of metals , \nsemiconductors and insulators allows for  a less empirical modeling of this refractive index.  \nINTRODUCTION  \nOptical devices are ubiquitous in our environment. They use many materials which interact \nwith light thro ugh a comp lex, frequency depende nt property  : their refractive  indices. These have \nbeen widely measured and tabulated . Several ways of empirical mod eling have  been proposed, \nusually  limit ed to a category of materials  and implying intricate formulas[ 1-5]. We have noticed that \nthe “polarizability ” computed by applyin g blindly the Lorentz -Lorenz formula can be fitted much \nmore accurately. Indeed the imaginary part can be very well described by a few Gaussian functions \nand the real part computed using Kramers -Kronig relationship.  \nMODEL AND FITS  \nThe dielectric constant ε of an assembly of polarizable point particle is well described by the \nClausius -Mossoti  formula  \n \n𝜖−1\n𝜖+2=𝐶 𝜌𝛼 \n \nwhere ρ is the density of particles  α their polarizability  and C a constant . Both ε and  α are complex \nnumbers and frequency dependent. In the optical domain this equation can be rewritten as  \n𝑛2−1\n𝑛2+2=𝐶 𝜌𝛼 \n \nknown as Lorentz -Lorenz formula. This formula has no basis for being applied to metals, but an \nempir ical use reveals an interesting phenomenon.  The data we use originate from the book of \nPalik [6]. \nThe most striking case is that of silver. The real and imaginary part of the so computed \n“polarizability” are shown in fig 1 a s a function of the wavevector (the data are plotted in red, the fit \nin blue) :   \nFigure 1 : Real (top) and imaginary part of the polarizabilty calculated by application of the Lorentz -Lorenz \nformula to the refractive index of silver  as a function of the wavevector (in cm-1) \n \n The imaginary  part is readily seen as the sum of two features : one shar p and intense near \n2.8 104 cm-1 and a broad one centered near 4.5 104 cm-1 . These can be quite accurately fitted by \nGauss curves. The fit parameters for the center wavevector (s0), the width (larg) and the amplitude \n(amp) of these Gauss functions (G1 and G2) are shown beside the plots.  The real part cannot be \ndeduced from the imaginary one using the Kramers -Kronig relation  (the K -K transform of a Gaussian \nis a Dawson function)  because we have to take into account at least one other line far enough in the \nUV not to appear in the figure. By convention we ascribe it a width of 1 and the other parameters are \nreported labeled as D1. The refractive index of transparent media is due to such a line. We shall see \nbelow that the polarizability of many dense material s can be described accurately by just a small \nnumber of Gaussian shaped absorption lines. Those lying out of the observed range are seen only to \nthe long tail of their real part. Their width cannot be determined and has been fixed to 1.  \nThe case of gold ha s some similarities :  \n \nFigure 2 Polarizability of gold  \nA sharp peak at low wavevector and a broad one at the high side.  However the sharp peak is \nmuch less intense and a wide continuum appears between them which can be accurately fitted only \nby introducing two additional gaussians. The same holds for copper  \n \nFigure 3 Polarizability of copper  \nThe f it is a little worse. It could perhaps be improved by adding more lines but their relevance \nis questionable. A common feature of these three spectra is a range of zero α” at small wavevector.  \nPlatinum and nickel can be described by one and two lines coveri ng the visible range  : \n \nFigure 4 Polarizability of platinum  \n \nFigure 5 Polarizability of nickel  \nSemiconductors also can be fitted with 3 lines fo r silicon and 4 for germanium  : \n \nFigure 6 : Polarizability of crystalline silicon.  \non \n \nFigure 7 Polarizability of germanium  \n  \nAs noted above a transparent medium such as silica is due to a UV band  \n \nFigure 8 Polarizability  of silica glass. The imaginary part is zero  \nWhile for zirconia (  ZrO2 ) we have also to take into account an infrared one  : \n \nFigure 9 Polarizability of zirconia glass. The imaginary part is zero  \nCONCLUSION  \nThe refractive index of dielectrics, semiconductors and metals can be interpreted and \ndescribed by transforming their refractive index into a polarizability using the Lorentz -Lorenz \nformula. The imaginary part of this polarizability appears as a sum of a f ew Gaussian functions, the \nparameters of wh ich can be readily fitted. S mall distort ions may be attributed to weaker bands but \ntheir relevance is questionable. The real part can be deduced from the Kramers -Kronig transform of \nthe imaginary part but one has t o take into account lines in the UV or in the IR, invisible in the \nimaginary part but who have a significant but smooth contribution in the real part. In some cases the \nfit for the imaginary part is good but not that of the real part. An apparent violation  of the Kramers \nKronig relation might signal imperfect measurements.  \nREFERENCES  \n1 W. Sellmeier  Annalen der Physik un d Chemie , 219, 272  (1871) \n2 R. Brendel, and D. Bormann, J. Appl. Phys. 71, 1 (1992).  \n3 D. Rioux, S. Vallieres, S. Besner, P. Munoz, E. Mazur, and M. Meunier, Adv Opt Mater 2, 176 (2014).  \n4 A.B. Djuri sic, E.H. Li , D. Raki c, M.L. Majewski , Appl. Phys. A 70, 29  (2000 ) \n5 M. Ruzi , C. Ennis , and E. G. Robertson, AIP Advances 7, 015042 (2017 ) \n6 Edward D. Palik , Handbook of Optical Constants of Solids , Elsevier (1997 ) \n "    },    {        "title": "1107.0753v2.Minimization_of_the_Switching_Time_of_a_Synthetic_Free_Layer_in_Thermally_Assisted_Spin_Torque_Switching.pdf",        "content": "arXiv:1107.0753v2  [cond-mat.mes-hall]  16 Sep 2011Applied Physics Express\nMinimizationof theSwitchingTime of aSyntheticFreeLayer inThermallyAssistedSpin\nTorqueSwitching\nTomohiroTaniguchiandHiroshi Imamura\nNanosystem Research Institute, AIST, 1-1-1 Umezono, Tsuku ba 305-8568, Japan\nWetheoreticallystudiedthethermallyassistedspintorqu eswitchingofasyntheticfreelayerandshowedthattheswit ching\ntimeisminimizedifthecondition HJ=|Hs|/(2α)issatisfied,where HJ,Hs,andαarethecouplingfieldoftwoferromagnetic\nlayers,theamplitudeofthespintorque,andtheGilbertdam pingconstant, respectively. Wealsoshowed thatthecoupli ng\nfieldof the synthetic freelayer can be determined from there sonance frequencies of thespin-torque diode effect.\nSpin random access memory (Spin RAM) using the tun-\nneling magnetoresistance (TMR) e ffect1,2)and spin torque\nswitching3,4)is one of the important spin-electronicsdevices\nfor future nanotechnology. For Spin RAM application, it is\nhighly desired to realize the magnetic tunnel junction (MTJ )\nwith high thermal stability ∆0, a low spin-torque switching\ncurrentIc, and a fast switching time. Recently, large ther-\nmalstabilitieshavebeenobservedinanti-ferromagnetica lly5)\nand ferromagnetically6)coupled synthetic free (SyF) layers\ninMgO-basedMTJs.Inparticular,theferromagneticallyco u-\npledSyFlayerisaremarkablestructurebecauseitshowsthe r-\nmalstabilityofmorethan100withalowswitchingcurrent.6)\nSincethecouplingbetweenthe ferromagneticlayersinthe\nSyF layer is indirect exchange coupling, we can systemati-\ncallyvarythesignandstrengthofthecouplingfieldbychang -\ning the spacer thickness between the two ferromagnetic lay-\ners. As shown in ref.7), the thermal switching probability of\ntheSyFlayerisadoubleexponentialfunctionofthecouplin g\nfield, anda tinychangein the couplingfield cansignificantly\nincreaseordecreasetheswitchingtime.Therefore,itisof in-\nteresttophysicalsciencetostudythedependenceofthethe r-\nmalswitchingtimeonthecouplingfield.\nIn this paper, we theoretically studied the spin-current-\ninduced dynamics of magnetizations in an SyF layer of an\nMTJ. We found the optimum condition of the coupling field,\nwhichminimizesthethermallyassistedspintorqueswitchi ng\ntime. We showedthat the couplingfield of the two ferromag-\nnetic layers in the SyF layer can be determined by using the\nspintorquediodee ffect.\nLet us first briefly describe the thermal switching of the\nSyF layer in the weak coupling limit, KV≫JS, where\nK,J,V, andSare the uniaxial anisotropy energy per unit\nvolume, the coupling energy per unit area, and the volume\nandcross-sectionalarea of the single ferromagneticlayer ,re-\nspectively.For simplicity,we assume that all the material pa-\nrameters of the two ferromagnetic layers (F 1and F2) in the\nSyF layer are identical. A typical MTJ with an SyF layer is\nstructured as a pinned layer /MgO barrier/ferromagnetic (F 1)\nlayer/nonmagnetic spacer /ferromagnetic (F 2) layer (see Fig.\n1), where the F 1and F2layers are ferromagneticallycoupled\ndueto the interlayerexchangecoupling.6)The F1and F2lay-\ners have uniaxial anisotropy along the zaxis and two energy\nminima at mk=±ez, wheremkis the unit vector pointing in\nthe direction of the magnetization of the F klayer. The spin\ncurrent injected from the pinned layer to the F 1layer exerts\nspin torque on the magnetization of the F 1layer.8)Then, the\nmagnetization of the F 1layer switches its direction due to\nthe spin torque,after which the magnetizationof the F 2layerelectron\n(positive current)p m1 m2Hz\nxy\nF1 layer F2 layer spacer MgO pinned layer\nFig. 1. Schematic view of the SyF layer. mkandpare the unit vectors\npointing in the directions of the magnetizations of the F kand pinned layers,\nrespectively. The positive current is defined as the electro n flow from the\npinned layer to the free layer. Hrepresents the applied field.\nswitchesits directiondueto coupling.By increasingthe co u-\npling field, the potential height of the F 1(F2) layer for the\nswitching becomes high (low), which makes the switching\ntime of the F 1(F2) layer long (short). Then, a minimum of\nthe totalswitchingtime appearsat a certaincouplingfield, as\nwe shallshowbelow.\nTheswitchingprobabilityfromtheparallel(P)toantipara l-\nlel(AP)alignmentofthepinnedandfreelayermagnetizatio ns\nisgivenby7)\nP=1−(νF1e−νF2t−νF2e−νF1t)/(νF1−νF2),(1)\nwhereνFk=fFkexp(−∆Fk)istheswitchingrateoftheF klayer.\nThe attempt frequency is given by fFk=f0δk, wheref0=\n[αγHan/(1+α2)]√∆0/π,δ1=[1−(H+HJ+Hs/α)2/H2\nan][1+(H+\nHJ+Hs/α)/Han],andδ2=[1−(H−HJ)2/H2\nan][1+(H−HJ)/Han].α,\nγ,H,Han=2K/M,HJ=J/(Md), and∆0=KV/(kBT) are the\nGilbert damping constant, gyromagnetic ratio, applied fiel d,\nuniaxialanisotropyfield,couplingfield,andthermalstabi lity,\nrespectively,and distheferromagneticlayerthickness. ∆Fkis\ngivenby7,9)\n∆F1=∆0[1+(H+HJ+Hs/α)/Han]2,(2)\n∆F2=∆0[1+(H−HJ)/Han]2. (3)\n∆F1is the potential height of the F 1layer before the F 2layer\nswitches its magnetization while ∆F2is the potential height\nof the F 2layer after the F 1layer switches its magnetization.\nHs=/planckover2pi1ηI/(2eMSd) is the amplitude of the spin torque in\nthe unit of the magnetic field, where ηis the spin polariza-\ntion of the current I. The positive current corresponds to the\nelectron flow from the pinned to the F 1layer; i.e., the nega-\ntive current I(Hs<0) induces the switching of the F 1layer.\nThe field strengthsshouldsatisfy |H+HJ+Hs/α|/Han<1and\n|H−HJ|/Han<1becauseeq.(1)isvalidinthethermalswitch-\ning region. In particular, |H+HJ+Hs/α|/Han<1 means that\n|I|<|Ic|. Theeffect of thefield like torqueis neglectedin Eq.\n(2) because its magnitude, βHswhere the beta term satisfies\nβ<1, is less than 1 Oe in the thermal switching region and\n12 Applied Physics Express\ncoupling field, H J (Oe)I=-8, -9, and \n     -10 (μA)solid : P=0.50\ndotted : P=0.95\n10 (μs)100 (μs)1 (ms)10 (ms)100 (ms)1 (s)switching time \n20 40 60 80 100\nFig. 2. Dependences of the switching time at P=0.50 (solid lines) and\nP=0.95 (dotted lines) on the coupling field HJwith currents I=−8 (yel-\nlow),−9 (blue), and−10 (red)µA.\nthus,negligible.\nFigure 2 shows the dependences of the switching times at\nP=0.50andP=0.95onthecouplingfieldwiththecurrents\n(a)−8, (b)−9, and (c)−10µA. The valuesof the parameters\nare taken to beα=0.007,γ=17.32 MHz/Oe,Han=200\nOe,M=995 emu/c.c.,S=π×80×35 nm2,d=2 nm, and\nT=300 K.6)The values of Handηare taken to be−65 Oe\nand 0.5,respectively.The value of His chosen so as to make\nthepotentialheightsfortheswitchinglowasmuchaspossib le\n(|H+HJ+Hs/α|/Han/lessorsimilar1 and|H−HJ|/Han/lessorsimilar1). As shown in\nFig. 2, the switching time is minimized at a certain coupling\nfield. We call this HJas the optimum coupling field for the\nfast thermallyassisted spintorqueswitching.\nLetusestimatetheoptimumcouplingfield.Forasmall HJ,\nthe switching time of the F 2layer is the main determinant of\nthe total switching time; thus, eq. (1) canbe approximateda s\nP≃1−e−νF2t. By increasing HJ,νF2increases and the switch-\ning time (∼1/νF2) decreases. Fast switching is achieved for\nνF2∼νF1in this region. On the other hand, for a large HJ,\nthe switching time of the F 1layer dominates, and eq. (1) is\napproximated as P≃1−e−νF1t. The switching time ( ∼1/νF1)\ndecreaseswithdecreasing HJ.Fastswitchinginthisregionis\nalso achieved forνF1∼νF2. The switching rate νFkis mainly\ndetermined by∆Fk. By putting∆F1=∆F2, the optimum cou-\nplingfield isobtainedas\nHJ=|Hs|/(2α). (4)\nThisisthemainresultofthispaper.Thevaluesobtainedwit h\neq.(4)for I=−8,−9and−10µAare53.7,60.5,and67.2Oe,\nrespectively,whichshowgoodagreementwithFig.2.\nTheconditionνF1≃νF2meansthatthemoste fficientswitch-\ning can be realized when two switching processes of the F 1\nand F2layers occur with the same rate. νF1>νF2means that\nthe magnetization of the F 1layer can easily switch due to a\nlargespintorque.However,thesystemshouldstayinthisst ate\nfor a long time because of a small switching rate of the F 2\nlayer. On the otherhand, when νF1<νF2, it takes a longtime\nto switch the magnetization of the F 1layer. Thus, when νF1\nandνF2are different, the system stays in an unswitched state\nof the F 1or F2layer for a long time, and the total switching\ntimebecomeslong.Forthermallyassistedfieldswitching,w e\ncannot find the optimum condition of the switching time be-\ncause the switching probabilities of the F 1and F2layers are\nthe same. Factor 2 in eq. (4) arises from the fact that HJaf-\nfectstheswitchingsofboththeF 1andF2layers,while Hsas-\nsiststhatofonlytheF 1layer.When HJ≪|Hs|/(2α),thetotalswitchingtimeisindependentofthecurrentstrength,beca use\nthetotalswitchingtimeinthisregionismainlydetermined by\ntheswitchingtimeoftheF 2layer,whichisindependentofthe\ncurrent. In the strong coupling limit, KV≪JS, two magne-\ntizations switch simultaneously,7)and the switching time is\nindependentofthecouplingfield.\nFor the AP-to-P switching, the factors δkand∆Fkare\ngiven byδ1=[1−(H−HJ+Hs/α)2/H2\nan][1−(H−\nHJ+Hs/α)/Han],δ2=[1−(H+HJ)2/H2\nan][1−(H+\nHJ)/Han],∆F1= ∆0[1−(H−HJ+Hs/α)/Han]2, and∆F2=\n∆0[1−(H+HJ)/Han]2.Inthiscase,apositivecurrent( Hs>0)\ninducestheswitching.Bysetting ∆F1=∆F2,theoptimumcou-\npling field is obtained as HJ=Hs/(2α). Thus, for both P-\nto-AP and AP-to-Pswitchings, the optimumcouplingfield is\nexpressedas HJ=|Hs|/(2α).\nInthecaseoftheanti-ferromagneticallycoupledSyFlayer ,\nH+HJandH−HJineqs.(2)and(3)shouldbereplacedby H+\n|HJ|and−H−|HJ|,respectively,wherethesignofthecoupling\nfieldisnegative( HJ<0).Theoptimumconditionisgivenby\n|HJ|=−H+|Hs|/(2α),wherethenegativecurrentisassumedto\nenhancethe switching of the F 1layer. For a sufficiently large\npositive field H>|Hs|/(2α),this conditioncannot be satisfied\nbecauseνF1isalwayssmallerthan νF2.\nOne might notice that the condition ∆F1= ∆F2for the\nferromagnetically coupled SyF layer has another solution\n|Hs|/(2α)=H+Han, which is independent of the coupling\nfield. We exclude this solution because such HandHscan-\nnot satisfy the conditions for the thermal switching region s\n|H+HJ+Hs/α|<Hanand|H−HJ|<Hansimultaneously.Sim-\nilarly, for the anti-ferromagnetically coupled SyF layer, we\nexcludethesolution |Hs|/(2α)=Hanobtainedfrom∆F1=∆F2.\nThenaturalquestionfromtheabovediscussionishowlarge\nthe coupling field is. The coupling field of a large plane film\ncan be determinedfromtwo ferromagneticresonance(FMR)\nfrequencies10,11)corresponding to the acoustic and optical\nmodes,whichdependon HJ. Theantiferromagneticcoupling\nfield can also be determined by the magnetization curve,5)\nin which finite magnetization appears when the applied field\nexceeds the saturation field Hs=−2HJ. These methods are,\nhowever,not applicable to nanostructuredferromagnetssu ch\nas the Spin RAM cells becausethe signal intensity is propor-\ntional to the volume of the ferromagnet, and thus, the inten-\nsity from the Spin RAM cell is negligibly small. It is desir-\nable to measure the coupling field of each cell because HJ\nstrongly depends on the surface state and may di ffer signifi-\ncantlyamongthecellsobtainedfromasinglefilm plane.\nHere,weproposethatthecouplingfieldcanbedetermined\nby using the spin torque diode e ffect12–14)of the SyF layer.\nThis method is applicable to a nanostructured ferromagnet,\nalthough the basic idea is similar to that of FMR measure-\nment.\nThe spin torque diode e ffect is measured by applying an\nalternating current Ia.c.cos(2πft) to an MTJ, which induces\noscillating spin torque on the magnetization of the F 1layer.\nThe free layer magnetizations oscillate due to the oscillat ing\nspin torque and the coupling, which lead to the oscillation o f\nthe TMR RTMR=RP+(1−p·m1)∆R/2 and the d.c. voltage\nVd.c.. Here,∆R=RAP−RP, andRPandRAPcorrespond to the\nresistancesattheparallelandantiparallelalignmentsof pand\nm1, respectively. pis the unit vector pointingin the directionApplied Physics Express 3\n-0.080.04\n0d.c. voltage (mV) \ncurrent frequency (GHz)0 2 4 6 8 10 12 14 single free\nF coupled SyF\nAF coupled SyF-0.04\nFig. 3. Dependences of the spin torque diode voltage of the single fr ee\nlayer (solid), the ferromagnetically (F) coupled SyF layer (dotted), and the\nanti-ferromagnetically (AF) coupled SyF layer (dashed) on the applied cur-\nrent frequency.\nofthepinnedlayermagnetization. Vd.c.isgivenby\nVd.c.=1\nT/integraldisplayT\n0dtIa.c.cos(2πft)−∆R\n2p·m1,(5)\nwhereT=1/f. The SyF layer shows large peaksof d.c. volt-\nage at the FMR frequencies of the acoustic facousticand opti-\ncalfopticalmodes. The couplingfield can be determined from\nthesefrequencies.\nThe resonancefrequencyof the ferromagneticallycoupled\nsystem is obtained as follows. The free energy of the SyF\nlayerisgivenby\nF\nMV=−H·(m1+m2)−Han\n2/bracketleftBig\n(m1·ez)2+(m2·ez)2/bracketrightBig\n+2πM/bracketleftBig\n(m1·ey)2+(m2·ey)2/bracketrightBig\n−HJm1·m2,(6)\nwhere the first, second, third, and fourth terms are the Zee-\nmanenergy,uniaxialanisotropyenergy,demagnetizationfi eld\nenergy, and coupling energy, respectively. The yandzaxes\nare normal to the plane and parallel to the easy axis, re-\nspectively. The applied field, H=H(sinθHex+cosθHez), lies\nin thexzplane with angleθHfrom the zaxis. The equilib-\nrium point is located at m1=m2=m(0)=(sinθ0,0,cosθ0),\nwhereθ0satisfiesHsin(θ0−θH)+Hansinθ0cosθ0=0. We\nemploy a new XYZcoordinate in which the YandZaxes\nare parallel to the yaxis andm(0), respectively, and denote a\nsmall component of the magnetization around m(0)asδmk=\n(mkX,mkY,0).The magnetizationdynamicsis desribedbyus-\ning the Landau-Lifshitz-Gilbert (LLG) equation d mk/dt=\n−γmk×Hk+αmk×(dmk/dt), whereHk=−(MV)−1∂F/∂mk\nis the field acting on mk. By assuming the oscillating solu-\ntion (∝e2πi˜ft) ofmkXandmkY, keeping the first-order terms\nofmkXandmkY, and neglecting the damping term, the LLG\nequations can be linearized as M(m1X,m1Y,m2X,m2Y)t=0.\nThe nonzero components of the coe fficient matrix are M11=\nM22=M33=M44=2πi˜f/γ,M12=M34=[Hcos(θH−θ0)+\nHancos2θ0+HJ+4πM],M21=M43=−[Hcos(θH−θ0)+\nHancos2θ0+HJ], andM14=−M23=M32=−M41=−HJ.\nThe FMR resonance frequencies are obtained under the con-\nditiondet[ M]=0,andaregivenby facoustic=γ√h1h2/(2π)and\nfoptical=γ√(h1+2HJ)(h2+2HJ)/(2π), whereh1=Hcos(θH−\nθ0)+Hancos2θ0andh2=Hcos(θH−θ0)+Hancos2θ0+4πM.\nHJcan be determined from these frequencies. For the anti-\nferromagneticallycoupledsystem, m1/nequalm2inequilibriumin\ngeneral, and the resonance frequencies are obtained by solv -ingthe4×4matrixequation.\nFigure 3 showsthe dependencesof the d.c. voltage Vd.c.of\nthe single free layer (solid) and the ferromagnetically (do t-\nted) and anti-ferromagnetically (dashed) coupled SyF laye rs\non the applied current frequency calculated by solving the\nLLG equations of the F 1, F2, and pinned layers. The spin\ntorque term,γHsm1×(p×m1)+γβHsp×m1, is added to\nthe LLG equation of the F 1layer. Here the field like torque\nis taken into account because it a ffects the shape of Vd.c.\nsignificantly.12)The magnetic field acting on pis given by\nHpin=H−4πMpyey+(Hanpz+Hp)ez, whereHpis the pinning\nfield due to the bottom anti-ferromagnetic layer.6)In Fig. 3,\nIa.c.=0.1 mA,∆R=400Ω,H=200 Oe,|HJ|=100 Oe,\nHp=2 kOe,θH=30◦andβ=0.3.12)Thefacousticandfoptical\noftheferromagneticallycoupledSyFlayerareestimatedto be\n5.98and7.50GHz,respectively,whichshowgoodagreement\nwith the peak points in Fig. 3. These results indicate that th e\nspin torquediode e ffect is useful in determiningthe coupling\nfield.\nInsummary,wetheoreticallystudiedthedependenceofthe\nthermally assisted spin torque switching time of a SyF layer\nonthecouplingfield.Wefoundthattheswitchingtimeismin-\nimized if the condition of HJ=|Hs|/(2α) is satisfied. We\nshowed that the coupling field can be determined from the\nresonancefrequencyofthespin torquediodee ffect.\nThe authors would like to acknowledge H. Kubota, T.\nSaruya, D. Bang, T. Yorozu, H. Maehara, and S. Yuasa of\nAISTfortheirsupportandthe discussionstheyhadwithus.\n1) S.Yuasa,T.Nagahama,A.Fukushima,Y.Suzuki,andK.Ando :Nature\nMaterials 3(2004) 868.\n2) S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughe s, M.\nSamant, and S.H. Yang: Nature Materials 3(2004) 862.\n3) J.C.Slonczewski:Phys.Rev.B 39,6995(1989);J.Magn.Magn.Mater.\n159(1996) L1.\n4) L.Berger: Phys.Rev. B 54(1996) 9353.\n5) J.Hayakawa, S.Ikeda,K.Miura,M.Yamanouchi, Y.M.Lee,R .Sasaki,\nM. Ichimura, K. Ito, T. Kawahara, R. Takemura, T. Meguro, F. M at-\nsukura, H. Takahashi, H.Matsuoka, and H.Ohno: IEEE.Trans. Magn.\n44(2008) 1962.\n6) S. Yakata, H.Kubota, T.Sugano, T.Seki, K. Yakushiji, A.F ukushima,\nS.Yuasa, and K.Ando: Appl. Phys.Lett. 95(2009) 242504.\n7) T.Taniguchi and H.Imamura: Phys.Rev. B 83(2011) 054432.\n8) Private communication with Hitoshi Kubota. It was experi mentally\nshown that the critical current in the CoFeB /Ru/CoFeB spin valve is\none order of magnitude larger than that in CoFeB /MgO/CoFB MTJs\n(unpublished). This result means that the spin torque arisi ng between\nthe free layers is negligible compared with that arising fro m the spin\ncurrent injected from thepinned layer.\n9) In ref.7),∆F1is expressed as∆0[1+(Happl+HJ)/Han]2(1−I/Ic)2,\nwhich is equivalent to eq. (2).\n10) Z.Zhang, L.Zhou,and P.E.Wigen: Phys.Rev. B 50(1994) 6094.\n11) J.Lindner,U.Wiedwald,K.Baberschke,andM.Farle:J.V ac.Sci.Tech-\nnol. A23(2005) 796.\n12) A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. M aehara,\nK.Tsunekawa,D.D.Djayaprawira, N.Watanabe, andS.Yuasa: Nature\n438(2005) 339.\n13) H. Kubota, A. Fukushima, K. Yakushiji, T. Nagahama, S. Yu asa, K.\nAndo, H. Maehara, Y. Nagamine, K. Tsunekawa, D. D. Djayapraw ira,\nN.Watanabe, and Y. Suzuki: Nature Physics 4(2008) 37.\n14) J. C. Sankey, Y.-T. Cui, J. Z. Sun, J. C. Slonczewski, R. A. Buhrman,\nand D.C. Ralph: Nature Physics 4,(2008) 67."    },    {        "title": "2007.08414v3.Thermal_noise_effects_on_the_magnetization_switching_of_a_ferromagnetic_anomalous_Josephson_junction.pdf",        "content": "Thermal noise effects on the magnetization switching of a\nferromagnetic anomalous Josephson junction\nC Guarcelloa,<, FS Bergeretb,c\naDipartimento di Fisica “E.R. Caianiello”, Università di Salerno, Via Giovanni Paolo II, 132, I-84084 Fisciano (SA), Italy\nbCentro de Física de Materiales, Centro Mixto CSIC-UPV/EHU, Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain\ncDonostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain\nARTICLE INFO\nKeywords :\nFerromagnetic Josephson junction\nAnomalous Josephson effect\nMagnetization reversal phenomenon\nLandau–Lifshitz–Gilbert (LLG) equa-\ntion\nResistivelyshuntedjunction(RSJ)model\nThermal noise effectsABSTRACT\nWediscusstheeffectsofthermalnoiseonthemagneticresponseofalateralferromagneticJosephson\njunctionwithspin-orbitcouplingandout-of-planemagnetization. Thedirectionofthemagneticmo-\nmentintheferromagneticlayercanbeinvertedbyusingcontrolledcurrentpulses. Thisphenomenon\nis due to the magnetoelectric effect that couples the flowing charge current and the magnetization of\nthe ferromagnet. We investigate the magnetization reversal effect versus intrinsic parameters of the\nferromagnet, such as the Gilbert damping and strength of the spin-orbit coupling. We estimate the\nmagnetization reversing time and find the optimal values of the parameters for fast switching. With\nthe aim of increasing the operation temperature we study the effects induced by thermal fluctuations\non the averaged stationary magnetization, and find the conditions that make the system more robust\nagainst noise.\n1. Introduction\nIn the past few years many efforts have been devoted to\nthe theoretical study of the magnetic response of ferromag-\nnetic anomalous Josephson junctions (JJs) [30, 52, 40, 48,\n50, 3, 51, 49, 39, 2, 41, 19, 44, 37], thus offering a path\nof concrete applications based on the electrical control of\nthe magnetization in a so-called '0–junction. A realization\nof such junction consists essentially of a superconductor-\nferromagnet-superconductor(SFS)Josephsonjunctionwith\nan intrinsic spin-orbit coupling (SOC). Its ground state cor-\nresponds to a finite phase shift, 0<'0<\u0019, in the current-\nphase-relation. Recently, such anomalous phase has been\nobserved experimentally in hybrid Josephson devices fabri-\ncated with a topological insulator Bi2Se3and Al/InAs het-\nerostructuresandnanowires[56,1,36,55]. Asdemonstrated\ntheoretically[9,28,52],themagnetizationoftheFlayercan\nbe electrically controlled. In fact, in a '0–junction, due to\nthe magnetoelectric effect, a charge current induces an in-\nplanemagneticmoment[12,13,35,5,29,7],whichinturn\nacts as a torque on the out-of plane magnetization of the F\nlayer,inducingeventuallyitsswitching. Alternatively,inthe\nplace of an electric current, the magnetization reversal can\nbe driven by a magnetic field in a SQUID setup [50], i.e.,\nadevicewidelyusedfordetectinganomalousJosephsonef-\nfects [56, 1, 36, 55, 38, 20].\nThe magnetization reversal phenomenon might eventu-\nallyfindanapplicationindifferentfieldsofsuperconducting\nspintronics[34,14,18]. Nevertheless,anyconcreteapplica-\ntionsbasedonthemagnetizationofa '0junctionhastodeal\nwiththeeffectsonthemagneticresponsestemmingfromthe\nunavoidable thermal fluctuations. In fact, the temperature\n<Corresponding author\ncguarcello@unisa.it (C. Guarcello); sebastian_bergeret@ehu.eus\n(F. Bergeret)\nORCID(s): 0000-0002-3683-2509 (C. Guarcello); 0000-0001-6007-4878\n(F. Bergeret)at which the system resides can significantly influence the\nevolutionofthemagneticmoment,eveninducingunwanted\ntransitions or hindering the inversion of the magnetization.\nTheimpactofstochasticthermalfluctuationsonthemag-\nnetization reversal phenomenon was addressed in Ref. [19].\nThatworkdescribesalsothefeasibility,initiallysuggestedin\nRef. [52], to employ a current-biased SFS Josephson junc-\ntion as a memory element, with the information encoded in\nthe magnetization direction of the F layer.\nInthepresentwork,wetakeacuefromRef.[19]andfo-\ncus in more detail on the noise effects on the magnetization\ndynamics, studying how the value of some system parame-\nterscaninfluencetherobustnessagainstthermalfluctuations\nofthecurrent-inducedmagnetizationreversalphenomenon.\nIn particular, we demonstrate that the values of the Gilbert\ndamping parameter and strength of the spin-orbit coupling\ncanbeconvenientlychosentomakethesystemmorestable,\nto increase the range of suitable working temperatures.\nThe work is organized as follows: In Sec. 2, 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. 3 we discuss the mag-\nnetic configuration of the junction after applying a current\npulse. We look the stationary magnetization as a function\noftheGilbertdampingparameterandtheSOCstrength;ad-\nditionally, we show a simple way to predict the overall re-\nsponseofthestationarymagnetization. Moreover,weinves-\ntigate the full temporal evolution of the magnetization and\nwe demonstrate the possibility to minimize the magnetiza-\ntionswitchingtimebyadjustingspecificsystemparameters.\nInSec.3.1,weshowtheeffectsofstochasticthermalfluctu-\nations on the magnetization dynamics. Finally, in Sec. 4 we\npresent our conclusions.\n:Preprint submitted to Elsevier Page 1 of 10Figure 1: S/F/S Josephson junction driven by a rectangular\nbias current pulse, Ibias. The cartoon depicts the inversion of\nthez-component of the magnetization of the F layer after the\npassage of the current pulse Ibiasthrough the junction.\n2. The model\nThe setup that we consider consists of an SFS junction,\nsee Fig. 1, with a thin ferromagnetic film with an out-of-\nplane magnetic anisotropy and a Rashba-like SOC [19].\nDue to the interplay between the exchange field and the\nSOC, the current-phase relation of the SFS Josephson junc-\ntion readsI'=Icsin.'*'0/, whereIcis the critical cur-\nrentofthejunction, 'istheJosephsonphasedifference,and\n'0is the anomalous phase shift. The latter depends on sev-\neralparametersofthesystem,suchustheRashbacoefficient\n\u000b[45,10],thetransparencyofS/Finterfaces,thespinrelax-\nation, and the disorder degree of the system. For our pur-\nposes,theexactdependenceof '0ontheseparametersisnot\nas important as the geometry of the device. If we assume a\ntwo-dimensional SOC with momenta in the plane of the F\nfilm, and the charge current flows in x-direction, the phase\nshift'0is proportional to the y-component of the magnetic\nmoment according to [9, 28, 29, 5]\n'0=rMy\nM; (1)\nwhereM=t\nM2\nx+M2\ny+M2\nzisthemodulusofthemag-\nnetization vector, and the parameter rquantifies the SOC\nstrength and encloses the \u000b-dependence. Equation (1) es-\ntablishes a direct coupling between the magnetic moment\nand the supercurrent, which eventually makes the current-\ninduced inversion of the magnetization possible.\nThetimeevolutionofthemagnetizationcanbedescribed\nintermsoftheLandau–Lifshitz–Gilbert(LLG)equation[32,\n16]\ndM\nd\u001c=\r\nM\u0018\nMdM\nd\u001c\u0019\n*grMHeff; (2)\nwheregrdenotes the gyromagnetic ratio. The first term on\ntheright-handsideaccountsthedissipationthroughthephe-\nnomenologicaldimensionlessGilbertdampingparameter \r,\nwhilethesecondtermrepresentstheprecessionaround Heff,\nwhich components can be calculated as [33]\nHeff;i=*1\nV)F\n)Mi;withi=x;y;z: (3)HereVisthevolumeoftheFlayerand Fisthefreeenergy\nof the system, which reads as follow\nF=*EJ'Ibias+Es.';'0/+EM: (4)\nHere,EJ= \b0Ic_.2\u0019/(with\b0being the flux quantum),\nIbiasistheexternalcurrentinunitsof Ic,Es.';'0/=EJ[1*\ncos.'*'0/], andEM= *KV\n2\u0018Mz\nM\u00192\nis the magnetic en-\nergy that depends on the anisotropy constant K. In the fol-\nlowing, we indicate the ratio between the energy scales of\nthe system with the parameter \"=EJ_.KV/.\nFrom Eq. (3), we obtain the effective magnetic field\nHeff=K\nM\u0004\"rsin\u0000'*rmy\u0001 y+mz z\u0005; (5)\nwheremx;y;z=Mx;y;z_Marethenormalizedcomponentsof\nthemagnetizationthathavetosatisfythecondition m2\nx+m2\ny+\nm2\nz=1. The LLG equations can be conveniently expressed\ninsphericalcoordinates[47],sothat mx;y;zcanbewrittenin\nterms of the polar and azimuthal angles \u0012and\u001eas\nmx.\u001c/ = sin\u0012.\u001c/cos\u001e.\u001c/\nmy.\u001c/ = sin\u0012.\u001c/sin\u001e.\u001c/ (6)\nmz.\u001c/ = cos\u0012.\u001c/:\nWe can also define the \u0012and\u001ecomponents of the normal-\nized effective field as\nHeff;\u0012=\"rsin.'*rmy/cos\u0012sin\u001e*mzsin\u0012(7)\nHeff;\u001e=\"rsin.'*rmy/cos\u001e: (8)\nThus,bynormalizingthetimetotheinverseoftheferromag-\nnetic resonance frequency !F=grK_M, that ist=!F\u001c,\ntheLLGequationsinsphericalcoordinatesreducetothefol-\nlowing two coupled equations [47]\nd\u0012\ndt=1\n1+\r2\u0018\nHeff;\u001e+\rHeff;\u0012\u0019\n(9)\nsin\u0012d\u001e\ndt=1\n1+\r2\u0018\n\rHeff;\u001e*Heff;\u0012\u0019\n:(10)\nThedynamicsofanoverdampedSFSJosephsonjunction\ncan be described in terms of the resistively shunted junc-\ntion (RSJ) model [4, 26, 54, 27] generalized to include the\nanomalous phase shift '0=rmy[19]. Including also ther-\nmal fluctuations accounted by the Gaussianly distributed,\ndelta correlated stochastic term, Ith.t/, the dynamics of the\nJosephsonphase '.t/canbedescribedbytheequation[19]:\nd'\ndt=!\u0004Ibias.t/*sin\u0000'*rmy\u0001+Ith.t/\u0005+rdmy\ndt:(11)\nHere, the time is still normalized to the inverse of the ferro-\nmagnetic resonance frequency and !=!J_!F, with!J=\n2\u0019IcR_\b0beingthecharacteristicfrequency[4]ofthejunc-\ntion with a normal-state resistance R.\n:Preprint submitted to Elsevier Page 2 of 10r=0.1\n0.0 0.1 0.2 0.3 0.4 0.50.0450.0500.0550.060\nγΔUγ=0.10.0 0.1 0.2 0.3 0.4 0.5rFigure 2: Average washboard potential barrier, \u0001U.\r;r/, cal-\nculated during the current pulse with intensity Imax=0:9and\nduration\u0001t= 10, versus\rat a fixedr= 0:1(blue-dashed\ncurve) and versus rat a fixed\r= 0:1(red-solid curve). The\nhorizontal gray dashed line indicates the washboard potential\nbarrier\u0001U.ib= 0:9/. The other parameters are: \"= 10and\n!=1.\nThe noise term Ith.t/is a sort of “thermal current” with\nthe usual white-noise statistical properties that, in normal-\nized units, can be expressed as [4, 23, 21]\nêIth.t/ë= 0 (12)\u0010Ith.t/Ith.t¨/\u0011= 2DI\u000e\u0000t*t¨\u0001; (13)\nwhere we introduced the dimensionless amplitude of ther-\nmal current fluctuations defined as\nDI=kBT\nR!F\nI2\nc=1\n!kBT\nEJ: (14)\nIn this work we assume only thermal fluctuations directly\naffectingthephasedynamics. Instead,thecaseofastochas-\ntic “thermal field” Hth, with an intensity DH, included also\nin Eq. (2) [8, 11, 47, 42] was discussed in Ref. [19]. Since\nthe normalized intensities of the two noise sources are pro-\nportional, i.e.,DH= .\r\"!/DI[19], one could, in princi-\nple, optimize the system parameters in such a way to make\ntheimpactofthethermalfieldnegligiblewithrespecttothe\nthermal current.\nInourwork,weassumeanSFSjunctionbiasedbyarect-\nangular current pulse, Ibias, centered at tc:\nIbias.t/=<\nImax; tc*\u0001t_2ftftc+\u0001t_2\n0; elsewhere:(15)\nHere,\u0001tis the width and Imaxis the intensity, in units of\nIc,ofthepulse,sothatthecondition Imax<1meansabias\ncurrent lower than the critical value, Ic.\nOften,itisusefultodepicttheresponseofaJJintermsof\ntheevolutionofaparticle,representingthesuperconducting\nphase difference 'across the JJ, in a cosine “washboard”\npotentialU.ib/ =EJ\u00041*cos.'/*ib'\u0005, with the current\nibflowing through the junction being the slope of this po-\ntential. The resulting activation energy barrier \u0001U.ib/ =\nFigure 3: (a) Stationary value, mst\nz, [computed by numeri-\ncal solution of Eqs. (9)-(11)] and (b) expected behavior, mth\nz,\n[calculated through Eq. (17)] of thez-component of the mag-\nnetization as a function of rand\r, in the absence of noise\nfluctuations, DI=0. The other parameters are: \"=10,!=1,\nImax=0:9,\u0001t=10, andmz.t=0/=+1 . The colored circles\nin panel (a) indicate the regions around the .\r;r/combinations\nused to obtain the curves in Fig. 4. The legend in panel (b)\nrefers to both panels.\n24t\n1*i2\nb*ibarcsin.ib/5\nconfinesthephase 'inapoten-\ntial minimum.\nForcompleteness,weobservethatthemagnetizationterm\nincluded in the modified RSJ model affects also the height\nof the potential barrier. In fact, the total current contribu-\ntion in Eq. (11) can be assumed to be the sum of the cur-\nrent pulseIbias.t/and the additional term stemming from\nthe time derivative of the ycomponent of the magnetiza-\ntion,Imy.t/ =r\n!dmy\ndt. This means that during the current\npulse, the phase particle “sees” a time-dependent potential\nbarrier that depends on \randr. In Fig. 2 we show the av-\neragepotentialbarrier, \u0001U.\r;r/,calculatedduringacurrent\npulse with intensity Imax= 0:9and duration \u0001t= 10, as a\nfunction of\randr. It is evident that Imy.t/tends to reduce\ntheaveragebarrierheightwithrespecttothevalue \u0001U.Imax/\ncalculated neglecting the additional magnetic contribution,\nsee the gray dashed line. This means that, by increasing \r\nandr, the phase particle would experience an effective po-\ntentialbarrierslightlyreducedfortheenhancingofthetotal\nbias current due to the magnetic contribution. This mecha-\nnismshouldbetakenintoaccountwhencomparingthenoise\nintensity at which the system becomes unstable with the ef-\nfectivepotentialbarrier. Infact,thismodulationofthewash-\nboardpotentialaccountstheresponsetonoisyfluctuationsat\nlow values of \randr, see Sec. 3.1.\nIn the following, we investigate how thermal noise af-\nfects the magnetization reversal, setting specific combina-\ntions of the system parameters. Specifically, we explore the\nresponse of the magnetization, with and without taking into\naccount noise effects, by varying \randrin suitable ranges.\nThe energy and timescales ratios are fixed to the values \"=\n10and!=1, respectively.\n:Preprint submitted to Elsevier Page 3 of 10Figure 4: Time evolution of Josephson phase and current, top panels, and magnetization\ncomponents, bottom panels, at different .\r;r/values, in the absence of noise, DI=0. The\nyellow shaded regions indicate the time window in which the current pulse, with Imax=0:9\nand\u0001t=10, is switched on. Legends in panel (a) refer to all panels.\n3. Results\nInthissection,wefirststudythemagneticresponseinthe\nabsenceofnoise, i.e.,weimposeDI=0. Weassumethatthe\nmagnetizationinitiallypointstowardsthe z-direction,thatis\nM=.0;0;1/att=0. With this initial condition, we solve\nEqs. (9)-(11) self-consistently, choosing different values of\nthe system parameters. From the solution we determine the\nmagnetization direction after the current pulse, and in par-\nticular we focus on the stationary magnetization mst\nz, that is\nthe value of mzlong after the current pulse is turned off (at\nthe timetmax=100).\nInthefollowingweanalyzealsothenormalizedmagne-\ntizationswitchingtime, \u001cSW,whichisdefinedasthetime(in\nunits of!*1\nF) thatmztakes to switch from the initial value\n+1to*1, after the current is turned off. This characteristic\ntimecloselydependsonthevaluesof rand\r;inparticularit\nispossibletofindspecificcombinationsoftheseparameters\nmaking\u001cSWshorter. Thismeansthepossibilitytominimize\nthe magnetization switching time by adjusting specific sys-\ntemparameters,suchas rand\r,bychoosingmaterialswith\nthe proper characteristics, for instance with a larger SOC to\nmakerlarger. Thisaspectbecomescrucialdesigningappli-\ncations based on the magnetization reversal phenomenon in\na'0–junction,inwhichahighresponsespeedcanplayakey\nrole,e.g., superconducting memory applications [22, 19].\nIn Fig. 3(a) we show the overall behavior of mst\nzas a\nfunction of both the Gilbert damping parameter, \r, and the\nSOC strength, r, in response to a current pulse with inten-\nsityImax= 0:9and width\u0001t= 10. This contour plot is\nformedbyadark-bandspattern,namely,weobserveregions\nofthe.\r;r/parametricspaceinwhichthemagnetizationre-\nversal systematically occurs, i.e., in whichmst\nz= *1, andother regions in which this effect is systematically lacking,\ni.e.,mst\nz=+1. In other words, when increasing rat a fixed\n\r, we observe a regular sequence of mst\nz=+1andmst\nz=*1\nvalues. Similar mst\nzpatterns were observed and discussed\nalso in Refs. [2, 19]. Here we go a step forward, and clarify\nhow the magnetic response depends on the choice of .\r;r/\ncombinations by analyzing different regions of the density\nplot in Fig. 3(a). Essentially, we proceed as follows: i) first,\nwechoose.\r;r/layingindifferentregionsofthedensityplot\n(i.e., we fix\rand changer) andii) then, we focus on .\r;r/\npoints situated on the same region ( i.e., we fixrand change\n\r). For more clarity, in Fig. 3(a) we also highlight with col-\nored circles the regions around the specific .\r;r/combina-\ntions chosen to fully characterize the temporal evolution of\nthe magnetization and to explore noise effects.\nThefulltimeevolutionoftheobservablesofinterest, i.e.,\nthe Josephson phase, the supercurrent, and the magnetiza-\ntion components, in response to a current pulse with ampli-\ntudeImax=0:9and width\u0001t=10is shown in Fig. 4. We\nconvenientlyexaminefirstthebehaviorofthephaseandthe\nJosephson current. In all panels of Fig. 4, we observe that\nduring the current pulse, i.e., within the yellow shaded re-\ngion,theJosephsonphasefirstincreases,andthenitgoesto\nzeroasthepulseisturnedoff. Infact,inthewashboard-like\npicture [4], the tilting imposed by the current pulse, even\nconsidering the Imy.t/contribution, is not enough to allow\nthe “particle” to overcome the nearest potential barrier and\nto switch the system to the finite voltage “running” state.\nThis means that the phase-particle remains confined within\na potential minimum, so that when the current is turned off,\nthe tilting of the washboard potential goes back to zero and\nthe phase restores its initial position, i.e.,'0. Also\nthe Josephson current follows a similar evolution, which is\n:Preprint submitted to Elsevier Page 4 of 10characterizedbyarapidexponentialincrease(decrease),ap-\nproaching the value Imax(0), when the pulse is switched on\n(off).\nTheanalysisofthetemporalevolutionofallthecompo-\nnents of magnetization can give us a deeper understanding\nofhow,andwhy,thephenomenonofreversalofmagnetiza-\ntionoccurs. Thus,inthetoppanels(a)-(d)ofFig.4,weshow\nhowthemagnetizationswitchingdevelopsbychoosing .\r;r/\ncombinations lying on different dark bands of the contour\nplotinFig.3(a). ThismeanstokeepfixedtheGilbertdamp-\ningparameter(inparticular,weset \r=0:1)andtoadjustthe\nrvalue,e.g., we impose r=^0:15;0:23;0:3;0:38`. During\nthe current pulse, i.e., within the yellow shaded regions, mx\nandmzundergoexponentiallydampedoscillationsarounda\nzero value. Instead, myfollows an exponential growth rip-\npled by small, damped oscillations. In other words, the cur-\nrent pulse temporarily causes the precession of the magne-\ntization around the y-direction. Subsequently, as the pulse\nisswitchedoff,thedynamicsofthemagnetizationmoments\nchange,indeedboth mxandmyundergodampedoscillations\naround a zero value, while the z-component, after a tran-\nsient regime, flips definitively to the value mz= *1. We\nalso observe that the oscillating behavior of myis reflected\nintheevolutionoftheJosephsonphase ',whichshowsmin-\nima/maxima in the same position as my.\nFrom the top panels of Fig. 4, it is evident that, as rin-\ncreases, also the oscillation frequency of both mxandmz\nincreases. Furthermore, at a larger r, the curve of myap-\nproaches earlier the value 1and its oscillations are more\ndampened. Thus, we can speculate that the dark bands in\nFig. 3(a) “differ” essentially in the number of oscillations\nmade bymzbefore reversing completely.\nNow, in order to understand how the Gilbert damping\nparameter affects the magnetization response, we consider\n.\r;r/combinations lying on the same band of the contour\nplot in Fig. 3(a), see panels of (e)-(h) of Fig. 4. To this aim,\nwe imposerí0:1and\r=^0:1;0:2;0:3;0:4`. An increase\nof\rtends to dampen more the oscillations of both mxand\nmy,especiallywhenthecurrentisturnedoff. Moreover,the\nvalue of\raffects also the time the magnetization needs to\nfully reverse. In fact, from Fig. 4 one can also estimate the\nnormalized magnetization switching time, \u001cSW.\nInFig.5(a)and(b)weillustratethebehaviorof \u001cSWasa\nfunctionofratafixed\r=0:1andasafunctionof \ratafixed\nr=0:1, respectively. The gray shaded fringes in this figure\nhighlighttherangesof rand\rvalueswithinwhichthemag-\nnetizationreversaltakesplace. Inotherwords,eachgrayre-\ngioncorrespondstoadarkbandinFig.3(a). InFig.5(a)we\ncannotethat,withineachfringe, \u001cSWbehavesnon–monotonically,\ninasmuch it grows significantly at the edges of the gray re-\ngionandreachesalocalminimumroughlyinitsmiddlepoint.\nMoreover, the value of this \u001cSWminimum tends to increase\npassing from a fringe to the next one at higher rvalues. A\nnon-monotonic trend emerges also looking the behavior of\n\u001cSWas a function of \r, see Fig. 5(b). In particular, the \u001cSW\nversus\rcurveisasymmetricandreachestheminimumvalue\n\u001cmin\nSWô11at\rô0:3. Thus,byassumingatypicalferromag-\n0.1 0.2 0.3 0.4 0.5203040506070\nrτsw\nγ=0.1\n0.1 0.2 0.3 0.4 0.510152025303540\nγτsw\nr=0.1(a)\n(b)Figure 5: Magnetization switching time \u001cSW, as a function of\nrat a fixed\r= 0:1, (a), and as a function of \rat a fixed\nr=0:1, (b). In both panels, the gray shaded bands represent\nthe range of rand\rvalues within which the magnetization\nreversal phenomenon takes place.\nnetresonancefrequencyequalto !Fô10GHz[51],weob-\ntainaminimumswitchingtimeoftheorderofnanoseconds.\nFinally, Fig. 4 suggests also that the stationary value of\nthe magnetization depends on the state of mzwhen the cur-\nrentpulseisswitchedoff. Infact,themagnetizationreversal\nseems to occur when mzis in the first half of its oscillation\nperiod just when current is turned off. This simple obser-\nvation allows us to roughly predict the overall response of\nthe stationary magnetization without solving Eqs. (9)-(11)\nnumerically. In fact, according to the dynamics illustrated\nin Fig. 4, one could in principle assume for the x– andz–\ncomponentsofthemagnetizationanoscillating,dampedtime\nevolution,suchas mx.t/ôe*t\n\u001cdsin.\nt/andmz.t/ôe*t\n\u001cdcos.\nt/\n(where\nistheoscillationfrequencyand \u001cdistheexponential-\ndecay constant). Inserting these tentative solutions in the\nequation for  mythat can be derived from Eq. (2), we obtain\nthe following equation for the frequency, \n.\r;r/\n\r\n1+\r2sin.2\nt/\n2+\n=Imax\"r\n1+\r2: (16)\nToobtainthisequationweassumethat \u001cd¸1andsin\u0000'*rmy\u0001í\nImax. Thesetwoassumptionscorrespond,essentially,tosup-\nposeasmall randapulsedurationlongerthantheexponen-\ntialriseoftheJosephsoncurrent,seeFig.4. Finally,bycom-\nparingtheoscillationperiod, T\n.\r;r/=2\u0019_\n.\r;r/,andthe\npulse duration \u0001t, we can give an estimate of the stationary\n:Preprint submitted to Elsevier Page 5 of 1010-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0mxstr=0.15-γ=0.1\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0myst\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\nDImzst10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0r=0.23-γ=0.1\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\nDI10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0r=0.3-γ=0.1\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\nDI10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0r=0.38-γ=0.1\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\n10-40.001 0.010 0.100 1 10-1.0-0.50.00.51.0\nDI(a) (b) (c) (d)Figure 6: Averagestationarymagnetizations, mst\nx,mst\ny, andmst\nz, asafunctionofthethermal\nnoise intensity, DI, at the.\r;r/-values used to obtain the top panels of Fig. 4, calculated\nby averaging over Nexp=103independent numerical repetitions. The vertical dashed lines\nindicate the average washboard potential barrier \u0001U.\r;r/.\nmagnetization as following\nmth\nz.\r;r/=h\nn\nl\nnj*1if Mod\u001c\n\u0001t;T\n.\r;r/\u001d\nfT\n.\r;r/_2\n+1if Mod\u001c\n\u0001t;T\n.\r;r/\u001d\n>T\n.\r;r/_2;\n(17)\nwhere Mod[a;b]gives the remainder on division of abyb.\nThe behavior of mth\nzas a function of \randris shown in\nFig. 3(b), at Imax= 0:9,\u0001t= 10, and\"= 10. This con-\ntour plot recalls closely that obtained solving Eqs. (9)-(11)\nnumerically and shown in Fig. 3(a), especially at high .\r;r/\nvalues. Thismeansthatthesimpleassumptionsmadetoob-\ntain Fig. 3(b) allow to grasp, in our case, the essential fea-\ntures behind the magnetization reversal phenomenon. As a\nclosingremark,weobservethatananalyticalsolutionforthe\nmagnetization dynamics induced by a current pulse was re-\ncently proposed in Ref. [37]. In this work, Mazanik et al.\nformulate criteria for magnetization reversal in a '0junc-\ntions,obtainingagoodagreementbetweennumericalresults\nandanalyticalprediction,inspecificrangesofthesystempa-\nrameters.\n3.1. Noise effects\nInthissectionwediscusstheeffectsproducedbyanon-\nnegligible thermal noise source on the magnetization dy-\nnamics. The temperature can influence significantly the re-\nsponseofthesystem,sincethermalfluctuationsmayeventu-\nallyinduceanunwantedmagnetizationswitchorpreventthe\nmagnetizationreverse. Weinvestigatetheeffectsofstochas-\nticthermalfluctuationsinthephasedynamics,byincluding\na Gaussian noise source in the RSJ model, see Eq. (11).\nIn the following, we focus on the average components\nof the stationary magnetization, mst\nx,mst\ny, andmst\nz, which arecomputed by averaging over Nexp= 103independent nu-\nmerical repetitions in the presence of a non-negligible ther-\nmal noise intensity, DI0.\nFigure 6 shows the behavior of the average magnetiza-\ntion as a function of the noise intensity DI, obtained by fix-\ning\r=0:1and varying r; in particular we choose the .\r;r/\ncombinations used to obtain the curves in the top panels of\nFig. 4. In this way we explore the noise effects by focusing\nonthedifferentgraybandsinFigs.3and5atagiven \r. We\nindicate with D0\nIthe noise amplitude at which mst\nzstarts to\ndeviatesignificantlyfromthevalue *1andwithDIthenoise\namplitudeatwhichtheswitchingprocessisfullysuppressed,\nthat is when mst\nzapproaches the zero value. In all panels of\nFig.6wealsomarkwithadashedverticallinethe DIvalue\ncoinciding with the average barrier height, \u0001U.\r;r/.\nr=0.133\nr=0.135\nr=0.14\nr=0.15\nr=0.16\nr=0.166r=0.169\n10-40.001 0.010 0.100-1.0-0.8-0.6-0.4-0.20.0\nDImzstγ=0.1\nFigure 7: Average stationary magnetization, mst\nz, as a function\nof the thermal noise intensity, DI, at\r=0:1andrË.0:133*\n0:17/, calculated by averaging over Nexp= 103independent\nnumerical repetitions. Lines in the figure are guides for the\neye.\n:Preprint submitted to Elsevier Page 6 of 1010-1.0-0.50.00.51.0mxstr=0.09-γ=0.1\n10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0myst\n10-4 0.001 0.01   0.1  1 10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\nDImzst10-1.0-0.50.00.51.0r=0.1-γ=0.2\n10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\n10-4 0.001 0.01   0.1  1 10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\nDI10-1.0-0.50.00.51.0r=0.11-γ=0.3\n10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\n10-4 0.001 0.01   0.1  1 10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\nDI10-1.0-0.50.00.51.0\n10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\n10-4 0.001 0.01   0.1  1 10-1.0-0.510-4 0.001 0.01   0.1  1\n1.0\n0.5\n0.0\nDI(a) (b) (c) (d)r=0.12-γ=0.4Figure 8: Average stationary magnetizations, mst\nx,mst\ny, andmst\nz, as a function of the\nthermal noise intensity, DI, at the.\r;r/-values used to obtain the bottom panels of Fig. 4,\ncalculated by averaging over Nexp=103independent numerical repetitions. The vertical\ndashed lines indicate the average washboard potential barrier \u0001U.\r;r/.\nFrom Fig. 6(a), which is obtained for r=0:15and\r=\n0:1, one sees that mst\nzô *1only forDI¿ D0\nI= 0:01.\nFor higher noise intensities both mst\nzand the error bars in-\ncrease, approaching a zero value of mst\nzexactly forDIô\n\u0001U.\r;r/=0:051. Byincreasingfurtherthenoiseintensity,\ni.e., forDIÀDI,mst\nzstill remains close to zero, showing\nhowever quite large error bars that indicate a highly fluctu-\natingresponsetotallydrivenbynoise. Thevaluesof mst\nxand\nmst\nyhover around zero at each noise intensity, despite their\nerror bars tend to enlarge when DIÀDI.\nIncreasingr, both the value of D0\nIandDIreduces sig-\nnificantly, see Fig. 6(b-d), so that the greater r, the moreDI\ndeviatesfrom \u0001U. Thismeansthat,increasing r,thesystem\nis more sensitive to noise and the maximum temperature at\nwhichitcanreside,withoutsignificanteffectsonthestation-\nary magnetization, reduces. In other words, the robustness\nagainstthermalfluctuationsofthemagnetizationreversalef-\nfect is damaged by a high rvalue.\nToobtainthecurvesinFig.6,wefixed \rwhilerchanges\ninsuchawaytoexplorethesystemresponseinthedifferent\ngraybandsofFig.5(a),morespecifically,wechosethe rval-\nueslyingexactlyinthemidpointofeachband. Ontheother\nhand, we can demonstrate that, if we restrict now to just a\nsingle gray band, the impact of thermal noise can change\nsignificantly even for a small variation of r. For instance,\nin Fig. 7 we show how mst\nzversusDImodifies by setting\nrË.0:13;0:17/,with\r=0:1,thatiswefocusonthedarkest\nfringeinFig.5(a). Weobservethatimposing r=0:15noise\neffects are kept at a minimum, while thermal fluctuations\naffect more the magnetization switching if ris slightly in-\ncreased,ordecreased. Inparticular,thevalueof DIdependslittleonr,unlikethevalueof D0\nIwhichchangessignificantly\nby changing r. Specifically, for r=^0:15,0:14, and0:135`\nwe obtain the values D0\nIô^10,3, and0:5`10*3, respec-\ntively. If we suppose a temperature–dependent critical cur-\nrent, withIc= 10\u0016A at low temperatures1, these noise\nintensitiesD0\nIcorrespond to the normalized temperatures\nT_Tcô^0:85,0:58,and0:12`,respectively. Insummary,at\nafixed\r,theoptimalvalueof rcorrespondstothemidpoint\nofagrayband,whereasjustasmallchangeof risenoughto\nundermine the stability of the system.\nFinally, we discuss how an increase of \rcan influence\nthe magnetization reversal. In Fig. 8 we present the average\nstationary magnetizations mst\nx,mst\ny, andmst\nzversusDI, im-\nposing the.\r;r/combinations used to obtain the curves in\nthe bottom panels of Fig. 4. For \r= 0:1,mst\nzapproaches a\nzero value only for DIô\u0001U.\r;r/ = 0:054, see Fig. 8(a) .\nForhighernoiseintensities, mst\nzremainsclosetozero,show-\ning quite large error bars. The same happens for the xand\ny*components of the magnetization. The \rcoefficient acts\nasafrictiononthemagnetizationdynamics,sothatalarger\n\rmeans a system more “stiff” and, therefore, less sensitive\nto noisy disturbances. This is why the noise intensity DI,\nat whichmst\nzapproaches a zero value, increases with \r, see\nFigs.6(b)-(d). Inparticular,athigh \rvalues,weobservethat\nDIis always well above \u0001U.\r;r/. This means that, in view\nof a possible application based on a '0–junction, a larger \r\ncould allow a higher working temperature, still preserving\nthe magnetization reversal phenomenon. In principle, one\n1In the case of weak proximity effect and large exchange field in F,\nthe critical current temperature-dependence is proportional to Ic.T/ ×\n\u0001.T/tanh\u0004\u0001.T/_.kBT/\u0005[6], where\u0001.T/is the superconducting gap.\nThus, to find the temperature corresponding to a given noise intensity, we\nuse this relation, with a zero-temperature value Ic.0/=10\u0016A.\n:Preprint submitted to Elsevier Page 7 of 10couldthinktooptimizethesystemparametersinsuchaway\ntomake,forinstance,theswitchingtimeminimal,seeFig.5,\nstill keeping the device at a suitable working temperature.\n4. Conclusions\nInconclusion,inthispaperwediscussthebistablemag-\nnetic response of a current-biased '0–junction, that is a su-\nperconductor–ferromagnet–superconductorJosephsonjunc-\ntionwithaRashba-likespin-orbitcoupling. Thedirectionof\nthemagnetizationoftheferromagneticlayercanbeinverted\nvia controlled current pulses. We study the temporal evolu-\ntion of all the components of the magnetization in different\nconditions. We determine the values of intrinsic system pa-\nrameters, such as the Gilbert damping and strength of the\nspin-orbitcoupling,correspondingtoaminimumswitching\ntime of the magnetization. We also suggest a way to grasp\nreadilytheessentialfeaturesbehindthemagnetizationrever-\nsalphenomenonthroughsimpleassumptions,withoutfacing\nthenumericalsolutionsofthedifferentialequationsforboth\nthe magnetization and the Josephson dynamics.\nWe also explore how the magnetization switching time\nstrongly depends on the .\r;r/values. We observe a switch-\ningtimesofordernanoseconds,whichcanbeacceptablefor\nqubits[46],butcouldbetooslowforconventionallowtem-\nperature electronics. Also the specific shape of the driving\npulse and the orientation of the switching field can signifi-\ncantly affect the magnetization switching time [43]. How-\never, in this work, we aim to demonstrate that the switch-\ning time can strongly depend on the value of the parame-\nter.\r;r/. From the other side, we assumed fixed energy\nand timescales ratios. Usually, the energy ratio \"ranges\nfrom\"í 100[28], if the magnetic anisotropy is weak, to\n\"í 1[51], in the case of a stronger anisotropy. In our\ncalculation we choose an intermediate value, \"= 10. The\ntypical ferromagnet resonance frequency is !Fí 10GHz,\nwhilethecharacteristicJosephsonfrequencyisusuallyinthe\nrange!JË [10*100] GHz. In our work, we conserva-\ntively choose !=!J_!F= 1, but we could reasonably\nimposealsoahighervalueforthefrequencyratio !. Inthis\ncase, one can expect a strong suppression of the magneti-\nzation switching time (e.g., see Ref. [37]. Please note that\nin this paper the frequency ratio is defined as w=!F_!J,\nso thatw=!*1). Since!J= 2\u0019_\b0IcR, the magne-\ntization switching time could be reduced by adjusting the\nvalues of the junction parameters IcandR. Moreover, in\nthisworkweestimateamagnetizationswitchingtimeofthe\norderofnanosecondsbyassumingaferromagnetresonance\nfrequency equal to !Fí10GHz; since!F×K_M(with\nKandMbeing the anisotropy constant and the modulus of\nthe magnetization vector, respectively) one could envisage\ntoreducefurtherthisratiotoobtainashortermagnetization\nswitching time.\nFinally,weexploretherobustnessofthecurrent-induced\nmagnetizationreversalagainstthermalfluctuations,inorder\nto find the regime of system parameters in which the mag-\nnetization switching induced by a current pulse is more sta-ble. In particular, we demonstrate that the choice of a low r\nand/orahigh \rvaluecanbeconvenienttokeepthermalfluc-\ntuationsatbay,inordertoincreasethetemperatureatwhich\nthe system can reside still preserving the magnetization re-\nversal effect.\nTheinvestigationofthermaleffectsonhybridsupercon-\nductive/ferromagneticstructuresisimportantinapplications\nofnovelelectronicdevices,suchasspintronics[34,17],qubits\nandsuperconductinglogicelements[15],anddetectorswith\nelectron cooling [31]. Moreover, the role of noise can be-\ncomecrucialinhigh-speedswitchingelectronics,wheresta-\nbilizationeffectsduetonoisecanleadtoenhancementofthe\nswitchingtime,theso-callednoise-delayedswitchingeffect.\nIndeed, noise-enhanced stabilization effects can play a rele-\nvant role by reducing the size of the magnetic element [53]\nand are also demonstrated to be important in the switching\ndynamics of Josephson devices [57, 24, 25].\n5. Acknowledgments\nF.S.B. acknowledge funding by the Spanish Ministerio\ndeCiencia,InnovaciónyUniversidades(MICINN)(Project\nNo. FIS2017-82804-P), partial support by Grupos Consol-\nidados UPV/EHU del Gobierno Vasco (Grant No. IT1249-\n19),andbyEU’sHorizon2020researchandinnovationpro-\ngram under Grant Agreement No. 800923 (SUPERTED).\nReferences\n[1] Assouline, A., Feuillet-Palma, C., Bergeal, N., Zhang, T., Mot-\ntaghizadeh, A., Zimmers, A., Lhuillier, E., Eddrie, M., Atkinson, P.,\nAprili, M., et al., 2019. Spin-orbit induced phase-shift in Bi2Se3\nJosephson junctions. Nat. Commun. 10, 126. URL: https://doi.\norg/10.1038/s41467-018-08022-y , doi: 10.1038/s41467-018-08022-y .\n[2] Atanasova,P.K.,Panayotova,S.A.,Rahmonov,I.R.,Shukrinov,Y.M.,\nZemlyanaya, E.V., Bashashin, M.V., 2019. Periodicity in the ap-\npearance of intervals of the reversal of the magnetic moment of a '0\nJosephson junction. JETP Letters 110, 722–726. 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URL: https://\nlink.aps.org/doi/10.1103/PhysRevB.89.214510 ,doi: 10.1103/PhysRevB.89.214510 .\n:Preprint submitted to Elsevier Page 10 of 10"    },    {        "title": "1107.2165v3.Spin_and_charge_transport_induced_by_gauge_fields_in_a_ferromagnet.pdf",        "content": "arXiv:1107.2165v3  [cond-mat.mes-hall]  29 Sep 2011Spin and charge transport induced by gauge fields in a ferroma gnet\nJunya Shibata1,∗and Hiroshi Kohno2,†\n1Faculty of Science and Engineering, Toyo University, Kawag oe, Saitama, 350-8585, Japan\n2Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan\n(Dated: May 30, 2022)\nWe present a microscopic theory of spin-dependent motive fo rce (“spin motive force”) induced\nby magnetization dynamics in a conducting ferromagnet, by t aking account of spin relaxation of\nconduction electrons. The theory is developed by calculati ng spin and charge transport driven by\ntwo kinds of gauge fields; one is the ordinary electromagneti c fieldAem\nµ, and the other is the effective\ngauge field Az\nµinduced by dynamical magnetic texture. The latter acts in th e spin channel and gives\nrise to a spin motive force. It is found that the current induc ed as a linear response to Az\nµis not\ngauge-invariant in the presence of spin-flip processes. Thi s fact is intimately related to the non-\nconservation of spin via Onsager reciprocity, so is robust, but indicates a theoretical inconsistency.\nThis problem is resolved by considering the time dependence of spin-relaxation source terms in\nthe “rotated frame”, as in the previous study on Gilbert damp ing [J. Phys. Soc. Jpn. 76, 063710\n(2007)]. This effect restores the gauge invariance while kee ping spin non-conservation. It also gives\na dissipative spin motive force expected as a reciprocal to t he dissipative spin torque (“ β-term”).\nPACS numbers: 72.25.Pn, 72.15.Gd, 75.76.+j, 75.78.Fg\nI. INTRODUCTION\nManipulation of magnetization by electric currents1–3\nhas been studied intensively for a decade because of\npromising spintronic applications.4Among them, it was\ndemonstrated theoretically5and experimentally6that\nan electric current in a conducting ferromagnet can\ndrive magnetic textures such as domain walls and vor-\ntices. This is understood as due to spin torques that a\ncurrent exerts on magnetization through a microscopic\nexchange interaction. They include the spin-transfer\ntorque,7–10which is based on the conservation of total an-\ngular momentum, and its dissipative correction called β-\nterm,11–17which arises in the presence of spin-relaxation\nprocesses in the electron system.\nIn 1986, Berger predicted a reciprocal effect that a\nmoving domain wall accompanied by a periodic rota-\ntion of magnetization generates an electromotive force, in\nanalogy with the Josephson effect of superconductivity.18\nThis effect is now understood as a motive force acting in\nspin channel, hence called spin motive force,19–27which\ndrives majority-spin and minority-spin electrons in mu-\ntually opposite directions. It is also understood to arise\nfrom a time-dependent magnetic texture in general. Re-\ncently, it was experimentally detected by Yang el al.26for\na vortex wall in a ferromagnetic nanowire. Similar phe-\nnomena have also been studied in systems with interface\nor nanoparticles.28–33\nA theoretical framework for studying spin motive force\nin ferromagnets was presented by Volovik,19or earlier by\nKorenmann et al.34To treat electrons in a spin (or mag-\nnetization) texture, they introduced a local spin frame\nwhose quantization axis coincides with the local spin\ndirection,35n; then there arises naturally an effective\nU(1) gauge field, Az\nµ, acting in electron’s spin channel,which gives rise to an effective ‘electric’ field19,24\nE0\ns,i=/planckover2pi1\ne(∂iAz\n0−∂0Az\ni) =/planckover2pi1\n2en·(∂in×˙n),(1)\nor a spin motive force, Fs=−eEs(−e: electron charge).\nRecently, it was pointed out that it acquires a dissipative\ncorrection23,24\nEdis\ns,i=β/planckover2pi1\n2e˙n·∂in, (2)\nin the presence of spin relaxation of conduction electrons.\nThe total field is then given by Es=E0\ns+Edis\ns. These\ntwo terms are reciprocals to the spin-transfer torque and\nthe spin torque β-term, respectively,22–24and the di-\nmensionless parameter βis the same as that of spin\ntorque.11–17\nA spin motive field Esinduces an electric current\nj=σ↑Es+σ↓(−Es) =σsEs, (3)\nwhereσ↑(σ↓) is a conductivity of majority- (minority-)\nspin electrons, and σs=σ↑−σ↓is the ‘spin conductiv-\nity’. In most theoretical studies, this relation is used to\nidentify a spin motive force.22–24In the presence of spin-\norbit coupling, it induces in addition a charge Hall cur-\nrent,σSHn×E0\ns, whereσSHis a spin Hall conductivity,36\nand as a reciprocal to this, a spin Hall current induced by\nexternal electric field will exert a spin-transfer torque.37\nEnhancement of magnetization damping due to induced\nspin current was also discussed.38,39\nThe purpose of this paper is to develop a microscopic\ntheory of spin motive force basing on the gauge field\nmentioned above. For this, we found it instructive to\ntreat spin and charge channels in parallel. We thus study\nspin and charge transport induced by two kinds of gauge\nfields, one acting in charge channel (ordinary electromag-\nnetic field) and the other acting in spin channel (spin2\nmotive field). Particular attention is paid to the effects\nof spin relaxation of conduction electrons.\nIn the first part of this paper, we study spin and charge\ntransport in a uniformly magnetized state induced by an\nordinary electromagnetic field. Our calculation is equiva-\nlent to the well-studied two-current model,40–43but some\ninteresting crossover is pointed out in diffusion modes.\nIn the second part, we study a spin motive force by\ncalculating electric and spin currents induced by mag-\nnetization dynamics. We encounter a difficulty that the\ncurrent induced as a linear response to the effective gauge\nfieldAz\nµcontains gauge non-invariant terms in the pres-\nence of spin-flip processes. This difficulty is resolved by\nnoting that there is another contribution from the source\nterm of spin relaxation, as realized in the study of Gilbert\ndamping.17We also found that such additional contribu-\ntion reproduces the dissipative spin motive force.\nSuch additional contributions may look tricky, but\ntheir necessity can be understood on general grounds.\nIn the present gauge-field formalism, in which spin and\ncharge channels are treated equally, spin conservation\nand gauge invariance (in the spin channel) are equiva-\nlent at the linear-response level because of Onsager reci-\nprocity. However, the former is violated by spin-flip pro-\ncesses whereas the latter should always hold in order for\nthe theory to be consistent. These contradictory aspects\ncan only be reconciled by some additional contributions.\nThe paper is organized as follows. After describing a\nmodel in Sec. II, we examine in Sec. III the density and\ncurrent response to the ordinary electromagnetic field,\nAem\nµ. Here the magnetization is assumed to be static\nand uniform. In Sec. IV, we consider the case that the\nmagnetization varies in space and time. By introducing\nanother gauge field, Az\nµ, which expresses the effects of\nmagnetic texture and dynamics, we examine the density\nand current within the linear response to Az\nµ, with an\nunpleasant, gauge-dependent result. This problem is re-\nsolved in Sec. V, where a dissipative correction to spin\nmotive force is also obtained. Results and discussion are\ngiven in Sec. VI, and summary is given in Sec. VII. Cal-\nculational details are given in Appendices.\nII. MODEL\nWe consider a ferromagnetic conductor consisting of\nconducting s-electrons and localized d-spins. We assume\nthat the s-electrons are degenerate free electrons subject\nto impurity scattering, and localized d-spins are classi-\ncal, which are mutually coupled via the s-dexchange\ninteraction. The Lagrangian for s-electrons is given by\nL=Lel−Hsd:\nLel=/integraldisplay\ndrc†/bracketleftbigg\ni/planckover2pi1∂\n∂t+/planckover2pi12\n2m∇2+εF−Vimp/bracketrightbigg\nc,(4)\nHsd=−M/integraldisplay\ndr n·(c†σc)x, (5)wherec†(x) = (c†\n↑(x),c†\n↓(x))is the electron creation op-\nerator at x= (t,r),εFis the Fermi energy, Mis thes-d\nexchange coupling constant, nis a unit vector represent-\ning the direction of d-spin,35andσis a vector of Pauli\nspin matrices. The impurity potential is modeled by\nVimp(r) =ui/summationdisplay\niδ(r−Ri)+us/summationdisplay\njδ(r−R′\nj)Sj·σ,(6)\nwhereuiandRiare the strength and position of normal\nimpurities, which introduce momentum relaxation pro-\ncesses, and usandR′\njare those of quenched magnetic\nimpurities with spin Sj, which introduce spin-relaxation\nprocesses.14,17We take a quenched average for the impu-\nrity spin direction as Sα\ni= 0and44\nSα\niSβ\nj=δijδαβ×/braceleftbigg\nS2\n⊥(α,β=x,y)\nS2z(α,β=z)(7)\nas well as for the impurity positions, R′\niandR′\nj. When\nthe magnetization is uniform and static, n= ˆz, the\nimpurity-averaged Green’s function is given by\nGkσ(z) =1\nz−εk+εFσ+iγσsgn(Imz), (8)\nwherekis a wavevector, εk=/planckover2pi12k2/2m, andεFσ=\nεF+σM. The subscript σ=↑,↓represents the major-\nity and minority spins, respectively, and corresponds to\nσ= +1,−1in the formula (and to ¯σ=↓,↑or−1,+1).\nTreating Vimpas perturbation, the damping rate γσis\nevaluated in the first Born approximation as\nγσ=/planckover2pi1\n2τσ=π(˜Γ1νσ+˜Γ2ν¯σ), (9)\nwhereνσ=mkFσ/2π2/planckover2pi12is the density of states at εFσ\nwithkFσ=√2mεFσ//planckover2pi1and\n˜Γ1=niu2+nsu2\nsS2z, (10)\n˜Γ2= 2nsu2\nsS2\n⊥, (11)\nwithniandnsbeing the concentration of normal and\nmagnetic impurities, respectively. The first and second\nterms in Eq. (9) come from spin-conserving and spin-flip\nscattering processes, respectively.\nIn this paper, we assume γσ≪εFσand focus on diffu-\nsive transport induced by slowly-varying external pertur-\nbations (electromagnetic fields or time-dependent mag-\nnetic texture). Let qandωbe wavenumber and frequency\nof the perturbation, and define\nXσ= (Dσq2−iω)τσ, (12)\nwith a diffusion constant Dσ. Then our assumption\nthroughout the paper is expressed as γσ≪εFσand\n|Xσ| ≪1.3\nIII. SPIN AND CHARGE TRANSPORT IN\nUNIFORMLY MAGNETIZED STATE\nA. Linear response to electromagnetic field\nLet us examine the density and current response in\nthe charge channel, jµ= (ρ,j), and spin channel,\njs,µ= (ρs,js), to the external electromagnetic field,\nAem\nµ= (−φem,Aem).44,45HereφemandAemare scalar\nand vector potentials, respectively, and the time and\nspace components of the four currents are given by\nρ=−ec†c( =j(0)\n0), (13)\nj=j(0)+e\nmρAem,j(0)=−e/planckover2pi1\n2mic†↔\n∇c, (14)\nρs=−ec†σzc( =j(0)\ns,0), (15)\njs=j(0)\ns+e\nmρsAem,j(0)\ns=−e/planckover2pi1\n2mic†σz↔\n∇c,(16)\nwithc†↔\n∇c=c†∇c−(∇c†)c. We have defined ρsandjsto\nhave the same dimensions as ρandj, respectively. The\ncoupling to the external fields is given by\nHem=/integraldisplay\ndr(ρφem−j(0)·Aem)\n=−/integraldisplay\ndrj(0)\nµAem\nµ. (17)\nThe currents, jµandjs,µ, are evaluated in the linear\nresponse to Aem\nµas\n∝an}b∇acketle{tjµ(q)∝an}b∇acket∇i}htω=e2Kcc\nµν(q,ω+i0)Aem\nq,ν(ω),(18)\n∝an}b∇acketle{tjs,µ(q)∝an}b∇acket∇i}htω=e2Ksc\nµν(q,ω+i0)Aem\nq,ν(ω),(19)\nwhereAem\nq,ν(ω)is a Fourier component of Aem\nν(x). The\nresponse functions Kcc\nµνandKsc\nµνare obtained from\ne2Kcc\nµν(q,iωλ) =/integraldisplay1/T\n0dτ eiωλτ∝an}b∇acketle{tTτj(0)\nµ(q,τ)j(0)\nν(−q)∝an}b∇acket∇i}ht\n+e\nm∝an}b∇acketle{tρ∝an}b∇acket∇i}htδµν(1−δν0), (20)\ne2Ksc\nµν(q,iωλ) =/integraldisplay1/T\n0dτ eiωλτ∝an}b∇acketle{tTτj(0)\ns,µ(q,τ)j(0)\nν(−q)∝an}b∇acket∇i}ht\n+e\nm∝an}b∇acketle{tρs∝an}b∇acket∇i}htδµν(1−δν0), (21)\nby the analytic continuation, iωλ→/planckover2pi1ω+i0, whereωλ=\n2πλT(λ: integer) is a bosonic Matsubara frequency. In\nthis paper, we focus on absolute zero, T= 0. The average\n∝an}b∇acketle{t···∝an}b∇acket∇i}htis taken in the equilibrium state determined by L.\nThe Fourier components of the currents are given by\nj(0)\nµ(q) =−e/summationdisplay\nk,σvµc†\nk−,σck+,σ, (22)\nj(0)\ns,µ(q) =−e/summationdisplay\nk,σσvµc†\nk−,σck+,σ (23)=(a)\nËû\n÷v÷ vö +\n=û û\nû û ûöû\nûûö\n+\n+û û\nû û ûöû\nûûö\n+Kö÷\n(b)= v÷ vö\nv÷ v÷ v÷\nv÷v÷Èà1 Èà2\nÈà1 Èà2cc\nFIG. 1: (a) Diagrammatic expression of Kcc\nµν. The thick (thin)\nsolid line represents an electron line carrying Matsubara f re-\nquencyiεn+iωλ(iεn). The shaded part represents the vertex\nfunction, Λσ\nν. (b) Dyson equation for Λσ\nν. The dotted lines\nrepresent impurity scattering, either with ( ˜Γ2) or without ( ˜Γ1)\nspin-flip scattering.\nwith\nvµ=/braceleftbigg\n1 ( µ= 0)\n/planckover2pi1ki/m(µ=i= 1,2,3)(24)\nandk±=k±q/2.\nThe response functions are evaluated with the ladder-\ntype vertex corrections46[Fig. 1(a)]. Deferring the details\nto Appendix A, we give the results in the next subsection.\nThe results are concisely expressed with the quantities\nYσ=Dσq2−iω, (25)\nZ=Y↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht, (26)\nand a notation, ∝an}b∇acketle{t···∝an}b∇acket∇i}ht, meaning to sum over σ=↑,↓; for\nexample, ∝an}b∇acketle{tν∝an}b∇acket∇i}ht=ν↑+ν↓,∝an}b∇acketle{tσν∝an}b∇acket∇i}ht=ν↑−ν↓,∝an}b∇acketle{tDν∝an}b∇acket∇i}ht=D↑ν↑+\nD↓ν↓, and∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht=D↑ν↑−D↓ν↓. By defining (¯Y)σ=\nY¯σ, we may also use ∝an}b∇acketle{tDν¯Y∝an}b∇acket∇i}ht=D↑ν↑Y↓+D↓ν↓Y↑and\n∝an}b∇acketle{tσDν¯Y∝an}b∇acket∇i}ht=D↑ν↑Y↓−D↓ν↓Y↑.\nB. Result\n1. Charge channel\nThe response functions Kcc\nµν(q,ω+i0)[Eq. (20)] for\nthe electric density/current are obtained as\nKcc\n00=q2K, (27)\nKcc\ni0=Kcc\n0i=qiωK, (28)\nKcc\nij=iω/braceleftbigg\n∝an}b∇acketle{tDν∝an}b∇acket∇i}ht/parenleftbigg\nδij−qiqj\nq2/parenrightbigg\n−iωKqiqj\nq2/bracerightbigg\n,(29)\nwhere\nK=∝an}b∇acketle{tDν¯Y∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tν∝an}b∇acket∇i}ht∝an}b∇acketle{tDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht. (30)4\nThe following properties are seen.\n(i) Gauge invariance47and charge conservation are sat-\nisfied,\nKcc\nµνqν= 0, qµKcc\nµν= 0, (31)\nwhereqµ= (−ω,q)is a four wavevector.45\n(ii) For˜Γ2= 0(without spin-flip scattering), we have\nK=∝an}b∇acketle{tDν¯Y∝an}b∇acket∇i}ht\nY↑Y↓=/summationdisplay\nσDσνσ\nDσq2−iω. (32)\nThis means that up- and down-spin electrons diffuse in-\ndependently, and there are two independent diffusion\nmodes.\n(iii) For ˜Γ2∝ne}ationslash= 0, and in the long-wavelength and low-\nfrequency limit, τ−1\nsf≡2π˜Γ2∝an}b∇acketle{tν∝an}b∇acket∇i}ht//planckover2pi1≫ |Yσ|, we have\nK=∝an}b∇acketle{tν∝an}b∇acket∇i}ht∝an}b∇acketle{tDν∝an}b∇acket∇i}ht\n∝an}b∇acketle{tDν∝an}b∇acket∇i}htq2−iω∝an}b∇acketle{tν∝an}b∇acket∇i}ht=σc/e2\nDeffq2−iω(33)\nwhere\nDeff=∝an}b∇acketle{tDν∝an}b∇acket∇i}ht\n∝an}b∇acketle{tν∝an}b∇acket∇i}ht=D↑ν↑+D↓ν↓\nν↑+ν↓(34)\nis the effective diffusion constant, and\nσc=e2∝an}b∇acketle{tDν∝an}b∇acket∇i}ht=e2/summationdisplay\nσDσνσ (35)\nis the electrical conductivity. There is only one diffusion\nmode owing to the spin mixing ˜Γ2. In the opposite limit,\nτ−1\nsf≪ |Yσ|, we have the behavior (32).\nFinally, the charge density ρ≡ ∝an}b∇acketle{tj0(q)∝an}b∇acket∇i}htωand the cur-\nrent density j≡ ∝an}b∇acketle{tji(q)∝an}b∇acket∇i}htωare given by\nρ=−e2KdivE, (36)\nj=σcE+e2∝an}b∇acketle{tD2ν¯Y∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tDν∝an}b∇acket∇i}ht2\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht∇(divE),(37)\nwhereE(q,ω)is a Fourier component of the electric field:\nE(q,ω) =−iqφem(q,ω)+iωAem(q,ω)withdivE=iq·\nEand∇(divE) =iq(iq·E).\n2. Spin channel\nThe response functions Ksc\nµν(q,ω+i0)[Eq. (A8)] for\nspin density/currents are obtained as\nKsc\n00=q2(Ks+∆Ks), (38)\nKsc\n0i=qiω(Ks+∆Ks), (39)\nKsc\ni0=qiωKs, (40)\nKsc\nij=iω/braceleftbigg\n∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht/parenleftbigg\nδij−qiqj\nq2/parenrightbigg\n−iωKsqiqj\nq2/bracerightbigg\n,(41)with\nKs=∝an}b∇acketle{tσDν¯Y∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tν∝an}b∇acket∇i}ht∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht,(42)\nKs+∆Ks=∝an}b∇acketle{tσDν¯Y∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tσν∝an}b∇acket∇i}ht∝an}b∇acketle{tDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht.(43)\nThe difference\n∆Ks= 2π˜Γ2∝an}b∇acketle{tσν∝an}b∇acket∇i}ht∝an}b∇acketle{tDν∝an}b∇acket∇i}ht−∝an}b∇acketle{tν∝an}b∇acket∇i}ht∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht\n= 2π˜Γ2(σcν−−σsν+)/Ze2\n= 2π˜Γ2ν+σc(Pν−Pj)/Ze2(44)\narises if˜Γ2∝ne}ationslash= 0(andPν∝ne}ationslash=Pj). In Eq. (44),\nσs=e2∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht=e2/summationdisplay\nσσDσνσ (45)\nis the ‘spin conductivity’, and Pν=ν−/ν+andPj=\nσs/σcrepresent spin asymmetry in the density of states\nand in current density, respectively, which are different\nin general. The following properties are seen.\n(i) Gauge invariance is satisfied,\nKsc\nµνqν= 0, (46)\nbut spin conservation is not,\nqµKsc\nµν=−/parenleftbig\nq2δν0+ωqiδνi/parenrightbig\nω∆Ks∝ne}ationslash= 0, (47)\nif˜Γ2∝ne}ationslash= 0, whereiis a space component.48\n(ii) Depending on the relative magnitude of τ−1\nsfand\n|Yσ|, there are two regimes similarly to the charge chan-\nnel. More interestingly, however, for τ−1\nsf≫ |Yσ|, the\nmagnitudes of ρsandjscan be independent, governed,\nrespectively, by asymmetry in density of states and by\nasymmetry in conductivity; ρs∝Pνσcandjs∝σs.\nFinally, the spin density ρs≡ ∝an}b∇acketle{tjs,0(q)∝an}b∇acket∇i}htωand the spin-\ncurrent density js≡ ∝an}b∇acketle{tjs,i(q)∝an}b∇acket∇i}htωare given by\nρs=−e2(Ks+∆Ks)divE, (48)\njs=σsE+e2∝an}b∇acketle{tσD2ν¯Y∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tDν∝an}b∇acket∇i}ht∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht∇(divE).\n(49)\n3. Spin-resolved channel\nFrom Eqs. (36), (37), (48) and (49), we obtain the\n“spin-resolved” density and current,\nρσ=−e2KσdivE, (50)\njσ=σσE+e2DσKσ∇(divE), (51)\nwhere\nKσ=DσY¯σ+2π˜Γ2∝an}b∇acketle{tDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}htνσ. (52)5\nFrom Eqs. (50) and (51), we may derive\njσ=σσE−Dσ∇ρσ (53)\nwhere\nσσ=e2Dσνσ (54)\nis the “spin-resolved” conductivity. Further discussion\nwill be given in Sec. VI.\nIV. SPIN AND CHARGE TRANSPORT IN\nTIME-DEPENDENT SPIN TEXTURE\nIn the previous section, we studied spin and charge\ntransport in a ferromagnetic conductor in its uniformly\nmagnetized state. In the second part of this paper, which\nconsists of Sec. IV and Sec. V, we consider a more gen-\neral case in which the magnetization varies in space and\ntime. This magnetic texture and dynamics induce den-\nsity change and current even if Aem\nµis absent, which are\ncalculated in this paper in the first order in both spatial\ngradient and time derivative.\nA. Transformation to local spin frame\nTo treat the effects of space- and time-dependent mag-\nnetization, we introduce a local spin frame where the spin\nquantization axis of s-electrons is taken to be the d-spin\ndirection n(x)at each space-time point.19,34,49The orig-\ninal spinor cis then transformed to a spinor ain the\nnew frame (rotated frame) as c=Ua, whereUis a 2\n×2 unitary matrix satisfying c†(n·σ)c=a†σza. It is\nconvenient to take U=m·σwith\nm=/parenleftbigg\nsinθ\n2cosφ,sinθ\n2sinφ,cosθ\n2/parenrightbigg\n, (55)\nwhereθandφare ordinary spherical angles parametriz-\ningn. From space/time derivatives, ∂µc=U(∂µ+iAµ)a,\nthere arises an SU(2) gauge field\nAµ=−iU†∂µU=Aα\nµσα. (56)\nThis is an effective gauge field, which represents\nspace/time variations of magnetization. The Lagrangian\nin the rotated frame is then given by L=˜Lel−He−A,\n˜Lel=/integraldisplay\ndra†/bracketleftbigg\ni/planckover2pi1∂\n∂t+/planckover2pi12\n2m∇2+εF−˜Vimp+Mσz/bracketrightbigg\na,\n(57)\nHe−A=−/planckover2pi1\ne/integraldisplay\ndr˜jα\nµAα\nµ+/planckover2pi12\n2m/integraldisplay\ndrAα\niAα\nia†a,(58)\nwhere˜jα\nµ= (˜ρα,˜jα)is a four current representing spin\nand spin-current densities (“paramagnetic” component)in the rotated frame,\n˜ρα=−ea†σαa( =˜jα\n0), (59)\n˜jα=−e/planckover2pi1\n2mia†σα↔\n∇a. (60)\nThe spin part of the impurity potential ˜Vimpis ex-\npressed as Sα\nj(c†σαc) =˜Sα\nj(t)(a†σαa), where ˜Sα\nj(t) =\nRαβ(R′\nj,t)Sβ\njis the impurity spin in the rotated frame17\nwith\nRαβ= 2mαmβ−δαβ(61)\nbeing a3×3orthogonal matrix representing the same\nrotation as U. Hereafter, the anisotropy axis of impurity\nspins is defined in reference to the rotated frame\n˜Sα\ni˜Sβ\nj=δijδαβ×/braceleftbigg\nS2\n⊥(α,β=x,y)\nS2z(α,β=z).(62)\nB. Effective U(1) gauge field\nThere is some arbitrariness in the choice of the rotated\nframe; one could take c=U′a′withU′=Ue−iσzχ/2,\nwhereχis an arbitrary function of x. This arbitrariness\nis a gauge degree of freedom in the sense that physical\nquantities should not depend on it. It is in fact expressed\nas the gauge transformation on aandAµ,\na′=e−iσzχ/2a, (63)\nA′\nµ=−i(U′)†∂µU′\n=eiσzχ/2Aµe−iσzχ/2−σz∂µχ/2, (64)\nor, in componentwise,\nA′x\nµ+iA′y\nµ=e−iχ/parenleftbig\nAx\nµ+iAy\nµ/parenrightbig\n, (65)\nA′z\nµ=Az\nµ−∂µχ/2. (66)\nNote that its zcomponent Az\nµtransforms like a gauge\npotential in ordinary electromagnetism, hence can be re-\ngarded as a U(1) gauge field. In the following, when we\nrefer to gauge transformation, it means Eqs. (63)-(66). In\nthe next subsection, we study spin and charge transport\ndriven by magnetization dynamics as a linear response\nto this effective gauge field Az\nµ.\nGenerally, one can do a gradient expansion in terms\nofAα\nµ. The expansion parameter is qvFστσandωτσ\n(forAz\nµ),36whereq−1andωare characteristic length\nand frequency, respectively, of the magnetic texture. In\nthis work, we consider only the lowest nontrivial or-\nder in the expansion by assuming qvFστσ≪1and\nωτσ≪1. This condition coincides with the condition,\n|Xσ|=|Dσq2−iω|τσ≪1, declared below Eq. (12).\nIn typical experiments with Permalloy ( vFσ∼105m/s,\nτσ∼10−14s)50,q−1∼100 nm, ω∼100 MHz26, we have\nDσq2τ∼10−4andωτσ∼10−6, and the above conditions\nare satisfied quite well.6\nC. Linear response to Aem\nµandAz\nµ.\nLet us examine the density/current response to the\ntwo gauge fields, Aem\nµandAz\nµ. Spin density and currents\nconsidered here are the ones whose spin is projected on\nn(orˆzin the rotated frame), i.e.,ρs= ˜ρzand˜js=˜jz.\nThe total current densities contain the gauge fields,\njµ= (ρ,˜j+(eρAem+/planckover2pi1˜ραAα)/m), (67)\njs,µ= (ρs,˜js+(eρsAem+/planckover2pi1ρAz)/m),(68)\nfor charge and spin channels, where ρ=−ea†aand˜j=\n(−e/planckover2pi1/2mi)a†↔\n∇a. By generalizing Eqs. (18) and (19), we\nmay write\n∝an}b∇acketle{tjµ(q)∝an}b∇acket∇i}htω=e2˜Kcc\nµνAem\nν+e/planckover2pi1˜Kcs\nµνAz\nν, (69)\n∝an}b∇acketle{tjs,µ(q)∝an}b∇acket∇i}htω=e2˜Ksc\nµνAem\nν+e/planckover2pi1˜Kss\nµνAz\nν. (70)\nThe response functions, ˜Kcc\nµνand˜Ksc\nµν, are obtained from\nEqs. (20) and (21) by replacing the electron operators in\nthe original frame, c(c†), by those in the rotated frame,\na(a†), and are already calculated as Kcc\nµνandKsc\nµνin\nSec. III. Thus the response to Aem\nµin Eqs. (69) and (70)\nexactly follows the results there.\nLet us then focus on the response to Az\nµ, in particu-\nlar, on˜Kcs\nµν. (˜Kss\nµνwill be presented in Appendix D.)\nFrom the definition (linear-response formula), one can\nshow that the Onsager’s reciprocity relations hold,\n˜Kcs\nµν(q,iωλ) =˜Ksc\nνµ(−q,−iωλ), (71)\nor\n˜Kcs\nµν(q,ω+i0) =˜Ksc\nνµ(−q,−ω−i0). (72)\nFrom this, we see that\nqµ˜Kcs\nµν=˜Ksc\nνµqµ= 0, (73)\nnamely, the charge conservation is satisfied also in the\nresponse to Az\nµ. On the other hand, if ˜Γ2∝ne}ationslash= 0, spin\nis not conserved, qν˜Ksc\nνµ∝ne}ationslash= 0as seen before. This fact,\ncombined with Eq. (72), implies that ˜Kcs\nµνis not gauge\ninvariant,\n˜Kcs\nµνqν=qν˜Ksc\nνµ∝ne}ationslash= 0, (74)\nif˜Γ2∝ne}ationslash= 0. The gauge non-invariant terms in Eq. (69)\nmay be extracted as48\nj′\nµ(q,ω) =e/planckover2pi1∆Ks{q2δµ0+qiωδµi}Az\nq,0. (75)\nTo summarize, the calculation based on the gauge field\nAz\nµfails to respect gauge invariance in the presence of\nspin-flip scattering. Stated more explicitly, the density\nand current calculated as a linear response to Az\nµare not\ngauge invariant.51V. CAREFUL TREATMENT OF SPIN\nRELAXATION EFFECTS\nA. Restoration of gauge invariance\nThe lack of gauge invariance encountered in Sec. IV-\nC is due to an oversight of some contributions. We re-\ncall that the quenched magnetic impurities in the origi-\nnal frame become time-dependent in the rotated frame,\n˜Sj(t) =Rαβ(R′\nj,t)Sβ\nj. Therefore, we should treat the\nspin part of the impurity potential\nHs=us/summationdisplay\nj/integraldisplay\ndr˜Sj(t)δ(r−R′\nj)·(a†σa)x (76)\nas a time-dependent perturbation. The same situation\nwas met in the calculation of Gilbert damping.17\nSince the first-order (linear) response vanishes,\n˜Sα\nj(t) = 0, let us consider the second-order (nonlinear)\nresponse,\n∆jµ(q,ω)\n=−ensu2\ns/integraldisplay∞\n−∞dω′\n2πχαβ\nµ(q;ω,ω′)/bracketleftbig˜Sα(ω−ω′)˜Sβ(ω′)/bracketrightbig\nq\n(77)\nwhere˜Sα\np(ω)is the Fourier component of/summationtext\nj˜Sα\nj(t)δ(r−\nR′\nj), andχαβ\nµis the nonlinear response function.17To\ncalculate it, it is simpler to use the path-ordered Green’s\nfunction.52The contribution represented in Fig. 2 are\ngiven by\nχαβ\nµ(q;ω,ω′) =/summationdisplay\nk,k′/integraldisplay∞\n−∞dε\n2πi\n×tr[(vµ+Λµ)Gk+(ε+)σαGk′(ε+ω′)σβGk−(ε)]<\n(78)\nwhereε+=ε+ω. The Green’s function Gk(ε)now\nstands for a path-ordered one, whose lesser component is\ngiven by\nG<\nk(ε) =f(ε)(GA\nk(ε)−GR\nk(ε)), (79)\nwithf(ε)being the Fermi distribution function. In\nEq. (78), we adopt a matrix notaion, (G)σ,σ′=Gσδσσ′,\n(Λµ)σ,σ′= Λσ\nµδσσ′withΛσ\nµgiven by Eq. (A6), and ‘tr’\nmeans trace in spin space.\nWe expand χαβ\nµ(q;ω,ω′)with respect to ωandω′as\nχαβ\nµ(q;ω,ω′) =Aαβ\nµ−iωBαβ\nµ−iω′Cαβ\nµ+··· (80)\nwhereAαβ\nµ,Bαβ\nµandCαβ\nµare the expansion coefficients.\nSubstituting Eq. (80) into Eq. (77), we have\n∆jµ(q,ω) =−ensu2\ns/bracketleftBig\nBαβ\nµ∂t(˜Sα˜Sβ)+Cαβ\nµ˜Sα∂t˜Sβ/bracketrightBig\nq,ω\n(81)7\nvö +vöûë\nûì\"+!0\"+!\n\"ûë\nûì\"+!0\"+!\n\"ÿëì\nö=\nFIG. 2: Diagrammatic expression of χαβ\nµ. The wavy line\nrepresents scattering from impurity spins, which are time-\ndependent in the rotated frame. The shaded part represents\nthe vertex function Λσ\nµ.\nwhere˜S=˜S(t)is time dependent. (We have dropped\na term containing Aαβ\nµ, which does not reflect the time\ndependence of ˜S(t).) From\n˜Sα∂t˜Sβ= (S2\n⊥δαγ\n⊥+S2zδαzδγz)(R∂tR)γβ,(82)\nwhereδαβ\n⊥=δαβ−δαzδβz, and the relation17\n(R∂µR)αβ= 2εαβγAγ\nµ, (83)\nwe see that Eq. (81) describes a response to Aγ\n0. The\ncoefficients are calculated as48(see Appendix B)\nBαβ\nµ=−1\n2Cαβ\nµ,\n=πν↑ν↓∝an}b∇acketle{tσY∝an}b∇acket∇i}htδµ0+iqi∝an}b∇acketle{tσD¯Y∝an}b∇acket∇i}htδµi\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}htεαβ(84)\nwhereεαβ=εαβz, and we have dropped unimportant\nterms proportional to δαβ\n⊥orδαzδγz. We thus have\n∆jµ(q,ω) =e/planckover2pi1∆˜Kcs\nµνAz\nq,ν, (85)\nwith48\n∆˜Kcs\nµν=−∆Ks{q2δµ0+qiωδµi}δν0. (86)\nThis new contribution cancels the gauge-dependent\nterms, Eq. (75), and restores the gauge invariance,\n(˜Kcs\nµν+∆˜Kcs\nµν)qν= 0. (87)\nNote that it does not affect the charge conservation since\nqµ∆˜Kcs\nµν= 0, nor the spin non-conservation ( qµ˜Ksc\nµν∝ne}ationslash= 0)\nsince it does not contribute to ˜Ksc\nµν.\nThe gauge-invariant result for the charge density\nρsmf (1)(q,ω)and current density jsmf (1)(q,ω)induced\nby magnetization dynamics is summarized as\nρsmf (1)=−e2KsdivE0\ns, (88)\njsmf (1)=σsE0\ns\n+e2∝an}b∇acketle{tσD2ν¯Y∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tDν∝an}b∇acket∇i}ht∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket���i}ht∇(divE0\ns).\n(89)v\nvûë\nûàì\"+!0\"+!\n\"\nûë\nûì\"+!0\"+!\n\"=\nm+ûë\nûì\"+!0\"+!\n\"1v\nvûë\nûì\"+!0\"+!\n\"\nûë\nûì\"+!0\"+!\n\"++\n+ÿëìí\nöi ö ö\nö öûívi\nîöiûívi\nûívi ûívi\nûí\nFIG. 3: Diagrammatic expression of χαβγ\nµi. The gray circle\nrepresents the interaction with Aγ\nµ.\nThe first term on the right-hand side of Eq. (89) has\nthe form of Eq. (3), and implies the existence of spin-\ndependent motive force described by the effective ‘elec-\ntric’ field E0\ns. The second term of Eq. (89) represents a\ndiffusion current arising from charge imbalance induced\nbyE0\ns, as made clear in Sec. VI. This term implies the ex-\nistence of nonlocal spin-transfer torque as the reciprocal\neffect, whose study will be left to the future.\nB. Dissipative correction\nIt is important to note that there is one more con-\ntribution within the same order in gradient expansion.\nIt is essentially given by Eq. (77), but with one more\nfactor of Aα\nµ. The response function, denoted by χαβγ\nµi,\nis obtained from Eq. (78) by further extracting Aα\nµvia\nEq. (58). These are expressed as [Fig. 3]\njsmf (2)\nµ(q,ω) =−e/planckover2pi1nsu2\ns/summationdisplay\nq′/integraldisplay∞\n−∞dω′\n2πχαβγ\nµi(q;ω,ω′)\n×/bracketleftbig˜Sα(ω−ω′)˜Sβ(ω′)/bracketrightbig\nq−q′Aγ\nq′,i,(90)\nwhere\nχαβγ\nµi(q;ω,ω′) =/summationdisplay\nk,k′/integraldisplay∞\n−∞dε\n2πitr/bracketleftBig\n(vµ+Λµ)\n× {v+\niGk+(ε+)σγGk+(ε+)σαGk′(ε+ω′)σβGk−(ε)\n+v−\niGk+(ε+)σαGk′(ε+ω′)σβGk−(ε)σγGk−(ε)}\n+1\nmδµiσγGk+(ε+)σαGk′(ε+ω′)σβGk−(ε)/bracketrightBig<\n,(91)\nwithv±\ni= (ki±qi/2)/m. We have put q′=0in Eq. (91),\nbut retained qandω. Note that the terms with γ=z\ncancel out, and Az\nµdoes not contribute. In the same way\nas Sec. V-A, we expand χαβγ\nµiwith respect to ωandω′8\nasχαβγ\nµi=Aαβγ\nµi−iωBαβγ\nµi−iω′Cαβγ\nµi+···and focus on\nthe coefficients Bαβγ\nµiandCαβγ\nµi. Deferring the details to\nAppendix C, we cite the result\nBαβγ\nµi=−1\n2Cαβγ\nµi\n= (δαzεβγ−δβzεαγ)ν+\n4M/summationdisplay\nσσ(Lσ\niµ)RA,(92)\nwhereν+=ν↑+ν↓, andLσ\niµ’s are given by Eqs. (A15)\nand (A17). Note the order of the subscripts, iµ. We thus\nhave\njsmf (2)\nµ(q,ω) =βe/planckover2pi1\nπ/summationdisplay\nσσ(Lσ\niµ)RA(A⊥\ni·A⊥\n0)q,ω(93)\nwhereA⊥\nµ=Aµ−ˆz(ˆz·Aµ), and\nβ=π\nMnsu2\ns(S2\n⊥+S2z)(ν↑+ν↓) (94)\nis a measure of spin relaxation. With the relation\nA⊥\ni·A⊥\n0=1\n4˙n·∂in, (95)\nwhich is gauge-invariant under (65), we finally obtain\nρsmf (2)=−e2KsdivEdis\ns, (96)\njsmf (2)=σsEdis\ns\n+e2∝an}b∇acketle{tσD2ν¯Y∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tDν∝an}b∇acket∇i}ht∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht∇(divEdis\ns),\n(97)\nwhereEdis\nsis given by Eq. (2). Since Edis\nscontains βas\na prefactor, Eqs. (96) and (97) come from spin-relaxation\nprocesses. This βis exactly the same as the coefficient\nof theβ-term of current-induced torque,14,17consistent\nwith the fact that these are reciprocal to each other.23,24\nVI. RESULTS AND DISCUSSION\nThe results obtained in this paper are summarized as\nρ=−∝an}b∇acketle{tDν¯YF∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tν∝an}b∇acket∇i}ht∝an}b∇acketle{tDνF∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht, (98)\nj=σcE+σsEs\n+∝an}b∇acketle{tD2ν¯Y∇F∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tDν∝an}b∇acket∇i}ht∝an}b∇acketle{tDν∇F∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht,(99)\nρs=−∝an}b∇acketle{tσDν¯YF∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tσν∝an}b∇acket∇i}ht∝an}b∇acketle{tDνF∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht, (100)\njs=σsE+σcEs\n+∝an}b∇acketle{tσD2ν¯Y∇F∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht∝an}b∇acketle{tDν∇F∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht,(101)whereFc=e2divE,Fs=e2divEs, andFσ=Fc+σFs.\nThe notations are as before; for example, ∝an}b∇acketle{tDν∇F∝an}b∇acket∇i}ht=\nD↑ν↑∇F↑+D↓ν↓∇F↓. From these relations [or Eqs. (104)\nand (105) below], we identify the spin motive field to be\nEs,i=/planckover2pi1\n2e{−n·(˙n×∂in)+β(˙n·∂in)}.(102)\nThe spin-resolved density and current are given by\nρσ=−e2div(KσE+K′\nσEs), (103)\njσ=σσEσ−Dσ∇ρσ, (104)\nEσ=E+σEs, (105)\nwhereEσis the total field felt by spin- σelectrons. The\ncoefficient Kσis given by Eq. (52), and K′\nσby\nK′\nσ=σDσY¯σ+2π˜Γ2∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}htνσ. (106)\nThere are two characteristic regimes depending on the\nrelative magnitude of τ−1\nsf≡2π˜Γ2∝an}b∇acketle{tν∝an}b∇acket∇i}ht//planckover2pi1and|Yσ|. For\nτ−1\nsf≪ |Yσ|, Eq. (103) becomes\nρσ≃ −σσ\nDσq2−iωdivEσ, (107)\nmeaning that the spin- σelectrons respond only to Eσ,\nnot toE¯σ, and the two spin components ( ↑and↓) be-\nhave independently. In particular, the response to a spin\nmotive field Es(setE=0for simplicity) is opposite in\nsign between ↑and↓electrons. In the opposite limit,\nτ−1\nsf≫ |Yσ|, Eq. (103) becomes\nρσ≃ −νσ/∝an}b∇acketle{tν∝an}b∇acket∇i}ht\nDeffq2−iωdiv(σ↑E↑+σ↓E↓),(108)\nwhereDeff=∝an}b∇acketle{tDν∝an}b∇acket∇i}ht/∝an}b∇acketle{tν∝an}b∇acket∇i}ht. In this case, the density of spin-\nσelectrons is affected not only by Eσbut also by E¯σ.\nThis is due to the strong spin mixing; as an elementary\nprocess, ρσis induced solely by Eσ, notE¯σ, but subse-\nquent spin-flip processes tends to equilibrate ρ↑andρ↓.\nNote that ↑electrons and ↓electrons respond to Eswith\nthe same sign. (The common sign is determined by that\nofσ↑−σ↓.)\nThe above features oppose the picture of two indepen-\ndentcurrents, but they are actually described within the\nconventional two-current model.40–43This is best demon-\nstrated by the relation\n∂\n∂tρσ+divjσ=−/parenleftbiggρσ\nτsf,σ−ρ¯σ\nτsf,¯σ/parenrightbigg\n, (109)\nwhere\nτ−1\nsf,σ= 2π˜Γ2ν¯σ//planckover2pi1 (110)\nis the spin-flip rate for spin- σelectrons. The right-\nhand side of Eq. (109) represents a coupling between ↑\nand↓electrons. In deriving Eq. (109), we have used9\nEqs. (103), (104), (106) and (52), and the relations,\n∝an}b∇acketle{tσK/ν∝an}b∇acket∇i}ht=∝an}b∇acketle{tσD¯Y∝an}b∇acket∇i}ht/Zand∝an}b∇acketle{tσK′/ν∝an}b∇acket∇i}ht=∝an}b∇acketle{tD¯Y∝an}b∇acket∇i}ht/Z. Note that\nρσ, being given by Eq. (103), represents a deviation from\nthe equilibrium value. One may define the deviation of\nchemical potential, δµσ, from equilibrium by\nρσ=−eνσδµσ. (111)\nThen Eq. (109) can be put in a familiar form42,43\n∂\n∂tρσ+divjσ=σσ\ne·δµσ−δµ¯σ\nℓ2σ. (112)\nwhereℓσ=/radicalbig\nDστsf,σis the spin diffusion length for spin-\nσelectrons.\nThe present work is therefore within the two-current\npicture. This fact was implicitly used in identifying the\nspin motive force on the basis of Eq. (3).\nVII. SUMMARY\nIn this paper, we have studied spin and charge trans-\nport in a conducting ferromagnet driven by two kinds of\ngauge fields, Aem\nµandAz\nµ, which act in charge channel\nand spin channel, respectively. In particular, we have\ngiven a microscopic calculation of spin motive force by\ntaking spin-relaxation effects into account.\nIn the first part, we calculated density and current in\nboth spin and charge channels in response to the ordi-\nnary electromagnetic field Aem\nµin a uniformly magne-\ntized state. We observed a crossover from two diffusion\nmodes to a single mode as the spin-flip rate is increased\n(for a fixed frequency/wavenumber of the disturbance),\nor as the frequency/wavenumber is decreased (for a fixed\nspin-flip rate). However, if expressed in terms of spin-\nresolved density and current, the so-called two-current\nmodel is shown to hold irrespective of the strength of\nspin-flip scattering.\nIn the second part, we have developed a microscopic\ntheory of spin motive force in the framework of gauge-\nfield method. We readily encountered the problem of\ngauge non-invariance; the current calculated as a linear\nresponse to Az\nµdepends on the gauge (choice of local\nspin frame). This fact is intimately related to the non-\nconservation of spin (due to spin-flip scattering) by On-\nsager reciprocity, hence is robust. This theoretical puzzl e\nwas resolved by noting the fact that the spin-dependent\nscattering terms (quenched impurity spins) are time-\ndependent in the rotated frame. By calculating the\nsecond-order (nonlinear) response to this time-dependent\nperturbation, we could recover a gauge-invariant result\nwhile keeping the spin non-conservation. The dissipative\ncorrection to the ordinary spin motive force, which is the\ninverse to the spin-torque β-term, is also obtained.\nNote added: After submitting the manuscript, we be-\ncame aware of a closely related work by Kim et al.55Acknowledgments\nThe authors would like to thank G. Tatara for dis-\ncussions, and K.-W. Kim for informing us of Ref. 55.\nThis work is partially supported by a Grant-in-Aid from\nMonka-sho, Japan.\nAppendix A: Calculation of response functions Kcc\nµν\nandKsc\nµν\nIn this Appendix, we evaluate the electromagntic re-\nsponse functions in the ladder approximation shown in\nFig. 1(a). From Eqs. (20) and (21), they are written as\nKcc\nµν(q,iωλ) =−T/summationdisplay\nn,σLσ\nµν(q;iεn+iωλ,iεn),(A1)\nKsc\nµν(q,iωλ) =−T/summationdisplay\nn,σσLσ\nµν(q;iεn+iωλ,iεn),(A2)\nwith\nLσ\nµν= Πσ\nµν+Πσ\nµ0Λσ\nν, (A3)\nΠσ\nµν=/summationdisplay\nkvµvνGk+,σ(iεn+iωλ)Gk−,σ(iεn),(A4)\nwhereεn= (2n+1)πT(n: integer) is a fermionic Matsub-\nara frequency. The vertex function Λσ\nνsatisfies [Fig. 1(b)]\nΛσ\nν=λσ\nν+˜Γ1ΠσΛσ\nν+˜Γ2Π¯σΛ¯σ\nν, (A5)\nwhereΠσ= Πσ\n00, andλσ\nν=˜Γ1Πσ\n0ν+˜Γ2Π¯σ\n0νis the lowest-\norder contribution. The equation (A5) is solved as\nΛσ\nν=λσ\nν−Π¯σ(˜Γ1λσ\nν−˜Γ2λ¯σ\nν)\n1−˜Γ1(Π↑+Π↓)+(˜Γ2\n1−˜Γ2\n2)Π↑Π↓.(A6)\nPerforming the analytic continuation, iωλ→ω+i0and\nretaining terms up to the first order in ω, we obtain\nKcc\nµν(q,ω+i0) =ν+δµ0δν0+iω\n2π/summationdisplay\nσ(Lσ\nµν)RA,(A7)\nKsc\nµν(q,ω+i0) =ν−δµ0δν0+iω\n2π/summationdisplay\nσσ(Lσ\nµν)RA,(A8)\nwhereν±=ν↑±ν↓. The function (Lσ\nµν)RAis obtained\nvia the analytic continuation, i(εn+ωλ)→ε+ω+i0\nandiεn→ε−i0, as indicated by the superscript “RA”.\nWe assume γσ≪εFσ, and discard (Lσ\nµν)RRand(Lσ\nµν)AA\nas in usual calculations of transport coefficients. The k-10\nintegrals are evaluated up to O(|Xσ|)orO(|Xσ|0)as\n(Πσ)RA=/summationdisplay\nkGR\nk+,σ(ω)GA\nk−,σ(0)\n≃2πνστσ(1−Xσ), (A9)\n(Πσ\ni0)RA=/summationdisplay\nkviGR\nk+,σ(ω)GA\nk−,σ(0)\n≃ −2πiqiDσνστσ, (A10)\n(Πσ\nij)RA=/summationdisplay\nkvivjGR\nk+,σ(ω)GA\nk−,σ(0)\n≃2πDσνσδij. (A11)\nwhereDσ=v2\nFστσ/3,vFσ=/planckover2pi1kFσ/m, andXσ=Yστσ\nwithYσ=Dσq2−iω. Using these formulas, we obtain\n(Λσ\n0)RA=Y¯σ+2π˜Γ2∝an}b∇acketle{tν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht·1\nτσ, (A12)\n(Λσ\ni)RA=−iqiDσY¯σ+2π˜Γ2∝an}b∇acketle{tDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht·1\nτσ,(A13)and thus\n(Lσ\n00)RA= 2πνσY¯σ+2π˜Γ2∝an}b∇acketle{tν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht, (A14)\n(Lσ\ni0)RA=−2πiqiνσDσY¯σ+2π˜Γ2∝an}b∇acketle{tν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht,(A15)\n(Lσ\n0i)RA=−2πiqiνσDσY¯σ+2π˜Γ2∝an}b∇acketle{tDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht,(A16)\n(Lσ\nij)RA= 2πνσDσ/braceleftbigg/parenleftbigg\nδij−qiqj\nq2/parenrightbigg\n−iωqiqj\nq2Y¯σ+2π˜Γ2∝an}b∇acketle{tν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht/bracerightBigg\n.(A17)\nAppendix B: Calculation of Cαβ\nµ\nThe nonlinear response function χαβ\nµin Eq. (78) is written as\nχαβ\nµ(q;ω,ω′) =/summationdisplay\nσ,σ′/bracketleftbig\n(δαβ\n⊥+iσεαβ)δσ′¯σ+δαzδβzδσ′σ/bracketrightbig/integraldisplay∞\n−∞dε\n2πi(Lσ\n0µ(q;ε+ω,ε)Iσ′(ε+ω′))<, (B1)\nwhereLσ\n0µis given by Eq. (A3), and Iσ(ε) =/summationtext\nkGkσ(ε). Following the Langreth’s method,53,54the lesser component\nofLσ\n0µ(q;ε+ω,ε)I¯σ(ε+ω′)≡LIis calculated as\n(LI)<=f(ε)(LRA−LRR)IR+f(ε+ω′)LRA(IA−IR)+f(ε+ω)(LAA−LRA)IA. (B2)\nNote that the ordering of Green’s functions in LIisG(ε+ω)G(ε+ω′)G(ε)[see Eq. (78)]. The superscripts RA, A\netc. specify the analytic branch; for example, LRA(ε+ω,ε) =L(ε+ω+i0,ε−i0),IA(ε) =I(ε−i0), etc. Thus the\ncoefficients in the expansion χαβ\nµ=Aαβ\nµ−iωBαβ\nµ−iω′Cαβ\nµ+···are obtained as\nBαβ\nµ=1\n2π/summationdisplay\nσ,σ′/bracketleftbig\n(δαβ\n⊥+iσεαβ)δσ′¯σ+δαzδβzδσ′σ/bracketrightbig\n(Lσ\n0µ(q;ω,0))RAIA\nσ′(0), (B3)\nCαβ\nµ=i\nπ/summationdisplay\nσ,σ′/bracketleftbig\n(δαβ\n⊥+iσεαβ)δσ′¯σ+δαzδβzδσ′σ/bracketrightbig\n(Lσ\n0µ(q;ω,0))RAImIR\nσ′(0). (B4)\nWe have retained only the lowest-order term in γσ. Substituting Eqs. (A14) and (A16) together with IR\nσ(0) =−iπνσ\n(whose real part is dropped consistently with the selfenerg y) into Eq. (B4), we obtain Eq. (84).11\nAppendix C: Calculation of Cαβγ\nµi\nConsider the nonlinear response function χαβγ\nµigiven by Eq. (91). As in Appendix B, we take a lesser component ,\nextract the ω′-linear term, and retain terms containing both GRandGAto obtain Bαβγ\nµi=−(1/2)Cαβγ\nµiand\nCαβγ\nµi=i∂\n∂ω′χαβγ\nµi(ω,ω′)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω′=0\n≃ −i/summationdisplay\nktr/bracketleftbig\n(vµ+ΛRA\nµ)vi{GR\nk+σγGR\nk+σαˆνσβGA\nk−+GR\nk+σαˆνσβGA\nk−σγGA\nk−}/bracketrightbig\n−i\nmδµi/summationdisplay\nktr/bracketleftbig\nσγGR\nk+σαˆνσβGA\nk−/bracketrightbig\n. (C1)\nHere(ΛRA\nµ)σσ′= (Λσ\nµ)RAδσσ′is given by Eqs. (A12)-(A13), and ˆν=/summationtext\nk′(GA\nk′−GR\nk′)/2πiis a matrix of density of\nstates,(ˆν)σσ′=νσδσσ′. In Eq. (C1), all G’s are evaluated at ε= 0except for those in Λµin which q,ωare retained.\nEquation (C1) is written as\nCαβγ\nµi=i/summationdisplay\nσ/bracketleftbig\nδαz(σδβγ\n⊥−iεβγ)ν¯σ−δβz(σδαγ\n⊥−iεαγ)νσ/bracketrightbig/braceleftbig\nMσ\nµi(q,ω)+¯M¯σ\nµi(q,ω)/bracerightbig\n, (C2)\nwhere\nMσ\nµi(q,ω) =Qσ\nµi(q)+(Λσ\nµ)RAQσ\n0i(q), (C3)\n¯Mσ\nµi(q,ω) =¯Qσ\nµi(q)+(Λσ\nµ)RA¯Qσ\n0i(q), (C4)\nQσ\nµi(q) =/summationdisplay\nkvµviGR\nk+,σGR\nk−,¯σGA\nk−,σ/vextendsingle/vextendsingle\nε=0= [¯Qσ\nµi(−q)]∗, (C5)\nIn the lowest order in γσ, we see that\nMσ\nµi(q,ω) =¯Mσ\nµi(q,ω) =−σ\n2M(Lσ\niµ)RA, (C6)\nwhere(Lσ\niµ)RAis given by Eqs. (A15) and (A17). Noting that Mσ\nµi+¯M¯σ\nµi=−/summationtext\nσσ(Lσ\niµ)RA/2Mis independent of\nσ, we obtain the leading term as\nCαβγ\nµi=−(δαzεβγ−δβzεαγ)ν+\n2M/summationdisplay\nσσ(Lσ\niµ)RA. (C7)12\nAppendix D: Spin current induced by spin motive\nforce\nThe response function ˜Kss\nµνin Eq. (70) is evaluated as\n˜Kss\nµν=ν+δµ0δν0+iω\n2π/summationdisplay\nσσ(Lσ\ns,µν)RA,(D1)\nLσ\ns,µν=σΠσ\nµν+Πσ\nµ0Λσ\ns,ν. (D2)\nThe spin-current vertex function Λσ\ns,µ, which satisfies\nΛσ\ns,ν=λσ\ns,ν+˜Γ1ΠσΛσ\ns,ν−˜Γ2Π¯σΛ¯σ\ns,ν, (D3)\nwithλσ\ns,ν=σ(˜Γ1Πσ\n0ν−˜Γ2Π¯σ\n0ν), is given by\n(Λσ\ns,0)RA=σY¯σ+2π˜Γ2∝an}b∇acketle{tσν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht1\nτσ, (D4)\n(Λσ\ns,i)RA=−iqiσDσY¯σ+2π˜Γ2∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht1\nτσ.(D5)Hence, we have\n(Lσ\ns,00)RA= 2πνσσY¯σ+2π˜Γ2∝an}b∇acketle{tσν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht, (D6)\n(Lσ\ns,i0)RA=−2πiqiDσνσσY¯σ+2π˜Γ2∝an}b∇acketle{tσν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht, (D7)\n(Lσ\ns,0i)RA=−2πiqiνσσDσY¯σ+2π˜Γ2∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht, (D8)\n(Lσ\ns,ij)RA= 2πDσνσ/braceleftBigg\nσδij−qiqjσDσY¯σ+2π˜Γ2∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht/bracerightBigg\n.\n(D9)\nNote that ˜Kss\nµν’s thus obtained do not satisfy spin con-\nservation nor gauge invariance, qµ˜Kss\nµν=˜Kss\nνµqµ∝ne}ationslash= 0, if\n˜Γ2∝ne}ationslash= 0.\nAs in Sec. V, time-dependent magnetic impurities, Eq. (76), in the rotated frame also induce a spin current\n∆js,µ(q,ω) =−ensu2\ns/integraldisplay∞\n−∞dω′\n2πχαβ\ns,µ(q;ω,ω′)/bracketleftbig˜Sα(ω−ω′)˜Sβ(ω′)/bracketrightbig\nq\n−e/planckover2pi1nsu2\ns/summationdisplay\nq′/integraldisplay∞\n−∞dω′\n2πχαβγ\ns,µi(q;ω,ω′)/bracketleftbig˜Sα(ω−ω′)˜Sβ(ω′)/bracketrightbig\nq−q′Aγ\nq′,i, (D10)\nwhere\nχαβ\ns,µ(q;ω,ω′) =/summationdisplay\nk,k′/integraldisplay∞\n−∞dε\n2πitr[(vµσz+Λs,µ)Gk+(ε+ω)σαGk′(ε+ω′)σβGk−(ε)]<, (D11)\nχαβγ\ns,µi(q;ω,ω′) =/summationdisplay\nk,k′/integraldisplay∞\n−∞dε\n2πitr/bracketleftbig\n(vµσz+Λs,µ)v+\niGk+(ε+ω)σγGk+(ε+ω)σαGk′(ε+ω′)σβGk−(ε)/bracketrightbig<\n+/summationdisplay\nk,k′/integraldisplay∞\n−∞dε\n2πitr/bracketleftbig\n(vµσz+Λs,µ)v−\niGk+(ε+ω)σαGk′(ε+ω′)σβGk−(ε)σγGk−(ε)/bracketrightbig<\n+1\nmδµiδγz/summationdisplay\nk,k′/integraldisplay∞\n−∞dε\n2πitr/bracketleftbig\nGk+(ε+ω)σαGk′(ε+ω′)σβGk−(ε)/bracketrightbig<, (D12)\nwithv±\ni= (ki±qi/2)/m. We have put q′=0in\nEq. (D12). By taking the lesser component and extract-\ning theω- andω′-linear terms, we have\n∆js,µ=e/planckover2pi1∆˜Kss\nµνAz\nν+βe/planckover2pi1\nπ/summationdisplay\nσσ(Lσ\ns,iµ)RA(A⊥\ni·A⊥\n0),\n(D13)\nwith\n∆˜Kss\nµν=−4π˜Γ2ν↑ν↓∝an}b∇acketle{tY∝an}b∇acket∇i}htδµ0−iqi∝an}b∇acketle{tD¯Y∝an}b∇acket∇i}htδµi\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}htδν0.\n(D14)The first term in Eq. (D13) corrects (the first two of)\nthe following response functions,\n˜Kss\n00+∆˜Kss\n00=q2K1, (D15)\n˜Kss\ni0+∆˜Kss\ni0=iqi/braceleftbig\n∝an}b∇acketle{tDν∝an}b∇acket∇i}ht−q2K2/bracerightbig\n, (D16)\n˜Kss\n0i=qiωK1, (D17)\n˜Kss\nij=iω/braceleftbig\n∝an}b∇acketle{tDν∝an}b∇acket∇i}htδij−qiqjK2/bracerightbig\n,(D18)13\nwhere\nK1=∝an}b∇acketle{tDν¯Y∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tσν∝an}b∇acket∇i}ht∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht, (D19)\nK2=∝an}b∇acketle{tD2ν¯Y∝an}b∇acket∇i}ht+2π˜Γ2∝an}b∇acketle{tσDν∝an}b∇acket∇i}ht2\nY↑Y↓+2π˜Γ2∝an}b∇acketle{tYν∝an}b∇acket∇i}ht, (D20)\nand restores the gauge invariance. This leads to a spin-\ncurrent density,\njsmf (1)\ns,µ(q,ω) =e2\n2π/summationdisplay\nσσ(Lσ\ns,iµ)RAE0\ni.(D21)The second term in Eq. 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B 84, 054462 (2011)."    },    {        "title": "2309.11152v1.Evaluating_Gilbert_Damping_in_Magnetic_Insulators_from_First_Principles.pdf",        "content": "Evaluating Gilbert Damping in Magnetic Insulators from First Principles\nLiangliang Hong,1, 2Changsong Xu,1, 2and Hongjun Xiang1, 2,∗\n1Key Laboratory of Computational Physical Sciences (Ministry of Education), Institute of Computational Physical Sciences,\nState Key Laboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai 200433, China\n2Shanghai Qi Zhi Institute, Shanghai 200030, China\n(Dated: September 21, 2023)\nMagnetic damping has a significant impact on the performance of various magnetic and spin-\ntronic devices, making it a long-standing focus of research. The strength of magnetic damping is\nusually quantified by the Gilbert damping constant in the Landau-Lifshitz-Gilbert equation. Here\nwe propose a first-principles based approach to evaluate the Gilbert damping constant contributed\nby spin-lattice coupling in magnetic insulators. The approach involves effective Hamiltonian mod-\nels and spin-lattice dynamics simulations. As a case study, we applied our method to Y 3Fe5O12,\nMnFe 2O4and Cr 2O3. Their damping constants were calculated to be 0 .8×10−4, 0.2×10−4,\n2.2×10−4, respectively at a low temperature. The results for Y 3Fe5O12and Cr 2O3are in good\nagreement with experimental measurements, while the discrepancy in MnFe 2O4can be attributed\nto the inhomogeneity and small band gap in real samples. The stronger damping observed in Cr 2O3,\ncompared to Y 3Fe5O12, essentially results from its stronger spin-lattice coupling. In addition, we\nconfirmed a proportional relationship between damping constants and the temperature difference\nof subsystems, which had been reported in previous studies. These successful applications suggest\nthat our approach serves as a promising candidate for estimating the Gilbert damping constant in\nmagnetic insulators.\nI. INTRODUCTION\nRecent decades have witnessed rapid developments in\nmagnetics and spintronics [1–3]. A long-time pursuit in\nspintronics is to actively control and manipulate the spin\ndegrees of freedom in solid-state systems. Related fun-\ndamental studies involve spin transport, spin dynamics\nand spin relaxation [4]. Within these domains, magnetic\ndamping often plays a crucial role. Generally, stronger\ndamping enables a faster writing rate for magnetic mem-\nories, while lower damping leads to a longer propagation\ndistance of spin waves. Therefore, it is always essential\nto accurately evaluate the magnetic damping in different\nmaterials. For instance, yttrium iron garnet (YIG) is a\nhighly promising spintronic material due to its ultra-low\nmagnetic damping [5–7]. However, the intrinsic mecha-\nnism behind its unique property has yet to be fully eluci-\ndated, which partly motivates us to carry out this work.\nAt present, magnetic damping is typically represented\nby a phenomenological term in the well-known Landau-\nLifshitz-Gilbert (LLG) equation, which has been widely\nemployed to simulate magnetization dynamics [8, 9]. A\nbasic form of this equation can be written as,\n∂ ⃗ m\n∂t=−γ ⃗ m×⃗B+α\nm⃗ m×∂ ⃗ m\n∂t(1)\nwhere ⃗Brepresents the total magnetic field acting on the\nlocal dipole ⃗ m,mdenotes the norm of ⃗ m,γis the gyro-\nmagnetic ratio, and αis the Gilbert damping constant.\nThe second term on the right side, as we mentioned, leads\n∗hxiang@fudan.edu.cndirectly to the relaxation process, in which the rate of en-\nergy dissipation is determined by the damping constant.\nGiven the importance of αin magnetization dynamics,\nits origin has been extensively studied in the literature\n[10–13]. To the best of our knowledge, both intrinsic and\nextrinsic mechanisms contribute to the damping. Specif-\nically, the intrinsic factors include spin-lattice and spin-\nelectron couplings, while the extrinsic contributions pri-\nmarily involve lattice imperfections and eddy currents\n[14, 15].\nTwo types of first-principles based methods have been\ndeveloped to calculate the damping constants in the past.\nOne approach involves the breathing Fermi surface model\n[16, 17] and the torque correlation model [18, 19], while\nthe other is based on the scattering theory from linear\nresponse [20–22]. These methods have demonstrated re-\nmarkable success in studying the magnetic damping in\ntransition metals such as Fe, Co, and Ni. Despite be-\ning free from complicated experiments, which are mostly\nbased on ferromagnetic resonance, these theoretical ap-\nproaches still exhibit several limitations. Firstly, when\ndealing with complex systems, we often have to spend a\nsignificant amount of computing resources on the first-\nprinciples calculations. In addition, these methods are\nmore suitable for calculating the electronic contribution\nto Gilbert damping in metallic magnets, thus rarely tak-\ning the effect of spin-lattice coupling into consideration\n[14, 23].\nRecently, spin-lattice dynamics (SLD) simulations [24]\nhave been adopted as an alternative method to evaluate\nthe Gilbert damping parameters. In Ref. [23], the au-\nthors constructed an empirically parameterized Hamil-\ntonian model for a cobalt cluster. They coupled a pre-\nheated lattice with a fully ordered spin state, then per-\nformed SLD simulation. During the relaxation process,arXiv:2309.11152v1  [cond-mat.mtrl-sci]  20 Sep 20232\nthe energy of lattice and spin subsystems were recorded\nand fitted to the following logistic functions,\nUlat=Ulat\n0−∆U0\n1 + exp[ −η∆U0t−Θ](2)\nUmag=Umag\n0+∆U0\n1 + exp[ −η∆U0t−Θ](3)\nfrom which they extracted the relaxation rate Γ = η∆U0\nand calculated the damping constant α=ηµS/γ. Here,\nµSdenotes the magnitude of magnetic moments. In Ref.\n[25], the authors also built an empirical potential model\nfor a periodic bcc Fe system. They firstly applied an ex-\nternal magnetic field in the z-direction and thermalized\nthe system to a finite temperature. Then, the magnetiza-\ntion orientation of each atom was rotated artificially by\na same angle. Afterwards, the system would relax back\nto equilibrium, during which the averaged z component\nof atomic magnetization was recorded and fitted to the\nfollowing function,\nmz(t) = tanh\u0014α\n1 +α2γBext(t+t0)\u0015\n(4)\nwhere αwas exactly the Gilbert damping parameter to\nbe estimated. Since these works selected transition met-\nals as the research object, their results were both orders\nof magnitude smaller than the experimental values. In\naddition, the use of empirically parameterized models re-\nduced the accuracy of their simulated results.\nIn this work, we combine SLD simulations with first-\nprinciples based effective Hamiltonian models to evalu-\nate the damping constants in magnetic insulators, where\nthe dominant contribution results from spin-lattice cou-\nplings. Compared to the previous studies, our work has\nmade improvements mainly in two aspects. Firstly, the\nutilization of first-principles based Hamiltonian models\nin simulations enhances the reliability of our conclusions.\nBesides, the better choice of research objects allows for\ndemonstrating the superiority of SLD simulations. In\nparticular, the microscopic origin of low damping in YIG\nwill be investigated. The paper is organized as follows.\nIn Sec. II, we introduce our effective Hamiltonian model,\nparameterization methods, and a scheme for evaluating\nGilbert damping parameters. Then, both the validation\nand application of our method are presented in Sec. III.\nFinally, we summarize this work and give a brief outlook\nin Sec. IV.\nII. MODEL AND METHODS\nThis section is split into three parts. Firstly (in Sec.\nII A), we introduce a generic form of our effective Hamil-\ntonian model. Then, methods involving the calculation\nof model parameters are presented in Sec. II B. At the\nlast part (Sec. II C), we propose a novel scheme to de-\ntermine the Gilbert damping constant through dynamics\nsimulations.A. The Hamiltonian Model\nSince our purpose is to evaluate the contribution of\nspin-lattice coupling to magnetic damping, the effective\nHamiltonian model must incorporate both spin and lat-\ntice degrees of freedom. A concise and generic formula\nthat meets our basic requirements consists of the three\nterms as follows:\nH=HL({ui,α}) +HS({⃗ sj}) +HSLC({ui,α,⃗ sj}) (5)\nwhere αabbreviates three orthogonal axes, ui,αrepre-\nsents the displacement of atom i, and ⃗ sjis a unit vector\nthat represents the direction of spin j.\nThe first term HLin Hamiltonian model describes the\ndynamical behavior of individual phonons. Technically,\nwe take the atomic displacements as independent vari-\nables and expand the Hamiltonian to the second order\nwith Taylor series. Then, we have the form as,\nHL=1\n2X\nijX\nαβKij,αβui,αuj,β+1\n2X\ni,αMi˙ui,α˙ui,α(6)\nwhere Kij,αβ denotes the force constant tensor and Mi\nrepresents the mass of atom i.\nSimilarly, the second term HSdescribes the dynami-\ncal behavior of individual magnons. For simplicity but\nno loss of accuracy, we only considered the Heisenberg\nexchange interactions between neighbor magnetic atoms\nin this work, though more complex interactions could be\ntaken into account in principle. Therefore, this term can\nbe expressed as,\nHS=X\n⟨i,j⟩Jij⃗Si·⃗Sj (7)\nwhere Jijdenotes the isotropic magnetic interaction co-\nefficient.\nThe third term HSLCrepresents the coupling of spin\nand lattice subsystems, and is expected to describe the\nscattering process between phonons and magnons. As\nan approximation of the lowest order, this term can be\nwritten as,\nHSLC=X\n⟨i,j⟩X\nkα\u0012∂Jij\n∂uk,αuk,α\u0013\n⃗Si·⃗Sj (8)\nAccording to the theory of quantum mechanics, this\ncoupling term provides a fundamental description of the\nsingle-phonon scattering process, which is believed to be\ndominant among all scatterings in the low-temperature\nregion. This type of relaxation mechanism in ferromag-\nnetic resonance was systematically studied by Kasuya\nand LeCraw for the first time [26]. It’s worth noting that\na higher order of Taylor expansion could have been con-\nducted to improve the accuracy of Hamiltonian models\ndirectly. For instance, the scattering between individual\nphonons can be adequately described by the anharmonic\nterms. However, as one always has to make a trade-off3\nbetween the precision and complexity of models, in this\nwork we choose to neglect the high order terms since the\nanharmonic effects in current investigated systems are\nnot important.\nIn this study, we adopted the symmetry-adapted clus-\nter expansion method implemented in the Property Anal-\nysis and Simulation Package for Materials (PASP) [27]\nto build the Hamiltonian model presented above. This\npackage can identify the nonequivalent interactions and\nequivalent atom clusters in a crystal system by analyz-\ning its structural properties based on the group theory.\nA significant benefit of working with PASP is we are en-\nabled to describe the target system with the least number\nof parameters. In the next section, we will discuss how\nto calculate the model parameters for different materials.\nB. Calculation of Model Parameters\nFirstly, the Heisenberg exchange coefficients Jijand\nspin-lattice coupling constants ∂Jij/∂uk,αcan be calcu-\nlated with the four-state method [28, 29]. The basic flow\nis to construct four artificially designated spin states of\nthe target system, calculate the corresponding energies\nand forces based on the density functional theory (DFT),\nthen determine the parameters by proper combination of\nthose results. At the last step, the following formulas will\nbe used,\nJij=E↑↑+E↓↓−E↑↓−E↓↑\n4S2(9)\n∂Jij\n∂uk,α=F↑↑\nk,α+F↓↓\nk,α−F↑↓\nk,α−F↓↑\nk,α\n4S2(10)\nwhere Sis the spin quantum number of magnetic atoms,\nEis the total energy of system and Fk,αrefers to one\ncomponent of the force on atom k. The superscripts ( ↑↑,\n↓↓,↑↓,↓↑) specify the constrained spin states of system\nin the calculation. More technical information about the\nfour-state method can be found in the references [28, 29].\nCompared to other approaches, the four-state method of-\nfers an obvious advantage in that no additional DFT cal-\nculations are needed to determine the coupling constants\n∂Jij/∂uk,αonce the exchange coefficients Jijhave been\nobtained. This is because the energy and forces are typ-\nically provided simultaneously by one DFT calculation.\nSince atomic masses Mican be directly obtained from\nthe periodic table, more efforts are needed to deal with\nthe force constant tensor Kij,αβ. Currently, there are two\ncommonly adopted ways to calculate the force constant\ntensor: density functional perturbation theory (DFPT)\nand finite displacement method. Both of these methods\nare applicable to our task.\nHowever, we cannot directly take the force constant\ntensor obtained from first-principles calculations as the\nmodel parameter. This is because in dynamics simula-\ntions we usually expand crystal cells to reduce the un-\ndesired influence of thermal fluctuations, which leads toa conflict between the periodic boundary condition and\nthe locality (also known as nearsightedness [30, 31]) of\nmodels. To be more specific, when calculating the con-\ntribution of one atom or spin to the total energy, we tend\nto set a well designed cutoff radius and ignore the inter-\nactions beyond it. This step is essential when dealing\nwith a large-scale system, otherwise we will suffer from\nthe model complexity and the computational cost. Nev-\nertheless, if we set the elements of Kij,αβ that represent\nout-of-range interactions to be zero and leave the others\nunchanged, we may violate the so-called acoustic sum-\nmation rules:\nX\niKij,αβ = 0 for all j, α, β. (11)\nIt should be pointed out that a straightforward en-\nforcement of the acoustic summation rules, achieved by\nsubtracting errors uniformly from force constants, will\nbreak the inherent crystal symmetry inevitably, which is\nthe technique employed in phonopy [32]. To address the\nabove issues, we adopted a more appropriate method in\nthis work. Before a detailed introduction, it’s necessary\nto recall that not every element of the force constant ten-\nsor serves as an independent variable due to the crystal\nsymmetries. Taking the cubic cell of Y 3Fe5O12(contain-\ning 160 atoms) for example, there are 230400 elements in\nthe tensor. After symmetry analyses, we find that only\n597 independent variables {pn}are needed to adequately\ndetermine all the tensor elements {Kij,αβ({pn})}, where\nthe effect of locality is already considered. Afterwards,\nour method is to set a correction factor xnfor each vari-\nablepnand minimize the deviation of parameters under\nthe constraints of Eq. (11). A mathematical reformula-\ntion of this method can be written as,\nmin\n{xn}X\nn(xn−1)2,with\nX\niKij,αβ({xnpn}) = 0 for all j, α, β.(12)\nIn the case of Y 3Fe5O12, there are only 18 linearly inde-\npendent constraints, which allow the extremum problem\nto be solved rigorously. The modified force constant ten-\nsor restores positive definiteness and translational sym-\nmetry while maintaining the crystal symmetries. There-\nfore, the modified tensor meets the requirements for dy-\nnamics simulations. In Sec. III B, the effectiveness of this\napproximate method will be demonstrated through a spe-\ncific example.\nAll the first-principles calculations mentioned in this\nsection are carried out using the Vienna ab initial simu-\nlation package (VASP) [33–35]. The force constants and\nphonon spectra are obtained by phonopy [32]. The opti-\nmizations formulated in (12) are accomplished with the\nfunction optimize.minimize implemented in SciPy [36].4\nC. Evaluation of Damping Constants\nAfter the construction and parameterization of Hamil-\ntonian models, we are finally able to perform spin-lattice\ndynamics simulations. Before the evaluation of Gilbert\ndamping constants, we briefly introduce the framework\nof SLD to cover some relevant concepts. In practice, the\nmotion of magnetic moments follows the stochastic Lan-\ndau–Lifshitz–Gilbert (SLLG) equation [14],\nd⃗ mi\ndt=−γL⃗ mi×\u0010\n⃗Bi+⃗Bfl\ni\u0011\n−γLα⃗ mi\n|⃗ mi|×h\n⃗ mi×\u0010\n⃗Bi+⃗Bfl\ni\u0011i\n(13)\nwhere γLis the renormalized gyromagnetic ratio, ⃗Bi=\n−∂H/∂ ⃗ m iis the effective local magnetic field and ⃗Bfl\ni\nrefers to a stochastic field introduced by Langevin ther-\nmostat. At the same time, the motion of atoms obeys\nthe Newton’s equation,\nd˙ui,α\ndt=1\nMi\u0010\n⃗Fi,α+⃗Ffl\ni,α\u0011\n−ν˙ui,α (14)\nwhere νis the damping constant and ⃗Ffl\ni,αrefers to a\nstochastic force caused by thermal fluctuations. In this\nwork, ⃗Bfl\niand⃗Ffl\ni,αare modeled as normally distributed\nnoises with temperature-dependent variances,\nBfl\ni,β∼N\u0010\n0,p\n2αkBTS/γ|⃗ mi|δt\u0011\n(15)\nFfl\ni,β∼N\u0010\n0,p\n2νMikBTL/δt\u0011\n(16)\nwhere TSandTLrefer to the equilibrium temperature of\nspin and lattice subsystems respectively. During simula-\ntions, we can also measure the transient temperature of\neach subsystem with the following formulas [37],\nTS=P\ni|⃗ mi×⃗Bi|2\n2kBP\ni⃗ mi·⃗Bi, TL=1\n2kBNX\ni,αMi˙u2\ni,α (17)\nIn this work, the LLG equation is numerically solved\nwith the semi-implicit SIB method proposed by Mentink\net al. [38]. The Newton’s motion equation is integrated\nusing the Grønbech-Jensen-Farago Verlet-type method\n[39]. To ensure the stability of those algorithms, a step\nlength of 0 .5 or 0 .2 fs is adopted [40], where the shorter\none is used in energy-conserving simulations.\nBased on the combination of atomistic spin dynamics\n(ASD) and SLD simulations, a new scheme is proposed\nto evaluate the damping constant in magnetic materials.\nHere is the basic flow of this method and more details of\na specific application are presented in Sec. III B.\n1. Freeze the spin degree of freedom and thermalize\nthe lattice from 0 to TLin the simulation.\n2. Fix atomic positions and raise the temperature of\nspin to TS> TL. Compared to TL> TS, this type\nof nonequilibrium state is more common in actual\nscenarios.3. Perform an energy-conserving SLD simulation to\nrelax the system. Normally, the spin temperature\nwill decrease to the same as lattice and stay there\ntill the end.\n4. Conduct a series of ASD simulations with different\nGilbert damping constants. The initial states are\nthe same as in step 3 and the equilibrium temper-\natures are set to be TL.\n5. Compare the cooling rates ∂TS/∂tof spin system\nbetween SLD and ASD simulations to evaluate the\nequivalent Gilbert damping constant contributed\nby spin-lattice coupling.\nThe key point behind step 5 is that the cooling rates\nobserved in ASD simulations are related to the assigned\ndamping constants, while in SLD simulation the cooling\nrate is determined by the strength of spin-lattice cou-\npling. Note that the former relation can be viewed as a\nnatural deduction of the LLG equation,\n∂TS\n∂t=1\nCV∂Emag\n∂t∝ −1\nCV\u0012∂ ⃗ m\n∂t·⃗B\u0013\n∝ −1\nCV\u0014\u0012α\nm⃗ m×∂ ⃗ m\n∂t\u0013\n·⃗B\u0015\n∝α (18)\nwhere we have used Eq. (1) and simplified the formula of\nmagnetic energy as Emag∝ −⃗ m·⃗B.\nIII. RESULTS\nThis section is divided into four parts. In Sec. III A,\nseveral test results are presented to validate the accu-\nracy of SLD simulations, which are implemented in the\nPASP package. Subsequently, detailed calculations on\nthree magnetic materials, namely Y 3Fe5O12, MnFe 2O4\nand Cr 2O3, are discussed in the rest parts.\nA. Validations\nIn order to guarantee the reliability of our conclusions\nobtained from dynamics simulations, a series of pretests\nwere carried out. We select some representative results\nand present them in Fig. 1, where Cr 2O3is taken as the\nobject to be studied.\nFirstly, we set the ground state of Cr 2O3as the ini-\ntial state and performed a NVT simulation with Tset=\n150K. As shown in Fig. 1(a), the temperature of spin\nand lattice subsystems increased to 150 Kin less than 5\nps and stayed there till the end. Since we can approxi-\nmate Ek= 0.5ELandEp= 0.5EL+ES, Fig. 1(b) also\nindicates that the contribution of phonons and magnons\nto the excited state energy is around 87.5% and 12.5%\nrespectively. This result could be verified from another\nperspective. Note that there are totally 10 atoms in the5\nFIG. 1. NVT and NVE relaxations of a spin-lattice coupled system (Cr 2O3) within the framework of spin-lattice dynamics.\nThe top row plots the time evolution of temperatures and the bottom row shows the variation of potential, kinetic and total\nenergies. (a) & (b): NVT thermalization from TL=TS= 0KtoTL=TS= 150 K. (c) & (d): NVE relaxation with TL= 30K,\nTS= 175 Kinitially. (e) & (f): NVE relaxation with TL= 180 K,TS= 30Kinitially.\nunit cell of Cr 2O3, which contribute 30 kBto the heat ca-\npacity. Meanwhile, the 4 magnetic atoms will contribute\nanother 4 kBin the low temperature region. Therefore,\nwe can estimate that the contribution of magnons to the\ntotal heat capacity is close to 11.8%, which is consistent\nwith the result from dynamics simulations.\nIn Figs. 1(c) & 1(d), the initial state was set to be a\nnonequilibrium state with TL= 30KandTS= 175 K. As\nwe expected, the total energy was well conserved when\nthe system evolved to equilibrium. In addition, the final\ntemperature fell within the range of 48 K∼55K, which\nagrees with our previous analysis of the heat capacities.\nLastly, we simulated the relaxation process using an-\nother nonequilibrium excited state with TL= 180 Kand\nTS= 30Kas the initial state. As shown in Figs. 1(e) &\n1(f), the temperature of spin system increased gradually\nto equilibrium with the total energy conserved through-\nout the simulation. Also, the final temperature is around\n160K, which matches well with our analysis. It should be\npointed out that there exist two notable differences be-\ntween this case and the previous. Firstly, the subsystems\nultimately evolved to a same temperature in a finite time,which alleviated our concerns about the accuracy of SLD\nsimulations. Besides, the relaxation time ( τ2) was much\nlonger than that ( τ1) in Fig. 1(c). For this phenomenon,\na qualitative explanation is presented below.\nBased on the theory of second quantization, the Hamil-\ntonian model presented in Sec. II A can be expressed in\nthe following form [41, 42],\nHL=X\nqpℏωqp(b†\nqpbqp+ 1/2) (19)\nHS=X\nλϵλa†\nλaλ+Const. (20)\nHSLC=X\nλ,qpMλ,qpa†\nλ−qaλ\u0000\nb†\nqp−b−qp\u0001\n(21)\nwhere bqpdenotes the annihilation operator of phonons\nwith wave vector qin branch p, and aλrepresents the an-\nnihilation operator of magnons with wave vector λ. All\nthe parameters, namely ωqp,ϵλandMλ,qp, can be deter-\nmined from the effective Hamiltonian model in principle.\nAccording to the Fermi’s golden rule, we have\nW{nλ−q, nλ, Nqp→nλ−q+ 1, nλ−1, Nqp+ 1}=2π\nℏ|Mλ,qp|2(nλ−q+ 1)( nλ)(Nqp+ 1)δ(ϵλ−q−ϵλ+ℏωqp) (22)\nW{nλ−q, nλ, N−qp→nλ−q+ 1, nλ−1, N−qp−1}=2π\nℏ|Mλ,qp|2(nλ−q+ 1)( nλ)(N−qp)δ(ϵλ−q−ϵλ−ℏω−qp) (23)6\nFIG. 2. (a) The primitive cell of Y 3Fe5O12. The golden balls\nrepresent iron atoms, the cyan ball stand for yttrium atoms,\nand the red balls refer to oxygen atoms. (b) The magnetic\nground state of YIG. The arrows of different colors represent\nthe spin directions of Fe atoms. (c) The density of states ob-\ntained by DFT calculations. (d) The temperature dependence\nof average magnetization measured in MC and ASD simula-\ntions. For YIG, the phase transition point from ferrimagnetic\nto paramagnetic lies in 530 K approximately.\nwhere Wrepresents the probability of one-phonon emis-\nsion or absorption, nλdenotes the occupation number of\nmagnons and Nqpstands for phonons. Both nλandNqp\ncan be evaluated approximately using the Bose–Einstein\ndistribution. According to the above formulas, the scat-\ntering rate Wgrows linearly with Nand quadratically\nwith n. Compared to Fig. 1(c), there are more phonons\nbut fewer magnons in the case of Fig. 1(e), thus leading\nto a lower transition probability and a longer relaxation\ntime. More technical details about the second quantiza-\ntion of interactions between phonons and magnons can\nbe found in Ref. [41, 42].\nB. Damping constants in Y 3Fe5O12\nIn the field of spintronics, Y 3Fe5O12(yttrium iron gar-\nnet, YIG) has gained much attention due to its ultra-low\nmagnetic damping [5–7]. The unique property of this\nmaterial motivated us to investigate the intrinsic mecha-\nnism behind it. The crystal structure of YIG is presented\nin Fig. 3(a). There are totally 80 atoms in the primitive\ncell, of which 12 Fe ions are located in the center of oxy-\ngen tetrahedrons while the other 8 Fe ions are sited in\noxygen octahedrons. The magnetic ground state of YIG\nis illustrated in Fig. 3(b). The Fe ions situated in differ-\nent chemical environments contribute spins in opposite\ndirections, which makes YIG a typical ferrimagnetic ma-\nterial.TABLE I. The Heisenberg exchange coefficients J of YIG,\nwhere an effective spin S= 1 is adopted. For the FeO−FeO\npairs, the Greek letters ( α&β) refer to different chemical\nenvironments. All the results are calculated with the four-\nstate method.\nSpin Pair. Distance (Angst) J (meV)\n1NN FeT−FeO3.445 47.414\n1NN FeT−FeT3.774 2.399\n1NN FeO−FeO(α) 5.337 0.538\n1NN FeO−FeO(β) 5.337 5.055\n2NN FeT−FeO5.555 0.285\n2NN FeT−FeT5.765 3.437\nIn order to evaluate the Gilbert damping constants in\nYIG, our first step is to prepare an effective Hamilto-\nnian model. Considering the balance between precision\nand efficiency, the cutoff radius of interactions was set\nto be 11.0 Bohr for atomic pairs and 6.7 Bohr for 3-\nbody clusters. After symmetry analyses, we identified\n612 nonequivalent interactions in total, which included\n6 Heisenberg exchange terms and 9 spin-lattice coupling\nterms.\nTo determine the interaction parameters, we carried\nout a series of first-principles calculations, where a cu-\nbic cell was adopted to reduce the interference between\nadjacent cells caused by periodic boundary conditions.\nFollowing the settings in Ref. [43], we utilized the pro-\njector augmented-wave (PAW) method [44] and revised\nPerdew-Burke-Ernzerhof exchange-correlation functional\nfor solids (PBEsol) [45] in our calculations. Besides, the\nDFT+U method in its simplified form [46] was employed\nwhere the effective Hubbard U parameter was set to be\n4 eV for the 3 delectrons of Fe ions. In addition, a cutoff\nenergy of 520 eV for plane wave basis and a Γ-centered\n2×2×2 mesh of k-points were used in the DFT calcu-\nlations.\nIn Figure 2(c), we present the density of states (DOS)\nfor YIG. With a band gap of 1.863 eV, there is hardly\nany electric current occurring in the low temperature re-\ngion. Moreover, the Heisenberg exchange coefficients of\nYIG is listed in Table I. To verify the accuracy of these\nparameters, we conducted both Monte Carlo (MC) and\nASD simulations. The temperature dependence of aver-\nage magnetization is shown in Fig. 2(d), which reveals\nthe critical temperature of YIG to be 530 K. This result\nis slightly lower than the measured Curie temperature,\nTC= 560 K[5], but falls within our tolerance. The cal-\nculated results of coupling constants are provided in the\nsupplementary material.\nNext, we come to deal with the force constant tensor.\nIn order to demonstrate the impact of locality and val-\nidate the effectiveness of our optimization method, we\npresent some results pertaining to the tensor of YIG in\nTable II. Here we use “VASP” to tag the original tensor7\nTABLE II. The force constant tensor of YIG. The columns\nlabeled by A represent the sorted absolute values ofP\niKij,αβ\nand the columns labeled by B list the sorted eigenvalues of\nKij,αβ. For the cubic cell of YIG, we obtained the original\ntensor with the VASP package. Then, we eliminated the el-\nements that represent interactions beyond the cutoff radius.\nThis step was done by PASP. Finally, the tensor was modified\nto meet the requirement of translational symmetry through\nthe optimization formulated in (12).\nVASP PASP Modified\nNo. A B A B A B\n1 0.000 0.000 1.587 -0.102 0.000 0.000\n2 0.000 0.000 1.587 -0.102 0.000 0.000\n3 0.000 0.000 1.587 -0.102 0.000 0.000\n4 0.000 1.065 1.587 0.643 0.000 0.444\n5 0.000 1.065 1.587 0.643 0.000 0.444\n6 0.000 1.065 1.587 0.643 0.000 0.444\nobtained from DFT calculations, “PASP” to label the\nmodified tensor in which interactions beyond the cutoff\nradius are eliminated, and “Modified” to label the tensor\nafter optimization of independent variables. As shown in\nTable II, the “PASP” tensor violated the acoustic sum\nrule and was not positive semi-definite, whereas these is-\nsues were resolved for the “Modified” tensor. Although\nan obvious difference existed between the “PASP” and\n“Modified” tensor in terms of their eigenvalues, we still\nassumed the target system could be reasonably described\nby the “Modified” tensor and the validity of this assump-\ntion would be verified by the calculated results of damp-\ning constants. Additional details regarding the selection\nof tensor elements and the deviation of phonon spectra\nare provided in Fig. 3. According to figure 3(b) and 3(c),\nthe major deviation in phonon spectra resulted from the\nelimination of tensor elements, rather than the subse-\nquent modification.\nCompleting the preparation of Hamiltonian model, we\napplied the scheme proposed in Sec. II C to our first ob-\nject, Y 3Fe5O12. An instance is presented in Figure 4. We\nsetTL= 30K,TS= 180 Kfor the initial nonequilibrium\nstate and adopted an expanded supercell which contains\n12800 atoms in the simulation. Fig. 4(a) shows the time\nevolution of spin temperature in different types of simu-\nlations. By comparing the curves, we could roughly esti-\nmate that the equivalent damping constant in SLD simu-\nlation fell within the range of 10−3∼10−4. To make the\nestimation more precise, we calculated the initial cool-\ning rates ∂TS/∂t|t=0through polynomial (or exponen-\ntial) fittings and plotted them in Fig. 4(b). Afterwards,\na linear regression was performed to determine the quan-\ntitative relation between lg( −∂TS/∂t|t=0) and lg( α). As\nwe expected, the cooling rates in ASD simulations were\nproportional to the assigned damping constants. Then,\nwe combined the results of SLD and ASD simulations toevaluate the equivalent damping constant. This step was\naccomplished by identifying the intersection of red and\nblue lines in Figure 4(b). Finally, the damping constant\nwas determined to be αf= (2.87±0.13)×10−4in this\ncase. To verify our method and result, we present a com-\nparison between SLD and ASD (where we set α=αf)\nsimulations in Fig. 4(c). The curves agree well with each\nother in the initial stage but deviate in the second half.\nThis phenomenon is within our expectation, because in\nthe SLD simulation the lattice heats up as the spin cools\ndown, thereby slowing the energy transfer between two\nsubsystems.\nIn addition to the above case, we have measured the\nequivalent damping constants under different conditions\nto investigate the temperature dependence of magnetic\ndamping. The final results are summarized in Figure 5.\nDetails about the estimation of uncertainties are given in\nthe supplementary material. For Y 3Fe5O12, the damping\nconstants at different temperatures stay on the order of\n10−4, which is in good agreement with the experimental\nresults (3 .2×10−4[47], 2 .2×10−4[48], 1 .2–1.7×10−4\n[49]). For example, the damping constant in bulk YIG\nwas reported as 0 .4×10−4in Ref. [50]. Meanwhile, our\ncalculations yielded α= (2.8±0.3)×10−5at ∆T= 15\nK and α= (7.0±0.7)×10−5at ∆T= 30 K, where both\nTL= 0 K. Therefore, the experimental value corresponds\nroughly to the temperature region of ∆ T= 15∼30 K in\nour study. We believe such extent of thermal excitation\nis quite common in all kinds of spintronics experiments.\nMoreover, Fig. 5 indicates that αis approximately pro-\nportional to the temperature difference between subsys-\ntems. This outcome is also consistent with some com-\nputational works in the past [23, 25]. By comparing the\nsubfigures in Figure 5, we found that αhas little depen-\ndence on the lattice temperature, although here TLcould\nbe viewed to some extent as the ambient temperature of\nthe spin system.\nAs a supplement to Sec. III A, we further validate our\nsimulations by analyzing the measured cooling rates in\nFig. 5(a). By subtracting Eq. (23) from Eq. (22), the\ntransfer rate of energy between magnon and phonon sys-\ntems can be expressed as,\n˙Q=X\nqpℏωqp⟨˙Nqp⟩=X\nλ,qpTλ,qp (24)\nwhere Tλ,qpdenotes different transfer channels,\nTλ,qp∝(nλ−nλ−q)Nqp+nλ−qnλ+ 1 (25)\nAccording to the Bose–Einstein distribution, the number\nof magnons and phonons can be expressed as,\nnλ=1\neϵλ/kBTS−1, Nqp=1\neℏωqp/kBTL−1(26)\nWhen TSis high enough and TLis close to zero, we can\napproximate nλ=kBTS/ϵλ∝TSandNqpclose to zero.\nUnder these conditions, we have ˙Q∝T2\nS. This relation8\nFIG. 3. (a) The selection of force constant tensor elements for the cubic cell of YIG. An 160 ×160 zero-one matrix is used\nto show the result of selection, in which ’1’ denotes the interactions within cutoff radius and ’0’ represents the elements that\nare artificially eliminated. (b) The phonon spectrum calculated from the force constant tensor before and after the elimination\nof tensor elements. (c) The phonon spectrum calculated from the force constant tensor before and after the optimization of\nindependent variables.\nFIG. 4. (a) The time evolution of spin temperature in SLD and ASD simulations. The gray line represents the SLD simulation\nwhile the others refer to the ASD simulations with different damping constants. (b) The initial cooling rates ∂TS/∂t|t=0with\nrespect to the damping constants α, where the scaling of axis is set to be logarithm. The gray squares refer to the results of\nASD simulations and the blue line acts as the linear regression. The red circle is plotted by intersection of the blue line and\nthe horizontal red dash line, which represents the initial cooling rate in the SLD simulation. Then we can obtain the equivalent\ndamping constant from the abscissa of the red circle. (c) The comparison between ASD and SLD simulations. In the ASD\nsimulation, the Gilbert damping constant is set to be α= 2.87×10−4, which is exactly the result of our evaluation from the\nSLD simulation.\nFIG. 5. The temperature dependence of Gilbert damping constants for Y 3Fe5O12. The label of abscissa axis ∆ Trefers to\nTS−TLof the initial state in dynamical simulations. Measurements on the magnetic damping are performed under different\ninitial conditions of the lattice temperature: (a) TL= 0, (b) TL= 30K, (c)TL= 60K.9\nFIG. 6. The relation between damping constants αand spin-\nlattice coupling constants ∂Jij/∂uk,αin YIG. Through a lin-\near fitting, the slope is determined to be 2 .01, which agrees\nwell with our theoretical predictions.\nis well verified by linear regressions and the details are\nprovided in the supplementary material.\nFurthermore, the accuracy of our simulations can also\nbe proved from another perspective. According to Eqs.\n(22) and (23), the scattering rate Wgrows quadratically\nwith the coupling parameters Mλ,qp. Based on the theory\nof second quantization, Mλ,qpshall be proportional to\nthe coupling constants ∂Jij/∂uk,α. Therefore, under a\ndefinite condition of temperature, we have:\nα∝˙Q∝∆W∝M2\nλ,qp∝(∂Jij/∂uk,α)2(27)\nIn order to verify this relation, we adjusted the spin-\nlattice coupling constants of YIG coherently while keep-\ning the other model parameters unchanged. Then, SLD\nsimulations were carried out to evaluate the correspond-\ning damping constants. The result is plotted in Fig. 6,\nwhere the x-label “slcc” stands for the spin-lattice cou-\npling constants and the subscript “0” refers to the orig-\ninal situation. From a linear fitting, the slope is deter-\nmined to be 2 .01, which agrees well with our prediction.\nC. Damping constants in MnFe 2O4\nAfter the calculation on YIG, we applied our method\nto MnFe 2O4(MFO), which was reported to possess a\nlarge Gilbert damping constant in the literature [13, 51].\nAs shown in Fig. 7(a), MnFe 2O4has a typical structure\nof spinels, where A sites are surrounded by four oxygen\natoms and B sites are located in octahedrons. Generally,\nspinels can be classified into normal and inverse struc-\ntures according to the distribution of divalent and triva-\nlent cations between A/B sites. In experiments, MFO\nusually crystallizes into a mixed phase where the normal\nstructure occupies the major part (80% in bulk MFO\n[52]). Here, we only considered its normal structure in\nthis work. Also, the magnetic ground state of MFO is\nshown in Fig. 22(b), where the magnetic moments are\nantiparallel between A/B sites.\nFIG. 7. (a) The cubic cell of MnFe 2O4. The purple balls rep-\nresent manganese atoms, the golden balls refer to iron atoms,\nand the red balls stand for oxygen atoms. (b) The magnetic\nground state of MFO. The arrows of different colors repre-\nsent the spin directions of Mn and Fe atoms separately. (c)\nThe density of states obtained by DFT calculations. (d) The\ntemperature dependence of average magnetization measured\nin MC and ASD simulations. For MnFe 2O4, the phase tran-\nsition point from ferrimagnetic to paramagnetic lies in 730K\napproximately.\nFirstly, we started to construct an effective Hamilto-\nnian model for MFO. With the same cutoff settings for\nYIG, we found 105 nonequivalent interactions, including\n4 Heisenberg exchange terms and 10 spin-lattice coupling\nterms. Subsequently, DFT calculations were carried out\nto determine the interaction parameters. In these calcu-\nlations, we adopted a cubic cell containing 56 atoms and\na Γ-centered 4 ×4×4 grid mesh in the reciprocal space.\nBesides, UMn= 3.3 eV and UFe= 3.6 eV were used as the\neffective Hubbard parameters [52]. With the exception of\naforementioned settings, all the relevant first-principles\ncalculations were performed under the same conditions\nas in Sec. III B.\nThe DOS of MnFe 2O4is plotted in Fig. 7(c), yielding\na calculated band gap of 0.612 eV. This value does not\nmatch with the result of transport experiments, which re-\nported a much smaller band gap (0 .04–0.06 eV) [53]. In\naddition, MC and ASD simulations were performed using\nthe Heisenberg exchange coefficients listed in Table III.\nThe temperature dependence of average magnetization,\nshown in Fig. 7(d), suggests the critical temperature to\nbe around 730 K. This result is significantly higher than\nthe measured value of 573 K [54]. Both of the above dis-\ncrepancies may be attributed to the inevitable difference\nbetween the ideal normal spinel structure in calculations\nand the partially disordered samples in reality. Despite\nthis problem, we proceeded to describe the target system\nwith our Hamiltonian model and expected to see how far\nthe calculated results of damping constants would differ10\nTABLE III. The exchange coefficients J of MnFe 2O4, where\nan effective spin S= 1 is adopted.\nSpin Pair. Distance (Angst) J (meV)\n1NN Fe-Fe 3.003 6.835\n1NN Mn-Fe 3.521 33.224\n1NN Mn-Mn 3.667 3.956\n2NN Fe-Fe 5.201 0.929\nfrom experimental values.\nAfter the preparation of Hamiltonian model, we con-\nducted dynamics simulations to evaluate the equivalent\ndamping parameters in MFO at different temperatures.\nA supercell containing 13440 atoms was adopted in the\nsimulation, and the results are summarized in Fig. 10.\nThe average of calculated damping constants is around\n8×10−5, which is much smaller than the measured value,\n1.0×10−2[13, 51]. Two factors may account for this in-\nconsistency. Firstly, the inhomogeneity in real MnFe 2O4\nsamples greatly enhances the scattering of magnons and\nphonons, thereby increasing the damping constants. Ad-\nditionally, due to the narrow band gap observed in ex-\nperiments, eddy currents can arise at finite temperatures,\nwhich leads to a rapid loss of energy in the form of joule\nheat. As the result of these factors, we failed to obtain a\nreasonable estimation of Gilbert damping constants for\nMnFe 2O4with our methodology. On the other side, the\ncontribution of different relaxation mechanisms to FMR\nlinewidth has been studied comprehensively for MnFe 2O4\nin Ref. [53], which further confirms our analyses.\nD. Damping constants in Cr 2O3\nChromia (Cr 2O3) is a well-known collinear magneto-\nelectric antiferromagnet, which holds great prospects in\nthe field of spintronics [55–57]. As shown in Fig. 8(a),\nthe primitive cell of Cr 2O3contains 10 atoms, with each\nchromium atom bonded to the six oxygen atoms around\nit. Additionally, Fig. 8(b) displays the magnetic ground\nstate of Cr 2O3, where the spins of two nearest neighbor-\ning Cr atoms are oriented in opposite directions.\nAs a preliminary step in constructing the Hamiltonian\nmodel, we set the cutoff radius of interactions to be 11.0\nBohr for atomic pairs and 7.0 Bohr for 3-body clusters.\nThrough symmetry analyses, we identified 319 nonequiv-\nalent interactions, including 5 Heisenberg exchange terms\nand 21 spin-lattice coupling terms.\nAfterwards, a series of first-principles calculations were\nperformed to determine the model parameters. Following\nthe settings in Ref. [58], we adopted a hexagonal cell of\nCr2O3which contained a total of 90 atoms in the calcula-\ntions. Additionally, we used the LSDA+U method in its\nfull spherically symmetric form [59]. As to the Hubbard\nparameters, Jwas fixed at its recommended value of 0.6\nFIG. 8. (a) The primitive cell of Cr 2O3. The dark blue balls\nrepresent chromium atoms, and the red balls stand for oxygen\natoms. (b) The magnetic ground state. The arrows of differ-\nent colors represent the spin directions of Cr atoms. (c) The\ndensity of states obtained by DFT calculations. (d) The tem-\nperature dependence of sublattice magnetization measured in\nMC and ASD simulations. For Cr 2O3, the phase transition\npoint from ferrimagnetic to paramagnetic lies in 310K approx-\nimately.\nTABLE IV. The exchange coefficients J of Cr 2O3, in which\nan effective spin S= 1 is adopted.\nSpin Pair. Distance (Angst) J (meV)\n1NN Cr-Cr 2.640 44.778\n2NN Cr-Cr 2.873 29.269\n3NN Cr-Cr 3.411 -0.182\n4NN Cr-Cr 3.635 0.007\n5NN Cr-Cr 4.137 -0.500\neV, and Uwas adjusted to fit the N´ eel temperature ob-\nserved in experiments [60]. We found U= 2.0 eV was the\noptimal value for 3 delectrons of Cr ions. Except for the\nsettings specified above, all the DFT calculations were\nconducted under the same conditions as in Sec. III C.\nThe DOS of Cr 2O3is plotted in Fig. 8(c), which yields\na calculated band gap of 1.935 eV. This value indicates\nthat the energy dissipation of electric currents can be ne-\nglected in this system. Additionally, we list the Heisen-\nberg exchange coefficients of chromia in Table IV. Both\nMC and ASD simulations were performed to investigate\nthe temperature dependence of sublattice magnetization.\nAccording to Fig. 8(d), the critical point was determined\nto be 310 K approximately, which was quite consistent\nwith experimental observations. Also, the force constants\nof Cr 2O3went through the modification formulated in\nSec. II B, and the spin-lattice coupling parameters are\nprovided in the supplementary material.\nAfter the construction of Hamiltonian model, we con-\nducted a series of dynamics simulations to evaluate the11\nFIG. 9. (a) The 1NN FeT-FeOpair in Y 3Fe5O12. (b) The\n1NN Cr-Cr pair in Cr 2O3. The steel blue arrow stands for\nthe orientation of ∂J/∂u and the red number along with it\nrepresents the magnitude in unit of meV/Angst.\nequivalent damping parameters in Cr 2O3. An expanded\nhexagonal cell containing 14400 atoms was adopted for\nthe simulation, and the results are summarized in Fig. 11.\nAs two specific cases, our calculation yielded α= (1.31±\n0.14)×10−4at ∆T= 15 K and α= (2.7±0.3)×10−4\nat ∆T= 30 K, where both TL= 0 K. Therefore, the\ncalculated damping constants within ∆ T= 15∼30 K\nare quite close to 2 ×10−4, which is the estimated value\nreported in Ref. [61].\nFurthermore, the damping constants in Cr 2O3exhibit\na significant non-linear relation with the temperature dif-\nference of subsystems. Through logarithmic fittings, we\ncalculated the power exponents for Figures 11(a) to 11(c),\nand the results were 1.17, 1.62, 1.38. If we disregard the\ndifference between ∆ TandTfor the moment, these val-\nues are in good agreement with the theoretical prediction\nof Kasuya and LeCraw [26]. According to their study, the\nrelaxation rate varies as Tnwhere n= 1∼2 while n= 2\ncorresponds to a larger regime of temperatures.\nCompared to YIG, the greater magnetic damping ob-\nserved in chromia can be attributed to its significantly\nstronger spin-lattice coupling. As shown in Fig. 9, the\nmagnitude of principal spin-lattice coupling constant in\nCr2O3is two or three times larger than that in YIG. This\ncould be explained by the fact that direct exchange in-\nteraction between two magnetic atoms decreases rapidlywith their distance [62]. Therefore, owing to the shorter\ndistance of Cr-Cr pair, the direct exchange interaction\nbetween neighboring Cr atoms is believed to have a great\ncontribution to the spin-lattice coupling in Cr 2O3.\nIV. CONCLUSIONS\nIn summary, we propose a scheme to evaluate the con-\ntribution of spin-lattice coupling to the Gilbert damp-\ning in insulating magnetic materials. Our methodology\ninvolves first-principles based Hamiltonian models and\nspin-lattice dynamics simulations. Following a series of\nvalidations, we applied our method to three magnetic ma-\nterials, namely Y 3Fe5O12, MnFe 2O4and Cr 2O3. Their\ndamping constants were estimated separately, and the\nresults show that, in general, αis approximately propor-\ntional to the temperature difference between spin and\nlattice subsystems. Under the condition of ∆ T= 30\nK, the calculated damping constants are averaged to be\n0.8×10−4for YIG, 0 .2×10−4for MFO and 2 .2×10−4\nfor Cr 2O3. The results for YIG and Cr 2O3are in good\nagreement with experimental measurements, while the\ndiscrepancy for MFO can be attributed to the inhomo-\ngeneity and small band gap in real samples. 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Xiang, Na-\nture of spin-lattice coupling in two-dimensional cri3 and\ncrgete3, SCIENCE CHINA-PHYSICS MECHANICS &\nASTRONOMY 64, 10.1007/s11433-021-1717-9 (2021)."    },    {        "title": "1701.03201v2.Dynamic_coupling_of_ferromagnets_via_spin_Hall_magnetoresistance.pdf",        "content": "arXiv:1701.03201v2  [cond-mat.mes-hall]  20 Mar 2017Dynamic coupling of ferromagnets via spin Hall magnetoresi stance\nTomohiro Taniguchi\nNational Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan\n(Dated: September 17, 2018)\nThe synchronized magnetization dynamics in ferromagnets o n a nonmagnetic heavy metal caused\nby the spin Hall effect is investigated theoretically. The di rect and inverse spin Hall effects near\nthe ferromagnetic/nonmagnetic interface generate longit udinal and transverse electric currents. The\nphenomenon is known as the spin Hall magnetoresistance effec t, whose magnitude depends on the\nmagnetization direction in the ferromagnet due to the spin t ransfer effect. When another ferro-\nmagnet is placed onto the same nonmagnet, these currents are again converted to the spin current\nby the spin Hall effect and excite the spin torque to this addit ional ferromagnet, resulting in the\nexcitation of the coupled motions of the magnetizations. Th e in-phase or antiphase synchronization\nof the magnetization oscillations, depending on the value o f the Gilbert damping constant and the\nfield-like torque strength, is found in the transverse geome try by solving the Landau-Lifshitz-Gilbert\nequation numerically. On the other hand, in addition to thes e synchronizations, the synchroniza-\ntion having a phase difference of a quarter of a period is also f ound in the longitudinal geometry.\nThe analytical theory clarifying the relation among the cur rent, frequency, and phase difference is\nalso developed, where it is shown that the phase differences o bserved in the numerical simulations\ncorrespond to that giving the fixed points of the energy suppl ied by the coupling torque.\nPACS numbers: 85.75.-d, 75.78.-n, 05.45.Xt, 72.25.-b\nI. INTRODUCTION\nDynamic coupling of ferromagnets has been of interest\ninthefieldofmagnetism. Thedipoleinteractionhasbeen\nthe basic interaction to excite the coupled motion of the\nmagnetizations, and is applied to magnetic recording1,2.\nAnother method to realizethe couplingis to usethe spin-\ntransfer effect3,4, where the applicationof an electric cur-\nrent to ferromagnetic/nonmagnetic multilayers results in\nthe magnetization switching and self-oscillation5–14. The\ncoupled dynamics through pure spin current generated\nin spin pumping15,16and nonlocal17geometries have also\nbeen observed. It should be emphasized that these cou-\nplings are strongly restricted by the characteristic length\nscales. Forexample, thedipolecouplingdecaysaccording\nto the inversecube detection law, whereas the spin trans-\nfer effect by a spin-polarized electric or pure spin current\noccurs in a system smaller than the spin diffusion length.\nRecently, physical phenomena, such as the spin\ntorque18–25and magnetoresistance effects26–34, due to\nthe spin Hall effect35–37in bilayers consisting of an insu-\nlating or metallic ferromagnet and a nonmagnetic heavy\nmetal have attracted much attention. The latter, known\nas the spin Hall magnetoresistance, originates from the\ncharge-spin conversion of an external electric current by\nthe direct and inverse spin Hall effects, and has been\nobserved by measuring the longitudinal and transverse\nelectric currents, which are given by\nJcx\nJ0= 1+χ′′+χm2\ny, (1)\nJcy\nJ0=−χmxmy−χ′mz, (2)respectively, where J0is the electric current density gen-\nerated by the external electric field. The definitions of\nthe dimensionless coefficients, χ,χ′, andχ′′, are given\nbelow. It should be emphasized that the currents given\nby Eqs. (1) and (2) depend on the magnetization direc-\ntionm= (mx,my,mz). When another ferromagnet is\nplaced onto the same nonmagnet, these currents will be\nconverted to spin current by the spin Hall effect again,\nand excite spin torque on this additional ferromagnet.\nThen, the magnetization dynamics of two ferromagnets\nwill be coupled through the angular dependencies of the\nelectric current given by Eqs. (1) and (2). This coupling\nis unavoidable whenever several ferromagnets are placed\nonto the same nonmagnet, and is not restricted by the\ndistance between the ferromagnets because it is carried\nby the electric current. Since the structure consisting\nof several ferromagnets on the nonmagnetic heavy metal\nwill be important from the viewpoints of both fundamen-\ntal physics and practical applications based on the spin\nHalleffect, suchasmagneticrandomaccessmemory, spin\ntorque oscillators, and bio-inspired computing38,39, it is\nof interest to clarify the role of this coupling.\nIn this paper, we investigate the coupled dynamics of\nmagnetizations in ferromagnets in the presence of the\nspin Hall effect by solving the Landau-Lifshitz-Gilbert\n(LLG) equation numerically for both longitudinal and\ntransversegeometries. Inadditiontothe externalelectric\ncurrent, the current contributing to the spin Hall mag-\nnetoresistance also excites the spin torque. The strength\nof this additional torque is estimated from the theory of\nthe spin Hall magnetoresistance extended to the system\nconsistingofseveralferromagnets. The conventionalspin\ntorque is proportional to the spin Hall angle ϑ, whereas\nthe new torque is on the order of ϑ3, and therefore, its\nvalue is two orders of magnitude smaller than the con-2\nx\nyzlongitudinal coupling\ntransverse coupling\nExF2\nF1 F3\nEx\n(b)(a)\nF1 F2\nσNEx+j1+j2SHE SHE ISHE ISHEN\nN\nFIG. 1: (a) Schematic view of the system in this study.\nThree ferromagnets F ℓ(ℓ= 1,2,3) are placed onto the same\nnonmagnet N. The external electric field is applied to the x\ndirection. (b) Schematic view of the generations of electri c\ncurrents by the direct and inverse spin Hall effect (SHE and\nISHE)in the longitudinal geometry. The total electric curr ent\nis the sum of the conventional electric current J0=σNExand\nthe current generated near the F 1/N and F 2/N interfaces, j1\nandj2.\nventional spin torque. Nevertheless, it is found that this\nadditional new torque affects the phase difference of the\nmagnetizationin the self-oscillationstate. The numerical\nsimulationrevealsthatthein-phaseorantiphasesynchro-\nnization is observed in the transverse geometry. On the\nother hand, in addition to them, the phase difference be-\ncomes a quarter of a period in the longitudinal geometry.\nIt is found that these phase differences depend on the\nvalues of the Gilbert damping constant and the dimen-\nsionless field-like torque strength. An analytical theory\nclarifying the relation among the current, frequency, and\nphase difference is also developed.\nThe paper is organized as follows. In Sec. II, we de-\nscribe the system under consideration, and discuss the\ntheoretical formula of the spin torque excited by the spin\nHalleffect inthepresenceoftheseveralferromagnetsand\nthe spin Hall magnetoresistance effect. In Sec. III, we\nstudy the phase differences in the synchronized state of\nthe magnetizations for both the longitudinal and trans-\nverse geometries by solving the LLG equation numeri-\ncally. InSec. IV,thetheoryclarifyingtherelationamong\nthe current, frequency, and phase difference is developed\nbased on the LLG equation averaged over constant en-\nergy curves. The summary of the paper is given in Sec.\nV.\nII. SYSTEM DESCRIPTION AND LLG\nEQUATION\nIn this section, we describe the system adopted in this\nstudy, and show the spin torque formulas including the\ncoupling torques between the ferromagnets.A. System description\nThe system we consider is schematically shown in Fig.\n1(a), where three ferromagnets F ℓ(ℓ= 1,2,3) are placed\nonto the same nonmagnet N. We assume that the ma-\nterial parameters in the ferromagnets are identical, for\nsimplicity. The external electric field, Ex, is applied\nto thexdirection, inducing the electric current density\nJ0=σNEx, where σNis the conductivity of the non-\nmagnet. The direct and inverse spin Hall effects produce\nelectriccurrentsinthelongitudinal( x)andtransverse( y)\ndirections. These electric currents are converted to the\nspin current and injected into the F 2and F 3layers due\nto the spin Hall effect, resulting in the excitation of the\nspin torque. Then, the magnetization dynamics in the F ℓ\n(ℓ= 2,3)layerisaffectedbythatintheF 1layer,andvice\nversa. We call the coupling between the F 1and F2layers\nthe longitudinal coupling, whereas that between the F 1\nand F 3layers the transverse coupling. We assume that\nboth the ferromagnet and nonmagnet are metallic be-\ncause metallic bilayers are generally used to measure the\nmagnetization switching and oscillation by the spin Hall\neffect18–21,24. Although the spin Hall magnetoresistance\nwas originally studied for insulating ferromagnets26–29, a\nlarge spin Hall magnetoresistance in metallic system has\nalso been reported recently32–34.\nThe dimensionless coefficients, χ,χ′, andχ′′, in Eqs.\n(1) and (2) for single ferromagnets have been derived for\nan insulating30or metallic34,40ferromagnet, which are\ngiven by\nχ=ϑ2λN\ndN/bracketleftbigg\nReg↑↓\ngN+g↑↓coth(dN/λN)−g∗\ngN/bracketrightbigg\ntanh2/parenleftbiggdN\n2λN/parenrightbigg\n,\n(3)\nχ′=−ϑ2λN\ndNImg↑↓\ngN+g↑↓coth(dN/λN)tanh2/parenleftbiggdN\n2λN/parenrightbigg\n,\n(4)\nχ′′=2ϑ2λN\ndNtanh/parenleftbiggdN\n2λN/parenrightbigg\n−ϑ2λN\ndNReg↑↓\ngN+g↑↓coth(dN/λN)tanh2/parenleftbiggdN\n2λN/parenrightbigg\n,\n(5)\nwheredNandλNare thickness and spin diffusion\nlength of the nonmagnet, respectively, whereas gN/S=\nhσN//parenleftbig\n2e2λN/parenrightbig\nwith the cross section area of the ferro-\nmagnetic/nonmagnetic interface S. The dimensionless\nmixing conductance g↑↓=g↑↓\nr+ig↑↓\niconsists of its real\nand imaginary parts41–43, andg∗is defined as\n1\ng∗=2\n(1−p2g)g+1\ngFtanh(dF/λF)+1\ngNtanh(dN/λN).\n(6)\nHere,g=g↑↑+g↓↓is the sum of the conductances\nof the spin-up and spin-down electrons, whereas pg=3\n(g↑↑−g↓↓)/gis its spin polarization. The ferromag-\nnetic/nonmagnetic interface resistance ris related to\ngviag/S= (h/e2)/r. We also introduce gF/S=\nh/parenleftbig\n1−p2\nσ/parenrightbig\nσF//parenleftbig\n2e2λF/parenrightbig\n, whereσFis the conductivity of\nthe ferromagnet and pσis its spin polarization. The\nthickness and spin diffusion length of the ferromagnet\nare denoted as dFandλF, respectively. The term related\ntog∗is neglected when the ferromagnetis an insulator30,\ni.e.,r→ ∞. The following quantities correspond to\nthe effective spin polarizations of the damping-like (or\nSlonczewski3) torque and the field-like torque, respec-\ntively,\nϑR(I)=ϑtanh/parenleftbiggdN\n2λN/parenrightbigg\nRe(Im)g↑↓\ngN+g↑↓coth(dN/λN).\n(7)\nFor the later discussion, we introduce\nβ=−ϑI\nϑR, (8)\nwhich corresponds to the ratio of the field-like torque to\nthe damping-like torque.\nThe values of the parameters used in the following\ncalculations are derived from recent experiments on the\nW/CoFeB heterostructure34, where ρF= 1/σF= 1.6\nkΩnm,pσ= 0.72,λF= 1.0 nm,ρN= 1/σN= 1.25\nkΩnm,λN= 1.2 nm, and ϑ= 0.27, whereas the thick-\nnesses areassumed to be dF= 2 nm and dN= 3 nm. The\ninterfaceconductanceswerenotevaluatedinRef.34byas-\nsuming a transparent interface. Instead, we use typical\nvalues of the interface conductances obtained from the\nfirst-principles calculations43,r= 0.25 kΩnm2,pg= 0.5,\nandg↑↓\nr/S= 25 nm−2. We note that the imaginary part\nof the mixing conductance, g↑↓\ni, is either positive or neg-\native, depending on the material and thickness43. The\nsign ofg↑↓\nidetermines those of χ′andϑI, or equivalently,\nβ. For example, when g↑↓\ni/S= 1 nm−2,χ≃0.010,\nχ′≃ −0.0002,χ′′≃0.035,θR≃0.167, and θI≃0.002\n(β≃ −0.010); see Appendix A. In the following calcula-\ntions, we study the magnetization dynamics for several\nvalues of β.\nB. Spin torques in longitudinal and transverse\ngeometries\nEquation (1) was derived for a system having single\nferromagnet. In this case, J0is the external electric cur-\nrent density, whereas ( χ′′+χmy)J0is the current density\ngenerated as a result of the charge-spinconversionby the\ndirect and inverse spin Hall effects. In the longitudinal\ngeometryinthepresentstudy, ontheotherhand, twofer-\nromagnetic/nonmagnetic interfaces, i.e., F 1/N and F 2/N\ninterfaces, contribute to the generation of the longitudi-\nnal current through the direct and inverse spin Hall ef-\nfects, as schematically shown in Fig. 1(b). Let us denote\nthe electric current density generated by these effects\nnear the F ℓ/N interface as jℓx(ℓ= 1,2). This currentdensity is determined by the conservation law of the elec-\ntric current, as follows. Considering in a similar manner\nto the case of the single ferromagnet, the electric current\nJ0+j1xis converted to the spin current by the spin Hall\neffect near the F 2/N interface, and this spin current pro-\nduces an additionalelectric current ( χ′′+χm2\n2y)(J0+j1x)\nby the inverse spin Hall effect. Therefore, the total lon-\ngitudinal electric current density near the F 2/N interface\nis (1+χ′′+χm2\n2y)(J0+j1x). Similarly, the electric cur-\nrent density near the F 1/N interface can be expressed\nas (1+χ′′+χm2\n1y)(J0+j2x). These currents should be\nequal to the total electric current density, J0+j1x+j2x,\naccording to the conservation law of the electric current.\nThen, we find that jℓxis given by\njℓx=/parenleftBig\nχ′′+χm2\nℓy/parenrightBig/parenleftBig\n1+χ′′+χm2\nℓ′y/parenrightBig\n1−/parenleftBig\nχ′′+χ′m2\nℓy/parenrightBig/parenleftBig\nχ′′+χm2\nℓ′y/parenrightBigJ0\n≃/parenleftbig\nχ′′+χm2\nℓy/parenrightbig\nJ0,(9)\nwhere we neglect the higher order terms of ϑ. Then, the\nspin torque excited on the F ℓlayer by the spin Hall effect\nin the longitudinal geometry is obtained by replacing the\nexternal electric current density J0=σNExin the previ-\nous work30withJ0+jℓ′x≃/parenleftBig\n1+χ′′+χm2\nℓ′y/parenrightBig\nJ0, and is\ngiven by\nTL\nℓ=−γ/planckover2pi1ϑRJ0\n2eMdF/parenleftbig\n1+χ′′+χm2\nℓ′y/parenrightbig\nmℓ×(ey×mℓ)\n−γ/planckover2pi1ϑIJ0\n2eMdF/parenleftbig\n1+χ′′+χm2\nℓ′y/parenrightbig\ney×mℓ,\n(10)\nwhere (ℓ,ℓ′) = (1,2) or (2,1). The unit vector pointing\nin the magnetization direction of the F ℓlayer ismℓ, and\nγ,M, anddare the gyromagnetic ratio, the saturation\nmagnetization, and the thickness of the ferromagnet, re-\nspectively.\nNote that the terms,\nT(0)\nℓ=−γ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)−γ/planckover2pi1ϑIJ0\n2eMdFey×mℓ,\n(11)\nin Eq. (10) are the conventional spin torques generated\nby the external electric current J0and the spin Hall ef-\nfect, and are often referred to as the damping-like torque\nandthefield-liketorque, respectively. Ontheotherhand,\nthe terms\n−γ/planckover2pi1ϑRJ0\n2eMdF/parenleftbig\nχ′′+χm2\nℓ′y/parenrightbig\nmℓ×(ey×mℓ)\n−γ/planckover2pi1ϑIJ0\n2eMdF/parenleftbig\nχ′′+χm2\nℓ′y/parenrightbig\ney×mℓ,(12)\nin Eq. (10) originate from the current jℓ′x, and are newly\nintroduced in this study. It should be emphasized that\nEq. (12) depends on the magnetization direction of the4\nother ferromagnet F ℓ′(ℓ′= 1 or 2), mℓ′, resulting in the\ncoupling between the F 1and F2layers.\nLet us next show the spin torque formulasin the trans-\nverse geometry. We denote the electric current den-\nsity flowing in the ydirection generated near the F ℓ/N\n(ℓ= 1,3) interface by the inverse spin Hall effect as jℓy.\nThe conservation law of the total electric current density,\nj1y+j3y, gives (see also Appendix B)\njℓy=−/bracketleftbigg(χmℓxmℓy+χ′mℓz)\n1−(χ′′+χm2\nℓx)(χ′′+χm2\nℓ′x)\n+/parenleftbig\nχ′′+χm2\nℓx/parenrightbig\n(χmℓ′xmℓ′y+χ′mℓ′z)\n1−(χ′′+χm2\nℓx)(χ′′+χm2\nℓ′x)/bracketrightBigg\nJ0\n≃ −(χmℓxmℓy+χ′mℓz)J0.(13)\nInadditiontotheconventionalspintorque,Eq. (11), this\ntransverse electric current also excites the spin torque on\nthe other ferromagnet. In total, the spin torque acting\non the F ℓlayer is given by [( ℓ,ℓ′) = (1,3) or (3,1)]\nTT\nℓ=−γ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)−γ/planckover2pi1ϑIJ0\n2eMdFey×mℓ\n−γ/planckover2pi1ϑR(χmℓ′xmℓ′y+χ′mℓ′z)J0\n2eMdFmℓ×(ex×mℓ)\n−γ/planckover2pi1ϑI(χmℓ′xmℓ′y+χ′mℓ′z)J0\n2eMdFex×mℓ,\n(14)\nThelasttwotermsrepresentthe couplingtorquebetween\nthe F1and F 3layers. Note that the direction of this\ncoupling torque is different from that of the conventional\ntorque because the currents J0andjℓyflow in different\ndirections.\nIn the following, the torques related to χ,χ′, andχ′′\nin Eqs. (10) and (14) are referred to as coupling torque.\nThe ratio of these new torques to the conventional spin\ntorque is on the order of χ∝ϑ2∼10−2. Since the\nconventional spin torque due is proportional to the spin\nHall angle ϑR∝ϑ, the coupling torque is proportional\ntoϑ3. Although the spin Hall angle is usually a small\nquantity, it will be shown that the coupling torques play\na non-negligible role on the magnetization dynamics, as\nshown below.\nC. LLG equation\nIn the following sections, we study the magnetization\ndynamics excited by the spin torque given by Eq. (10)\nor (14) both numerically and analytically. We neglect\nthe transverse coupling when the role of the longitudinal\ncoupling is studied, and vice versa, for simplicity, which\ncorresponds to considering a system consisting of the F 1\nand F 2layers, or F 1and F 3layers. The magnetization\ndynamics in the F ℓ(ℓ= 1,2,3) is described by the LLG\nequation,\ndmℓ\ndt=−γmℓ×Hℓ+αmℓ×dmℓ\ndt+TL,T\nℓ,(15)where the Gilbert damping constant is denoted as α.\nIn the following calculations, we use the values of the\nmaterial parameters, γ= 1.764×107rad/(Oe s) and\nα= 0.005, derived from the experiments44. For the later\ndiscussion, it is useful to show the explicit forms of the\nLLG equation in the longitudinal and transverse geome-\ntries. Equation (15) for the longitudinal geometry is\n/parenleftbig\n1+α2/parenrightbigdmℓ\ndt=−γmℓ×Hℓ\n−γ(1+χ′′)/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)−αγmℓ×(mℓ×Hℓ)\n−γχm2\nℓ′y/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)\n−γ(1+χ′′)(α+β)/planckover2pi1ϑRJ0\n2eMdFmℓ×ey\n+γ(1+χ′′)αβ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)\n−γ(α+β)χm2\nℓ′y/planckover2pi1ϑRJ0\n2eMdFmℓ×ey\n+γαβχm2\nℓ′y/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ),\n(16)\nwhereas that for the transverse geometry is\n/parenleftbig\n1+α2/parenrightbigdmℓ\ndt=−γmℓ×Hℓ\n−γ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)−αγmℓ×(mℓ×Hℓ)\n−γ(χmℓ′xmℓ′y+χ′mℓ′z)/planckover2pi1ϑRJ0\n2eMdFmℓ×(ex×mℓ)\n−γ(α+β)/planckover2pi1ϑRJ0\n2eMdFmℓ×ey\n+γαβ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)\n−γ(α+β)(χmℓ′xmℓ′y+χ′mℓ′z)/planckover2pi1ϑRJ0\n2eMdFmℓ×ex\n+γαβ(χmℓ′xmℓ′y+χ′mℓ′z)/planckover2pi1ϑRJ0\n2eMdFmℓ×(ex×mℓ).\n(17)\nIII. NUMERICAL ANALYSIS OF\nSYNCHRONIZATION\nIn this section, we study the magnetization dynam-\nics in the ferromagnets in the presence of the coupling\ntorques by solving Eq. (15) numerically. The self-\noscillation of the magnetization provides an interesting\nexample to understand the role of the coupling torque.\nNote that the coupling torques are proportional to the\nparameters χ,χ′, andχ′′, and their products to other pa-\nrameters such as αβχ, the values of which are relatively\nsmall, as can be seen in Eqs. (16) and (17). Nevertheless,\nthe coupling torques lead to the phase synchronization,\nas shown below.5\nThe self-oscillation of the magnetization in single fer-\nromagnets by the spin Hall effect has been observed for\nin-plane magnetized ferromagnets19, and is induced by\nthe conventional spin torque given by Eq. (11). There-\nfore, in the following, we assume that the magnetic field,\nHℓ=HKmℓyey−4πMmℓzez, (18)\nconsists of an in-plane anisotropy field HKalong the y\ndirection and a demagnetization field 4 πMin thezdi-\nrection, which we assumeas HK= 200Oe and M= 1500\nemu/c.c.34in the following calculations. It is known for\nthe case of the single ferromagnet45that the in-plane\nself-oscillation can be excited when the electric current\ndensityJ0is in the range of Jc< J0< J∗, where\nJc=2αeMd F\n/planckover2pi1ϑR(HK+2πM), (19)\nJ∗=4αeMd F\nπ/planckover2pi1ϑR/radicalbig\n4πM(HK+4πM),(20)\nwhich in this study are Jc≃26 andJ∗≃33 MA/cm2.\nA. Transverse geometry\nLet us first investigate the magnetization dynamics\nin the transverse geometry because this geometry pro-\nvides a simple example of the coupled motion. We start\nwith solving the LLG equation given by Eq. (17) for\nthe F1and F 3layers. Figure 2(a) shows an exam-\nple of the trajectory of the magnetization dynamics ob-\ntained by solving Eq. (17) numerically, where J0= 30\nMA/cm2. As shown, the in-plane oscillation is observed\nin the steady state. The initial conditions set for m1\nandm3are different as m1(0) = (cos80◦,sin80◦,0) and\nm3(0) = (cos95◦,sin95◦,0). Therefore, the dynamics of\nm1andm3near the initial time are different, as shown\nin Fig. 2(b), where the time evolutions of m1x(t) and\nm3x(t) in 0≤t≤1 ns are shown. Nevertheless, the dy-\nnamics of m1andm3synchronize gradually, and finally,\nsynchronizationof m1x(t) =m3x(t) andm1y(t) =m3y(t)\nis realized, as shown in Figs. 2(c) and 2(d). We empha-\nsize here that the dynamics shown in these figures are\nobtained for β=−0.01.\nThe mutual, as well as self, synchronization of the\nspin torque induced magnetization oscillation by using\nspin waves, electric current, microwave field, or dipole\ncoupling has been an exciting topic from the viewpoints\nof both nonlinear science and practical applications46–62.\nThe key quantity of the synchronization is the phase dif-\nference ∆ ϕbetween each magnetization to enhance the\nemission power from the spin torque oscillators and to\ninvestigate new practical applications such as neuromor-\nphic architectures38,39. The synchronization found here,\ni.e.,mℓx(t) =mℓ′x(t), is called the in-phase synchro-\nnization. We should emphasize here that, although the(a)\nmxmymz\n1.00-1.0\n-1.01.0\n0-1.001.0\ntime (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1\n:F3:F1\n:F3\n01.0\n-1.0 01.0\n0.5mx\nmy(c) (d)\ntime (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1\n:F301.0\n-1.0 01.0\n0.5mx\nmy(e) (f)time (ns)0 0.2 0.4 0.6 0.8 1.001.0\n-1.0mx(b)\nFIG. 2: (a) A typical in-plane self-oscillation trajectory of\nthe magnetization obtained by solving Eq. (15) numerically\nfor the transverse geometry. The dotted lines indicate the\noscillation direction. (b) The time evolutions of m1x(t) and\nm1y(t) near the initial state. The time variations of the x\nandycomponents of the magnetizations in the steady state\nare shown in (c) and (d) for β=−0.01 and (e) and (f) for\nβ= +0.01. The solid red lines correspond to time evolution\nin F1, and dotted blue lines to those in F 3for (b) through (f).\nresults shown in Figs. 2(b) and 2(c) are shown for one\ncertain initial condition, the in-phase synchronizations\nare confirmed for the present set of the parameters even\nwhen the initial conditions are changed.\nOn the other hand, it was shown for the case of the\ncurrent-injectionlockingofthespin torqueoscillatorthat\nthe phase difference between the magnetization oscilla-\ntion and the alternative current depends on the strength\nof the field-like torque53. The field-like torque in the\npresent system can be either positive or negative, as\nmentioned above. These facts motivate us to study the\nmagnetization dynamics for different values of β. When\nβ= +0.01, synchronized dynamics is observed in a sim-\nilar manner, but in this case, the phase difference is an-\ntiphase, i.e., m1x(t) =−m3x(t), as shown in Figs. 2(e)\nand 2(f).\nFigure 3 summarizes the dependences of the phase dif-\nference, ∆ ϕ, between the magnetizations on the field-like\ntorque strength βfor the several values of the damping\nconstant α. The vertical axis in this figure represents\nthe phase difference in the unit of the oscillation period;\ni.e., ∆ϕ= 0correspondsto the in-phase synchronization.\nOn the other hand, ∆ ϕ= 0.50 means that the phase dif-6\nfield-like torque, βphase difference , Δφ\n-0.030 -0.020 -0.010 0.010 0.020 0.030 000.250.50\nα=0.005, 0.010, 0.015,\n0.020, 0.025, 0.030\nFIG. 3: Dependencesofthephase differences, ∆ ϕ, for several\nvalues of αon the field-like torquestrength βin the transverse\ngeometry. ∆ ϕ= 0 and 0 .5 correspond to the in-phase and\nantiphase, respectively. The values of the current density ,J0,\nis increased linearly, where J0= 30 MA/cm2forα= 0.005.\nference is half of a period, and thus, is antiphase. The\nalgorithm evaluating ∆ ϕin the numerical simulation is\nsummarized in Appendix A. The results indicate that\nthe phase difference is antiphase for positive β, whereas\nit becomes in-phase when βbecomes smaller than a cer-\ntainvalue, exceptforthenarrowintermediateregionnear\nβ∼ −α/2, where the phase difference is in between in-\nphase and antiphase.\nB. Longitudinal geometry\nNext, we study the magnetization dynamics in the lon-\ngitudinal geometry between F 1and F 2layers by solv-\ning Eq. (16) numerically. Figure 4(a) and 4(b) show\nmℓx(t) andmℓy(t) in a steady state, where β=−0.01.\nThe initial conditions in these figures are m1(0) =\n(cos80◦,sin80◦,0) and m2(0) = (cos95◦,sin95◦,0).\nAn antiphase synchronization is observed in this case.\nWe notice, however, that the in-phase synchronization\ncan also be realized when the initial conditions are\nchanged. Figure 4(c) and 4(d) show such an exam-\nple, where m1(0) = (cos80◦,sin80◦,0) andm2(0) =\n(cos85◦,sin85◦,0) are assumed. These numerical results\nindicate that both the in-phase and antiphase synchro-\nnizations are stable in this case, and whether the phase\ndifference finally becomes in-phase or antiphase depends\non the initial conditions, material parameters, and cur-\nrent magnitude. We also notice that the phase difference\nis changed when the value of βis changed. Figures 4(e)\nand 4(f) show mℓxandmℓyforβ= +0.01. In this case,\nthe phase difference of the magnetizations is a quarter of\na precession period.\nThe dependences of the phase difference on the field-\nlike torque strength βfor several values of the Gilbert\ndamping constant αare summarized in Fig. 5, where\n∆ϕ= 0.25 in this figure means that the phase differencetime (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1\n:F201.0\n-1.0 01.0\n0.5mx\nmy(a) (b)\ntime (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1\n:F201.0\n-1.0 01.0\n0.5mx\nmy(c) (d)\ntime (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1\n:F201.0\n-1.0 01.0\n0.5mx\nmy(e) (f)\nFIG. 4: The variations of the xandycomponents of the\nmagnetizations in the F 1(solid, red) and F 2(dotted, blue) in\nthe longitudinal geometry are shown in (a), (c) and (b), (d)\nrespectively, where β=−0.01. The initial conditions of m2\nare different between (a), (b) and (c), (d). The magnetizatio n\ndynamics for β= +0.01 are shown in (e) and (f)\nis a quarter of a period. The phase difference is found to\nbecome a quarter of the period for positive β, whereas it\nbecomes in-phase or antiphase for negative β, depending\non the initial states of the magnetizations.\nC. Summary of numerical simulations\nLet us summarize the results of the numerical simu-\nlations here. In the transverse geometry, the coupling\ntorque induces the synchronized oscillation of the mag-\nnetizations and finally stabilizes the configuration in the\nin-phase or antiphase state, depending on the values of\nthe field-liketorque strength βand the damping constant\nα. The phase difference in the stable synchronized state\nin the longitudinal geometry also depends on the values\nofβandα, as well as the initial conditions. In this case,\nhowever, in addition to the in-phase or antiphase state, a\nphase difference with a quarter of a period is generated.\nThe phase difference can be measured from the spin\nHallmagnetoresistanceeffect. AccordingtoEq. (13), the\ntotal electric current density in the transverse direction7\nfield-like torque, β-0.030 -0.020 -0.010 0.010 0.020 0.030 000.250.50\nα=0.005 α=0.010\nα=0.015\nα=0.020α=0.030α=0.025phase difference , Δφ\nFIG. 5: Dependencesofthephase differences, ∆ ϕ, for several\nvalues of αon the field-like torque strength βin the longitu-\ndinal geometry, where ∆ ϕ= 0 and 0 .5 correspond to the\nin-phase and antiphase, respectively. ∆ ϕ= 0.25 means that\nthe phase difference is a quarter of a period. The values of\nthe current density, J0, is increased linearly, where J0= 30\nMA/cm2forα= 0.005.\ntime (ns)999.0 999.2 999.4 999.6 999.8 100000.3\n0.2\n0.1\n-0.3-0.2-0.1 current density (106 A/cm2)(a)\ntime (ns)999.0 999.2 999.4 999.6 999.8 10003233current density (106 A/cm2)(b)\nFIG. 6: (a) The transverse electric current densities given\nby Eq. (21) for in-phase synchronization (solid line) and an -\ntiphase synchronization (dotted line). (b) The longitudin al\nelectric current densities given by Eq. (22) for in-phase or\nantiphase synchronization (solid line) and phase differenc e of\na quarter of a period (dotted line).\nis\nJT\nc=j1y+j3y\n=−(χm1xm1y+χ′m1z)J0\n−(χm3xm3y+χ′m3z)J0.(21)\nThen, an oscillating current appears in the transverse di-\nrection for in-phase synchronization, whereas the trans-\nversecurrentbecomeszeroforantiphasesynchronization,\nas shown in Fig. 6(a). The Fourier transformation of\nEq. (21) for in-phase synchronization has peaks at the\nfrequencies of fn= (2n−1)f0(n= 1,2,3,···), where f0\nis the lowest frequency; see Appendix C. Similarly, the\ntotal electric current density in the longitudinal direction\nis\nJL\nc=J0+j1x+j2x\n=J0+/parenleftbig\nχ′′+χm2\n1y/parenrightbig\nJ0+/parenleftbig\nχ′′+χm2\n2y/parenrightbig\nJ0.(22)\nThe oscillation frequency of this current is different for\nthe synchronizations having the phase difference of in-\nphase or antiphase and that of a quarter of a period,as shown in Fig. 6(b). The Fourier transformation of\nEq.(22) has the peaks at fn= 2nf0for in-phase or an-\ntiphase and fn= 4nf0when the phase difference is a\nquarter of a period.\nAn interesting question regarding these numerical re-\nsults is to clarify the reason why the phase difference\nfinally becomes a certain value for a given set of the\nparameters, i.e., which phase difference is an attrac-\ntor of the limit cycle. It is, however, difficult to an-\nswer this question directly due to the following rea-\nson. We note that the present model includes sev-\neral small-valued parameters, α,β,χ,χ′, andχ′′,\nas shown in Eqs. (16) and (17), and is complicated.\nThe torque related to the lowest order of βin these\nequations is the conventional field-like torque given by\n[γβ/planckover2pi1ϑRJ0/(2eMdF)]mℓ×ey. It should be noted that\nthe phase difference is not determined solely by this term\nbecause this lowest order term of the field-like torque\ndoes not include the coupling between the magnetiza-\ntions. Similarly, the attractor is not determined solely\nby the lowest order term of the coupling torque, which\nis [γ/planckover2pi1ϑRJ0/(2eMdF)]χm2\nℓ′ymℓ×(ey×mℓ) in Eq. (16)\nand [γ/planckover2pi1ϑRJ0/(2eMdF)]χmℓ′xmℓ′ymℓ×(ex×mℓ) in Eq.\n(17), because this torque does not include the field-like\ntorque. Their combinations or the higher order terms\nincluding both βand the coupling torques related to χ,\nχ′, andχ′′should be taken into account to answer the\nquestion, which is difficult due to the nonlinearity and\ncomplexity of the LLG equation.\nNevertheless, it is possible to reveal the relation be-\ntween the current and frequency in the synchronized os-\ncillation state by assuming a certain value of the phase\ndifference between the magnetizations. The current-\nfrequency relation has been often measured in the ex-\nperiments, and therefore, it will be useful to develop a\ntheory clarifying the role of the coupling on the current-\nfrequency relation. In the next section, we will investi-\ngate this subject.\nIV. THEORETICAL ANALYSIS OF\nCURRENT-FREQUENCY RELATION\nThe purpose of this section is to develop an analyti-\ncal theory of the synchronization revealing the relation\namongthe current, frequency, and the phase difference of\nthe magnetizations in the synchronized oscillation state.\nA. Basis of analysis\nHere, let us mention the basis of our theoretical anal-\nysis. It is difficult to solve the LLG equation exactly\nbecause of its nonlinearity. Instead, we employ the aver-\naging technique of the LLG equation on constant energy\ncurves63. This approach has been used to study the spin\ntorque switching in thermally activated regions64–68and\nspin torque oscillators69–77, as well as the microwave as-8\nsisted magnetization reversal78,79, but has not been ap-\nplied to the coupled system. This approach is valid when\nthe magnetic energy is changed slowly compared with\nthe oscillation period. We note that only the lowest or-\nder terms in the LLG equation is necessary to derive the\ncurrent-frequency relation, as far as several parameters\nsuch asβare small. Thus, we use the simplified LLG\nequationmaintainingonlythedominantterms. TheLLG\nequation used in this section for the longitudinal geome-\ntry is\ndmℓ\ndt≃−γmℓ×Hℓ−αγmℓ×(mℓ×Hℓ)\n−γ/planckover2pi1ϑRJ0\n2eMdF/parenleftbig\n1+χ′′+χm2\nℓ′y/parenrightbig\nmℓ×(ey×mℓ),\n(23)\nwhereas that for the transverse geometry is\ndmℓ\ndt≃−γmℓ×Hℓ−αγmℓ×(mℓ×Hℓ)\n−γ/planckover2pi1ϑRJ0\n2eMdFmℓ×(ey×mℓ)\n−γ/planckover2pi1ϑRJ0\n2eMdFχmℓ′xmℓ′ymℓ×(ex×mℓ).(24)\nIn the self-oscillation state, the damping torque dur-\ning the precession is balanced with the spin torque, and\nthetorqueduetothemagneticfield, correspondingtothe\nfirst term on the right hand side of Eq. (15), becomes the\ndominantterm determining the magnetizationdynamics.\nThe dynamic trajectory given by this field torque corre-\nsponds to constant energy curves of the energy density,\nE=−M/integraltext\ndmℓ·Hℓ, where its explicit form is\nE=−MHK\n2m2\nℓy+2πM2m2\nℓz. (25)\nThe minimum and saddle points of Eq. (25) are mmin=\n±eyandmsaddle=±ex, where the corresponding energy\ndensities are Emin=−MHK/2 andEsaddle= 0. The\nsolution of mℓprecessing on a constant energy curve is\ndescribed by the Jacobi elliptic function80as (see also\nAppendix C summarizing the derivations)\nmx=/radicalbigg\n1+2E\nMHKsn/bracketleftbigg4K(k)\nτ(E)t+ϕ0,k/bracketrightbigg\n,(26)\nmy=/radicalBigg\n4πM−2E/M\nHK+4πMdn/bracketleftbigg4K(k)\nτ(E)t+ϕ0,k/bracketrightbigg\n,(27)\nmz=/radicalBigg\nHK+2E/M\nHK+4πMcn/bracketleftbigg4K(k)\nτ(E)t+ϕ0,k/bracketrightbigg\n,(28)\nwhereϕ0is the initial phase. The period of mxandmzis\nτ(E), whereasthat of myisτ(E)/2, wherethe precession\nfrequency f(E) = 1/τ(E) is given by\nf(E) =γ/radicalbig\nHK(4πM−2E/M)\n4K(k),(29)whereK(k) is the first kind of complete elliptic integral\nwith the modulus\nk=/radicalBigg\n4πM(HK+2E/M)\nHK(4πM−2E/M). (30)\nNote that Eq. (29) reproduces the ferromagnetic res-\nonance (FMR) frequency, γ/radicalbig\nHK(HK+4πM)/(2π), in\nthe limit of E→Emin. Identifying Eandϕ0corresponds\nto the determination of the initial condition.\nThe averagedtechnique investigates the energy change\nduring a precession on a constant energy curve, which is\nobtained from the LLG equation as\n/contintegraldisplay\ndtdE\ndt=Ws+WL,T\ns+Wα, (31)\nwhere Wsis the energy change by the conventional spin\ntorque due to the spin Hall effect, whereas Wαis the dis-\nsipation due to the damping torque. The integral range\nis over the precession period. The explicit forms of Ws\nandWαare given by67\nWs=/contintegraldisplay\ndtγ/planckover2pi1ϑRJ0\n2edF[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)]\n=π/planckover2pi1θRJ0(HK+2E/M)\nedF/radicalbig\nHK(HK+4πM),(32)\nWα=−/contintegraldisplay\ndtαγM/bracketleftBig\nH2\nℓ−(mℓ·Hℓ)2/bracketrightBig\n=−4αM/radicalBigg\n4πM−2E/M\nHK/bracketleftbigg2E\nMK(k)+HKE(k)/bracketrightbigg\n,\n(33)\nwhereE(k)isthesecondkindofcompleteellipticintegral.\nOn the other hand, WL,T\nsrepresents the work done\nby the coupling torque in the longitudinal or transverse\ngeometry, corresponding to the last term in Eq. (23)\nor (24). The explicit forms of WL\nsandWT\nsare shown\nin the following sections. For both cases, the relation\nbetween the current and frequency is clarified as follows.\nInthe self-oscillationstate, sincethespin torquebalances\nthe damping torque, the following condition should be\nsatisfied,\n/contintegraldisplay\ndtdE\ndt= 0. (34)\nThe current density J0satisfying Eq. (34) is the current\nnecessaryto excite the self-oscillationon the constanten-\nergy curve of E, and is denoted as J0(E). The relation\nbetween the current and frequency in the self-oscillation\nstate is given by this J0(E) andf(E) given by Eq. (29).\nIt should be emphasized that the current density J0(E)\ndepends on the phase difference between the magnetiza-\ntions through WL,T\ns. We will therefore study the relation\nbetween the phase difference and the current-frequency\nrelation in line with this deduction.9\nB. Transverse geometry\nIn this section, we investigate the current-frequency\nrelation in the transverse geometry. The work done by\nthe coupling torque is defined as\nWT\ns=/contintegraldisplay\ndtγ/planckover2pi1ϑRχJ0\n2edFmℓ′xmℓ′y[ex·Hℓ−(mℓ·ex)(mℓ·Hℓ)].\n(35)\nBefore advancing the calculation, let us briefly men-\ntion the definition of the phase difference in the present\napproach. If the oscillation trajectory is a circle, the\nphase difference is easily defined, i.e., the antiphase cor-\nresponds to ∆ ϕ=π, whereas ∆ ϕ= 0 is the in-phase.\nIn the present case, on the other hand, the oscillation\ntrajectory is described by the elliptic function, as shown\nin Eqs. (26), (27), and (28). In this case, the phase dif-\nference is defined using the elliptic integral K(k), where\n∆ϕ= 0 for the in-phase synchronization, and the an-\ntiphase synchronization corresponds to ∆ ϕ= 2K(k). It\nis useful to note that ∆ ϕ= 2K(k) becomes πin the limit\nofk→0, corresponding to the case that the oscillation\ntrajectory becomes a circle. Similarly, ∆ ϕ=K(k) means\nthat the phase difference is a quarter of a period.\nEquation (35) for an arbitrary phase difference is eval-\nuated by numerically calculating the integral with the\nsolutions of mℓandmℓ′shown in Appendix D. It is,\nhowever, useful to derive the analytical solutions of Eq.\n(35) for specific values of the phase difference. Equa-\ntion (35) for both the in-phase (∆ ϕ= 0) and antiphase\n[∆ϕ= 2K(k)] becomes\nWT\ns=∓π/planckover2pi1ϑRχJ0\n2edF/radicalbig\nHK(HK+4πM)/parenleftbigg\n−2E\nM/parenrightbigg/parenleftbigg\n1+2E\nMHK/parenrightbigg\n,\n(36)\nwhere the double sign means the upper for the in-phase\n(∆ϕ= 0)synchronizationandthelowerfortheantiphase\n[∆ϕ= 2K(k)] synchronization. Equation (36) is zero at\nE=EminandEsaddle, and is negative (positive) for the\nenergy density Ein the rage of Emin< E < E saddle\nwhen ∆ϕ= 0 [2K(k)]. This means that the coupling\ntorque acts as a damping (an antidamping) torque when\nthe phase differenceis in-phase(antiphase). We alsonote\nthatWT\ns= 0 when ∆ ϕ=K(k); i.e., the phase difference\nis a quarter of a period. The calculations necessary to\nobtain these specific values of WT\nsare also summarized\nin Appendix D. We note that the sign changeof WT\nswith\nrespect to the phase difference is related to the fact that\nthe coupling torque in the transverse geometry, Eq. (14),\nhas the angular dependence of mℓ′xmℓ′ymℓ×(ex×mℓ).\nBecause of this angular dependence, the coupling torque\nacts as an anti-damping (a damping) torque when mℓx\nandmℓ′xhave the opposite (same) signs, resulting in the\nincrease (decrease) of the energy supplied to the ferro-\nmagnets by the coupling torque.\nIn summary, the work done by the coupling torque,\nWT\ns, isnegativeandminimizedatthe in-phase(∆ ϕ= 0),\nzero for ∆ ϕ=K(k), and positive and maximized at the\nantiphase [∆ ϕ= 2K(k)].32\n30\n28\n26\n6.05.04.03.02.0\n01.02.03.04.0current density, J 0(E) (106 A/cm 2)(a)\nfrequnecy (GHz)\nphase difference, Δφ/K(k)\n31\n29\n27\n25current density, J 0(E) (106 A/cm 2)(c) (d)\nphase difference, Δ φ/K(k)0 1.0 2.0 3.0 4.027.0127.0227.03current density, J 0(E) (106 A/cm 2)\n6.05.04.03.02.0\n01.02.03.04.0\nfrequnecy (GHz)\nphase difference, Δφ/K(k)(b)\nphase difference, Δ φ/K(k)0 1.0 2.0 3.0 4.028.0028.1028.2028.30current density, J 0(E) (106 A/cm 2)\nFIG. 7: (a) The current density, J0(E), necessary to excite\nthe self-oscillation in the transverse geometry as a functi on of\nthe oscillation frequency f(E) and the phase difference ∆ ϕ\nof the magnetizations. The phase difference is in the unit\nofK(k). (b) Dependence of J0(E) for the transverse geome-\ntry, on ∆ ϕatf(E) = 4.6 GHz. The dotted line represents\nJ0(E) in the absence of the coupling. (c) The relation among\nJ0(E),f(E), and ∆ ϕin the longitudinal geometry. (d) The\ncurrent density J0(E) forf(E) = 4.6 GHz in the longitudinal\ngeometry.\nThe current J0in the transverse geometry is defined\nas\nJ0(E) =2αeMd F\n/planckover2pi1ϑRN\nDT, (37)\nwhereNandDTare defined as\nN=γ/contintegraldisplay\ndt/bracketleftBig\nH2\nℓ−(mℓ·Hℓ)2/bracketrightBig\n, (38)\nDT=γ/contintegraldisplay\ndt[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)]\n+γχ/contintegraldisplay\ndtmℓ′xmℓ′y[ex·Hℓ−(mℓ·ex)(mℓ·Hℓ)].\n(39)\nThe explicit form of Nis obtained from Eq. (33) as\nN= 4/radicalBigg\n4πM−2E/M\nHK/bracketleftbigg2E\nMK(k)+HKE(k)/bracketrightbigg\n.(40)\nOn the other hand, DTfor the in-phase or antiphase is\nobtained from Eqs. (32) and (36) as\nDT=2π(HK+2E/M)/radicalbig\nHK(HK+4πM)\n∓πχ(−2E/M)[1+2E/(MHK)]/radicalbig\nHK(HK+4πM),(41)10\nwhere the double sign means the upper for the in-phase\nsynchronization and the lower for the antiphase synchro-\nnization.\nFigure 7(a) shows J0(E) as functions of the oscilla-\ntion frequency f(E) and the phase difference ∆ ϕ. The\ncurrent-frequencyrelationinthetransversegeometrycan\nbe obtained from this figure. To reveal the role of the\nphase difference more clearly, we show J0(E) for a cer-\ntain value of f(E)(= 4.6GHz) in Fig. 7(b). Note that\nJ0(E) is smaller than that in the absence of the coupling,\nwhich is shown by the dotted line, and minimized when\n∆ϕ= 2K(k), i.e., theantiphase. Thisisbecausethework\ndone by the coupling torque is positive and maximized at\nthe antiphase. On the other hand, J0(E) is maximized\nat the in-phase, and is larger than that in the absence\nof the coupling because the work done by the coupling\ntorqueis negativeand minimized at the in-phase. We no-\ntice that the phase differences observed in the numerical\nsimulation in Sec. III, i.e., the in-phase and antiphase,\ncorrespond to ∆ ϕsatisfying\n∂J0\n∂∆ϕ= 0, (42)\nor equivalently,\n∂WT\ns\n∂∆ϕ= 0. (43)\nIn other words, the phase differences observed in the nu-\nmerical simulations correspond to those giving the ex-\ntrema of J0(WT\ns).\nC. Longitudinal geometry\nLet us investigate the theoretical relation between the\ncurrent and frequency in the longitudinal geometry. In\nthis case, the averaged LLG equation is given by\n/contintegraldisplay\ndtdE\ndt= (1+χ′′)Ws+WL\ns+Wα,(44)\nwhere WsandWαare given by Eqs. (32) and (33). On\nthe other hand, WL\nsrepresenting the energy change due\nto the longitudinal coupling is defined as\nWL\ns=/contintegraldisplay\ndtγ/planckover2pi1ϑRχJ0\n2edFm2\nℓ′y[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)].\n(45)\nFor both the in-phase and antiphase, Eq. (45) becomes\n(see Appendix D)\nWL\ns=π/planckover2pi1ϑRχJ0\n2edF/radicalbig\nHK(HK+4πM)\n×(HK+2E/M)[4πM(HK−2E/M)−2HK(2E/M)]\nHK(HK+4πM).\n(46)Onthe other hand, when the phasedifference isaquarter\nof a period [∆ ϕ=K(k)], Eq. (45) becomes\nWL\ns=π/planckover2pi1ϑRχJ0\nedFHK+2E/M\nHK(HK+4πM)/radicalBigg\n−2E\nM/parenleftbigg\n4πM−2E\nM/parenrightbigg\n.\n(47)\nItshouldbeemphasizedthat WL\nsisalwayspositiveforan\narbitrary phase difference. This is because the coupling\ntorqueinEq. (10)alwaysactsasananti-dampingtorque.\nWe can determine the current density J0(E) satisfying\nEq. (34) in the longitudinal geometry, as in the case of\nthe transverse geometry, by replacing DTin Eq. (37)\nwith\nDL=γ(1+χ′′)/contintegraldisplay\ndt[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)]\n+γχ/contintegraldisplay\ndtm2\nℓ′y[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)].\n(48)\nThe explicit form of DLfor both the in-phase and an-\ntiphase is obtained from Eqs. (32) and (46) as\nDL=2π(1+χ′′)(HK+2E/M)/radicalbig\nHK(HK+4πM)\n+πχ(HK+2E/M)[4πM(HK−2E/M)−2HK(2E/M)]\n[HK(HK+4πM)]3/2,\n(49)\nwhereas that when the phase difference is a quarter of a\nperiod is obtained from Eqs. (32) and (47) as\nDL=2π(1+χ′′)(HK+2E/M)/radicalbig\nHK(HK+4πM)\n+2πχ(HK+2E/M)\nHK(HK+4πM)/radicalBigg\n−2E\nM/parenleftbigg\n4πM−2E\nM/parenrightbigg\n.\n(50)\nFigure 7(c) shows J0(E) as functions of f(E) and ∆ϕ.\nThe current-frequency relation in the longitudinal geom-\netry can be obtained from this figure. It is noted that\nJ0(E) is always smaller than that in the absence of the\ncoupling because the coupling torque in the longitudinal\ngeometry always points to the anti-damping direction,\nand therefore, the work done by the coupling torque is\nalways positive. Figure 7(b) shows J0(E) as a function\nof ∆ϕat a certain value of f(E). As shown, J0(E) has\nminima at both the in-phase (∆ ϕ= 0) and the antiphase\n[∆ϕ= 2K(k)], whereas it is maximized when the phase\ndifference is a quarter of a period [∆ ϕ=K(k)]. We again\nnotice that these phase differences found in the numer-\nical simulations in Sec. III correspond to ∆ ϕsatisfying\n∂J0(E)/∂∆ϕ= 0, or equivalently,\n∂WL\ns\n∂∆ϕ= 0. (51)11\nD. Phase differences in stable synchronization and\nfixed points of effective potential\nEquation (31) describes the slow change of the mag-\nnetic energy in the oscillation state. The magnetization\ndynamics is regarded as a motion of a point particle in\nan effective potential given by its right-hand side. Equa-\ntions (43) and (51) correspond to the stability conditions\nofthe point particlein this effective potential. Therefore,\nthe phase difference found in the numerical simulation fi-\nnally converges to one of these values satisfying Eq. (43)\nor Eq. (51). Whether the in-phase, antiphase, or the\nphase difference with a quarter of a period becomes the\nattractor depends on the higher order terms of the small\nparameters, as well as the initial states of the magnetiza-\ntions, asmentioned at the end ofSec. III. This discussion\nis beyond the scope of this paper.\nE. Instability threshold\nAt mentioned at the beginning of Sec. III, the in-plane\nself-oscillation for a single ferromagnet is stabilized when\nthe currentdensity is in the rangeof Jc< J0< J∗, where\nJcandJ∗are given by Eqs. (19) and (20), respectively.\nAt the end of this section, let us briefly discuss the effect\nof the coupling on these scaling currents.\nLet us remindthe readersthat Jcis the currentdensity\nnecessarytodestabilizethemagnetizationinequilibrium,\nwhereas J∗is the current necessary to overcome the en-\nergy barrier, Esaddle−Emin. These current densities are\ntheoretically defined as67\nJc= lim\nE→EminJ0(E), (52)\nJ∗= lim\nE→EsaddleJ0(E). (53)\nIt is confirmed that Eqs. (19) and (20) are reproduced\nby substituting Eqs. (32) and (33) in the definition of\nJ0(E) in the absence of the coupling.\nOn the other hand, in the presence of the transverse\ncoupling, it is confirmed from Eq. (36) that a factor\n[1−(χ/2)] should be multiplied to the denominator of\nEq. (19) when the phase difference between the mag-\nnetizations is in-phase, whereas this factor is replaced\nby [1+( χ/2)] when the phase difference is antiphase.\nThe other scaling current, J∗, is unchanged for these\nphase differences. In the longitudinal geometry, we see\nfrom Eqs. (46) and (47) that the factor (1+ χ+χ′′)\nshould be multiplied to the denominator of Eq. (19)\nwhen the phase difference is in-phase, antiphase, or a\nquarter of a period, whereas, for J∗, the factor becomes\n1+χ′′+(χ/2)[4πM/(HK+4πM)] for in-phase and an-\ntiphase, and 1+ χ′′when the phase difference is a quarter\nof a period.V. CONCLUSION\nIn conclusion, the coupled magnetization dynamics in\nthe ferromagnets through the spin Hall magnetoresis-\ntance effect was investigated. The coupling appears in\nboth the longitudinal and transverse directions of the\nalignment of the ferromagnets. The in-phase or an-\ntiphase synchronization of the magnetization oscillation\nwas found in the transversegeometry by solving the LLG\nequation numerically. On the other hand, in addition to\nthem, the synchronization having the phase difference of\na quarter of a period is also found in the longitudinal ge-\nometry. It wasshownthatthesephasedifferencesdepend\non the values of the damping constant and the field-like\ntorque strength. The analytical theory revealing the re-\nlationamongthe current, frequency, and phasedifference\nwas also developed. It was shown that the phase differ-\nences observed in the numerical simulations correspond\nto that giving the fixed points of the energy supplied by\nthe coupling torque.\nAcknowledgement\nThe author is grateful to Takehiko Yorozu for his con-\ntributions to the analytical calculations, and to Hitoshi\nKubota, Sumito Tsunegi, Yoichi Shiota, Shingo Tamaru,\nTazumi Nagasawa, Kiwamu Kudo, and Yoji Kawamura\nfor valuable discussions. The author is also thankful to\nSatoshiIba, AurelieSpiesser,AtsushiSugihara, Takahide\nKubota, Hiroki Maehara, and Ai Emura for their sup-\nport and encouragement. This work was supported by\nJSPS KAKENHI Grant-in-Aid for Young Scientists (B)\n16K17486.\nAppendix A: Values of parameters in numerical\nsimulations\nThe exact values of the parameters used in the sim-\nulations, evaluated from the parameters found in the\nexperiment34, areϑR= 0.16680863, β=−0.00973617,\nχ= 0.009525272, χ′=−0.000152995, and χ′′=\n0.03516089for g↑↓\ni/S= 1.0 nm−2. In the main text, β=\n−0.01 andβ= +0.01 correspond to β=−0.00973617\nandβ= +0.00973617, respectively. Strictly speaking,\nthe change of the value of g↑↓\niaffects not only βbut\nalso other quantities such as ϑR,χ, andχ′. We, how-\never, change the value of βonly in the numerical simu-\nlation, for simplicity, because the results do not change\nsignificantly unless |g↑↓\ni/g↑↓\nr| ≪1. The LLG equation\nwith these parameters is solved by using the fourth-order\nRunge-Kutta method from t= 0 tot= 1µs with the\ntime step of ∆ t= 20 fs; i.e., the number of the time\nmesh isNt= 5×107.\nThe presentsystem has twostable states at mℓ=±ey.\nFor convention, we assume that the magnetizations ini-\ntially stay near one equilibrium, mℓ= +ey. For in-phase12\nsynchronizations, such as those shown in Figs. 2(c) and\n2(d), the zcomponents are also synchronized with in-\nphase, i.e., mℓz(t) =mℓ′z(t). On the other hand, for\nantiphase synchronization shown in, for example in Figs.\n2(e) and 2(f), the zcomponents are also synchronized\nwith antiphase, mℓz(t) =−mℓ′z(t).\nThe algorithm evaluating the phase differences shown\nin Figs. 3 and 5 from the discrete numerical data\nis as follows. We gathered Ni= 216= 65536 data\nofmℓ(t) (ℓ= 1,2,3) from t= (Nt−Ni+1)∆tto\nt=Nt∆t= 1µs. Then, the averaged periods\nTℓof the oscillation of each magnetization were eval-\nuated from the peaks of mℓ(t) in this time range\nasTℓ= [(tℓ,a−tℓ,a−1)+···+(tℓ,2−tℓ,1)]/(Nℓ−1) =\n(tℓ,a−tℓ,1)/(Nℓ−1), where Nℓis the number of the\npeaks in mℓ(t), whereas tℓ,ais the time corresponding\nto thea-th peak. Then, the phase difference is evalu-\nated as ∆ ϕ=/summationtextN′\na=1|tℓ,a−tℓ′,a|//parenleftbig\nN′¯T/parenrightbig\n, where N′=\nmin[Nℓ,Nℓ′] and¯T= (Tℓ+Tℓ′)/2 with (ℓ,ℓ′) = (1,3)\nfor the transverse geometry, whereas that is (1 ,2) for\nthe longitudinal geometry. For the in-phase synchro-\nnization, this ∆ ϕis zero because tℓ,a=tℓ′,a. When\nthe phase difference is antiphase, ∆ ϕ= 0.50 because\n|tℓ,a−tℓ′,a|=¯T/2 in this case. Similarly, ∆ ϕis 0.25\nwhen the phase difference is a quarter of a period.\nNote that the critical current density to excite the self-\noscillation, given by Eq. (19), is proportional to the\ndamping constant α. Therefore, the value of the current\ndensity should be increased to observe the self-oscillation\nwhenαis varied, as in the case of Figs. 3 and 5. In these\nfigures,J0isassumedas n×30MA/cm2forα=n×0.005\n(n= 1−6).\nThe numerical simulation in Fig. 2(e) indicates that\nthe antiphase synchronization is an attractor for β=\n+0.01. An exception is that if the initial conditions are\nset to be identical, the final state becomes in-phase syn-\nchronization due to the symmetry of the LLG equation\nwith respect to the change of ( ℓ,ℓ′)→(ℓ′,ℓ). Since Eq.\n(43) is satisfied, the phase difference is fixed to in-phase\neven if it is unstable. Similar situations occur in other\ncases for such specific initial conditions.\nAppendix B: Derivation of coupling torque in\ntransverse geometry\nIn a ferromagnetic/nonmagnetic bilayer, the spin cur-\nrent density flowing in the idirection ( i=x,y,z) with\nthe spin polarization in the νdirection is related to the\nelectrochemical potential ¯ µNand the spin accumulation\nδµNvia\nJsiν,N=−/planckover2pi1σN\n2e2∂iδµN,ν−/planckover2pi1ϑσN\n2e2ǫiνj∂j¯µN,(B1)\nwhere∂j¯µN/eis the electric field in the jdirection, and\ntherefore, σN∂j¯µN/eis the electric current density. Weassume that this equation is extended to\nJ(ℓ)\nszν,N=−/planckover2pi1σN\n2e2∂zδµ(ℓ)\nN,ν+/planckover2pi1ϑ\n2e(J0δνy−jℓ′yδνx),(B2)\nin the transverse geometry, where J(ℓ)\nszν,Nis the spin cur-\nrent density flowing near the F ℓ/N interface in the zdi-\nrection with the spin polarization in the νdirection. The\nspin accumulation obeys the diffusion equation, and the\nboundary conditions of the diffusion equation are given\nby the spin current density at the boundaries. Using Eq.\n(B2), the solution of the spin accumulation is given by40\nδµ(ℓ)\nN,ν=2π\n(gN/S)sinh(dN/λN)/braceleftbigg\n−JFℓ/N\nszνcosh/parenleftbiggz+dN\nλN/parenrightbigg\n−/planckover2pi1ϑ\n2e(J0δνy−jℓ′yδνx)/bracketleftbigg\ncosh/parenleftbiggz\nλN/parenrightbigg\n−cosh/parenleftbiggz+dN\nλN/parenrightbigg/bracketrightbigg/bracerightbigg\n,\n(B3)\nwhere we assume that the nonmagnet is in the region of\n−dN≤z≤0. The spin current density, JFℓ/N\nszν, at the\nFℓ/N interface is given by30,40\nJFℓ/N\ns=/planckover2pi1ϑg∗\n2egNtanh/parenleftbiggdN\n2λN/parenrightbigg\n(J0mℓy−jℓ′ymℓx)mℓ\n+/planckover2pi1\n2eJ0[ϑRmℓ×(ey×mℓ)+ϑIey×mℓ]\n−/planckover2pi1\n2ejℓ′y[ϑRmℓ×(ex×mℓ)+ϑIex×mℓ],\n(B4)\nwhere the vector notation in boldface represents the di-\nrection of the spin polarization, whereas the spatial di-\nrection of Eq. (B4) is defined as the positive direction,\ni.e., from the nonmagnet to the ferromagnet.\nOn the other hand, the electric current density in the\nnonmagnet flowing in the idirection is given by\nJci,N=σN\ne∂i¯µN−ϑσN\neǫijν∂jδµN,ν.(B5)\nIn the present case, the electric current density near the\nFℓ/N interface flowing in the ydirection becomes\nJ(ℓ)\ncy,N=jℓ′y−ϑσN\ne∂zδµ(ℓ)\nN,x. (B6)\nSubstituting Eqs. (B3) and (B4) into Eq. (B6),\nand averaging along the zdirection as J(ℓ)\ncy,N=\n(1/dN)/integraltext0\n−dNJ(ℓ)\ncy,Ndz, we find that\nJ(ℓ)\ncy,N=/parenleftbig\n1+χ′′+χm2\nℓx/parenrightbig\njℓ′y\n−(χmℓxmℓy+χ′mℓz)J0.(B7)\nThe conservation law of the electric current along the y\ndirection requires that J(ℓ)\ncy,N=jℓy+jℓ′y. Solving this\nequation for ℓ= 1 and 3, we obtain Eq. (13). The spin\ntorque is defined from Eq. (B4) as\nTℓ=−γ\nMdFmℓ×/parenleftBig\nJFℓ/N\ns×mℓ/parenrightBig\n.(B8)13\nAppendix C: Analytical solution of magnetization on\na constant energy curve\nIn this appendix, we shows the derivation of the ana-\nlyticalsolutionofthe magnetizationonaconstantenergy\ncurve. Forsimplicity, weremovethesubscript ℓ(= 1,2,3)\ndistinguishing the ferromagnets. The magnetization dy-\nnamics on a constant energy curve is described by the\nLandau-Lifshitz (LL) equation\ndm\ndt=−γm×H. (C1)\nThe magnetic field, Eq. (18), is related to the magnetic\nenergy density EviaE=−M/integraltext\ndm·H, as mentioned\nin the main text. Using the relation m2\nx+m2\ny+m2\nz= 1,\nEq. (25) is rewritten as\nm2\nz+HK\nHK+4πMm2\nx=2E/M+HK\nHK+4πM.(C2)\nThisequationindicatesthat mzandmxcanbe expressed\nasmz=v′cosuandmx= (v′/v)sinu, respectively,\nwherevandv′are defined as v2=HK/(HK+4πM)\nandv′= (2E/M+HK)/(HK+4πM). Then, du/dt=\n(du/dsinu)(dsinu/dt) = (1/cosu)[d(v/v′)mx/dt] =\n(v/mz)(dmx/dt), which becomes, from Eq. (C1),\ndu\ndt=γ(HK+4πM)vmy. (C3)\nIntroducing new variable w= sinu, this equation gives\ndw/radicalbig\n(1−w2)(1−k2w2)=γ(HK+4πM)v/radicalbig\n1−v′2dt,\n(C4)\nwhere the modulus kis given by Eq. (30). The mod-\nulus monotonically varies from 0 to 1 by changing the\nenergy density Efrom its minimum Eminto saddle\nEsaddle. We also notice that ( HK+4πM)v√\n1−v′2=/radicalbig\nHK(4πM−2E/M). Equation (C4) indicates that w\nis given by\nw= sn/bracketleftBig\nγ/radicalbig\nHK(4πM−2E/M)t+ϕ0,k/bracketrightBig\n,(C5)\nwheresn( u,k)istheJacobiellipticfunction, and ϕ0isthe\ninitial phase determined by the initial condition. Using\nthe relations sn2(u,k) + cn2(u,k) = 1 and dn2(u,k) =/radicalbig\n1−k2sn2(u,k), we find that the solution of mon the\nconstant energy curve is given by Eqs. (26), (27), and\n(28).\nThe peak frequencies of the Fourier transformation of\nEq. (21) in the transverse geometry are discussed as fol-\nlows. We note that Eq. (21) for in-phase synchronization\nis proportional to mℓx(t)mℓy(t) andmℓz(t). Substituting\nthe following formulas80,\nsn(u,k) =2π\nkK(k)∞/summationdisplay\nm=0qm+1/2\n1−q2m+1sin/bracketleftbigg(2m+1)πu\n2K(k)/bracketrightbigg\n,\n(C6)cn(u,k) =2π\nkK(k)∞/summationdisplay\nm=0qm+1/2\n1+q2m+1cos/bracketleftbigg(2m+1)πu\n2K(k)/bracketrightbigg\n,\n(C7)\ndn(u,k) =π\n2K(k)\n+2π\nK(k)∞/summationdisplay\nm=0qm+1\n1+q2(m+1)cos/bracketleftbigg(m+1)πu\nK(k)/bracketrightbigg\n,\n(C8)\nto Eqs. (26), (27), (28), where q=\nexp/bracketleftbig\n−πK/parenleftbig√\n1−k2/parenrightbig\n/K(k)/bracketrightbig\n, it is found that the\npeak frequencies of Eq. (21) appear at fn= (2n−1)f0\n(n= 1,2,3,···), where the lowest frequency f0is given\nby Eq. (29).\nIn the longitudinal geometry, Eq. (22) is proportional\ntom2\n1y+m2\n2y. When the phase difference of the magne-\ntizations is in-phase or antiphase, it becomes 2 m2\n1y. In\nthis case, using the formula81\ndn2(u,k) =E(k)\nK(k)+2π2\nK2(k)∞/summationdisplay\nm=1mqm\n1−q2mcos/bracketleftbiggmπu\nK(k)/bracketrightbigg\n,\n(C9)\nit is found that the Fourier transformation of Eq.\n(22) has the peaks at fn= 2nf0. On the other\nhand, when the phase difference is a quarter of a pe-\nriod, Eq. (22) is proportional to g(u)≡dn2(u,k) +\ndn[u+K(k),k] = dn2(u,k) +/bracketleftbig/parenleftbig\n1−k2/parenrightbig\n/dn2(u,k)/bracketrightbig\n. We\nnotice that g[u+K(k)] =g(u), indicating that the\nFourier transformation of Eq. (22) in this case has the\npeaks at fn= 4nf0.\nAppendix D: Details of calculations of Eqs. (35) and\n(45)\nEquations (35) and (45) can be calculated by substi-\ntuting the solution of mℓon a constant energy curve to\nthe integrals. As emphasized in the main text, the phase\ndifference ∆ ϕbetween the magnetizations is an impor-\ntant quantity. According to Eqs. (26), (27), and (28), we\nsetmℓandmℓ′as\nmℓx=/radicalbigg\n1+2E\nMHKsn/bracketleftbigg4K(k)\nτ(E)t,k/bracketrightbigg\n,(D1)\nmℓy=/radicalBigg\n4πM−2E/M\nHK+4πMdn/bracketleftbigg4K(k)\nτ(E)t,k/bracketrightbigg\n,(D2)\nmℓz=/radicalBigg\nHK+2E/M\nHK+4πMcn/bracketleftbigg4K(k)\nτ(E)t,k/bracketrightbigg\n,(D3)\nand\nmℓ′x=/radicalbigg\n1+2E\nMHKsn/bracketleftbigg4K(k)\nτ(E)t+∆ϕ,k/bracketrightbigg\n,(D4)14\nmℓ′y=/radicalBigg\n4πM−2E/M\nHK+4πMdn/bracketleftbigg4K(k)\nτ(E)t+∆ϕ,k/bracketrightbigg\n,(D5)\nmℓ′z=/radicalBigg\nHK+2E/M\nHK+4πMcn/bracketleftbigg4K(k)\nτ(E)t+∆ϕ,k/bracketrightbigg\n.(D6)\nThe value of ∆ ϕvaries in the rage of 0 ≤∆ϕ <4K(k).\n∆ϕ= 0 corresponds to the in-phase synchronization,\nwhereas ∆ ϕis 2K(k) for the antiphase synchronization.\nThe analytical formulas of Eqs. (35) and (45) for\nthe in-phase and antiphase synchronizations can be ob-\ntained as follows. First, since the elliptic functions sat-\nisfy sn[u+2K(k),k] =−sn(u,k), cn[u+2K(k),k] =\n−cn(u,k), and dn[ u+2K(k),k] = dn( u,k),WT\nshas\nthe same magnitude but different sign for ∆ ϕ= 0 and\n∆ϕ= 2K(k), whereas WL\nsis the same for the in-phase\nand antiphase. Therefore, it is sufficient to calculate WT\ns\nandWL\nsfor the in-phase case. In this case, it is unneces-\nsary to distinguish mℓandmℓ′. Next, it should be noted\nthat Eq. (35) includes the following two integrals,\n/contintegraldisplay\ndtm2\nℓxm3\nℓy∝/integraldisplay\ndusn2(u,k)dn3(u,k),(D7)\n/contintegraldisplay\ndtm2\nℓxmℓym2\nℓz∝/integraldisplay\ndusn2(u,k)cn2(u,k)dn(u,k).\n(D8)\nBy replacing the integral variable from uwithx=\nsn(u,k), and noting that du=dx//radicalbig\n(1−x2)(1−k2x2),\nthese integrals are calculated as\n/integraldisplay\ndusn2(u,k)dn3(u,k) =/integraldisplay\ndxx2(1−k2x2)√\n1−x2\n=x√\n1−x2/bracketleftbig\n−4+k2/parenleftbig\n3+2x2/parenrightbig/bracketrightbig\n+/parenleftbig\n4−3k2/parenrightbig\nsin−1x\n8,\n(D9)\n/integraldisplay\ndusn2(u,k)cn2(u,k)dn(u,k) =/integraldisplay\ndxx2/radicalbig\n1−x2\n=x√\n1−x2/parenleftbig\n−1+2x2/parenrightbig\n+sin−1x\n8.\n(D10)\nUsing these integrals, Eq. (36) is obtained. On the other\nhand, Eq. (45) includes the following three integrals,\n/contintegraldisplay\ndtm3\nℓy∝/integraldisplay\ndudn3(u,k)\n=/integraldisplay\ndx1−k2x2\n√\n1−x2\n=k2x√\n1−x2+/parenleftbig\n2−k2/parenrightbig\nsin−1x\n2,(D11)\n/contintegraldisplay\ndtm5\nℓy∝/integraldisplay\ndudn5(u,k)\n=/integraldisplay\ndx/parenleftbig\n1−k2x2/parenrightbig2\n√\n1−x2\n=k2x√\n1−x2/bracketleftbig\n8−k2/parenleftbig\n3+2x2/parenrightbig/bracketrightbig\n+/parenleftbig\n8−8k2+3k4/parenrightbig\nsin−1x\n8,\n(D12)/contintegraldisplay\ndtm3\nℓym2\nℓz∝/integraldisplay\ndudn3(u,k)cn2(u,k)\n=/integraldisplay\ndx/radicalbig\n1−x2/parenleftbig\n1−k2x2/parenrightbig\n=x√\n1−x2/bracketleftbig\n4+k2/parenleftbig\n1−2x2/parenrightbig/bracketrightbig\n+/parenleftbig\n4−k2/parenrightbig\nsin−1x\n8.\n(D13)\nUsing these integrals, Eq. (46) is obtained.\nWhen the phase difference is a quarter\nof a period (∆ ϕ=K(k)), the relations,\nsn[u+K(k),k] = cn(u,k)/dn(u,k), cn[u+K(k),k] =\n−√\n1−k2cn(u,k)/dn(u,k), dn[ u+K(k),k] =√\n1−k2/dn(u,k), are used to evaluate Eqs. (35)\nand (45). In this case, we notice that all the integrands\nin Eq. (35) become odd functions of t, and the integrals\nover 0≤t≤τare zero. Therefore, WT\ns= 0 for\n∆ϕ=K(k). On the other hand, the following integrals\nare necessary to obtain Eq. (47),\n/contintegraldisplay\ndtm2\nℓ′ymℓy∝/integraldisplaydu\ndn(u,k)\n=/integraldisplay1\n0dx√\n1−x2(1−k2x2)\n=tan−1/bracketleftBig/radicalbig\n(1−k2)/(1−x2)x/bracketrightBig\n√\n1−k2,(D14)\n/contintegraldisplay\ndtm2\nℓ′ym3\nℓy∝/integraldisplay\ndudn(u,k)\n=/integraldisplaydx√\n1−x2\n= sin−1x,(D15)\n/contintegraldisplay\ndtm2\nℓ′ymℓym2\nℓz∝/integraldisplay\nducn2(u,k)\ndn(u,k)\n=/integraldisplay\ndx√\n1−x2\n1−k2x2\n=sin−1x−√\n1−k2tan−1/bracketleftBig/radicalbig\n(1−k2)/(1−x2)x/bracketrightBig\nk2.\n(D16)15\n1A. 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Bauer1, 3\n1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n2Institute for Advanced Studies in Basic Science, 45195 Zanjan, Iran\n3Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan\n(Dated: July 11, 2018)\nWe theoretically investigate pumping of phonons by the dynamics of a magnetic film into a non-\nmagnetic contact. The enhanced damping due to the loss of energy and angular momentum shows\ninterferencepatternsasafunctionofresonancefrequencyandmagneticfilmthicknessthatcannotbe\ndescribed by viscous (“Gilbert”) damping. The phonon pumping depends on magnetization direction\nas well as geometrical and material parameters and is observable, e.g., in thin films of yttrium iron\ngarnet on a thick dielectric substrate.\nThe dynamics of ferromagnetic heterostructures is at\nthe root of devices for information and communication\ntechnologies [1–5]. When a normal metal contact is at-\ntached to a ferromagnet, the magnetization dynamics\ndrives a spin current through the interface. This effect\nis known as spin pumping and can strongly enhance the\n(Gilbert) viscous damping in ultra-thin magnetic films\n[6–8]. Spin pumping and its (Onsager) reciprocal, the\nspin transfer torque [9, 10], are crucial in spintronics, as\nthey allow electric control and detection of magnetiza-\ntion dynamics. When a magnet is connected to a non-\nmagnetic insulator instead of a metal, angular momen-\ntum cannot leave the magnet in the form of electronic or\nmagnonic spin currents, but they can do so in the form\nof phonons. Half a century ago it was reported [11, 12]\nand explained [13–16] that magnetization dynamics can\ngenerate phonons by magnetostriction. More recently,\nthe inverse effect of magnetization dynamics excited by\nsurface acoustic waves (SAWs) has been studied [17–20]\nand found to generate spin currents in proximity normal\nmetals [21, 22]. The emission and detection of SAWs was\ncombined in one and the same device [23, 24], and adia-\nbatic transformation between magnons and phonons was\nobserved in inhomogeneous magnetic fields [25]. The an-\ngular momentum of phonons [26, 27] has recently come\ninto focus again in the context of the Einstein-de Haas\neffect [28] and spin-phonon interactions in general [29].\nThe interpretation of the phonon angular momentum in\ntermsoforbitalandspincontributions[29]hasbeenchal-\nlenged [30], a discussion that bears similarities with the\ninterpretation of the photon angular momentum [31]. In\nour opinion this distinction is rather semantic since not\nrequired to arrive at concrete results. A recent quantum\ntheory of the dynamics of a magnetic impurity [32] pre-\ndicts a broadening of the electron spin resonance and a\nrenormalized g-factor by coupling to an elastic contin-\nuum via the spin-orbit interaction, which appears to be\nrelated to the enhanced damping and effective gyromag-\nnetic ratio discussed here.\nA phonon current generated by magnetization dynam-\nics generates damping by carrying away angular momen-\ntum and energy from the ferromagnet. While the phonon\nphonon sinkzmagnet\nnon-magnet0\nphononsmHFigure 1. Magnetic film (shaded) with magnetization mat-\ntached to a semi-infinite elastic material, which serves as an\nideal phonon sink.\ncontribution to the bulk Gilbert damping has been stud-\nied theoretically [33–38], the damping enhancement by\ninterfaces to non-magnetic substrates or overlayers has\nto our knowledge not been addressed before. Here we\npresent a theory of the coupled lattice and magnetiza-\ntion dynamics of a ferromagnetic film attached to a half-\ninfinite non-magnet, which serves as an ideal phonon\nsink. We predict, for instance, significantly enhanced\ndamping when an yttrium iron garnet (YIG) film is\ngrown on a thick gadolinium gallium garnet (GGG) sub-\nstrate.\nWe consider an easy-axis magnetic film with static ex-\nternal magnetic field and equilibrium magnetization ei-\nther normal (see Fig. 1) or parallel to the plane. The\nmagnet is connected to a semi-infinite elastic material.\nMagnetization and lattice are coupled by the magne-\ntocrystalline anisotropy and the magnetoelastic interac-\ntion, giving rise to coupled field equations of motion in\nthe magnet [39–42]. By matching these with the lattice\ndynamics in the non-magnet by proper boundary con-\nditions, we predict the dynamics of the heterostructure\nas a function of geometrical and constitutive parameters.\nWe find that magnetization dynamics induced, e.g., by\nferromagnetic resonance (FMR) excites the lattice in the\nattachednon-magnet. Inanalogywiththeelectroniccase\nwecallthiseffect“phononpumping” thataffectsthemag-\nnetization dynamics. We consider only equilibrium mag-\nnetizations that are normal or parallel to the interface,\nin which the pumped phonons are pure shear waves that\ncarry angular momentum. We note that for general mag-arXiv:1804.07080v2  [cond-mat.mes-hall]  16 Jul 20182\nnetization directions both shear and pressure waves are\nemitted, however.\nWe consider a magnetic film (metallic or insulating)\nthat extends from z=\u0000dtoz= 0. It is subject to suffi-\nciently high magnetic fields H0such that magnetization\nis uniform, i.e. M(r) =M:For in-plane magnetizations,\nH0> Ms, where the magnetization Msgoverns the de-\nmagnetizing field [43]. The energy of the magnet|non-\nmagnet bilayer can be written\nE=ET+Eel+EZ+ED+E0\nK+Eme;(1)\nwhich are integrals over the energy densities \"X(r). The\ndifferent contributions are explained in the following.\nThe kinetic energy density of the elastic motion reads\n\"T(r) =(\n1\n2\u001a_u2(r); z> 0\n1\n2~\u001a_u2(r);\u0000d<z< 0; (2)\nand the elastic energy density [44]\n\"el=(\n1\n2\u0015(P\n\u000bX\u000b\u000b(r))2+\u0016P\n\u000b\fX2\n\u000b\f(r); z> 0\n1\n2~\u0015(P\n\u000bX\u000b\u000b(r))2+ ~\u0016P\n\u000b\fX2\n\u000b\f(r);\u0000d<z< 0;\n(3)\nwhere\u000b;\f2fx;y;zg,\u0015and\u0016are the Lamé parameters\nand\u001athe mass density of the non-magnet. The tilded\nparameters are those of the magnet. The strain tensor\nX\u000b\fis defined in terms of the displacement fields u\u000b(r),\nX\u000b\f(r) =1\n2\u0012@u\u000b(r)\n@r\f+@u\f(r)\n@r\u000b\u0013\n: (4)\nEZ=\u0000\u00160VM\u0001Hextis the Zeeman energy for Hext=\nH0+h(t), where h(t)is time-dependent. ED=\n1\n2\u00160VMTDMis the magnetostatic energy with shape-\ndependent demagnetization tensor DandVthe volume\nof the magnet. For a thin film with zaxis along the sur-\nface normal n0,Dzz= 1while the other components van-\nish.E0\nK=K1V(m\u0002n0)2is the uniaxial magnetocrys-\ntalline anisotropy in the absence of lattice deformations,\nwhere m=M=MsandK1is the anisotropy constant.\nThe magnetoelastic energy Emecouples the magnetiza-\ntion to the lattice, as discussed in the following.\nThe magnetoelastic energy density can be expanded as\n\"me(r) =1\nM2sX\n\u000b;\fM\u000b(r)M\f(r)\n\u0002[B\u000b\fX\u000b\f(r) +C\u000b\f\n\u000b\f(r)]:(5)\nFor an isotropic medium the magnetoelastic constants\nB\u000b\fread [45]\nB\u000b\f=\u000e\u000b\fBk+ (1\u0000\u000e\u000b\f)B?: (6)\nRotational deformations as expressed by the tensor\n\n\u000b\f(r) =1\n2\u0012@u\u000b(r)\n@r\f\u0000@u\f(r)\n@r\u000b\u0013\n(7)are often disregarded [39–42, 46], but lead to a position\ndependence of the easy axis n(r)from the equilibrium\nvalue n0=ezand an anisotropy energy density [29, 47,\n48]\n\"K(r) =K1\nM2s[M\u0002n(r)]2: (8)\nTo first order in the small deformation\n\u000en(r) =n(r)\u0000n0=0\n@\nxz(r)\n\nyz(r)\n01\nA; (9)\n\"K(r) =\"0\nK+ 2K1(n0\u0000mzm)\u0001\u000en(r):(10)\nFrom \n\u000b\f=\u0000\n\f\u000bit follows that (for non-chiral crystal\nstructures) C\u000b\f=\u0000C\f\u000b. For the uniaxial anisotropy\nconsidered here Cxz=Cyz=\u0000K1. The magnetoelastic\ncoupling due to the magnetocrystalline anisotropy thus\ncontributes [47]\n\"K\nme(r) =\u00002K1\nM2sMz(r) [Mx(r)\nxz(r) +My(r)\nyz(r)]:\n(11)\nPureYIGismagneticallyverysoft, sothemagnetoelastic\nconstants are much larger than the anisotropy constant\n[49, 50]\nBk= 3:48\u0002105Jm\u00003; B?= 6:96\u0002105Jm\u00003;\nK1=\u00006:10\u0002102J m\u00003; (12)\nbut this ratio can be very different for other magnets.\nWe find below that for the Kittel mode dynamics both\ncoupling processes cannot be distinguished, even though\nthey can characteristically affect the magnon-phonon\ncoupling for finite wave numbers.\nThe magnetization dynamics within the magnetic film\nis described by the Landau-Lifshitz-Gilbert (LLG) equa-\ntion [51, 52]\n_m=\u0000\r\u00160m\u0002He\u000b+\u001c(\u000b)\nm; (13)\nwhere\u0000\ris the gyromagnetic ratio, the effective mag-\nnetic field which includes the magnetoelastic coupling\nHe\u000b=\u0000rmE=(\u00160VMs); (14)\nand the Gilbert damping torque [52]\n\u001c(\u000b)\nm=\u000bm\u0002_m: (15)\nThe equation of motion of the elastic continuum reads\n[44]\nu(r;t) =c2\nt4u(r;t) + (c2\nl\u0000c2\nt)r[r\u0001u(r;t)];(16)\nwith longitudinal and transverse sound velocities\ncl=s\n\u0015+ 2\u0016\n\u001a; ct=r\u0016\n\u001a; (17)3\nwhere elastic constants and mass density of non-magnet\nand magnet can differ.\nA uniform precession of the magnetization interacts\nwith the lattice deformation at the surfaces of the mag-\nnetic film [13, 14] and at defects in the bulk. The present\ntheorythenholdswhenthethicknessofthemagneticfilm\nd\u001cp\nA, whereAis the cross section area. The Kittel\nmode induces lattice distortions that are uniform in the\nfilm planeu\u000b(r) =u\u000b(z)[14]. The elastic energy density\nis then affected by shear waves only:\n\"el(z) =(\n\u0016\n2\u0000\nu02\nx(z) +u02\ny(z)\u0001\n; z> 0\n~\u0016\n2\u0000\nu02\nx(z) +u02\ny(z)\u0001\n;\u0000d<z< 0;(18)\nwhereu0\n\u000b(z) =@u\u000b(z)=@z. The magnetic field\nHext=\u0000hx(t); hy(t); H 0\u0001Twith monochromatic drive\nhx;y(t) = Re\u0000\nhx;ye\u0000i!t\u0001\nand static component H0along\nthezaxis. At the FMR frequency !?=!H+!Awith\n!H=\r\u00160H0and!A=\r(2K1=Ms\u0000\r\u0016Ms). The equi-\nlibrium magnetization is perpendicular for !?>0. The\nmagnetoelastic energy derived above then simplifies to\nEz\nme=(B?\u0000K1)A\nMsX\n\u000b=x;yM\u000b[u\u000b(0)\u0000u\u000b(\u0000d)];(19)\nwhichresultsinsurfaceshearforces F\u0006(0) =\u0000F\u0006(\u0000d) =\n\u0000(B?\u0000K1)Am\u0006, withF\u0006=Fx\u0006iFy. These forces\ngenerate a stress or transverse momentum current in the\nzdirection (see Supplemental Material)\nj\u0006(z) =\u0000\u0016(z)u0\n\u0006(z); (20)\nwith\u0016(z) =\u0016forz >0and\u0016(z) = ~\u0016for\u0000d < z < 0,\nandu\u0006=ux\u0006iuy, which is related to the transverse mo-\nmentump\u0006(z) =\u001a( _ux(z)\u0006i_uy(z))by Newton’s equa-\ntion:\n_p\u0006(z) =\u0000@\n@zj\u0006(z): (21)\nThe boundary conditions require momentum conserva-\ntion and elastic continuity at the interfaces,\nj\u0006(\u0000d) = (B?\u0000K1)m\u0006;(22)\nj\u0006(0+)\u0000j\u0006(0\u0000) =\u0000(B?\u0000K1)m\u0006;(23)\nu\u0006(0+) =u\u0006(0\u0000): (24)\nWe treat the magnetoelastic coupling as a small pertur-\nbation and therefore we approximate the magnetization\nm\u0006entering the above boundary conditions as indepen-\ndent of the lattice displacement u\u0006. The loss of angular\nmomentum (see Supplemental Material) affects the mag-\nnetization dynamics in the LLG equation in the form of a\ntorque, which we derive from the magnetoelastic energy\n(19),\n_m\u0006jme=\u0006i!c\nd[u\u0006(0)\u0000u\u0006(\u0000d)]\n=\u0006i!cRe(v)m\u0006\u0007!cIm(v)m\u0006;(25)where!c=\r(B?\u0000K1)=Ms(for YIG:!c= 8:76\u0002\n1011s\u00001) andv= [u\u0006(0)\u0000u\u0006(\u0000d)]=(dm\u0006). We can\ndistinguish an effective field\nHme=!c\n\r\u00160Re(v)ez; (26)\nand a damping coefficient\n\u000b(?)\nme=\u0000!c\n!Imv: (27)\nThe latter can be compared with the Gilbert damping\nconstant\u000bthat enters the linearized equation of motion\nas\n_m\u0006j\u000b=\u0006i\u000b_m\u0006=\u0006\u000b!m\u0006: (28)\nWith the ansatz\nu\u0006(z;t) =(\nC\u0006eikz\u0000i!t; z> 0\nD\u0006ei~kz\u0000i!t+E\u0006e\u0000i~kz\u0000i!t;\u0000d<z< 0;\n(29)\nwe obtain\nv=Ms!c\n!\rd~\u001a~ct2h\ncos(~kd)\u00001i\n\u0000i\u001act\n~\u001a~ctsin(~kd)\nsin(~kd) +i\u001act\n~\u001a~ctcos(~kd);(30)\nand the damping coefficient for perpendicular magneti-\nzation\n\u000b(?)\nme=\u0010!c\n!\u00112Ms\n\rd~\u001a~ct\u001act\n~\u001a~ct4 sin4\u0010~kd\n2\u0011\nsin2(~kd) +\u0010\n\u001act\n~\u001a~ct\u00112\ncos2(~kd);\n(31)\nwhere!=ctk= ~ct~k. The oscillatory behavior of the\ndamping\u000b(?)\nmecomes from the interference of the elastic\nwaves that are generated at the top and bottom surfaces\nof the magnetic film. When they constructively (destruc-\ntively) interfere at the FMR frequency, the damping is\nenhanced(suppressed), becausethemagnon-phononcou-\npling and phonon emission are large (small).\nWhen\u001act\u001c~\u001a~ct(soft substrate) or when acoustic\nimpedances are matched ( \u001act= ~\u001a~ct), damping at the\nresonance ~kd= (2n+ 1)\u0019withn2N0[14] simplifies to\n\u000b(?)\nme!\u0010!c\n!\u001124Ms\n\rd\u001act: (32)\nWhen\u001act\u001d~\u001a~ct(hard substrate), the magnet is acousti-\ncally pinned at the interface and the acoustic resonances\nare at ~kd= (2n+ 1)\u0019=2[14] with\n\u000b(?)\nme!\u0010!c\n!\u00112Ms\n\rd~\u001a~ct\u001act\n~\u001a~ct: (33)\nIn contrast to Gilbert damping, \u000b(?)\nmedepends on the\nfrequency and vanishes in the limits !!0and!!1.\nTherefore, it does not obey the LLG phenomenology4\nand in the non-linear regime does not simply enhance\n\u000bin Eq. (15). The magnetization damping \u000b0in bulk\nmagnetic insulators, on the other hand, is usually of\nthe Gilbert type. It is caused by phonons as well, but\nnot necessarily the magnetoelastic coupling. A theory\nof Gilbert damping [38] assumes a bottleneck process\nby sound wave attenuation, which appears realistic for\nmagnets with high acoustic quality such as YIG. In the\npresent phonon pumping model, energy and angular mo-\nmentum is lost by the emission of sound waves into an\nattached perfect phononwaveguide, so thepumpingpro-\ncess dominates. Such a scenario could also dominate the\ndamping in magnets in which the magnetic quality is rel-\natively higher than the acoustic one.\nWhen the field is rotated to Hext =\u0000\nhx(t); H 0; hz(t)\u0001T, the equilibrium magnetization\nis in the in-plane ydirection and the magnetoelastic\nenergy couples only to the strain uy,\nEy\nme=(B?\u0000K1)A\nMsMz[uy(0)\u0000uy(\u0000d)]:(34)\nThe FMR frequency for in-plane magnetization !k=\n!Hp\n1\u0000!A=!Hwith!A< !H. The magnetoelastic\ncoupling generates again only transverse sound waves.\nThe linearized LLG equation including the phononic\ntorques reads now\n_mx= (!H+!me)mz\u0000\r\u00160hz\u0000!Amz\n+ (\u000b+\u000bme) _mz; (35)\n_mz=\u0000!Hmx+\r\u00160hx\u0000\u000b_mx; (36)\nwhere\u000bmeis given by Eq. (27) and !me=\r\u00160Hmewith\neffective field Hme=Hme\u0001ezgiven by Eq. (26). Both\nHmeand\u000bmecontribute only to _mx. The phonon pump-\ning is always less efficient for the in-plane configuration:\n\u000b(k)\nme=1\n1 + (!k=!H)2\u000b(?)\nme: (37)\nAs an example, we insert parameters for a thin YIG\nfilm on a semi-infinite gadolinium gallium garnet (GGG)\nsubstrate at room temperature. We have chosen YIG\nbecause of its low intrinsic damping and high quality in-\nterface to the GGG substrate. Substantially larger mag-\nnetoelastic coupling in other materials should be offset\nagainst generally larger bulk damping. For GGG, \u001a=\n7:07\u0002103kg m\u00003,cl= 6411 m s\u00001, andct= 3568 m s\u00001\n[53]. For YIG, Ms= 1:4\u0002105A m\u00001,\r= 1:76\u0002\n1011s\u00001T\u00001,~\u001a= 5170 kg m\u00003,~cl= 7209 m s\u00001,~ct=\n3843 m s\u00001, and!c= 8:76\u00021011s\u00001[49, 50]. The ratio\nof the acoustic impedances ~\u001a~ct=\u001act= 0:79. The damp-\ning enhancement \u000b(?)\nmeis shown in Fig. 2 over a range\nof FMR frequencies and film thicknesses. The FMR fre-\nquencies!?=!H+!Aand!k=!Hp\n1\u0000!A=!Hfor\nthe normal and in-plane configurations are tunable by\nthe static magnetic field component H0via!H=\r\u00160H0.\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n<10−510−410−3\nα(⊥)\nmeFigure 2. Damping enhancement \u000b(?)\nmeby phonon pumping\nin a YIG film on a semi-infinite GGG substrate, as given by\nEq. (31).\nThe damping enhancement peaks at acoustic resonance\nfrequencies \u0017\u0019n~ct=(2d). The counter-intuitive result\nthat the damping increases for thicker films can be un-\nderstood by the competition between the magnetoelastic\neffect that increases with thickness at the resonances and\nwins against the increase in total magnetization. How-\never, with increasing thickness the resonance frequencies\ndecrease below a minimum value at which FMR can be\nexcited. For a fixed FMR frequency \u000bme!0ford!1:\nFor comparison, the Gilbert damping in nanometer thin\nYIG films is of the order \u000b\u001810\u00004[54] which is larger\nthan corresponding values for single crystals. We con-\ncludethattheenhanceddampingisatleastpartlycaused\nby interaction with the substrate and not by a reduced\ncrystal quality.\nThe resonances in the figures are very broad because\nthe\u001act\u0019~\u001a~ctimplies very strong coupling of the discrete\nphonons in the thin magnetic layer with the phonon con-\ntinuum in the substrate. When an acoustic mismatch\nis introduced, the broad peaks increasingly sharpen, re-\nflecting the increased lifetime of the magnon polaron res-\nonances in the magnet.\nThe frequency dependent effective magnetic field H(?)\nme\nis shown in Fig. 3. The frequency dependence of H(?)\nme\nimplies a weak frequency dependence of the effective gy-\nromagnetic ratio\n\r(?)\ne\u000b=\r \n1 +\r\u00160H(?)\nme\n!!\n: (38)\nIn the limit of vanishing film thickness, \u00160H(?)\nme!\n\u0000(B?\u0000K1)2=(Ms~\u0016).\nWe assumed that the non-magnet is an ideal phonon\nsink, which means that injected sound waves do not\nreturn. In the opposite limit in which the phonons\ncannot escape, i.e. when the substrate is a thin film\nwith high acoustic quality, the additional damping van-\nishes. This can be interpreted in terms of a phonon5\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n−60−45−30−15015304560\nµ0H(⊥)\nme[µT]\nFigure3. Effectivefield H(?)\nmegeneratedbythemagnetoelastic\ngeneration of phonons in a YIG film on a semi-infinite GGG\nsubstrate, as given by Eq. (26).\naccumulation that, when allowed to thermalize, gener-\nates a phonon chemical potential and/or non-equilibrium\ntemperature. The non-equilibrium thermodynamics of\nphonons in magnetic nanostructures is subject of our on-\ngoing research.\nThe damping enhancement by phonons may be com-\npared with that from electronic spin pumping [6–8],\n\u000bsp=\r~\n4\u0019dMsh\ne2g; (39)\nwhich is inversely proportional to the thickness dof the\nmagnetic film and does not depend on the FMR fre-\nquency, i.e. obeys the LLG phenomenology. Here, g\nis the spin mixing conductance per unit area at the in-\nterface. While phonons can be pumped into any elastic\nmaterial, spin pumping requires an electrically conduct-\ning contact. With a typical value of hg=e2\u00181018m\u00002\nthe damping enhancement of YIG on platinum is \u000bsp\u0018\n10\u00002nm=d:The physics is quite different, however, since\n\u000bsp;in contrast to \u000bme, does not require coherence over\nthe interface.\nIn conclusion, the pumping of phonons by magnetic\nanisotropy and magnetostriction causes a frequency-\ndependent contributions to the damping and effective\nfield of the magnetization dynamics. The generation\nof phonons by magnetic precession can cause significant\ndamping in a magnetic film when grown on an insulating,\nnon-magnetic substrate and partly explains the increased\ndampinginvariablyobservedforthinnerfilms. Theimpli-\ncations of further reaching ramifications, such as phonon-\ninduced dynamic exchange interactions, phonon accumu-\nlations and phonon spin Seebeck effect require additional\nresearch.\nThis work is financially supported by the Nederlandse\nOrganisatievoorWetenschappelijkOnderzoek(NWO)as\nwell as by the Grant-in-Aid for Scientific Research on In-\nnovative Area, ”Nano Spin Conversion Science” (Grant\nNo. 26103006). H. K. acknowledges support from theIran Science Elites Federation. We acknowledge use-\nful discussions with Yaroslav Blanter, Rembert Duine,\nAkashdeep Kamra, Eiji Saitoh, and Sanchar Sharma.\n[1] S. Bader and S. Parkin, Annu. Rev. Condens. Matter\nPhys. 1, 71 (2010).\n[2] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. Phys. 43, 264001 (2010).\n[3] F. Pulizzi, Nat. Mater. 11, 367 (2012).\n[4] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. Hillebrands, Nat. Phys. 11, 453 (2015).\n[5] S. A. Nikitov, D. V. Kalyabin, I. V. Lisenkov, A. 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Bodzenta, Phys. Status Solidi B\n146, 467 (1988).\n[54] H. Chang, P. Li, W. Zhang, T. Liu, A. Hoffmann,\nL. Deng, and M. Wu, IEEE Magn. Lett. 5, 1 (2014).\n[55] E. B. Magrab, Vibrations of Elastic Systems (Springer,\nDordrecht, 2012).1\nSupplemental Material\nInSecs.IandIIofthisSupplementwederivetheangu-\nlar and transverse momentum of transverse elastic waves\nand the corresponding momentum currents. In Sec. III\nwe give results for a magnet sandwiched between two\nnon-magnets and in Sec. IV we present a theory for the\nmagnetizationdampingenhancementfrompumpingflex-\nural waves into a thin beam.\nI. ANGULAR MOMENTUM\nThe magnetization M=Msmof an uniformly mag-\nnetized magnet with saturation magnetization Msand\nvolumeVis associated with the angular momentum\nS=\u0000MsV\n\rm; (S1)\nwhere\u0000\ris the gyromagnetic ratio. The angular mo-\nmentum density relative to the origin of an elastic body\nwith displacement field u(r;t)and constant mass density\n\u001areads\nl(t) =\u001a(r+u(r;t))\u0002_u(r;t): (S2)\nWith uniaxial anisotropy axis along z, FMR generates\nthe transverse elastic wave\nu(z;t) = Re2\n40\n@ux\nuy\n01\nAeikz\u0000i!t3\n5; (S3)\nwith dispersion relation !=ctk. Defining the time av-\nerage\nhf(t)i= lim\nT!11\nTZT\n0dtf(t); (S4)\nhli= (0;0;hlzi)can be expressed as\nhlzi=\u001ahux_uy\u0000_uxuyi\n=\u0000\u001a!\n4\u0010\nju+j2\u0000ju\u0000j2\u0011\n; (S5)\nwhereu\u0006=ux\u0006iuyand where we used\n\nRe(ae\u0000i!t)Re(be\u0000i!t)\u000b\n=1\n2Re (a\u0003b):(S6)\nThe non-magnet harbors a constant phonon angular mo-\nmentum density in the z-direction, which implies pres-\nence of a phonon angular momentum current Ahjz\nliat\nthe interface to the magnet with area Athat is ab-\nsorbed at the phonon sink. In our model the angular\nmomentum loss rate of the magnet by phonon pumping\nh_Szjmei=\u0000Ahjz\nliand\nD\n_Sz\f\f\f\nmeE\n=hfSz;Ez\nmegi=MsV\u000b(?)\nme!\n4\r\u0010\njm+j2\u0000jm\u0000j2\u0011\n(S7)wheref;gis the Poisson bracket and Ez\nmethe magnetoe-\nlastic energy Eq. (19) in the main text. From Eq. (31)\nand\nu\u0006=Cm\u0006; (S8)\nwith\nC=(B?\u0000K1)h\ncos(~kd)\u00001i\nik\u0016cos(~kd) +~k~\u0016sin(~kd); (S9)\nwe obtain the relation\nhjz\nli=cthlzi; (S10)\nwhichagreeswiththesimplephysicalpictureofanelastic\nwave carrying away its angular momentum density hlzi\nwith transverse sound velocity ct.\nII. TRANSVERSE MOMENTUM\nFor the transverse elastic wave (S3) in a magnet ex-\ntending from z=z0toz=z1(withz1> z0), the time\nderivative of the transverse momentum P\u0006=Px\u0006iPy\nreads\n_P\u0006=\u001aZ\nVd3ru\u0006(z;t)\n=\u0016A\u0002\nu0\n\u0006(z1;t)\u0000u0\n\u0006(z0;t)\u0003\n:(S11)\nThe change of momentum can be interpreted as a trans-\nverse momentum current density j\u0006(z0) =\u0000\u0016u0\n\u0006(z0)\nflowing into the magnet at z0and a current j\u0006(z1) =\n\u0000\u0016u0\n\u0006(z1)flowing out at z1. The momentum current\nis related to the transverse momentum density p\u0006(z) =\n\u001a_u\u0006(z)by\n_p\u0006(z) =\u0000@\n@zj\u0006(z); (S12)\nwhich confirms that\nj\u0006(z;t) =\u0000\u0016u0\n\u0006(z;t): (S13)\nThe instantaneous conservation of transverse momentum\nisaboundaryconditionsattheinterface. Itstimeaverage\nhj\u0006i= 0, but the associated angular momentum along z\nis finite, as shown above.\nIII. SANDWICHED MAGNET\nWhen a non-magnetic material is attached at both\nsides of the magnet and elastic waves leave the magnet\natz= 0andz=\u0000d, the boundary condition are\nj\u0006(\u0000d\u0000)\u0000j\u0006(\u0000d+) = (B?\u0000K1)m\u0006;(S14)\nj\u0006(0+)\u0000j\u0006(0\u0000) =\u0000(B?\u0000K1)m\u0006;(S15)\nu\u0006(0+) =u\u0006(0\u0000); (S16)\nu\u0006(\u0000d+) =u\u0006(\u0000d\u0000); (S17)2\nwithd\u0006=d\u00060+. Since the Hamiltonian is piece-wise\nconstant\nu\u0006(z;t) =8\n><\n>:C\u0006eikz\u0000i!t; z> 0\nD\u0006ei~kz\u0000i!t+E\u0006e\u0000i~kz\u0000i!t;\u0000d<z< 0\nF\u0006e\u0000ikz\u0000i!t: z< \u0000d;\n(S18)\nUsing the boundary conditions\nv=u\u0006(0)\u0000u\u0006(\u0000d)\ndm\u0006=Ms!c\n!\rd~\u001a~ct2\ni\u001act\n~\u001a~ct\u0000cot(~kd\n2);(S19)\nleading to the damping coefficient\n\u000b(?)\nme=\u0010!c\n!\u00112Ms\n\rd~\u001a~ct2\n\u001act\n~\u001a~ct+~\u001a~ct\n\u001actcot2\u0010~kd\n2\u0011:(S20)\nWhen ~\u001a~ct=\u001act,\n\u000b(?)\nme=\u0010!c\n!\u001122Ms\n\rd\u001actsin2 ~kd\n2!\n;(S21)\nwhich differs from the sin4\u0010\n~kd=2\u0011\ndependence obtained\nfor the magnet|non-magnet bilayer. This result can be\nexplained by the phonon angular momentum leaking at\ntwo interfaces that should enhance the damping for thin\nmagneticfilms. However, thephononpumpingisacoher-\nent process that couples both interfaces, so the damping\nis not increased simply by a factor of 2 as in case of in-\ncoherent spin pumping of a magnetic film sandwiched by\nmetals. The position of the resonances, ~kd= (2n+1)\u0019=2\nwithn2N0, are independent of the ratio \u001act=~\u001a~ctwith\n\u000bme=\u0010!c\n!\u001122Ms\n\rd\u001act: (S22)\n\u000b(?)\nmeand the effective magnetic field for YIG sandwiched\nbetween two infinitely thick GGG layers are shown in\nFigs. S1 and S2.\nIV. FLEXURAL (BENDING) WAVES IN THIN\nBEAMS\nIn the main text we focus on the generation of trans-\nverseorlongitudinalsoundwaves. Infree-standingstruc-\ntured samples such as cantilevers, additional modes be-\ncome important that can be excited by magnetization\ndynamics as well. This can be illustrated by a thin\ncylindrical elastic beam (see Fig. S3) with cross section\nareaA=\u0019r2, in which flexural waves are generated by\nthe magnet of volume V=Adattached to the top of\nthe beam. The elastic energy according to the Euler-\nBernoulli beam theory [44]\nEel=ZL\n0dz\u00141\n2\u001aA_u2(z;t) +1\n2EYI?u002(z;t)\u0015\n;(S23)\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n<10−510−410−3\nα(⊥)\nmeFigure S1. Phonon pumping-enhanced \u000b(?)\nmein a YIG film\nsandwiched between two infinitely thick GGG layers.\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n−40−30−20−10010203040\nµ0H(⊥)\nme[µT]\nFigure S2. Phonon pumping effective field H(?)\nmein a YIG film\nsandwiched between two infinitely thick GGG layers.\nleads to the equation of motion for the flexural waves [44]\n\u001aAu\u0006(z;t) +EYI?u(4)\n\u0006(z;t) = 0;(S24)\nwhereI?=R\ndAx2=\u0019r4=4and elastic modulus EY=\n\u0016(3\u0015+ 2\u0016)=(\u0015+\u0016). The dispersion relation of the flex-\nural waves is quadratic,\n!=s\nEYI?\n\u001aAk2: (S25)\nWhen the dimensions of the magnet are much smaller\nthan the wavelength of the elastic waves, the magne-\ntoelastic coupling is suppressed and magnetization and\nlattice are coupled exclusively by the magnetocrystalline\nand, in contrast to the bulk magnet, also the shape\nanisotropies,\nEA=EK+ED; (S26)\nwith\nEK=K1V(m\u0002n)2; (S27)\nED=1\n2\u00160VMTDM: (S28)3\nm\nH\nz 0magnet non-magnet\nflexural wave\nFigure S3. Thin film magnet (shaded) with magnetization m\nattached to a thin semi-infinite elastic beam.\nFor a thin magnetic film Dzz=D3= 1. When the\nmagnet become very thick ( d\u001dr), i.e. a needle with\nits point forming the contact, and a coordinate system\nwithzaxis along the surface normal n,Dxx=Dyy=\nD?=\u00001=2. All otherD\u000b\fvanish. In contrast to the\nextended bilayer treated in the main text nis now a\ndynamic variable with n\u0006=\u0000u0\n\u0006(0;t). The mechanical\ntorque exerted by the magnet on the elastic beam reads\n\u001c=_L=VMs\n\r_m+_J; (S29)\nwhere _J=\u00160VMsm\u0002Hextis the torque exerted by the\nexternal magnetic field on the total angular momentum.\nFor a magnet with equilibrium magnetization mkn0\n\u001c\u0006=\u0006if(m\u0006\u0000n\u0006); (S30)\nwheref=VMs!A=\rand\n!A=(\n2\rK1=Ms\u0000\r\u00160Ms;thin film\n2\rK1=Ms+1\n2\r\u00160Ms;needle:(S31)\nIn order to compute the angular momentum current\npumped into the attached non-magnet, _L\u0006=\u001c\u0006, we\nhave to specify four boundary conditions. Two are pro-\nvided by the assumption that the beam is infinitely long\nso that the are no reflections. The absence of shear forces\nat the boundary is expressed by u000(0;t) = 0, while the\nbending by the torque follows from the principle of least\naction [55]\nu00\n\u0006(0;t) =\u0006i\u001c\u0006\nEYI?: (S32)The general solution for the differential equation can be\nwritten\nu\u0006(z;t) =e\u0000i!t\u0000\nA\u0006eikz+B\u0006e\u0000kz\u0001\n;(S33)\nbecause there are no back-reflections in the semi-infinite\nbeam. We find\nn\u0006=wm\u0006\nwith\nw=\u00002f\nEYI?k\u0012\n1 +i\u00002f\nEYI?k\u0013\u00001\n;(S34)\nand the following source term in the LLG equation,\n_m\u0006jan=\u0007i!An\u0006\n=\u0006i!ARe(w)m\u0006\u0007!AIm(w)m\u0006;(S35)\nThe first term on the right-hand-side is a field-like torque\nequivalent to the effective field\n\u00160Han=!A\n\rRe(w)ez; (S36)\nand the second one a damping-like torque with damping\ncoefficient\n\u000ban=\u0000!A\n!Imw: (S37)\nSince for weak magnetoelastic coupling we expect \u000ban\u001c\n1and thereforejwj \u001c 1, which is equivalent to\n2f=(EYI?k)\u001c1, we may approximate\nw\u0019f(1\u0000i)\nEYI?k; (S38)\n\u000ban\u0019VMs!2\nA\n!k\rI?EY; (S39)\n\u00160Han\u0019\u0000VMs!2\nA\n\r2EYI?kez: (S40)\nThe damping enhancement scales as\n\u000ban/V\nA2!3\n2: (S41)\nFor a needle-shaped YIG magnet attached to GGG with\nEY= 2:5\u00021011Paand!A= 1:4\u00021010s\u00001\n\u000ban\u00198:6\u000210\u00006d=nm\n(\u0017=GHz)3\n2(r=nm)2;(S42)\n\u00160jHanj\u00193:1\u000210\u00007d=nm\n(\u0017=GHz)1\n2(r=nm)2T;(S43)\nwhich are very small numbers even at nanoscale dimen-\nsions."    },    {        "title": "2104.10918v1.Impact_of_Fe___80__B___20___insertion_on_the_properties_of_dual_MgO_perpendicular_magnetic_tunnel_junctions.pdf",        "content": "1\nImpact of Fe 80B20 insertion on the properties of dual-MgO perpendicular \nmagnetic tunnel junctions \nEnlong Liu1, Taeyoung Lee2 and Hyunsoo Yang1 \n1 Department of Electrical and computer Engineering, National University of Singapore, \n117576, Singapore \n2 GLOBALFOUNDRIES Singapore Pte. Ltd., Singapore 738 406, Singapore \nE-mail: eleyang@nus.edu.sg \n\nAbstract \nWe explore the impact of Fe 80B20 inserted at both Co 20Fe60B20/MgO interfaces of dual-MgO \nfree layers (FLs) in bottom-pinned magnetic tunne l junctions (MTJs). MTJ stacks are annealed \nfor 30 min at 350 °C and 400 °C in a vacuum after film deposition. Current-in-plane tunneling \nmeasurements are carried out to characteri ze magnetotransport properties of the MTJs. \nConventional magnetometry measurements and ferromagnetic resonance are conducted to \nestimate the saturation magnetization, the effective perpendicular anisotropy field and the Gilbert damping of dual-MgO FLs as a function of the Fe\n80B20 thickness and annealing \ntemperatures. With ultrathin Fe 80B20 (0.2  0.4 nm)  inserted, perpendicular magnetic \nanisotropy (PMA) of FLs increases with si milar tunnel magneto-resistance (TMR) and low \ndamping values. As Fe 80B20 layer thickness further increases (0.6  1.2 nm), both TMR and \nPMA degrade, and damping increases dramatically. This study demonstrates a novel approach \nto tune properties of MTJ stacks with dual-MgO FLs up to 400 °C annealing, which enables \nMTJ stacks for various applications. \n\n1. Introduction \nMagnetic tunnel junctions (MTJs) with perpendi cular magnetic anisotropy (PMA) have been \nstudied in the recent decades as the crucial el ement for next generation memory applications, \nsuch as spin-transfer-torque and spin-orb it-torque magnetic random access memory \n(STT/SOT-MRAM), due to their non-volatility, en ergy effectiveness, high endurance, and \nscalability [1–3]. Many efforts have been made in the engineering of the data-storage layer in MTJs, i.e. the free layer (FL), whose magnetic moment can be switched by the writing current. \nEspecially, dual-MgO FLs with a structure as MgO/CoFeB/spacer/CoFeB/MgO have been under intense development [4–6]. On the one hand, dual-MgO FLs can provide high PMA and \nlow damping to guarantee the high thermal stability and low switching current, after scaling-\ndown of MTJ devices [7]. On the other hand, the aformentioned properties of dual-MgO FLs \ncan be maintained after post-annealing up to 400 °C, which is required for the CMOS back-end-of-line (BEOL) process [8]. \nTo further optimize dual-MgO FLs performanc e, previous studies focused on different \ntopics. Among them, non-magnetic spacer engineering and element composition effects  have \ndrawn lots of interest. Researches on non-ma gnetic spacer sandwiched between two CoFeB \nlayers in dual-MgO FLs have been widely conducte d. Materials such as Mo [9–11], Ta [8,12], \nand W [13–15] were explored as the spacer to identify its impact on PMA and damping before 2\nand after annealing. Other works examined the effect of element (Fe or B) composition in \nCoFeB layers in MTJ stacks on several parameters, including tunnel magneto-resistance \n(TMR) [16], PMA [17–19], and annealing stability  [20]. It has been demonstrated that a \nthickness gradient in the B content can modify the properties of CoFeB/MgO bilayer system \nsuch as damping and anisotropy [21]. PMA of th e FL was also reported to be improved with \nincreasing the Fe composition in CoFeB, on which its TMR is almost independent [22].\nHowever, it is still an open question how a gradient of Fe in dual-MgO FLs impacts the overall \nproperties of MTJ stacks. In such a case, th e PMA of dual-MgO FLs would benefit from an \nincreased Fe concentration, while the TMR of the MTJs is expected to be improved due to the \nformation of Fe/MgO/Fe interface after annealing [23].  \nHere we propose an insertion of ultrathin Fe 80B20 (hereafter FeB) at the interface of \nCo20Fe60B20/MgO in dual-MgO FLs to achieve tunable magnetic properties and annealing \nstability. By optimizing the thickness of the FeB insertion layers at both CoFeB/MgO \ninterfaces, a large PMA can be obtained after 350 °C annealing and further improved at 400 \n°C, which is accompanied with a low damping constant. The result of a low saturation \nmagnetization, large anisotropy field and low damping at the same time in the FeB-inserted dual-MgO FLs makes it promising for low switching currents in MTJ devices without reducing \nthe thermal stability [24]. In addition, the tunability of FL performance by FeB insertion enlarges its potential for various spintronic appl ications where CoFeB-based MTJs are present, \nsuch as SOT-MRAM, spin logic devices and STT nano-oscillators. \n2. Experimental \nBottom-pinned perpendicular MTJs with [Co/Pt ] multilayers as a perpendicular synthetic \nantiferromagnet (p-SAF) were in-situ  deposited at room temperature by magnetron sputtering \non W/Ru/W/Ru/W bottom electrodes (BE) and ca pped by the Ta/Ru top electrode (TE) in a \nULVAC Magest S200 multi-chamber machine. All samples were first annealed with a 0.5 T \nmagnetic field perpendicular to film plane in a magnetic vacuum annealing oven at 350 °C for 30 min. The TMR and resistance-area product (RA) of the MTJ stack was measured via current-\nin-plane tunnelling method (CIPT) [25]. The hyst eresis loops of blanket stacks were measured \nby a vibrating sample magnetometer (VSM) with the magnetic field perpendicular to the \nsample plane. Field-modulated ferromagneti c resonance (FMR) measurements with the \nfrequency range of 10-25 GHz were conducted to ex tract the resonance field and linewidth of \nthe FL versus the frequency, from which the effective perpendicular anisotropy field and Gilbert damping of the FL can be estimated.  All FMR measurements were conducted with \nsamples placed film-side down on a coplanar waveguide in an electromagnet with a field range \nup to 0.5 T and perpendicular to the sample pl ane.  The same batch of samples were then \nannealed at 400 °C for 30 min to study the annealing impact. \n3. Results and discussion \n3.1 Stack characterization without FeB insertion \nThe detailed stack structure of MTJs used in this  study is provided schematically in figure 1(a). \nIt consists of (thickness in nm): \n Hard layer (HL): Pt (5)/[Co (0.25)/Pt (0.2)]/Co (0.6) \n Reference layer (RL): Co (0.6)/Pt (0.2)/ Co (0.3)/Pt (0.2)/Co (0.5)/W (0.3)/Co\n20Fe60B20 \n(0.8) 3\n Free layer (FL): Co 20Fe60B20 (1.2)/W (0.4)/ Co 20Fe60B20 (0.8). \n \nThe FL is sandwiched between the MgO tunnel barrier and the 2nd MgO layer to form a dual-\nMgO structure. The magnetic hysteresis  loop of  the full stack in figure 1(b) indicates good \nPMA in each functional layer after 350 °C annealing. From the minor hysteresis loop shown \nin figure 1(c), PMA of the FL is still maintained  after 400 °C annealing, and the coercive field \nalso increases slightly. The reduction of the saturation magnetization per area ( 𝑀௦∙𝑡) of the \nFL after 400 °C annealing can be attributed to magnetic dead layer formation, which will be \ndiscussed in the following sections.   \nAfter 400 °C annealing, a sloped plateau is observed around 500 mT in the red curve in \nfigure 1(b), indicating a decreased PMA in the RL  [26]. In addition, the TMR of the stack is \nreduced from 110% to 50%. Since the FL PMA is maintained, the TMR reduction is attributed \nto a PMA loss in the RL. Thus, the impact of Fe B insertion in dual-MgO FLs on TMR is studied \nin stacks after 350 °C annealing. \n3.2 Impact of FeB insertion on TMR and RA \nFigure 2(a) and (b) show schematically the FeB insertion position in the dual-MgO FL. Top \nFeB is inserted between the 2nd MgO layer and CoFeB above W spacer, while bottom FeB is \nbetween the MgO tunnel barrier and CoFeB below W spacer. To eliminate the difference in \nthe thickness of FL after insertion, the total thickness of FeB insertion plus remaining CoFeB \nis kept at 0.8 nm and 1.2 nm for layers above  and below W spacer, respectively. As such, the \nthickness of the top FeB insertion layer is chosen  as 0.2, 0.4, 0.6 and 0.8 nm. For the bottom \nFeB insertion layer, its thickness options are 0.2, 0.4, 0.6, 0.8, 1.0 and 1.2 nm. \nThe TMR and RA values as a function of the FeB insertion layer at different CoFeB/MgO \ninterfaces are summarized in figure 2(c) and (d), respectively. The trend for both TMR and RA \nis similar regardless of FeB insertion position. The TMR value is similar for the thickness of FeB < 0.4 nm, but the RA value reduces for thin FeB insertion < 0.6 nm and increases with \nthicker FeB. In addition, it is found that top FeB insertion leads to a more pronounced RA \nreduction. This phenomenon indicates that with top FeB insertion, the contribution to RA from \nthe 2\nnd MgO layer can be reduced. Since the MgO tunnel barrier (0.85 nm) is thicker than the \n2nd MgO layer (0.7 nm), its RA change due to bottom FeB insertion is not as significant as in \nthe top FeB case. Overall, a similar TMR value at low RA is realized when ultrathin FeB (0.2 \n 0.4 nm) is inserted, suggesting that th e formation of Fe/MgO interface benefits \nmagnetotransport properties of MTJ stacks. \nThe TMR starts to drop when FeB thicker than 0.6 nm was inserted at the bottom \nCoFeB/MgO interface, but the RA value does not show any significant change. This TMR drop \nis attributed mainly to a lower spin polarization when thicker FeB replaces CoFeB in the FL \n[27]. \n3.3 Impact of FeB insertion and annealing on FL magnetic properties \nIn figure 1(c), an example of hysteresis loop of  the FL is shown. The saturation magnetization \nper area of the FL can be estimated by using 𝑀௦∙𝑡ൌ𝑚 /𝐴, where 𝑚 is the magnetic moment, \n𝑡 is the thickness of FL, and 𝐴 is the sample area. The effective anisotropy field ( 𝜇𝐻) is \nderived from FMR measurements, as shown by exemplary data in figure 3(a) from dual-MgO \nFLs without FeB insertion, with 0.2 nm top FeB insertion and 0.2 nm bottom FeB insertion, all \nafter 350 C annealing. For each sample, the power absorption by FL versus the applied field 4\nscan is measured at various frequencies, fro m which the ferromagne tic resonance field ( 𝜇𝐻௦) \nand linewidth ( 𝜇Δ𝐻) is estimated. The relation between 𝜇𝐻௦ and frequency 𝑓 is described \nby the Kittel equation for the out-of-plane applied field [28]: \n𝑓ൌఊ\nଶగ൫𝜇𝐻௦𝜇𝐻൯                             (1) \nwhere 𝛾 is the gyromagnetic ratio. From figure 3(a), the x-intercept is the 𝜇𝐻. From 𝑀௦∙𝑡 \nand 𝜇𝐻, the effective perpendicular anisotropy energy can be calculated as \n𝐾∙𝑡ൌଵ\nଶ𝜇𝐻ሺ𝑀௦∙𝑡ሻ.                        (2)  \nFigure 4 summarized the results calculated by the above method. First, 𝑀௦∙𝑡 of FLs can be \ndescribed simply by 𝑀௦∙𝑡ൌ𝑀 ௦ி∙2.0െ𝑡ி∙ሺ𝑀௦ிെ𝑀௦ிሻ, where 𝑡ி is the \ninserted thickness of FeB. It is found in figure 4(a) and (b) that 𝑀௦∙𝑡 decreases with thicker \ninserted FeB, i.e. the slope is negative. Thus, in our stack 𝑀௦ி is smaller than 𝑀௦ி. Next, \nfor the top FeB insertion case in figure 4(a), the amount of change in 𝑀௦∙𝑡 at the same FeB \ninsertion thickness after 350 °C annealing is larger  than that of the bottom insertion case in \nfigure 4(b). This indicates that the top FeB inserted layer is more damaged than the bottom \ninserted FeB. Another major difference between top and bottom insertion cases is a larger 𝑀௦∙\n𝑡 reduction after 400 ºC annealing compared to that after 350 °C annealing, with thick bottom \nFeB inserted (> 0.4 nm). As the bottom-inserted FeB is less damaged, it still reduces the 𝑀௦∙𝑡 \nvalue on further annealing at 400 ºC. However, for the top inserted case, there is not much \nchange after 400 ºC annealing, supporting that the top FeB insertion layer is damanged, mainly \ndue to the 2nd MgO deposition.  \nThe FeB thickness dependence of  𝜇𝐻 and 𝐾∙𝑡 can be discussed together. In the top \nFeB insertion case after 350 ºC annealing, 𝜇𝐻 in figure 4(c) increases monotonically with \nFeB insertion. Due to the reduction in 𝑀௦∙𝑡, however, this increase cannot lead to a higher \nPMA. In figure 4(e), the PMA of FL after 350 ºC  annealing changes little, and even slightly \ndecreases with thicker FeB. However, after 400 ºC annealing, 𝜇𝐻 overall increases without \nany clear dependence on the FeB thickness, and the PMA of FL reaches the maximum value \nwhen 0.2 nm FeB is inserted.  \nOn the other hand, the behavior of 𝜇𝐻 and 𝐾∙𝑡 in bottom-inserted FeB cases is \ndifferent. In general, the PMA can be improved more significantly with bottom-inserted FeB \nthan top-inserted FeB cases. In 350 ºC  annealing cases, both 𝜇𝐻 and 𝐾∙𝑡 increase till \n1.0 nm FeB insertion as shown in figure 4(d) and figure 4(f), respectively. While in 400 ºC \nannealing cases, 𝐾∙𝑡 is significantly improved in thin FeB (0.2 – 0.6 nm) cases. To \nsummarize, the impact of FeB on the PMA of FL differs according to the insertion position. \nRegardless of annealing temperature, FeB insertion at the bottom CoFeB/MgO interface \ninduces a larger PMA, probably due to less damage and the formation of Fe/MgO interface. By changing the FeB insertion position and thickness, magnetic properties of dual-MgO FL \ncan be tuned in a wide range.  \nIt can be noticed that the vertical error bar of  anisotropy field is huge when the FeB thickness \nis 0.8 nm in the top insertion case, and 1.0 or 1.2 nm in bottom insertion cases. It reflects that the resonance field is difficult to be determined precisely in those cases, and thus the fits of \nEq.(1) contains large uncertainties. It is probabl y due to a large damping constant in thick FeB \ninsertion cases, which will be discussed in the following section.  \n3.4 Impact of FeB insertion layer on FL damping 5\n \nIn order to evaluate the effect of FeB insertion on the FL damping, FMR measurements are \nconducted. In figure 3(b), 𝜇Δ𝐻 versus the external excitation frequency for the three samples \nwas plotted. The linewidth of the resonance is linear in frequency: \n𝜇Δ𝐻 ൌ 𝜇 Δ𝐻ସగఈ\nఊ𝑓                          (3) \nwhere 𝜇Δ𝐻 is the inhomogeneous linewidth broadening and 𝛼 is the Gilbert damping \ncoefficient.  \nFigure 5 summarizes 𝛼 as a function of top or bottom FeB insertion thickness under different \nannealing conditions. For top FeB insertion (figure 5(a)), the influence of FeB insertion on 𝛼 \nshows a moderate dependence on its thickness after 350 C annealing. For bottom FeB insertion \n(figure 5(b)), however, 𝛼 reaches the minimum at 0.4 nm FeB insertion and increases \ndramatically beyond the measurement range w ith the FeB thickness in both annealing \nconditions. For those cases, the resonance is to o broadened to be resolved, reflecting a very \nlarge damping [21]. It also leads to a huge uncertainty in the resonance field and hence 𝜇𝐻 \ndetermination, as mentioned in the preivous section.  \nAn increase of 𝛼 is observed at 0.6 nm FeB in the 400 C annealing condition. However, a \nreduction in 𝛼 after 400 C annealing is obtained when ultrathin FeB (0.2  0.4 nm) is inserted \nat either interface. This suggests that the annealing treatment has different effects on 𝛼 of the \nsamples with various FeB insertion thicknesses. Perhaps different amount of FeB insertion leads to changes in the Fe concentration, micr ostructures and crystallization of dual-MgO FLs \nafter boron depletion upon annealing and thereby to different damping behaviors [29,30]. \nFinally, Table 1 summarizes and compares MTJ stacks after 350 C annealing with FeB \ninsertion at top, bottom, or both CoFeB/MgO interfaces. As the FeB insertion thickness and \nposition differ, the PMA of the dual-MgO FL can be tuned in a wide range, while the damping \nis almost independent. From the systematic studies on insertion thickness in the previous \nsections, top and bottom FeB are optimized to be 0.2 nm and 0.4 nm, respectively. As a result, \nthe MTJ stack with such FeB insertion at both interfaces in dual-MgO FLs can be engineered \nwith a high TMR, low RA, large 𝐾\n∙𝑡, low 𝑀௦∙𝑡, high 𝜇𝐻, and low damping constant.  \n4. Conclusion \nIn this paper, we explore the impact of Fe 80B20 layer inserted at two interfaces of \nCo40Fe60B20/MgO in dual-MgO FLs in MTJ stacks and its annealing stability. With ultrthin \nFeB (0.2  0.4 nm) inserted at the top or bottom CoFeB/MgO interface, the TMR can be \nmaintained with lower RA values, while the top-FeB insertion results in a more RA drop with \na similar TMR. In both cases, the FL saturation magnetization reduces with increasing the Table1. Comparison of magnetotransport and magnetic proper ties of dual-MgO FLs with different FeB insertion \nafter 350 C 30 min annealing. \n \nFL types TMR RA 𝑀௦∙𝑡 𝜇𝐻 𝐾∙𝑡  \n% m2 10-5 A mT mJ∙m2 10-3 \nMgO/CoFeB(1.2)/W/CoFeB(0.8)/MgO 109.3 8.5 184.0  2.7 320  1 0.294  0.004 12.5  2.6 \nMgO/CoFeB(1.2)/W/CoFeB(0.6)/ FeB(0.2) /MgO 114.9 6.9 177.7  3.1 340  5 0.302  0.007 15.0  4.9 \nMgO/ FeB(0.2) /CoFeB(1.0)/W/CoFeB(0.8)/MgO 108.6 8.3 181.2  1.7 349  2 0.316  0.006 11.3  5.0 \nMgO/ FeB(0.2) /CoFeB(1.0)/W/CoFeB(0.6)/ FeB(0.2) /MgO 108.2 7.5 175.7  2.6 381  2 0.335  0.005 12.1  0.7 \nMgO/ FeB(0.4) /CoFeB(0.8)/W/CoFeB(0.6)/ FeB(0.2) /MgO 110.5 8.6 168.3  2.9 434  2 0.365  0.006  4.4  0.7 \n 6\ninserted FeB thickness, while the FL effective anisotropy field increases. However, the PMA \nof dual-MgO FLs with FeB inserted at the bo ttom interface shows a larger improvement than \nits top FeB insertion counterpart, even after 400 C annealing. At the same time, the FeB (0.2 \n 0.4 nm) insertion at either interface reduces the damping constant in the FL. By optimizing \nthe FeB insertion layer thickness, the dual-MgO FL with a low saturation magnetization, high \neffective anisotropy field and low damping can be achieved after 400 C annealing. However, \nthe performance degrades if a thicker FeB is used to replace CoFeB in dual-MgO FLs. \nThis study demonstrates a novel approach to tune dual-MgO FL properties other than typical \nboron composition or non-magnetic spacer engineering. By using the FeB insertion layer at \nCoFeB/MgO interfaces, magnetic properties of the FL and the magnetotransportation of MTJs \ncan be engineered in a wide range, which enables MTJs to meet different performance requirements for various spintronic applications. \nAcknowledgements \nThis work was supported by NRF In vestigatorship (NRFI06-2020-0015). \nReferences \n[1] Ikeda S, Miura K, Yamamoto H, Mizunuma K, Gan H D, Endo M, Kanai S, Hayakawa \nJ, Matsukura F and Ohno H 2010 A perpendicular-anisotropy CoFeB-MgO magnetic \ntunnel junction. Nat. 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Express  6 073002 \n\n9\n \nFigure1. (a) Stack layout of blanket MTJs without FeB insertion. Free layer (FL), reference \nlayer (RL) and hard layer (HL) are indicated. Thicknesses of sublayers are shown in nm with \nparentheses. Blanket films were annealed first at 350 C and then at 400 C, both for 30 min. \n(b) Major loop and (c) minor loop of the stack in  (a) measured by VSM after different annealing \nconditions with the magnetic field perpendicular to the sample plane. \n\n \nFigure2. Schematic of FeB insertion at (a) top and (b) bottom CoFeB/MgO interface in dual-\nMgO FL in the stack shown in figure 1(a). The total thickness of inserted FeB plus remaining \nCoFeB is kept at 0.8 nm and 1.2 nm for layers above and below W spacer, respectively. The \nimpact of FeB insertion on TMR (c) a nd RA (d) of the MTJ stacks after 350 C annealing are \nshown. \n\n10\n \nFigure3. (a) The external excitation frequency as a function of ferromagnetic resonance field \nand (b) the linewidth versus frequency of FL in MTJ stacks without FeB insertion (black open \ncircles), with 0.2 nm top FeB insertion (red open triangles), and with 0.2 nm bottom FeB \ninsertion (blue open squares). Solid lines are fits. \n \n \nFigure4. Effect of FeB insertion and annealing conditions on 𝑀௦∙𝑡 ((a) and (b)), 𝜇𝐻 ((c) \nand (d)), and 𝐾∙𝑡 ((e) and (f)). (a), (c) and (e) show th e impact from FeB insertion at the \ntop CoFeB/MgO interface, while (b), (d) and (f) show the impact from bottom interface. \n11\n\n \nFigure5. Gilbert damping as a function of FeB insertion thickness at (a) top interface and (b) \nbottom interface under two annealing conditions. \n\n"    },    {        "title": "1805.01230v1.Exact_Intrinsic_Localized_Excitation_of_an_Anisotropic_Ferromagnetic_Spin_Chain_in_External_Magnetic_Field_with_Gilbert_Damping__Spin_Current_and_PT_Symmetry.pdf",        "content": "Exact Intrinsic Localized Excitation of an Anisotropic Ferromagnetic Spin Chain in External\nMagnetic Field with Gilbert Damping, Spin Current and PT-Symmetry\nM. Lakshmanan1,a)and Avadh Saxena2,b)\n1)Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University,\nTiruchirappalli - 620 024, India\n2)Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory,\nLos Alamos, NM 87545, USA\nWe obtain the exact one-spin intrinsic localized excitation in an anisotropic Heisenberg\nferromagnetic spin chain in a constant/variable external magnetic field with Gilbert damping\nincluded. We also point out how an appropriate magnitude spin current term in a spin\ntransfer nano-oscillator (STNO) can stabilize the tendency towards damping. Further, we\nshow how this excitation can be sustained in a recently suggested PT-symmetric magnetic\nnanostructure. We also briefly consider more general spin excitations.\na)Electronic mail: lakshman.cnld@gmail.com\nb)Electronic mail: avadh@lanl.gov\n1arXiv:1805.01230v1  [cond-mat.mes-hall]  3 May 2018I. INTRODUCTION\nThe study of dynamics of classical Heisenberg ferromagnetic spin chain with anisotropic inter-\naction is of considerable importance in applied magnetism1,2and from application point of view3,4.\nWhile several continuum versions are known to be completely integrable soliton systems5–7, such\nas the isotropic case, no discrete integrable case is known in the literature, except for a modified\nversion, namely the Ishimori lattice8. On the other hand, the present authors9have shown the exis-\ntence of several classes of exact solutions in terms of Jacobian elliptic functions which exist for the\ncase of the discrete lattice including onsite anisotropy and external magnetic field. Identifying such\ninteresting classes of solutions and their relevance in the context of appropriate physical situations\nconstitute one of the important areas of investigation in spin dynamics9,10.\nFrom another point of view, occurrence of intrinsic localized breathers/oscillations in suitable\nanisotropic ferromagnetic spin chains is of practical relevance11,12and is being explored for the past\nseveral years. Apart from many numerical studies, in recent times the present authors and Subash13\nhave obtained explicit analytical solutions for the Heisenberg anisotropic spin chain with additional\nonsite anisotropy and constant external magnetic field corresponding to excitations of one, two and\nthree spins and also investigated their stability. Additionally, relevant situations were pointed out\nwhere such excitations can be physically identified.\nInrecenttimes,onehasalsoseenthatatnanoscalelevelspintransfernano-oscillator(STNO)14,15,\nwhich essentially consists of a trilayer structure of two nanoscale ferromagnetic films separated by a\nnon-ferromagnetic but conducting layer, can lead to switching of spin angular momentum directions\nand allow for the generation of microwave oscillations16,17. The ferromagnetic film even when it is\nhomogeneous is dominated by anisotropic interactions besides the presence of external magnetic\nfields (both dc and ac) and spin current terms. The equation of motion defining the evolution of\nthe spins is the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation18where the spin current term\nis given by the Slonczewski form. One notices that the LLGS equation is a simple generalization of\nthe Landau-Lifshitz-Gilbert (LLG) equation which describes the nonlinear magnetization dynamics\n2in bulk materials as in the case of ferromagnetic lattices. Then it becomes important to ask what is\nthe influence of spin current term on the spin excitations, particularly intrinsic localized oscillations\n(ILOs) and identify the conditions under which damping effect can be off-set by the spin current\nterm.\nFrom yet another point of view, one may consider the possibility of designing a PT-symmetric\nferromagnetic nanoscale device by considering two nano-film structures interspersed by a nonmag-\nnetic but conducting thinner layer (i.e. a sandwich structure) as suggested by Lee, Kottos and\nShapiro19very recently. These authors have proposed a class of synthetic magnetic nanostructures\nwhich utilize natural dissipation (loss) mechanisms along with suitable chosen gain mechanism so\nas to control the magnetization dynamics. We will also explore how the spin ILOs can be identified\nin these structures.\nIn this paper we deduce an explicit one-spin excitation in an anisotropic ferromagnetic lattice\n(without onsite anisotropy, to start with) in the presence of external magnetic field and explore the\neffect of spin current term to maintain the oscillatory nature of the spin excitation. We then point\nout how this can be generalized to more general spin excitations and in PT-symmetric nanostruc-\ntures.\nThe organization of the paper is as follows. In Sec. 2 we deduce the dynamical equation for an\nanisotropic ferromagnetic spin in the presence of external magnetic field and set up the appropriate\nequation for a one-spin excitation in the presence of Gilbert damping. In Sec. 3, we deduce the\nexplicit one-spin excitation including the damping effect and analyze how the spin excitation gets\naffected by the damping. In Sec. 4, we incorporate the spin current term and point out how an\nappropriate strength of spin current can off-set the effect of damping so as to control the spin\noscillations. In Sec. 5, we point out how the above analysis can be extended to a PT-symmetric\nnanostructure. We briefly indicate how this study can be extended to consider more general spin\nexcitations in Sec. 6. Finally in Sec. 7, we present our conclusions.\n3II. DYNAMICS OF THE ANISOTROPIC SPIN CHAIN AND ONE-SPIN\nEXCITATION\nConsidering the evolution of spins of a one-dimensional anisotropic Heisenberg ferromagnetic\nspin chain modeled by the Hamiltonian12\nH=−N/summationdisplay\n{n}(ASx\nnSx\nn+1+BSy\nnSy\nn+1+CSz\nnSz\nn+1)−D/summationdisplay\nn(Sz\nn)2−/vectorH·/summationdisplay\nn/vectorSn, (1)\nwhere the spin components /vectorSn= (Sx\nn,Sy\nn,Sz\nn)are classical unit vectors satisfying the constant length\ncondition\n(Sx\nn)2+ (Sy\nn)2+ (Sz\nn)2= 1, n = 1,2,...,N. (2)\nHereA,B, andCare the exchange anisotropy parameters, Dis the onsite anisotropy parameter and\nthe external magnetic field /vectorH= (H,0,0)is chosen along the x-axis for convenience. By introducing\nthe appropriate spin-Poisson brackets and deducing the LLG spin evolution equation one can obtain\nthe equation for the spin lattice (1) as\nd/vectorSn\ndt=/vectorSn×/vectorHeff+α/vectorSn×(/vectorSn×/vectorHeff), (3)\nwhere\n/vectorHeff=A(Sx\nn+1+Sx\nn−1)ˆi+B(Sy\nn+1+Sy\nn−1)ˆj+C(Sz\nn+1+Sz\nn−1)ˆk+ 2DSz\nnˆk+/vectorH,(4)\nandαis the Gilbert damping parameter. In component form Eq. (3) with Eq. (4) reads as\ndSx\nn\ndt=CSy\nn(Sz\nn+1+Sz\nn−1)−BSz\nn(Sy\nn+1+Sy\nn−1)−2DSy\nnSz\nn+α/bracketleftbigg\nBSy\nnSz\nn(Sy\nn+1+Sy\nn−1)\n−A(Sx\nn+1+Sx\nn−1)((Sy\nn)2+ (Sz\nn)2) +CSz\nnSx\nn(Sz\nn+1+Sz\nn−1)−2DSx\nn(Sz\nn)2−H((Sx\nn)2+ (Sz\nn)2)/bracketrightbigg\n,(5)\ndSy\nn\ndt=ASz\nn(Sx\nn+1+Sx\nn−1)−CSx\nn(Sz\nn+1+Sz\nn−1) + 2DSx\nnSz\nn+HSz\nn+α/bracketleftbigg\nCSz\nnSy\nn(Sz\nn+1+Sz\nn−1)\n−B((Sx\nn)2+ (Sz\nn)2)(Sy\nn+1+Sy\nn−1) +ASx\nnSy\nn(Sx\nn+1+Sx\nn−1)−2DSy\nn(Sz\nn)2+HSx\nnSy\nn/bracketrightbigg\n, (6)\n4dSz\nn\ndt=BSx\nn(Sy\nn+1+Sy\nn−1)−ASy\nn(Sx\nn+1+Sx\nn−1)−HSy\nn+α/bracketleftbigg\nASx\nnSy\nn(Sx\nn+1+Sx\nn−1)\n−C((Sx\nn)2+ (Sy\nn)2)(Sz\nn+1+Sz\nn−1) +BSy\nnSz\nn(Sy\nn+1+Sy\nn−1) + 2DSz\nn((Sx\nn)2+ (Sy\nn)2) +HSx\nnSz\nn/bracketrightbigg\n.(7)\nNow looking for the one spin excitation for (1) as\n/vectorSn=...,(1,0,0),(1,0,0),(Sx\ni(t),Sy\ni(t),Sz\ni(t)),(1,0,0),(1,0,0),..., (8)\nwhere we have used nto denote a general spin in the lattice and used ito specify the localized spin\nexcitation, and redesignating (Sx\ni(t),Sy\ni(t),Sz\ni(t))as(Sx\n0(t),Sy\n0(t),Sz\n0(t)), the equation of motion\n(LLG equation) for the excited spin can be given as\ndSx\n0\ndt=−2DSy\n0Sz\n0−α/bracketleftbigg\n(2A+H)((Sy\n0)2+ (Sz\n0)2) + 2DSx\n0(Sz\n0)2/bracketrightbigg\n, (9)\ndSy\n0\ndt= (2A+H)Sz\n0+ 2DSx\n0Sz\n0+α/bracketleftbigg\n(2A+H)Sx\n0Sy\n0−2DSy\n0(Sz\n0)2/bracketrightbigg\n, (10)\ndSz\n0\ndt=−(2A+H)Sy\n0+α/bracketleftbigg\n(2A+H)Sx\n0Sz\n0+ 2D((Sx\n0)2+ (Sy\n0)2)Sz\n0/bracketrightbigg\n. (11)\nNote that from Eqs. (9) - (11), one can check that\nSx\n0dSx\n0\ndt+Sy\n0dSy\n0\ndt+Sz\n0dSz\n0\ndt= 0, (12)\nso that/vectorS2= (Sx\no)2+ (Sy\n0)2+ (Sz\n0)2=Constant = 1is conserved.\nNext, further confining to the case where the onsite anisotropy vanishes, D= 0, we have the\nLLG equation for the one-spin excitation,\ndSx\n0\ndt=−α(2A+H)(1−(Sx\n0)2), (13)\ndSy\n0\ndt= (2A+H)Sz\n0+α(2A+H)Sx\n0Sy\n0, (14)\ndSz\n0\ndt=−(2A+H)Sy\n0+α(2A+H)Sx\n0Sz\n0, (15)\nwith the constraint /vectorS2= (Sx\no)2+ (Sy\n0)2+ (Sz\n0)2= 1. The system (13) - (15) can be exactly solved\nas shown below.\n5III. EXPLICIT ONE-SPIN EXCITATION\nNow the above system of nonlinear differential equations can be straightforwardly solved. Inte-\ngrating (14) we obtain\nSx\n0(t) =c2e−2α(2A+H)t−1\nc2e−2α(2A+H)t+ 1, (16)\nwherecis an arbitrary constant. We also note that when α= 0, that is no damping, Sx\n0(t) =\n(c2−1)/(c2+ 1) =const =√\n1−a2as noted in ref. [13], Eq. (11). Also we note that Sx\n0(0) =\n(c2−1)/(c2+ 1)andSx\n0(∞) =−1, indicating a switching from a given initial value to the other\nground state, Sx\n0=−1.\nTo findSy\n0andSz\n0, we proceed as follows. Considering Eq. (14) and differentiating once with\nrespect toton both sides to obtain (d2Sy\n0/dt2), after making use of the forms of (dSx\n0/dt)and\n(dSz\n0/dt)from (13) and (15), respectively, we have\nd2Sy\n0\ndt2=−(2A+H)2(1 +α2)Sy\n0+ 2α(2A+H)2Sx\n0Sz\n0+ 2α2(2A+H)2(Sx\n0)2Sy\n0,(17)\nso that\nSz\n0(t) =1\n2α(2A+H)2Sx\n0/bracketleftBiggd2Sy\n0\ndt2+ (2A+H)2(1 +α2)Sy\n0−2α2(2A+H)2(Sx\n0)2Sy\n0/bracketrightBigg\n.(18)\nAlso from (14) we can write\nSz\n0(t) =1\n2A+H/bracketleftBiggdSy\n0\ndt−αSx\n0Sy\n0/bracketrightBigg\n. (19)\nEquating the right hand sides of (18) and (19), we obtain\nd2Sy\n0\ndt2=−2α(2A+H)Sx\n0dSy\n0\ndt+ (2A+H)2(1 +α2)Sy\n0= 0. (20)\nAfterastandardtransformationandtwointegrations(asindicatedinAppendixA),wecanexplicitly\nwrite the solution for Sy\n0as\nSy\n0=cexp(−α(2A+H)t)\nc2exp(−2α(2A+H)t) + 1ˆacos(Ωt+δ),Ω = 2A+H, (21)\n6where ˆais an arbitrary constant. Also from (14) we have\nSz\n0(t) =1\n(2A+H)/bracketleftBiggdSy\n0\ndt−αSx\n0Sy\n0/bracketrightBigg\n=−ˆasin(Ωt+δ)ce−α(2A+H)t\nc2e(−2α(2A+H)t+ 1. (22)\nNowinordertofixtheconstant ˆawedemandthatthespinlengthconstraint (Sx\n0)2+(Sy\n0)2+(Sz\n0)2= 1\nbe valid. This leads to\nˆa2= 4orˆa= 2, (23)\nso that we have now the complete solution of the excited spin as\nSx\n0(t) =c2e−2α(2A+H)t−1\nc2e−2α(2A+H)t+ 1, (24)\nSy\n0(t) =2ce−α(2A+H)t\nc2e−2α(2A+H)t+ 1cos[(2A+H)t+δ], (25)\nSz\n0(t) =−2ce−2α(2A+H)t\nc2e−2α(2A+H)t+ 1sin[(2A+H)t+δ]. (26)\nNote that the arbitrary constant corresponding to the undamped case ( α= 0) is\nˆa=c2−1\nc2+ 1. (27)\nIt is obvious from the above that for α= 0,Sx\n0=constant =c2−1\nc2+1, whileSy\n0andSz\n0are periodic\nfunctions of t. In this case Eqs. (14)-(15) are linear in Sy\n0andSz\n0so that the perturbation around\nthe origin in the ( Sy\n0−Sz\n0) plane admits pure imaginary eigenvalues. When α/negationslash= 0, they get damped\nas shown in Fig. 1 corresponding to the explicit forms (24)-(26). Note that in the above we have\nassumed the external magnetic field to be a constant in time. However, even in the case where the\nfield is a variable function of time, say\nH(t) =h0+h1cosωt, (28)\nwhereh0,h1andωare constants, we observe from the equations of motion of the spin components\n(13) - (15), that Hoccurs always as a linear combination 2A+H(t) = (2A+h0+h1cosωt).\nTherefore by redefining the time (2A+H)tas\nτ= (2A+h0)t−h1ωsinωt, (29)\n7all the previous analysis goes through. The final spin excitations are of the same form as (24) - (26)\nbut with the transformed time variable given by Eq. (29).\ntSx\n0(t)(i)\n2000 1500 1000 500 01\n0.5\n0\n-0.5\n-1\ntSy\n0(t)(ii)\n2000 1500 1000 500 01\n0.5\n0\n-0.5\n-1\ntSz\n0(t)(iii)\n2000 1500 1000 500 01\n0.5\n0\n-0.5\n-1\nFIG. 1. Damped spin excitation: One-spin excitation (Eqs. (24)-(26)) showing the three spin components\nfor the damped cases ( α= 0.005).\nIV. EFFECT OF SLONCZEWSKI SPIN CURRENT\nNext we consider the influence of spin current term in a trilayer structured STNO (see Fig. 2),\nwhere we consider the excitation of a single spin of magnetization in the outer uniformly magnetized\nlayer under anisotropic interaction and external magnetic field in the presence of spin current. The\ncorresponding spin excitation is given by the Landau-Lifshitz-Gilbert-Slonczewski equation for the\nspin as\nFIG. 2. A schematic representation of STNO.\nd/vectorSn\ndt=/vectorSn×/vectorHeff+α/vectorSn×(/vectorSn×/vectorHeff) +j/vectorSn×(/vectorSn×/vectorSp), (30)\n8where/vectorHeffis the effective field given by Eq. (4) and jis the magnitude of the spin current and\nthe polarization vector /vectorSpis\n/vectorSp= (1,0,0), (31)\ncorresponding to the flow of electrons in the x-direction. Consequently,\n/vectorSn×(/vectorSn×/vectorSp) =/vectorSn(/vectorSn·/vectorSp)−/vectorSp=−((Sy\nn)2+ (Sz\nn)2)ˆi+Sx\nnSy\nnˆj+Sx\nnSz\nnˆk, (32)\nwhere (ˆi,ˆj,ˆk)form the unit orthonormal trihedral. As a result, the equations for the one-spin\nexcitations get modified from (13) - (15) as\ndSx\n0\ndt=−[α(2A+H)−j](1−(Sz\n0)2), (33)\ndSy\n0\ndt= (2A+H)Sz\n0+ [α(2A+H)−j]Sx\n0Sy\n0, (34)\ndSz\n0\ndt=−(2A+H)Sy\n0+ [α(2A+H)−j]Sx\n0Sz\n0. (35)\nNow choosing the spin current as\nj=α(2A+H), (36)\none can check that\ndSx\n0\ndt= 0, (37)\ndSy\n0\ndt= (2A+H)Sz\n0, (38)\ndSz\n0\ndt=−(2A+H)Sz\n0. (39)\nConsequently, the spin vector evolves as\n/vectorS0=/parenleftBig√\n1−ˆa2,ˆacos(Ωt+δ),−ˆasin(Ωt+δ/parenrightBig\n, (40)\nwhere ˆa=constant and Ω = (2A+H), and the effect of damping is exactly offset by the spin\ncurrent term. Thus the spin current acts effectively as an “external magnetic field plus anisotropy\"\nand the system can generate microwave oscillations. When j <α (2A+H), damping will overtake\nasymptotically and the spin will switch its direction.\n9V.PT-SYMMETRIC MAGNETIC DEVICE\nRecentlyaclassofsyntheticmagneticnanostructuresthatmakesuseofthenatureofloss/dissipation\nmechanism together with appropriate amplification (gain) process has been suggested by Lee, Kot-\ntos and Shapiro19to control magnetization dynamics. The suggested arrangement consists of two\ncoupled nano-ferromagnetic films, n= 1,2(when separated by a spacer) in the presence of an\nexternal magnetic field along the x-axis, for example out-of-plane geometry (so that the z-axis is\nperpendicular to the films) as shown in Fig. 3.\nFIG.3. APT-symmetrictrilayerstructurecomprisingtwomagneticthinfilmsandaspacerlayersuggested\nby Lee, Kottos and Shapiro19.\nConsidering the effective instantaneous local fields as /vectorH1effand/vectorH2efffor the two layers 1 and 2,\nrespectively, associated with the homogeneous magnetization vectors /vectorS1= (/vectorM1/|/vectorM1|)and/vectorS2=\n(/vectorM2/|/vectorM2|), we have the associated dynamical equations\nd/vectorS1\ndt=/vectorS1×/vectorH1eff+k/vectorS1×/vectorS2+α/vectorS1×d/vectorS1\ndt, (41)\nd/vectorS2\ndt=/vectorS2×/vectorH2eff+k/vectorS2×/vectorS1−α/vectorS2×d/vectorS2\ndt, (42)\nwherekistheferromagneticcouplingand αisthedamping/gaincoefficient. Notethatthecombined\nsystems (41)-(42) are invariant under the simultaneous changes of the variables /vectorS1,2→ −/vectorS2,1,\n/vectorH1,2eff→−/vectorH2,1effandt→−t, which may be treated as equivalent to combined PT-symmetry\n10operation19. Now we choose the two layers such that\n/vectorS2=/vectorS1×/vectorSp,/vectorS1=−/vectorS2×/vectorSp, (43)\nwhere/vectorSp= (1,0,0)is a fixed polarization vector. Equation (43) implies that /vectorSpis perpendicular\nto the plane of /vectorS1and/vectorS2. Then, similar to the analysis in Sec. 4, we can choose the ferromagnetic\ncouplingksuch that for simple anisotropy as in Eqs. (13) - (15) and external magnetic field H, we\ncan choose\nk=α(2A+H), (44)\nso that the gain/loss terms are exactly cancelled by the ferromagnetic coupling, leaving out\nd/vectorS1\ndt=/vectorS1×/vectorH1eff, (45)\nd/vectorS2\ndt=/vectorS2×/vectorH2eff, (46)\nleading to spin oscillations and thereby to an effective control of magnetization oscillations.\nVI. MORE GENERAL SPIN EXCITATIONS\nOne can consider more general localized spin excitations like two, three, etc. spin excitations.\nFor example, in the case of localized two-spin excitations,\n/vectorSn=...,(1,0,0),(1,0,0),(Sx\ni,Sy\ni,Sz\ni),(Sx\ni+1,Sy\ni+1,Sz\ni+1),(1,0,0),(1,0,0),...\n=...,(1,0,0),(1,0,0),(Sx\n0,Sy\n0,Sz\n0),(Sx\n1,Sy\n1,Sz\n1),(1,0,0),(1,0,0),..., (47)\n11we obtain the dynamical equations from (5) - (7) as\ndSx\n0\ndt=CSy\n0Sz\n1−BSz\n0Sy\n1−2DSy\n0Sz\n0\n+α[BSx\n0Sy\n0Sy\n1−(A(Sx\n1+ 1) +H)((Sy\n0)2+ (Sz\n0)2) +CSx\n0Sz\n0Sz\n1−2DSx\n0(Sz\n0)2], (48)\ndSy\n0\ndt=ASz\n0(Sx\n1+ 1)−CSx\n0Sz\n1+ 2DSx\n0Sz\n0+HSz\n0\n+α[(A(Sx\n1+ 1) +H)Sx\n0Sy\n0−BSy\n1((Sx\n0)2+ (Sz\n0)2) +CSy\n0Sz\n0Sz\n1−2DSy\n0(Sz\n0)2], (49)\ndSz\n0\ndt=BSx\n0Sy\n1−ASy\n0(Sx\n1+ 1)−HSy\n0\n+α[(A(Sx\n1+ 1) +H)Sx\n0Sz\n0+BSy\n0Sz\n0Sy\n1−CSz\n1((Sx\n0)2+ (Sy\n0)2) + 2DSz\n0((Sx\n0)2+ (Sy\n0)2)],(50)\ndSx\n1\ndt=CSz\n0Sy\n1−BSy\n0Sz\n1−2DSy\n1Sz\n1\n+α[BSy\n0Sx\n1Sy\n1−(A(Sx\n0+ 1) +H)((Sy\n1)2+ (Sz\n1)2) +CSz\n0Sx\n1Sz\n1−2DSx\n1(Sz\n1)2], (51)\ndSy\n1\ndt=ASz\n1(Sx\n0+ 1)−CSx\n1Sz\n0+ 2DSx\n1Sz\n1+HSz\n1\n+α[(A(Sx\n0+ 1) +H)Sx\n1Sy\n1−BSy\n0((Sx\n1)2+ (Sz\n1)2) +CSz\n0Sy\n1Sz\n1−2DSy\n1(Sz\n1)2], (52)\ndSz\n1\ndt=BSx\n1Sy\n0−ASy\n1(Sx\n0+ 1)−HSy\n1\n+α[(A(Sx\n0+ 1) +H)Sx\n1Sz\n1+BSy\n0Sy\n1Sz\n1−CSz\n0((Sx\n1)2+ (Sy\n1)2) + 2DSz\n1((Sx\n1)2+ (Sy\n1)2)].(53)\nNote that the terms proportional to αare generalizations for the present two-spin excitation case\ncompared to those given in Eqs. (9) - (11). As such the system (48) - (53) does not seem to be\nanalytically solvable. In Fig. 4, we numerically integrate the system for both the undamped case\n(α= 0) and the damped case ( α/negationslash= 0) for nonzero Dand present the solutions in the undamped\nand damped cases to show the existence of more general internal localized excitations. The analysis\ncan be extended to even more general situations, which will be presented elsewhere.\nVII. CONCLUSION\nBy looking at the simplest internal localized excitations in an anisotropic Heisenberg ferromag-\nnetic spin chain in external magnetic field with additional Gilbert damping, we deduced the explicit\nsolutions which characteristically show the effect of damping. Then applying a spin current in an\n12tSx\n0(t)(i)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n0(t)(ii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n0(t)(iii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSx\n1(t)(iv)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n1(t)(v)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n1(t)(vi)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1Fig. 4 (a): Undamped two-spin excitations\ntSx\n0(t)(i)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n0(t)(ii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n0(t)(iii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSx\n1(t)(iv)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n1(t)(v)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n1(t)(vi)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\nFig. 4 (b): Damped two-spin excitations\nFIG. 4. Solution of Eqs. (48)-(53) for two-spin excitations S0andS1for the (a) undamped ( α= 0) and\n(b) damped cases ( α= 0.005), with the anisotropy parameters A= 0.1,B= 0.23,C= 1.0andD= 0.3\nand the magnetic field H= 113Oe.\nSTNO of appropriate magnitude, we pointed out how the tendency toward damping can be offset\nexactly and thereby sustaining the magnetic oscillations. Our prediction about a PT-symmetric\n13STNO could be tested in magnetic multilayer structures with carefully balanced gain and loss. We\nhave also pointed out how such controlled oscillations can be effected in a recently suggested nano-\nmagnetic trilayer device. It will be insightful to observe these oscillations in appropriate magnetic\nsystems experimentally. Finally, in a related context we note that nonreciprocal optical modes can\nexist at an interface between two PT-symmetric magnetic domains near a frequency corresponding\nto almost zero effective permeability20.\nVIII. ACKNOWLEDGMENTS\nThe authors wish to thank Dr. D. Aravinthan for his help in the numerical analysis. The\nresearch work of ML was supported by a NASI Senior Scientist Platinum Jubilee Fellowship\n(NAS 69/5/2016-17) and a DST-SERB Distinguished Fellowship (No.: SERB/F/6717/2017-18).\nML was also supported by a Council of Scientific and Industrial Research, India research project\n(No.: 03/1331/15/EMR-II) and a National Board for Higher Mathematics research project (No.:\n2/48(5)/2015/NBHM(R.P.)/R&D II/14127). ML also wishes to thank the Center for Nonlinear\nStudies, Los Alamos National Laboratory, USA for its warm hospitality during his visit in the\nsummer of 2017. This work was supported in part by the U.S. Department of Energy.\nAPPENDIX A\nHere we briefly point out how to solve Eq. (20). Introducing the transformation\nSy\n0(t) =eα(2A+H)/integraltext\nSx\n0dt·ˆSy\n0(t) (A. 1)\ninto Eq. (20), we obtain\nd2ˆSy\n0(t)\ndt2+ (2A+H)2ˆSy\n0(t) = 0. (A. 2)\nConsequently, we have\nˆSy\n0(t) = ˆacos(Ωt+δ),Ω = 2A+H, (A. 3)\n14where ˆaandδare arbitrary constants. Then, the prefactor on the right hand side of (21) can be\ndeduced as follows. Since\nI=/integraldisplay\nSx\n0dt=/integraldisplayc2exp(−2α(2A+H)t)−1\nc2exp(−2α(2A+H)t) + 1dt=−1\n2α(2A+H)log(c2exp(−2α(2A+H)t) + 1)2\nc2exp(−2α(2A+H)t,\n(A. 4)\nthe prefactor becomes\nexp/bracketleftbigg\nα(2A+H)/integraldisplay\nSx\n0dt/bracketrightbigg\n=cexp(−α(2A+H)t)\nc2exp(−2α(2A+H)t) + 1. (A. 5)\nCorrespondingly\nSy\n0=cexp(−α(2A+H)t)\nc2exp(−2α(2A+H)t) + 1ˆacos(Ωt+δ),Ω = 2A+H, (A. 6)\nwhich is Eq. (21).\nREFERENCES\n1B. Hillerbrands and K. Ounadjela, Spin Dynamics in Confined Magnetic Structures , Vols. I & II\n(Springer, Berlin) 2002.\n2M. Lakshmanan, Philos. Trans. R. Soc. A 369(2011) 1280.\n3B. Georges, V. Cros and A. Fert, Phys. Rev. B 73(2006) 0604R.\n4Z. Yang, S. Zhang and Y. C. Li, Phys. Rev. Lett. 99(2007) 134101.\n5M. Lakshmanan, Phys. Lett. A 61(1977) 53.\n6K. Nakamura and T. 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Mater. 159(1996) L261.\n19J. M. Lee, T. Kottos and B. Shapiro, Phys. Rev. B 91(2015) 094416.\n20J. Wang, H. Y. Dong, C. W. Ling, C. T. Chan and K. H. Fung, Phys. Rev. B 91(2015) 235410.\n16"    },    {        "title": "1107.0638v1.Influence_of_randomness_and_retardation_on_the_FMR_linewidth.pdf",        "content": "In\ruence of randomness and retardation on the FMR-linewidth\nThomas Bose and Ste\u000ben Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany\u0003\n(Dated: October 25, 2018)\nAbstract\nThe theory predicts that the spin-wave lifetime \u001cLand the linewidth of ferromagnetic resonance\n\u0001Bcan be governed by random \felds and spatial memory. To that aim the e\u000bective \feld around\nwhich the magnetic moments perform a precession is superimposed by a stochastic time dependent\nmagnetic \feld with \fnite correlation time. The magnetization dynamics is altered by inclusion\nof a spatial memory e\u000bect monitoring a non-local interaction of size \u0018. The underlying Landau-\nLifshitz-Gilbert equation (LLG) is modi\fed accordingly. The stochastic LLG is equivalent to a\nFokker-Planck equation which enables to calculate the mean values of the magnetization vector.\nWithin the spin-wave approximation we present an analytical solution for the excitation energy\nand its damping. The lifetime and the linewidth are analyzed depending on the strength of the\nrandom \feld Dand its correlation time \u001ccas well as the retardation strength \u0000 0and the size \u0018.\nWhereas\u001cLdecreases with increasing D, retardation strength \u0000 0and\u001cc, the lifetime is enhanced\nfor growing width \u0018of the spatial retardation kernel. In the same manner we calculate the exper-\nimentally measurable linewidth \u0001 Bis increased strongly when the correlation time \u001ccranges in\nthe nanosecond interval.\n\u0003thomas.bose@physik.uni-halle.de; ste\u000ben.trimper@physik.uni-halle.de\n1arXiv:1107.0638v1  [cond-mat.mes-hall]  4 Jul 2011I. INTRODUCTION\nFerromagnetic resonance (FMR) is a powerful technique to study magnetic materials, in\nparticular the inherent magnetization dynamics [1, 2]. So the observable FMR-linewidth is\nvery sensitive to the underlying dynamical processes as well as the real structure of the ma-\nterial like anisotropy. From a theoretical point of view the Landau-Lifshitz-Gilbert equation\n(LLG) [3, 4], see Eq. (1) in the present paper, is an appropriate tool to investigate magnetic\nexcitations and dissipative processes as the damping of the excitations. Although, the LLG is\nknown since a few decades it is still a standard model to analyze magnetodynamics. Recently\nthe Gilbert damping parameter was determined experimentally for ferromagnetic thin \flms\nin [5] and by \frst-principle calculations for itinerant ferromagnets in [6]. Obviously, the ap-\nplicability of the LLG depends on the physical situation in mind. In case the magnetization\nis not conserved the Landau-Lifshitz-Bloch (LLB) equations are more appropriate, in partic-\nular in the vicinity of the phase transition as demonstrated in [7]. The LLB equations were\nused to investigate magnetization switching near the Curie temperature in [8, 9]. Moreover,\nthe geometrical con\fguration of the sample are able to play an important role in measuring\nthe FMR-linewidth. Related to this fact the contribution of the Gilbert damping to the\nlinewidth can be superimposed by extrinsic e\u000bects as magnon-magnon scattering processes\n[10] which become of the same order of magnitude or even exceed the Gilbert damping. Es-\npecially for an in-plane con\fguration where the external \feld as well as the magnetization lie\nin the \flm plane the in\ruence of two magnon processes to the FMR-linewidth cannot be ne-\nglected [11, 12]. Those theoretical results predicting a nonlinear dependence of the linewidth\non the frequency were extended to the case when the magnetization is tipped out of plane\n[13]. Di\u000berent experimental \fndings emphasize the importance of extrinsic contributions for\nin-plane setups, see [14{17]. A quantitative separation of Gilbert damping and two magnon\nscattering contributions was carried out [14, 16, 17]. Contrary to these observations there are\nother investigations [18], which o\u000ber no qualitative di\u000berence between in-plane and normal-\nto-plane measurements. In both realizations the linewidth depends linearly on the frequency\neven for frequencies smaller than 10 GHz . Such theoretical and experimental works suggest,\namong others, that the FMR-linewidth is exclusively controlled by the Gilbert damping and\nexhibits a pure linear frequency dependence in a perpendicular con\fguration with respect\nto thin \flms measurements. Furthermore, the two magnon scattering is supposed to be of\n2less importance in bulk ferromagnets [16]. Thus the LLG equation seems still applicable\nto describe magnetization dynamics provided the physical situation is carefully analyzed as\npointed out in [2]. A more realistic magnetization dynamics requires a modi\fcation of the\nLLG. Recently, the anisotropic damping and its manifestation in the FMR-linewidth has\nbeen discussed by several authors [19{21]. An alternative formulation of Gilbert damping\nby means of scattering theory was discussed in [22]. In addition, ferromagnetic resonance\nmeasurements were used as well to investigate spin transport in magnetic single and double\nlayer structures [23]. Moreover, very recently it was shown that the transfer of spin angular\nmomentum can induce ferromagnetic resonance dynamics in a ferromagnetic \flm due to\nthe spin Hall e\u000bect in an adjacent \flm with strong spin-orbit scattering [24]. Related to\nthis phenomena it was reported on the direct time-resolved measurement of spin torque in\nmagnetic tunnel junctions to detect resonant magnetic precession due to an oscillating spin\ntorque [25]. A theory of ferromagnetic resonance in perpendicular magnetized nanodisks is\nsuggested in [26].\nTo push forward the theory stochastic forces and non-local interactions should be included\ninto the model to gain a more realistic description of magnetic materials and to reveal unex-\npected behavior as for example the noise suppression by noise behavior argued in [27]. The\ne\u000bects of noise in magnetic nanosystems obeying spin torque dynamics are investigated in\n[28, 29]. Experimentally, the role of noise in magnetic systems was prospected in [30, 31].\nThe present work is addressed to the in\ruence of randomness on the magnetization dynam-\nics. As the two new aspects the system considered is simultaneously subjected to feedback\ncoupling and to a stochastic \feld with colored noise. The starting point is the LLG equation\nwhich is generalized in a manner that both spatial memory e\u000bects and a temporal stochastic\n\feld with a \fnite correlation time is incorporated into the model. Previously the in\ruence\nof colored noise [32] and retardation e\u000bects [33] within the LLG were analyzed separately.\nOtherwise, both e\u000bects can occur simultaneously. Consequently we study a combined model\nconcerning both kind of impacts, feedback and randomness. As demonstrated in former\npapers there exits the possibility that the total damping, originated by the Gilbert damp-\ning and that one induced by memory e\u000bects are able to cancel by the distinct damping\nmechanisms. In this paper we are interested in the FMR-linewidth. The corresponding\nparameters range in such reasonable intervals where di\u000berent dissipation sources are not\nobservable. The main goal is to calculate the FMR-linewidth and to discuss its dependence\n3on the parameters characterizing randomness and retardation.\nLet us give a brief outline of the paper. In Sec. 2 we present the mathematical model and\nits underlying basic assumptions. The stochastic LLG equation is equivalent to a Fokker-\nPlanck equation which is derived approximately in Sec. 3. This equation enables to compute\nthe mean values of the magnetization. The results are discussed in detail in Sec. 4. Finally,\nwe conclude by summarizing the results and by an outlook in Sec. 5.\nII. MODEL\nAs already indicated in the introduction we are interested in micro- and nanosized mag-\nnets. Therefore a coarse-grained description is an appropriate tool to investigate magnetic\nmaterial. In this mesoscopic description the discrete magnetic moments are replaced by\na spatiotemporal vector \feld m(r;t). The interaction and the dynamics of the moments\nare formulated in a continuous approximation. The situation is schematically illustrated\nin Fig. 1. Here, the magnetization m(r1) represents the magnetic properties within the\nmesoscopic microscopic\nνm(r1)∝/summationtext\n{iln}∈d3r1siln\nd3r1\nsh11 sh1kshj1 shjk\n/Bullet\nx\nyzr1\nFIG. 1. Illustration of the of the coarse-grained mesoscopic model. The sirepresent microscopic\nmagnetic moments which are related to the magnetization m. Further explanation can be found\nin the text.\nvolume-element d3r1which is build around the position r1. The \feld m(r1) stands for the\ntotal set of microscopic spins which will be visible if one zooms into the microscopic struc-\n4ture. The huge number of microscopic degrees of freedom within d3r1are substituted by\na single degree of freedom, namely the mesoscopic quantity mwhich can be considered as\nthe sum over the microscopic spins located at equivalent crystal positions. Moreover, the\nmagnetization vector \feld m(r1) is assumed to be oriented continuously in space. The basics\nof our model consists in this mesoscopic description discussed before. Further, the system is\nsupposed to o\u000ber an uniaxial anisotropy where the direction of the anisotropy axis is denoted\nby\u0017. Our calculations refer to weak excitations which evolve as spin waves and possess a\n\fnite life time. Both quantities are found in the long wave-length limit qa\u001c1, whereqis\nthe amount of the wave vector and ais the lattice constant. This assumption re\rects the\nmesoscopic level of description. Experimentally the dynamic behavior of the magnetization\nmcan be detected for instance by means of ferromagnetic resonance (FMR). Because the\nmain goal of the paper is to put forward the modeling towards more realistic systems we\ndevelop a dynamic model for the magnetization \feld m(r) in which retardation e\u000bects as\nwell as stochastic \felds are included. In particular, the aim is to relate our \fndings for the\nmagnetic excitations and their damping to an experimentally accessible quantity, namely\nthe FMR-line width \u0001 B, cf. Eq. (23).\nAs underlying model we start from is the Landau-Lifshitz-Gilbert equation\n@m\n@t=\u0000\r\n1 +\u000b2m\u0002h\nBe\u000b+\u000b[m\u0002Be\u000b]i\n; (1)\nwhich will be generalized accordingly. In Eq. (1) the quantities \rand\u000bare the gyromagnetic\nratio and the dimensionless Gilbert damping parameter, respectively. In this description\nm(r;t) is the unit vector m=M=Mswith the magnetization Mthe saturation magneti-\nzation. The local e\u000bective \feld Be\u000b(r;t) causes the precession of the magnetization. In\ngeneral, the e\u000bective \feld Be\u000bis composed of di\u000berent contributions, an internal \feld due\nto the interaction of the spins, the magnetic anisotropy and an external \feld. This e\u000bective\n\feld can be derived from the Hamiltonian of the system by functional variation with respect\ntom\nBe\u000b=\u0000M\u00001\ns\u000eH\n\u000em: (2)\n5The Hamiltonian Hcan be expressed as [32, 34]\nH=Z\nd3rfwex+wan+wextg\nwex=1\n2Ms~J(rm)2;\nwan=1\n2MsKsin2\u0012 ; w ext=\u0000Bext\u0001M:(3)\nThe quantities ~J=Ja2andKdesignate the exchange energy density and the magneto-\ncrystalline anisotropy energy density. Here, Jis the coupling strength between nearest\nneighbors referring to the isotropic Heisenberg model [35] and ais the lattice constant.\nFurther, Bextis the static external magnetic \feld. The quantity \u0012is the angle between\nthe the local magnetization mand the anisotropy axis \u0017= (0;0;1). We assume that \u0017\npoints in the direction of the easy axis in the ground state. Therefore, K > 0 characterizes\nthe strength of the anisotropy. In deriving Eq. (3) we have used m2= 1. Let us stress\nagain that this assumption seems to be correct if the temperature is well below the Curie\ntemperature [7]. Our calculations based on the LLG suggest that other damping mechanism\nsuch as an extrinsic magnon-magnon scattering due to magnetic inhomogeneities should be\ninactive and hence they are irrelevant. In thin \flms this situation can be achieved when\nboth the magnetization and the static external \feld are perpendicular to the \flm plane.\nIn our model this situation is realized when both the easy axis of the anisotropy \u0017as well\nas the external \feld Bextpoint into the z-direction. Hence the equilibrium magnetization\nis likewise oriented parallel to the z-axis. This situation corresponds to a normal-to-plane\ncon\fguration. From here we conclude that the application of the LLG leads to reasonable\nresults. For a di\u000berent realization an alternative dynamical approach seems to be more\naccurate, see also the conclusions.\nTo proceed further, the vector mis decomposed into a static and a dynamic part termed\nas\u0016and = ( 1; 2; 3), respectively. In the frame of spin wave approximation we make\nthe ansatz\nm(r;t) =\u0016+ (r;t) =\u0016\u0017+ ; \u0016 = const:; (4)\nCombining Eqs. (2) and (3) yields the e\u000bective \feld\nBe\u000b=~Jr2 \u0000K 0+Bext; 0= ( 1; 2;0): (5)\n6It is appropriate to introduce dimensionless quantities:\nl2\n0=~J\nK=Ja2\nK; \f = (l0q)2+ 1;\n\n =\rK; \u0016t= \nt;jBextj\nK=\":(6)\nThe quantity l0is called the characteristic magnetic length [36] whereas the parameter\n\"re\rects the ration between the strengths of the external and the anisotropy \feld. For\nconvenience later we will substitute \u0016t!tagain. So far we have introduced the LLG in\nEq.(1) in its conventional form and incorporated our special basic model assumptions for\na ferromagnetic material below its Curie temperature. To proceed toward a more realistic\ndescription of magnets the LLG will be extended by the inclusion of retardation e\u000bects\nand random magnetic \felds. Whereas retardation is implemented by a memory kernel\n\u0000(r;r0;t;t0) a stochastic \feld \u0011(r;t) contributes additionally to the e\u000bective \feld, i.e.\nBe\u000b(r;t)!be\u000b(r;t) =Be\u000b(r;t) +\u0011(r;t): (7)\nTaking both e\u000bects into account we propose the following generalized LLG\n@m(r;t)\n@t=Zt\n0dt0Z\nddr0\u0000(r\u0000r0;t\u0000t0)\n\u0002\u001a\n\u00001\n1 +\u000b2m(r0;t0)\u0002\u0002\nbe\u000b(r0;t0)+\n+\u000b[m(r0;t0)\u0002be\u000b(r0;t0)]\u0003\u001b\n;(8)\nwhere the stochastic \feld is included in the dimensionless e\u000bective \feld as\nbe\u000b=l2\n0r2 \u0000 0+\"b0+\u0011(r;t): (9)\nThe unit vector b0indicates the direction of the external magnetic \feld. In general, the\nkernel should respect the retardation concerning temporal and spatial processes. More\nprecise, a change of the magnetic moment at position rshould in\ruence another moment at\nposition r0and vice versa. This in\ruence is thought to be an additional contribution which\nshould not be confused with parts of the exchange interaction in the e\u000bective \feld, i.g.\nthe length\u0018on which spatial retardation e\u000bects are relevant could be of a di\u000berent order of\nmagnitude in comparison with the lattice constant a. Insofar, a purely coordinate dependent\npart of the kernel re\rects a kind of non-local interaction. All moments within a radius \u0018\n7contribute to the interaction. Likewise a temporal feedback mechanism can be taken into\naccount due to the fact that the transport of information from one magnetic moment to its\nneighbors needs at least a \fnite albeit small time. Such an in-time retardation mechanism\nis considered already in [33]. Here we concentrate on instantaneous retardation in time\nwhereas the spatial part is realized for simplicity by a Gaussian shape\n\u0000(r;t) =\u000e(t)(\n\u00000\n(p\u0019\u0018)3exp\"\n\u0000\u0012r\n\u0018\u00132#)\n; (10)\nwhere \u0000 0and\u0018determine the strength and the size of the retardation, respectively. The \u000e-\nfunction in the last equation signalizes that all contribution to the interaction within a sphere\nwith radius \u0018contribute simultaneously to the interaction. As discussed below a typical value\nfor\u0018is assumed to be of the order 10\u00008m, i.e. the time for the signal propagation within \u0018\nis about 10\u000015\u000010\u000016s. Because this time is much smaller as the lifetime of the spin-waves,\nsee the discussion below, we conclude that delay e\u000bects within the region with radius \u0018can\nbe neglected. As indicated in Eq. (7) the noise \u0011(r;t) can also depend on space and time,\ni.e. in general random forces can e\u000bect the value of the magnetization at di\u000berent positions\nin a distinct manner while additionally their \ructuations are also time dependent. Such\na behavior maybe lead back to local in\fnitesimal temperature gradients or defects. The\nrandom \feld \u0011(r;t) is regarded as a colored noise the statistical properties of which obey\nthe following relations\nh\u0011\u000b(t)i=0;\n\u001f\u000b\f(t;t0) =h\u0011\u000b(t)\u0011\f(t0)i\n=D\u000b\f\n\u001c\u000b\fexp\u0014\n\u0000jt\u0000t0j\n\u001c\u000b\f\u0015\n\u001c\u000b\f!0\u0000\u0000\u0000\u0000! 2D\u000b\f\u000e(t\u0000t0):(11)\nThe components \u0011\u000b(t) have a zero mean and a \fnite correlation time. As an aside in\nthe limit\u001c!0 the usual white noise properties are recovered. However, we want to\nconcentrate on the more realistic colored noise case with \u001c > 0. In Eq. (11) we assume\n\u0011\u000b(r;t) =\u0011\u000b(t). In other words the total system is a\u000bected by the same random in\ruences.\nThis may be reasonable if we have a well controllable constant temperature over the whole\nsample and an ideal sample without defects. Let us brie\ry summarize the new properties\nof the model de\fned by Eqs. (8) and (9). The in\ruences of retardation and a multiplicative\n8noise as well are implemented in the conventional Landau-Lifshitz-Gilbert equation. After\na general incorporation into the model we had to limit the properties of both retardation\nand noise to an idealized situation in order to obtain analytical results in the subsequent\nsection. However, although each of the Eqs. (10) and (11) represents a simpli\fed version\nof a more general case the linking between both by means of the equation of motion for\nthe magnetization in Eq. (8) models a quite complex behavior which is partly indicated in\nFig. 2. While the exchange interaction is a short range coupling over a lattice constant a,\nsi si+1 si+2 si+n−1si+n∝Jfeedback ∝Γ0\na\nρ=n a\nFIG. 2. Schematic depiction of the di\u000berence between the exchange interaction Jand the coupling\ndue to retardation /\u00000. As is visible feedback mechanisms can range over a larger distance\n\u0018'\u001a=na, wherenis integer.\nthe interplay due to retardation with strength \u0000 0can cover a distance \u001awhich is a multiple\nof the lattice constant. If this distance \u001ais comparable to the characteristic length scale \u0018in\nEq. (10) retardation e\u000bects should be relevant. This microscopic picture can be transferred\nto a mesoscopic one and means a kind of non-local interaction. On the one hand at every\nspatial point the same kind of noise a\u000bects the magnetization. Otherwise, the magnetization\nm(r;t) takes di\u000berent values at distinct positions rand therefore, the impact of the noise\nmight be slightly di\u000berent, too. Although spatial alterations of the noise are not regarded in\nthe correlation function de\fned in Eq. (11) the memory kernel respects spatial correlations\nwithin\u0018as seen in Eq. (10). Insofar the e\u000bect of noise at di\u000berent spatial positions is\ntransmitted by the memory kernel \u0000( r;r0;t;t0). Another important hallmark is that the\nnoise-noise correlation function \u001f\u000b\f(t;t0) is featured by a \fnite lifetime, cf. Eq. (11). For\nthe forthcoming calculations we assume that \u001c\u000b\f=\u001cc\u000e\u000b\f. Likewise the matrix of the noise\n9correlation strength is supposed to be diagonal, i.e. Dkl=D\u000ekl. Hence, the two important\nstochastic parameters are the correlation time \u001ccand the correlation strength Dwhereas\nthe relevant parameters originated by the retardation are the retardation strength \u0000 0and\nthe retardation length \u0018, see Eq. (10). The results will be discussed in terms of the set of\nparameters D;\u001cc;\u00000and\u0018.\nIII. STATISTICAL TREATMENT\nEqs. (8) and (9) represents the stochastic LLG. Due to the coupling to the stochastic \feld\n\u0011(r;t) the magnetization \feld m(r;t) becomes a stochastic variable. To calculate the mean\nvalues of mone needs the probability distribution P(m;t). To that aim the current section\nis devoted to the derivation of an approximated Fokker-Planck equation which allows to\n\fnd the equations of motion for averaged quantities. To that purpose let us reformulate the\nmodel presented in Eqs. (4), (8) and (9). After Fourier transformation '(q;t) =FTf (r;t)g\nwe \fnd in linear spin-wave approximation\nd\ndt'\u000b(q;t) =A\u000b['(q;t)] + B\u000b\f['(q;t)]\u0011\f(t): (12)\nThe vector Aand the matrix Bposses the components\nA=f(q;\u0018)\n1 +\u000b20\nBBB@\u0000(\f\u0016+\u000f) (\u000b\u0016' 1+'2)\n(\f\u0016+\u000f) ('1\u0000\u000b\u0016' 2)\n01\nCCCA; (13)\nand\nB=f(q;\u0018)\n1 +\u000b20\nBBB@\u000b\u0016' 3'3\u0000('2+\u000b\u0016' 1)\n\u0000'3\u000b\u0016' 3'1\u0000\u000b\u0016' 2\n'2\u0000'1 01\nCCCA: (14)\nHere the function f(q;\u0018) is the Fourier transform of the memory kernel \u0000( r;t) de\fned in\nEq. (10) and \u0016and\fare introduced in Eqs. (4) and (6), respectively. Notice that fdepends\nonly on the absolute value qof the wave vector and takes the form\nf(q; \u0018) = \u0000 0exp\u0014\n\u00001\n4\u00182q2\u0015\n: (15)\nTo get the probability distribution function of the stochastic process determined by Eqs. (11)\nand (12)-(14) we de\fne according to [37, 38]\nP(';t) =h\u000e['(t)\u0000']i: (16)\n10Here the symbol <:::> means the average over all realizations of the stochastic process. As\nusual'(t) represents the stochastic process whereas 'are the possible realizations of the\nprocess at time t. Due to the colored noise the corresponding Fokker-Planck equation can be\nobtained only approximatively in lowest order of the correlation time. The time evolution\nof Eq. (16) can be written in the form\n@\n@tP(';t) =LP(';t): (17)\nIn deriving this expression we have used the time evolution of '(t) according to Eq. (12),\nthe Novikov theorem [39] and the correlation function given by Eq. (11) with \u001c\u000b\f=\u001cc\u000e\u000b\f,\nD\u000b\f=D\u000e\u000b\f. The form of the operator Lis given in a correlation time and cumulant\nexpansion while transient terms have been neglected [40{42]\nL(';\u001cc) =\u0000@\n@'\u000bA\u000b(') +@\n@'\u000bB\u000b\f(')@\n@'\r\n\u0002(\nD\u0002\nB\r\f(')\u0000\u001ccM\r\f(')\u0003\n+D2\u001cc\u0014\nK\r\f\u0016(')@\n@'\u0017B\u0017\u0016(')\n+1\n2B\r\u0016(')@\n@'\u0017K\u0017\f\u0016(')\u0015)\n;(18)\nwith\nM\r\f=A\u0017@B\r\f\n@'\u0017\u0000B\u0017\f@A\r\n@'\u0017\nK\r\u0017\f=B\u0016\f@B\r\u0016\n@'\u0016\u0000@B\r\f\n@'\u0016B\u0016\u0017:(19)\nNotice that summation over double-indices is understood. The single probability distribution\nis determined by the operator Lin Eq. (18) which enables us to \fnd the equation of motion\nfor the expectation values h'\u000bi. It follows\nd\ndth'\u000b(t)i=hA\u000bi+D\u001c@B\u000b\f\n@'\r\u0000\nB\r\f\u0000\u001ccM\r\f\u0001\u001d\n\u0000D2\u001cc(\u001c@\n@'\u0017\u0012@B\u000b\f\n@'\rK\r\f\u0014\u0013\nB\u0016\u0014\u001d\n+1\n2\u001c@\n@'\u0017\u0012@B\u000b\f\n@'\rB\r\u0014\u0013\nK\u0016\f\u0014\u001d)\n:(20)\nNotice that in the white noise case all terms /\u001ccwould vanish.\n11IV. RESULTS AND DISCUSSION\nWe \fnd an analytical solution for the colored noise problem in Eq. (20) by standard\nGreens' function technique and Laplace transformation. After performing the summation\nin Eq. (20) while making use of Eqs. (13), (14) and the expressions in Eq. (19) the result\nreads\nh'(t)i=0\nBBB@e\u0000\u000etcos(\nt)e\u0000\u000etsin(\nt) 0\n\u0000e\u0000\u000etsin(\nt)e\u0000\u000etcos(\nt) 0\n0 0 e\u0000\u001bt1\nCCCA\u0001h'0i; (21)\nwhereh'0i=h'(t= 0)iare the initial conditions. Physically, the parameters \u000e; \u001band \nplay the roles of the inverse magnon lifetimes and the frequency of the spin wave, respectively.\nThey are determined by\n\u000e=\u000b\u0016[\"+\f\u0016]f(q;\u0018)\n1 +\u000b2+D[2\u0000\u000b2\u00162]\u0012f(q;\u0018)\n1 +\u000b2\u00132\n+ 2D\u001cc\u000b\u0016[\"+\f\u0016]\u0012f(q;\u0018)\n1 +\u000b2\u00133\n+D2\u001cc[1\u00006\u000b2\u00162]\u0012f(q;\u0018)\n1 +\u000b2\u00134\n;\n\n =\u0000[\"+\f\u0016]f(q;\u0018)\n1 +\u000b2+ 3D\u000b\u0016\u0012f(q;\u0018)\n1 +\u000b2\u00132\n+D\u001cc[\u000b2\u00162\u00001] [\"+\f\u0016]\u0012f(q;\u0018)\n1 +\u000b2\u00133\n+D2\u001cc\n2\u000b\u0016[11\u00003\u000b2\u00162]\u0012f(q;\u0018)\n1 +\u000b2\u00134\n;\n\u001b=2Df(q;\u0018)\n1 +\u000b22\n\u0000[4D\u001cc\u000b\u0016(\"+\f\u0016)]f(q;\u0018)\n1 +\u000b23\n+D2\u001cc[3\u000b2\u00162+ 1]f(q;\u0018)\n1 +\u000b24\n:(22)\nNote that the parameters of the retardation mechanism, the strength \u0000 0and the length scale\n\u0018, are included in the function f(q;\u0018) de\fned in Eq. (15). The two important parameters\noriginated from the noise are the correlation time \u001ccand the correlation strength Dof the\nrandom force. Both a\u000bect the quantities in Eq. (22) as well. We proceed by studying the\nsystem under the variation of these four model parameters. To be comparable to FMR\nexperiments we refer to the following quantities\n\u001cL= (\u000e\rK )\u00001;\u0001B= 1:16\u000b!\n\r= 1:16\u000bK\n; (23)\n12i.e. the lifetime \u001cLof the spin waves and the FMR-linewidth \u0001 B, compare [1, 17], which\nare related to the dimensionless inverse lifetime \u000eand frequency \n from Eq. (22). Here the\nfrequency!is tantamount to the resonance frequency of the spin waves. The lifetime \u001cL\nand the linewidth \u0001 Bare given in SI-units. Notice that the frequency independent part\n\u0001B0, typically added on the right-hand side of the equation for \u0001 Bis already subtracted in\nEq. (23). The contribution \u0001 B0is supposed to take into account magnetic inhomogeneities.\nFor a quantitative evaluation we need to set the model parameters to reasonable values. In\ndoing so we also refer to Eq. (6). First let us start with \fxed values. For the Gilbert damping\nparameter we choose the bulk value for Co which was found to be \u000b'0:005 [43, 44]. A\nsimilar value ( \u000b'0:0044) was measured for a FE 4=V4multilayer sample in perpendicular\ncon\fguration where only intrinsic Gilbert damping is operative [16].\nThe anisotropy \feld Kis estimated as follows. Since the exchange interaction is typically\nabout 104times larger than relativistic interactions which are responsible for anisotropy\n[45] and the magnetic exchange \feld can adopt large values we estimate the anisotropy\nasK= 0:1 T. Since we are interested in small excitations transverse to the anisotropy\naxis\u0017we suppose \u0016= 0:9 for the time independent part of the magnetization pointing in\nthe direction of the anisotropy axis \u0017, compare Eq. (4). Moreover, the gyromagnetic ratio\n\r'1:76\u00021011(Ts)\u00001. The characteristic magnetic length de\fned in Eq. (6) is of the order\nl0'10\u00008m. For the calculations a static magnetic \feld of about 0 :5 T is taken into account\nwhich corresponds to the scaled external \feld \"= 5. Notice that the dispersion relation \nin Eq. (22) is q-dependent. In the following we assume a medial value q= 106m\u00001. The\nparameters which are altered in the upcoming analysis are the noise correlation strength D\nand the retardation strength \u0000 0. We investigate our model for both values ranging in the\ninterval [0,10]. After this estimation the two parameters, the noise correlation time \u001ccand\nthe retardation length \u0018are left over. For a comprehensive estimation we suggest that \u0018is\nranged in 10\u000012m<\u0018 < 10\u00006m. The lower limit is smaller than a typical lattice constant\na'10\u000010m where the upper limit is a few orders of magnitude larger than the lattice\nconstant. Likewise the correlation time \u001cccaptures a quite large interval. Remark that\nwe keep the notation \u001cc, especially with regard to Fig. 3, although we also designated the\ndimensionless correlation time as \u001cc. In order to cover a wide range the time interval is chosen\nin between atto- and nanoseconds. The results are depicted in Fig. 3 and Fig. 4. In Fig. 3 the\nbehavior of the FMR-linewidth \u0001 Bas well as the lifetime of the spin waves \u001cL, introduced\n13(a)\n(c)(b)\n(d)FIG. 3. The FMR-linewidth and the lifetime depending on: (a) the noise correlation strength Dfor\n\u001cc= 568 as, \u0000 0= 1,\u0018= 10\u00008m; (b) the noise correlation time \u001ccforD= 1, \u0000 0= 1,\u0018= 10\u00008m;\n(c) the retardation strength \u0000 0for\u001cc= 568 as,D= 1,\u0018= 10\u00008m; (d) the retardation length \u0018for\n\u001cc= 568 as, \u0000 0= 1, \u0000 0. The other parameters take l0= 10\u00008m,q= 106m\u00001,\"= 5,\u0016= 0:9 and\n\u000b= 0:005.\nin Eq. (23), are shown in dependence on the di\u000berent model parameters explained above.\nThe in\ruence of the correlation noise strength on \u0001 Band\u001cLis shown in Fig. 3(a). Whereas\nthe linewidth decreases only very weak linearly when the noise strength Dis increased,\nthe lifetime of the spin waves \u001cLreveals a strong dependency on D. This is indicated by\nthe fact that \u001cLis monotonic decaying while it covers several orders of magnitude with\ngrowing noise strength D. The curve shape for the lifetime \u001cLseems to be comprehensible\nbecause the stronger the stochastic forces are correlated and interact mutually the faster the\ncoherent motion of the spin moments is destroyed. This microscopic picture is reasonable\nunder the premise that the evolution of spin waves is based on the phase coherence between\nadjacent magnetic moments. Apparently the frequency and consequently the linewidth \u0001 B\nshow only a quite small e\u000bect, compare Eq. (23). Therefore, the variation of Dreveals\nno signi\fcant in\ruence on the frequency velocity of the moments. A distinct behavior is\ndepicted in Fig. 3(b) for \u0001 Band\u001cLas a function of the noise correlation time \u001cc. Both\nthe linewidth and the lifetime remain constant for large interval of the correlation time \u001cc,\n14roughly speaking for \u001ccranging from as to ps. If the correlation time is in between ps and ns\nthe linewidth \u0001 Bincreases about a factor of 20 and the lifetime \u001cLdecreases to a value about\n9-times smaller. Thus \u001cca\u000bects both \u0001 Band\u001cLin an opposite manner provided the noise-\nnoise correlations occur on time scales larger than ps. In this regime a growing correlation\ntime\u001ccimplicates likewise an enhancement of the resonance frequency of the spin waves\n!/\u0001B, see Eq. (23). Simultaneously the spin wave lifetime \u001cLdeclines strongly. Such a\nbehavior may be attributed to a 'stochastic acceleration' which on the one hand enhances\nthe frequency but on the other hand drives neighboring magnetic moments out of phase\ncoherence. Remark that for times \u001cc>1ns the linewidth \u0001 Btends to in\fnity. This e\u000bect is\nnot shown in the picture. Concerning the in\ruence of the retardation parameters we refer to\nFig. 3(c), which illustrates the in\ruence of the retardation strength \u0000 0. As recognizable the\nFMR-linewidth exhibits a seemingly linear dependence as function of \u0000 0while \u0001Bgrows\nwith increasing retardation strength. The lifetime \u001cLdecreases in a non-linear manner. The\ndecay covers a range of \u00193 orders of magnitude. We suggest the following mechanism\nbehind this e\u000bect: Let us consider two moments both localized at arbitrary positions within\nthe retardation length \u0018as schematically displayed in Fig. 2. The mutual coupling due to\nretardation between both characterized by \u0000 0leads to a phase shift between neighboring\nspins. Therefore, the phase coherence originated by the self-organized internal magnetic \feld\nis interfered in view of an interplay within the feedback coupling in coordinate space. The\nstronger this interaction \u0000 0is the faster is the damping of the spin waves. Accordingly, spin\nwave solutions for di\u000berent values of \u0000 0are plotted exemplary in Fig. 4. The retardation\nlength\u0018in\ruences \u0001 Band\u001cLas well as is visible in Fig. 3(d). Here the quantities \u0001 Band\n\u001cLremain constant for a retardation strength \u0018ranging within the pm regime and a few\ntenth\u0016m. For larger \u0018-values the linewidth \u0001 Bdecreases while the lifetime \u001cLincreases. In\nthe regime\u0018 >1\u0016m the linewidth \u0001 Btends to zero and the lifetime \u001cL!1 . This behavior\nis not depicted in Fig. 3(d). Notice that for reasonable values of \u0018which not exceed the\nsample size the mentioned situation is not realized. The shapes of the curves in Fig. 3(d)\nmay be explained as follows. This graph corresponds to a \fxed retardation strength \u0000 0\nwhile the retardation length \u0018is enlarged. Again we refer to the physical picture where\nthe internal \feld, originated by the mutual interaction of the moments, and the coupling\ndue to the retardation operate as opposite mechanisms. The interplay happens in such a\nmanner that an increasing retardation strength \u0000 0weakens or destroys the phase coherence\n15-0.6-0.4-0.20.00.20.40.60.81.0mean value\n0 10 20 30 40 50 60\ntime[ps]0= 2\n0= 3\n0= 5\n/angbracketleftϕ1/angbracketright\n/angbracketleftϕ0/angbracketrightFIG. 4. Evolution of spin waves for di\u000berent values of the retardation strength \u0000 0. The other\nparameters take l0= 10\u00008m,q= 106m\u00001,\"= 5,\u0016= 0:9 and\u000b= 0:005,\u001cc= 568 as,D= 0:5 and\n\u0018= 10\u00008m.\nbetween adjacent spins. Yet it is found that a growing retardation length \u0018counteracts the\ndamping of the spin waves. As a consequence we suppose that the more spins are involved\ninto the retardation e\u000bect, i.e. the larger the parameter \u0018becomes, the more the damping\nis reduced. In other words it seems that retardation e\u000bects can average out if su\u000eciently\nmany magnetic moments are involved.\nV. CONCLUSIONS\nIn the present paper we have studied a model on a mesoscopic scale realized by means\nof Landau-Lifshitz-Gilbert dynamics. The magnetization is driven by an e\u000bective magnetic\n\feld. This \feld consists of an internal \feld due to the exchange interaction, an anisotropy\n\feld and a static external \feld. Additionally, the e\u000bective \feld is supplemented by a time\n16depending random one obeying colored noise statistics. Moreover, the stochastic LLG is\ngeneralized by the introduction of a retardation kernel depending on the spatial coordinates\nonly. Such a kernel simulates a kind of non-local interaction of size \u0018. After deriving an\napproximated Fokker-Planck equation we were able to calculate the mean values of the\ncomponents of the magnetization in the linear spin wave approach. They depend strongly\non the parameters characterizing the retardation (strength \u0000 0, length\u0018) as well as the\nstochastic (strength D, correlation time \u001cc) processes. As a result of the analysis we found\nthat the increase of the retardation strength \u0000 0compared with the growth of the retardation\nlength\u0018can entail con\rictive e\u000bects on the lifetime \u001cL. The main results are depicted in\nFig. 3. There, in addition to the lifetime of the spin waves \u001cLthe FMR-linewidth \u0001 Bis\ndisplayed. In doing so we want to provide comparability to experimental investigations\nbased on ferromagnetic resonance for cases when the LLG is applicable. Let us remark\nthat also other mechanisms are able to contribute to the damping process. 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Suess1\nUniversity of Vienna, Physics of Functional Materials, Boltzmanngasse 5, 1090 Vienna,\nAustria\n(Dated: 12 July 2019)\nWe optimize the recording medium for heat-assisted magnetic recording by using a high/low\nTcbilayer structure to reduce AC and DC noise. Compared to a former work, small Gilbert\ndamping\u000b= 0:02 is considered for the FePt like hard magnetic material. Atomistic simu-\nlations are performed for a cylindrical recording grain with diameter d= 5 nm and height\nh= 8 nm. Di\u000berent soft magnetic material compositions are tested and the amount of hard\nand soft magnetic material is optimized. The results show that for a soft magnetic material\nwith\u000bSM= 0:1 andJij;SM= 7:72\u000210\u000021J/link a composition with 50% hard and 50% soft\nmagnetic material leads to the best results. Additionally, we analyse how much the areal\ndensity can be improved by using the optimized bilayer structure compared to the pure hard\nmagnetic recording material. It turns out that the optimized bilayer design allows an areal\ndensity that is 1 Tb/in2higher than that of the pure hard magnetic material while obtaining\nthe same SNR.\nI. INTRODUCTION\nHeat-assisted magnetic recording (HAMR) [1{7] is a\npromising recording technology to further increase the\nareal storage densities (ADs) of hard disk drives. Con-\nventional state-of-the-art recording technologies are not\nable to overcome the so-called recording trilemma [8]:\nHigher ADs require smaller grains. These grains need\nto have high uniaxial anisotropy to be thermally sta-\nble. However today's write heads are not able to pro-\nduce \felds that are strong enough to switch these high\nanisotropy grains. In the HAMR process a heat pulse\nis included in the recording process to locally heat the\nrecording medium. This leads to a drop of the coercivity,\nmaking the high anisotropy recording medium writeable.\nThe medium is then quickly cooled and the information\nreliably stored.\nTo reach high linear densities it is necessary to reduce\nAC and DC noise in recording media [9]. AC noise de-\ntermines the distance between neighboring bits in bit-\npatterned [10{12] media or the transition between grains\nin granular media. DC noise restricts the maximum\nswitching probability of grains away from the transition.\nIt has been shown, that pure hard magnetic grains do\nnot switch reliably [13] if bit-patterned media are con-\nsidered whereas non-optimized exchange coupled bilayer\nstructures [14{19] of hard and soft magnetic material ex-\nperience high AC noise [20]. A work to reduce noise\nin recording media by optimizing a high/low Tcbilayer\nstructure (see Ref. [21]) showed that an optimial bilayer\nstructure consists of 80% hard magnetic and 20% soft\nmagnetic material. However, in the former work the\nGilbert damping was assumed to be \u000bHM= 0:1 which\nis hard to achieve in a FePt like hard magnetic material\nin reality. In realistic hard magnetic recording materi-\nals, the damping constant is \u000b= 0:02, according to the\nAdvanced Storage Technology Consortium (ASTC) [22].\na)Electronic mail: olivia.muthsam@univie.ac.atSince it has been shown that the damping constant has\na strong in\ruence on the maximum switching probabil-\nity and the down-track jitter, we follow the optimization\napproach and optimize a bilayer structure for the ASTC\nparameters. After the optimization, we study how the\noptimized material di\u000bers from that with \u000bHM= 0:1.\nAdditionally, we investigate how much the areal storage\ndensity (AD) can be improved when using the optimized\nrecording material instead of the pure hard magnetic one.\nThis is done with the help of the signal-to-noise ratio\n(SNR), which gives the power of the signal over the power\nof the noise and is a good indicator for the quality of writ-\nten bits.\nThe structure of this work is as follows: In Section II,\nthe HAMR model and the material parameters are pre-\nsented. In Section III, the results are shown and they are\ndiscussed in Section IV.\nII. HAMR MODEL\nThe optimization simulations are performed with the\natomistic simulation program VAMPIRE [23] which\nsolves the stochastic Landau-Lifshitz-Gilbert (LLG)\nequation. In the simulations, a cylindrical recording\ngrain with a diameter d= 5 nm and a height h= 8 nm\nis used. It can be considered as one recording bit in\nbit-patterned media. A simple cubic crystal structure is\nused and only nearest neighbor interactions are consid-\nered. The e\u000bective lattice parameter aand the exchange\ninteraxtion Jijare adjusted in order to lead to the exper-\nimentally obtained saturation magnetization and Curie\ntemperature. [24; 25]. The write head is assumed to\nmove with a velocity of v= 15 m/s. A continuous laser\npulse is assumed with the Gaussian temperature pro\fle\nT(x;y;t ) = (Twrite\u0000Tmin)e\u0000x2+y2\n2\u001b2+Tmin (1)\n=Tpeak(y)\u0001e\u0000x2\n2\u001b2+Tmin (2)arXiv:1907.05027v1  [physics.app-ph]  11 Jul 20192\nwith\n\u001b=FWHMp\n8 ln(2): (3)\nThe full width at half maximum (FWHM) is assumed\nto be 60 nm. Both, the down-track position xand\nthe o\u000b-track position yare variable in the simulations.\nThe initial and \fnal temperature is Tmin= 300 K. The\napplied \feld is modeled as a trapezoidal \feld with\na write \feld duration of 0.57 ns and a \feld rise and\ndecay time of 0.1 ns. The \feld is applied at an angle of\n22 deg with respect to the normal. The \feld strength is\nassumed to be +0.8 T and -0.8 T in z-direction. Initially,\nthe magnetization of each grain points in + z-direction.\nThe trapezoidal \feld tries to switch the magnetization\nof the grain from + z-direction to\u0000z-direction. At the\nend of every simulation, it is evaluated if the bit has\nswitched or not.\nA. Material parameters\nThe material parameters for the hard magnetic\nmaterial can be seen in Table I. For the soft magnetic\nmaterial, the atomistic spin moment is assumed to be\n\u0016s= 1:6\u0016Bwhich corresponds to a saturation polariza-\ntionJs= 1:35 T. The uniaxial anisotropy constant ku;SM\nin the soft magnetic layer is initially set to 0 but later\nvaried. The Gilbert damping \u000bSMand the exchange\ninteraction Jij;SM within the soft magnetic material are\nvaried. Experimentally, it is possible to increase the\ndamping constant by doping the soft magnetic material\nwith transition metals like Gd or Os [26{30]. Thus, also\nenhanced damping constants \u000bSMlarger than 0 :02 are\nconsidered in the simulations.\nIII. RESULTS\nA. Hard magentic grain\nFirst, a switching probability phase diagram for the\npure hard magnetic material is computed where the\nswitching probability is depending on the down-track po-\nsitionxand the o\u000b-track position y. With eq. (2) each\no\u000b-track position ycan be transformed into an unique\npeak temperature Tpeak, if the write temperature Twriteis\n\fxed, and vice versa. Thus, the switching probability in\nFigure 1 is shown as a function of the down-track position\nxand the peak temperature Tpeakthat corresponds to y.\nThe resolution of the phase diagram in down-track direc-\ntion is \u0001x= 1:5 nm and that in temperature direction\nis \u0001Tpeak= 25 K. In each phase point, 128 trajectories\nare simulated with a simulation length of 1 :5 ns. Thus,\nthe phase diagram contains more than 30.000 switching\ntrajectories. From the phase diagram it can be seen that\nFIG. 1. Switching probability phase diagram of a pure FePt\nlike hard magnetic grain. The contour lines indicate the\ntransition between areas with switching probability less than\n1% (red) and areas with switching probability higher than\n99.2% (blue). The dashed lines mark the switching probabil-\nity curves of Figure 2.\nthe pure hard magnetic grain shows only two small ar-\neas with switching probability larger than 99 :2%. This\nthreshold is used, since 128 simulations per phase point\nare performed and a switching probability of 100% corre-\nsponds to a number of successfully switched trajectories\nlarger than 1\u00001=128 = 0:992.\nTo determine the down-track jitter \u001b, a down-track\nswitching probability curve P(x) for\u000020 nm\u0014x\u00146 nm\nat a \fxed temperature Tpeak= 760 K is determined for\npure hard magnetic material (see Figure 2). The switch-\ning probability curve is \ftted with a Gaussian cumulative\nfunction\n\b\u0016;\u001b2=1\n2(1 + erf(x\u0000\u0016p\n2\u001b2))\u0001P (4)\nwith\nerf(x) =2p\u0019Zx\n0e\u0000\u001c2d\u001c; (5)\nwhere the standard deviation \u001b, the mean value \u0016and\nthe mean maximum switching probability P2[0;1] are\nthe \ftting parameters. The standard deviation \u001bdeter-\nmines the steepness of the transition function and is a\nmeasure for the transition jitter. In the further course\nit will be called \u001bdown:The \ftting parameter Pis a\nmeasure for the average switching probability for su\u000e-\nciently high temperatures. The resulting \ftting parame-\nters of the hard magnetic material can be seen in Table V.\nNote, that the calculated jitter values only consider the\ndown-track contribution of the write jitter. The so-called\na\u0000parameter is given by3\nCurie temp. TC[K] Damping\u000bUniaxial anisotropy. ku\n[J/link]Jij[J/link] \u0016s[\u0016B]\n693.5 0.02 9:124\u000210\u0000236:72\u000210\u0000211.6\nTABLE I. Material parameters of a FePt like hard magnetic granular recording medium.\n−15 −10 −5 000.20.40.60.81\ndown-track x[nm]switching probability\nHM\nFIG. 2. Down-track switching probability curve P(x) at a\npeak temperature Tpeak = 760 K for a pure hard magnetic\ngrain.\na=q\n\u001b2\ndown+\u001b2g (6)\nwhere\u001bgis a grain-size-dependent jitter contribution\n[31]. The write jitter can then be calculated by\n\u001bwrite\u0019ar\nS\nW(7)\nwhereWis the reader width and S=D+Bis the\ngrain diameter, i.e. the sum of the particle size Dand\nthe nonmagnetic boundary B[32; 33].\nB. Media Optimization\nTo \fnd the best soft magnetic material composition,\ndown-track switching probability curves P(x) similar to\nFigure 2 are computed for 50/50 bilayer structures with\ndi\u000berent damping constants \u000bSMand di\u000berent exchange\ninteractions Jij;SM. The range in which the parameters\nare varied can be seen in Table II. Note, that P(x) is\ncomputed at di\u000berent peak temperatures for the di\u000berent\nexchange interactions, since there holds\nJij=3kBTC\n\u000fz; (8)\nwherekBis the Boltzmann constant, z is the number\nof nearest neighbors and \u000fis a correction factor from the\nmean-\feld expression which is approximately 0.86 [23].\nThe temperature at which P(x) is calculated is chosen\nto beTC+ 60 K. The down-track switching probability\nFIG. 3. Down-track jitter \u001bdown as a function of the damp-\ning constant and the exchange interaction. The contour line\nindicates the transition between areas with down-track jitter\nlarger than 0.5 nm (light red, blue) and areas with down-track\njitter smaller than 0.5 nm (dark red).\ncurves are then \ftted with eq. (4). The down-track jitter\nparameters as a function of the damping constant and\nthe exchange interaction can be see in Figure 3. The\nmaximum switching probability is 1 for \u000b\u00150:1.\nFrom the simulations it can be seen that a Gilbert\ndamping\u000bSM= 0:1 together with Jij;SM= 7:72\u0002\n10\u000021J/link leads to the best results with the smallest\ndown-track jitter \u001bdown = 0:41 nm and a switching proa-\nbilityP= 1.\nThe last soft magnetic parameter that is varied, is the\nuniaxial anisotropy ku;SM. It is known that the small-\nest coercive \feld in an exchange spring medium can be\nachieved if KSM= 1=5KHM[34; 35]. Here\nKi=natku;i\na3i2fSM;HMg (9)\nare the macroscopic anisotropy constants in J/m3\nwith the unit cell size a= 0:24 nm and the number of\natomsnatper unit cell. ku;SMis varied between 0 and\n1=2ku;HM= 4:562\u000210\u000023J/link. The damping constant4\nParameter min. value max.value\n\u000bSM 0.02 0.5\nJij;SM[J/link] 5:72\u000210\u0000219:72\u000210\u000021\nku;SM[J/link] 0 1=2ku;HM= 4:562\u000210\u000023\nTABLE II. Range in which the di\u000berent soft magnetic material parameters are varied.\nku;SM\u000210\u000023[J/link]\u001bdown[nm]P\n0 0.41 1.0\n0:562 0.919 1.0\n1:8428 [= 1=5ku;HM] 1.04 1.0\n3:124 0.898 1.0\n4:562 [= 1=2ku;HM] 1.01 1.0\nTABLE III. Resulting down-track jitter parameters and mean maximum switching probability values for soft magnetic materials\nwith di\u000berent uniaxial anisotropy constants ku;SM.\nis\u000bSM= 0:1. The resulting \ftting parameters are sum-\nmarized in Table III. It can be seen that the switching\nprobability is one for all varied ku;SM. However, the\ndown-track jitter increases for higher ku;SM. Since for\nku;SM= 0 J/link the jitter is the smallest, this value is\nchosen for the optimal material composition.\nIn conclusion, the material parameters of the optimized\nsoft magnetic material composition can be seen in Ta-\nble IV.\nNext, simulations for di\u000berent ratios of hard and soft\nmagnetic material are performed. Down-track switching\nprobability curves P(x) are computed for di\u000berent ratios\natTpeak= 780 K and the down-track jitter and the mean\nmaximum switching probability are determined. The re-\nsults are listed in Table V.\nIt can be seen that a structure with 50% hard magnetic\nand 50% soft magnetic materials leads to the smallest\njitter and the highest switching probability. This result\ndi\u000bers from the optimized material composition in Ref.\n[21], where the optimal composition consists of 80% hard\nmagnetic and 20% soft magnetic materials. In Figure 4,\na switching probability phase diagram of the optimized\nbilayer structure with 50% hard and 50% soft magnetic\nmaterial can be seen.\nIt is visible that the switching probability of the\nstructure is larger than 99 :2% for a bigger area of down-\ntrack positions and peak temperatures. This shows the\nreduction of DC noise in the optimized structure.\nC. Areal Density\nTo analyse the possible increase of areal density by us-\ning the optimized bilayer structure instead of the pure\nhard magnetic recording medium, the signal-to-noise ra-\ntio is calculated. With the help of an analytical model of\na phase diagram developed by Slanovc et al [33] it is pos-\nsible to calculate a switching probability phase diagram\nfrom eight input parameters. The input parameters are\nPmax,\u001bdown;the o\u000b-track jitter \u001bo\u000b;the transition cur-\nvature, the bit length, the half maximum temperature\nand the position of the phase diagram in Tpeak direc-\ntion and the position of the phase diagram in down-track\ndirection. The \u001bdown andPmaxvalues are those result-\nFIG. 4. Switching probability phase diagram of recording\ngrain consisting of a composition of 50% hard magnetic ma-\nterial and 50% soft magnetic material with ku;SM= 0 J/link\nandJij;SM= 7:72\u000210\u000021J/link. The contour lines indicate\nthe transition between areas with switching probability less\nthan 1% (red) and areas with switching probability higher\nthan 99.2% (blue).\ning from the simulations for pure hard magnetic material\nand the optimized bilayer structure. All other model in-\nput parameters are obtained by a least square \ft from\na switching probability phase diagram computed with\na coarse-grained LLB model [36]. The phase diagram\nis mapped onto a granular recording medium where the\nswitching probability of the grain corresponds to its po-\nsition. The writing process is repeated for 50 di\u000berent\nrandomly initialized granular media. The SNR is then\ncomputed from the read-back process with the help of a\nSNR calculator provided by SEAGATE [37].\nThe SNR is analysed for areal densities of 2 to 5 Tb/in2.\nFor the bitsize ( bs) at a certain areal density, there are\ndi\u000berent track width and bit length combinations ( t;b)5\nDamping\u000bSMUniaxial anisotropy. ku\n[J/link]Jij[J/link] \u0016s[\u0016B]\n0.1 0 7:72\u000210\u0000211.6\nTABLE IV. Resulting material parameters for the optimal soft magnetic material composition.\nHM/SM\u001bdown[nm]P\nHM 0.974 0.95\n90/10 1.06 0.969\n80/20 0.813 0.998\n70/30 0.6 0.988\n60/40 0.8 0.999\n50/50 0.41 1.0\nTABLE V. Resulting down-track jitter parameters and mean maximum switching probability values for hard magnetic material\nand three di\u000berent hard/soft bilayer structures with di\u000berent damping constants in the soft magnetic material.\nthat yield\nbs=t\u0001b: (10)\nTo compute the SNR for a certain ( t;b) combination,\nthe reader was scaled in both the down-track and the o\u000b-\ntrack direction according to the bit length and the track\nwidth, respectively. The reader resolution Rin down-\ntrack direction is scaled by\nR=R0\u0001b\nb0(11)\nwherebis the bit length, R0= 13:26 nm is the initial\nreader resolution and b0= 10:2 nm denotes the mean ini-\ntial bit length according to ASTC. In o\u000b-track direction,\nthe reader width is scaled to the respective track width\nt. The initial track width is 44 :34 nm. In Figure 5(a) and\n(b) the SNR is shown as a function of the bit length and\nthe track width for pure hard magnetic material and the\noptimized bilayer structure, respectively. Additionally,\nthe phase plots include the SNR curves for ( t;b) combina-\ntions that yield areal densities from 2 to 5 Tb/in2. From\nthe phase diagram it is visible that higher SNR values can\nbe achieved for the optimized structures than for the pure\nhard magnetic material in the same bit length \u0000track\nwidth range. For example, the SNR for an areal density\nof 2 Tb/in2for the bilayer structure is larger than 15 dB\nwhereas it is between 10 dB and 15 dB for pure hard mag-\nnetic material. For each AD there is a ( t;b) combination\nfor which the SNR is maximal and which is marked by\na dot in the phase plot. In Figure 6 the maximum SNR\nover the areal density is displayed for both structures.\nThe results show that the SNR that can be achieved with\nthe optimized structure is around 2 db higher than that\nof the hard magnetic material, if the same areal density\nis assumed. To get the same SNR, the optimized design\nallows for an areal density that is 1 Tb/in2higher than\nfor the hard magnetic one. Summarizing, the bit length\n\u0000track width combinations at which the maximum SNR\nis achieved are given in Table VI.\nFIG. 5. Signal-to-noise ratio (in dB) as a function of the bit\nlength and the track width for (a) pure hard magnetic ma-\nterial and (b) the optimized hard/soft bilayer structure. The\nred lines indicate the bit length \u0000track width combinations\nthat yield 2, 3, 4 and 5 Tb/in2areal density. The dots indicate\nthe combination at which the SNR is maximal.\nIV. CONCLUSION\nTo conclude, we optimized a recording medium with\nhigh/lowTCgrains for heat-assisted magnetic record-\ning with a low Gilbert damping in the hard magnetic\npart\u000bHM= 0:02. The simulations for a cylindrical\nrecording grain with d= 5 nm and h= 8 nm were6\nAD [Tb/in2]Max. SNR [dB] (HM) x[nm] (HM) y[nm] (HM) Max. SNR [dB] (HM/SM) x[nm] (HM/SM) y[nm] (HM/SM)\n2 13.85 10.0 32.26 16.08 8.06 40.02\n3 11.07 6.23 34.52 13.37 5.37 37.53\n4 9.46 5.0 32.26 11.55 5.0 32.26\n5 7.16 4.3 30.01 9.16 4.69 27.51\nTABLE VI. Resulting bit length xand track width ycombinations for the maximum SNR at di\u000berent areal densities (AD) for\npure hard magnetic material (HM) and the optimized bilayer structure (HM/SM).\nFIG. 6. Maximum SNR for di\u000berent areal densities for pure\nhard magnetic material and the optimized bilayer structure.\nperformed with the atomistic simulation program VAM-\nPIRE. The damping constant of the soft magnetic mate-\nrial was assumed to be enhanced by doping the soft mag-\nnetic material with transition metals. The simulations\nshowed that larger damping constants lead to smaller jit-\nter and higher switching probabilities. A damping con-\nstant\u000bSM= 0:1, in combination with an exchange in-\nteractionJij;SM= 7:72\u000210\u000021J/link and an uniaxial\nanisotropy constant ku;SM= 0 J/link, led to the best re-\nsults in terms of small down-track jitter and high switch-\ning probability in a wide range of down-track and o\u000b-\ntrack positions. Interestingly, the soft magnetic com-\nposition is almost the same as for the structure with\n\u000bHM= 0:1 obtained in a previous work [21].\nIn further simulations the amount of hard and soft\nmagnetic material was varied. Surprisingly, the results\nshowed that a higher amount of soft magnetic material\nleads to smaller down-track jitter. This is not as expected\nsince for\u000bHM= 0:1 an increase of the soft magnetic ma-\nterial led to larger AC noise [21]. However, it can be\neasily explained why a higher amount of soft magnetic\nmaterial leads to better jitter results. Studying the in-\n\ruence of the damping constant on the down-track jitter\nshows that an increase of the damping constant from 0 :02\nto 0:1 reduces the down-track jitter by almost 30%. Ad-\nditionally,the maximum switching probability increases\nto 1. Since it can be seen that higher damping leads to\nsmaller jitter and higher maximum switching probability,\nit is reasonable that a higher amount of soft magnetic\nmaterial with \u000bSM= 0:1 leads to a better recording per-formance. In the former work the improved performance\ndue to higher damping was not an issue since the damp-\ning constant was 0.1 in both layers. This explains the\ndi\u000berent ratios of hard and soft magnetic material.\nFurthermore, we analyzed the increase of the areal den-\nsity can be improved if the optimized bilayer structure\nis used instead of pure hard magnetic recording mate-\nrial. This was done by analyzing the signal-to-noise ra-\ntio (SNR). The results showed that the areal density of\nthe optimized bilayer structure could be increased by\n1 Tb/in2to achieve the same SNR as for the pure hard\nmagnetic structure. In other words, that means that at\na certain areal density, the SNR was increased by 2 dB\nby using the optimized structure. Concluding, the opti-\nmized bilayer structure is a promising design to increase\nthe areal storage density by just modifying the recording\nmaterial.\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. The\ncomputational results presented have been achieved us-\ning the Vienna Scienti\fc Cluster (VSC).\n1Hiroshi Kobayashi, Motoharu Tanaka, Hajime Machida, Takashi\nYano, and Uee Myong Hwang. Thermomagnetic recording .\nGoogle Patents, August 1984.\n2C. Mee and G. Fan. 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IEEE Transactions on Magnet-\nics, 53(2):1{6, February 2017."    },    {        "title": "2303.12380v2.Twisted_bilayer_graphene_reveals_its_flat_bands_under_spin_pumping.pdf",        "content": "arXiv:2303.12380v2  [cond-mat.mes-hall]  1 Sep 2023Twisted bilayer graphene reveals its flat bands under spin pu mping\nSonia Haddad1,2,3,∗Takeo Kato2, Jihang Zhu3, and Lassaad Mandhour1\n1Laboratoire de Physique de la Mati` ere Condens´ ee, Facult´ e des Sciences de Tunis,\nUniversit´ e Tunis El Manar, Campus Universitaire 1060 Tuni s, Tunisia\n2Institute for Solid State Physics, University of Tokyo, Kas hiwa, Chiba 277-8581, Japan\n3Max Planck Institute for the Physics of Complex Systems,\nN¨ othnitzer Strasse 38, Dresden 01187, Germany\n(Dated: September 4, 2023)\nThe salient property of the electronic band structure of twi sted bilayer graphene (TBG), at the\nso-called magic angle (MA), is the emergence of flat bands aro und the charge neutralitypoint. These\nbands are associated with the observed superconducting pha ses and the correlated insulating states.\nScanning tunneling microscopy combined with angle resolve d photoemission spectroscopy are usu-\nally used to visualize the flatness of the band structure of TB G at the MA. Here, we theoretically\nargue that spin pumping (SP) provides a direct probe of the fla t bands of TBG and an accurate\ndetermination of the MA. We consider a junction separating a ferromagnetic insulator and a het-\nerostructure of TBG adjacent to a monolayer of a transition m etal dichalcogenide. We show that\nthe Gilbert damping of the ferromagnetic resonance experim ent, through this junction, depends on\nthe twist angle of TBG, and exhibits a sharp drop at the MA. We d iscuss the experimental realiza-\ntion of our results which open the way to a twist switchable sp intronics in twisted van der Waals\nheterostructures.\nIntroduction. – Stacking two graphene layers with\na relative twist angle θresults in a moir´ e superstruc-\nture which is found to host, in the vicinity of the\nso-called magic angle (MA) θM∼1.1◦, unconven-\ntional superconductivity and strongly correlated insulat-\ning states1–3. There is a general consensus that such\nstrong electronic correlations originate from the moir´ e\nflat bands emerging at the MA around the charge neu-\ntrality point4–11. The tantalizing signature of the flat\nbands have been experimentally demonstrated by prob-\ning the corresponding peaks of the density of states us-\ning transport1–3,12,13, electroniccompressibilitymeasure-\nments14,15, scanning tunneling microscopy (STM) and\nspectroscopy (STS)16–23. The direct evidence of these\nflat bands has been reported by angle resolved photoe-\nmission spectroscopy (ARPES) measurements combined\nto different imaging techniques24–26. However, spectro-\nscopic measurements on magic-angle TBG raise many\ntechnical challenges related to the need of an accurate\ncontrol of the twist angle, and the necessity to have non-\nencapsulated samples which can degrade in air25.\nHere we propose a noninvasive method to probe the\nflat bands of TBG and accurately determine the MA.\nThis method is based on spin pumping (SP) induced by\nferromagnetic resonance (FMR)27–30, where the increase\nin the FMR linewidth, given by the Gilbert damping\n(GD) coefficient, provides insight into the spin excita-\ntions of the nonmagnetic (NM) material adjacent to the\nferromagnet31–33. SP is expected to be efficient if the\nNM has high spin-orbit coupling (SOC) strength34.\nIn our work, we consider spin injection from a ferromag-\nnetic insulator(FI) into aTBG alignedonamonolayerof\ntransition metal dichalcogenides (TMD) which are con-\nsidered as good substrate candidates to induce relatively\nstrong SOC in graphene and TBG35–58.\nWe theoretically study a planar junction of a FI and\na TBG adjacent to WSe 2(TBG/WSe 2) as depicted inFig.1. We consider the case where a microwave of a\nfrequency Ω is applied to this junction, and focus on the\ntwist angle dependence of the FMR linewidth59.\nFigure 1. Schematic representation of the junction between\na ferromagnetic insulator (FI) and a heterostructure of TBG\nadjacent to a monolayer of WSe 2. The labels (1) and (2) de-\nnote the graphene layers of TBG represented by the red and\nthe blue lines. The red arrow indicates the spin orientation\nof the FI characterized by an average spin /angbracketleftSFI/angbracketright= (S0,0,0),\nwritten in the coordinate frame of the FI magnetization. The\ngray lines represent the boron-nitride (hBN) layers encaps u-\nlating the TBG/WSe 2heterostructure.\nContinuum model. – In TBG with a twist angle θ, the\nHamiltonian hl(k) ofagraphenelayer l(l= 1,2), rotated\nat an angle θl, ishl(k) =eiθl\n2σzh(0)\nl(k)e−iθl\n2σz, where\nθ2=−θ1=θ\n2andh(0)\nl(k) is the unrotated monolayer\nHamiltonian. In the continuum limit, h(0)\n1(k) reduces to\nh(0)\n1(k) =−/planckover2pi1vFk·σ∗, wherevFis the Fermi velocity,\nσ∗= (ξσx,σy), andσi(i=x,y,z) are the sublattice-\nPauli matrices and ξis the valley index. We assume that\nthe SOC is only induced in the graphene layer adjacent\nto the TMD layer, since the SOC arises from overlaps\nbetween atomic orbitals50. This assumption is consistent\nwith recent studies on bilayer graphene and TBG aligned\non TMD layers50,56,60,61. Layer (2), in contact with the\nWSe2monolayer, is then descried by the Hamiltonian2\nh(0)\n2(k) =h(0)\n1(k) +hSOC+m\n2σz56, wherehSOCis given\nby\nhSOC=λI\n2ξsz+λR\n2(ξσxsy−σysx)+λKM\n2ξσzsz,(1)\nsi(i=x,y,z) are the spin-Pauli matrices, λI,λRand\nλKMcorrespond, respectively, to the Ising, Rashba and\nKane-Mele SOC parameters56. The variation ranges of\nthese parameters are λI∼1−5 meV,λR∼1−15 meV,\nwhileλKMis expected to be small37,44,50,51,62–64. The\nlast term in h(0)\n2(k) is due to the inversion symmetry\nbreaking induced by the TMD layer. Hereafter, we ne-\nglect this term regarding the small value of mcompared\nto the SOC parameters56.\nAs in the case of TBG65, the low-energy Hamiltonian of\nTBG/WSe 2reduces, at the valley ξ, to\nHξ,SOC(k) =\nh1(k)T1T2T3\nT†\n1h2,1(k) 0 0\nT†\n20h2,2(k) 0\nT†\n30 0 h2,3(k)\n.(2)\nHξ,SOC(k) is written in the basis Ψ =\n(ψ0(k),ψ1(k),ψ2(k),ψ3(k)) constructed on the four-component spin-sublattice spinor ψ0(k) andψj(k),\n(j= 1,2,3) corresponding, respectively, to layer (1)\nand layer (2) (see Secs. I and II of the Supplemental\nMaterial66and Refs.8,56,65,67–70). The momentum\nkis measured relatively to the Dirac point K1ξof\nlayer (1). In Eq. ( 2),Tjare the spin-independent\ninterlayer coupling matrices, h2,j(k) =h2(k+qjξ),\n(j= 1,2,3) where qjξare the vectors connecting\nK1ξto its three neighboring Dirac points K2ξof\nlayer (2) in the moir´ e Brillouin zone (mBZ)65, and\nare given by q1ξ=K1ξ−K2ξ,q2ξ=q1ξ+ξGM\n1,\nq3ξ=q1ξ+ξ/parenleftbig\nGM\n1+GM\n2/parenrightbig\n, where/parenleftbig\nGM\n1,GM\n2/parenrightbig\nis the\nmBZ basis (see Sec. I of the Supplemental Material66).\nIn the unrelaxed TBG, and choosing sublattice A as the\norigin of the unit cell in each layer, the Tjmatrices take\nthe form8T1=w(Iσ+σx),T2=w/parenleftig\nIσ−1\n2σx+ξ√\n3\n2σy/parenrightig\nandT3=w/parenleftig\nIσ−1\n2σx−ξ√\n3\n2σy/parenrightig\n66, where\nw∼110meV71is the interlayer tunneling ampli-\ntude and Iσis the identity matrix acting on the\nsublattice indices.\nUsing the perturbative approach of Ref.65, we de-\nrive, from Eq. ( 2), the effective low-energy Hamiltonian\nH(1)\nξ,SOC(k) of TBG/WSe 2(see Sec. II of the Supplemen-\ntal Material66). To the leading order in k,H(1)\nξ,SOC(k)\nreads as66\nH(1)\nξ,SOC(k) =/an}bracketle{tΨ|Hξ,SOC|Ψ/an}bracketri}ht\n/an}bracketle{tΨ|Ψ/an}bracketri}ht=ψ†\n0/bracketleftbig\nheff(k)+hSOC\neff/bracketrightbig\nψ0, (3)\nheff(k) =−/planckover2pi1vF\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/braceleftbigg\nkx/bracketleftbigg/parenleftbig\n1−3α2/parenrightbig\nξσxIs−3α2\n/planckover2pi1vFq0(ξλIσysz+λR(ξσysy−σxsx))/bracketrightbigg\n+ky/bracketleftbigg/parenleftbig\n1−3α2/parenrightbig\nσyIs−3α2\n/planckover2pi1vFq0(−λIσxsz+λR(σxsy+ξσysx))/bracketrightbigg/bracerightbigg\n, (4)\nhSOC\neff=3α2\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/bracketleftbigg\nξλIszIσ+λR\n2(sxσy−ξsyσx)/bracketrightbigg\n, (5)\nwhere/an}bracketle{tΨ|Ψ/an}bracketri}ht ∼1+6α2,α=w\n/planckover2pi1vFq0,q0=|qjξ|=4π\n3aθ,a\nis the graphene lattice constant and σi, (i=x,y,z) act\nnow on the band indices σ=±of the eigenenergies of\nH(1)\nξ,SOC, denotedEσ,±, and given to the leading orders in\nkandλI,R\n/planckover2pi1vFq0by\nE(k)σ,±=σ\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/radicalig\nf1(k)±6α2/radicalbig\nf2(k) (6)\nf1(k) = (/planckover2pi1vF)2/parenleftbig\n1−3α2/parenrightbig2||k||2+9\n2α4/parenleftbig\n2λ2\nI+λ2\nR/parenrightbig\nf2(k) = (/planckover2pi1vF)2/parenleftbig\n1−3α2/parenrightbig2||k||2/parenleftbigg\nλ2\nI+1\n4λ2\nR/parenrightbigg\n+9\n16α4λ4\nR.\nEquation 5shows that the SOC parameters λIandλRare renormalized by the moir´ e structure of TBG to\n˜λI∼6α2\n1+6α2λI,˜λR∼3α2\n1+6α2λR, (7)\nwhich increase by decreasing the twist angle.\nThe expression of H(1)\nξ,SOC[Eq. (3)] can be taken as a\nstarting point to unveil the role of SOC in the emer-\ngence of the stable superconducting phase observed, at\nθ∼0.8◦, in TBG adjacent to WSe 256.\nTo probe the validity of the effective Hamiltonian H(1)\nξ,SOC\n[Eq. (3)], we compared the corresponding eigenenergies\nwith the numerical band structure obtained within the\ncontinuum model and taking into account 148 bands per\nvalley and spin projection (see Sec. II of the Supplemen-\ntal Material66). The results show that H(1)\nξ,SOCdescribes3\ncorrectly the band structure of TBG/WSe 2down to a\ntwist angle θ∼0.7◦. At smaller angles, the effective\nFermi velocities of H(1)\nξ,SOCare overestimated. Such a\ndiscrepancy is expected since the lattice relaxation effect\nis important at small angles56. It is worth noting that,\nfor the sake of simplicity, we did not consider a relaxed\nTBG, since we are interested in the SP around the MA.\nGilbert damping. – In the absence of a junction,\nthe magnon Green function of the FI is defined as72–78\nG0(qm,iωn) =2S0//planckover2pi1\niωn−ωqm−αG|ωn|, whereωn= 2πn//planckover2pi1β\nare the Matsubara frequencies for bosons, S0is the am-\nplitude of the average spin per site, and αGis the GD\nstrength. The term −αG|ωn|describes the spin relax-\nation within the FI. In FMR experiments, the microwave\nexcitationinducesauniformspinprecession,whichlimits\nthe magnon self-energy to the processes with qm= 079.\nIn the presence of the interfacial coupling, a cor-\nrection,δαG(ω), to the GD term is induced by the\nadjacent heterostructure TBG/WSe 2.δαG(ω) can be\nexpressed in terms of the the self-energy ΣR\n0(ω)≡\nΣqm=0(iωn→ω+iδ), resulting from the interfacial ex-\nchange interactions, as79\nδαG(ω)≡ −2S0\n/planckover2pi1ωImΣR\n0(ω). (8)\nFor simplicity, we neglect the real part of ΣR\n0(ω) which\nsimply shifts the FMR line and did not affect the\nlinewidth, in which we are interested. The self-energy,\nin Eq. (8), includes the contributions of all the inter-\nfacial spin transfer processes and can be written as\nΣ0(iωn) =/summationtext\nqΣ0(q,iωn). Each process, described by\nthe self-energy Σ 0(q,iωn), is characterized by a momen-\ntum transfer qand a matrix element Tq,qm=0≡Tq,0.\nIn the second order perturbation, with respect to\nthe interfacial exchange interaction Tq,0, the self-energy\nΣ0(q,iωn), is written as78\nΣ0(q,iωn) =|Tq,0|2\n4β/summationdisplay\nk,iωmTr/bracketleftig\nσx′,−\nsˆg(k,ωm)\n×σx′,+\nsˆg(k+q,iωm+iωn)/bracketrightig\n.(9)\nσx′,±\nsare the electronic spin ladder operators written in\nthe coordinate system ( x′,y′,z′) of the FI magnetiza-\ntion characterized by an average spin /an}bracketle{tSFI/an}bracketri}ht= (S0,0,0).\nˆg(k,iωm) is the electronic Matsubara Green function\ngiven by ˆg(k,iωn) =/bracketleftig\niωnI−H(1)\nSOC(k)/bracketrightig−1\n, whereωn=\n(2n+ 1)π//planckover2pi1βare the fermionic Matsubara frequencies.\nIn the basis of the spin-band four-component spinor Ψ =\n(ψ+,↑,ψ+,↓,ψ−,↑,ψ−,↓)56, ˆg(k,iωn) reads as ˆg(k,iωn) =\nˆg0(k,iωn)Is+ˆ g(k,iωn)·s, where s= (sx,sy,sz) are\nthe spin-Pauli matrices; ˆ g= (ˆgx,ˆgy,ˆgz), ˆg0, and ˆgi\n(i=x,y,z) are expressed, to the leading order in the\nSOC, as a function of the band-Pauli matrices σi(see\nSec. III of the Supplemental Material66).Since the ferromagnetic peak, given by Im GR\n0, is sharp\nenough, namely αG+δαG≪1, one can replace the res-\nonance frequency ωqm=0by the FMR frequency Ω. The\nGD correction can then be expressed as66,78\nδαG(Ω) =−2S0\n/planckover2pi1ΩImΣR\n0(Ω). (10)\nIn general, the interfacialspin transferincludes cleanand\ndirty processes. The former (latter) take place with con-\nserved (non-conserved) electron momentum, which turns\nout to take q=0(q/ne}ationslash=0) in Eq. ( 979).\nWefirstconsideracleaninterface, forwhichananalytical\nexpression of the GD correction [Eq. 10] can be derived\n(see Sec. IV of the Supplemental Material66and refer-\nence78,79). The case of a dirty junction is discussed in\nthe next section.\nCarryingoutthe summationover ωminEq.(9), weob-\ntaintheanalyticalexpressionoftheinterfacialself-energy\n(see Sec. IV of the Supplemental Material66). The sum\nover the electronic states k= (k,ϕk) runs over the states\nincluded within a cutoff, kc∼q0/2, on the momentum\namplitudek, where the low-energy Hamiltonian [Eq. ( 3)]\nis expected to hold (see Sec. IV of the Supplemental\nMaterial66).\nIn the following, we discuss the behavior of the nor-\nmalized GD coefficient\nδαG/α0\nG=/parenleftbiggλ\n/planckover2pi1Ω/parenrightbigg2\n˜Σ(q=0,Ω), (11)\nwhere˜Σ is a dimensionless function depending on the\ntwist angle θ, temperature T, the chemical poten-\ntialµand the orientation of the FI magnetization,\nα0\nG= 2S0/parenleftig\n|T0|\nλ/parenrightig2\nandλ=λI+λR\n2is the average SOC\n(for details, see Sec. IV of the Supplemental Material66\nand reference80).\nDiscussion. – In Fig.2, we plotδαG/α0\nG[Eq. (11)],\nas a function of the twist angle θ, for the undoped\nTBG, at different temperatures and for a fixed FMR\nenergy/planckover2pi1Ω = 0.06meV which corresponds to the yttrium\niron garnet. The SOC parameters are λI= 3meV and\nλI= 4meV as in Ref. [ 56].\nFigure2shows that regardless of the temperature\nrange,δαGincreasesbydecreasing θbut dropssharplyat\nthe MA, where it exhibits a relatively small peak which\nis smeared out at low temperature.\nPutting aside its drop at the MA, the enhancement of\nδαG, by decreasing θ, can be, in a first step, ascribed to\nthe dependence of the self-energy [Eq. ( 9)] on the effec-\ntive SOC, given by Eq. ( 7), which increase by decreasing\nθ. However, to understand the behavior of δαGat the\nMA one needs to go back to the band structure, Eσ,±(k)\n[Eq. (6)], of the continuum Hamiltonian of TBG/WSe 2,\nwhich is depicted in Fig. 3at different twist angles.\nThe arrows indicate the out-of-plane electronic spin4\nFigure 2. Normalized GD, δαG/α0\nG[Eq. (11)], as a function of\nthe twist angle at different temperature ranges. Calculatio ns\nare done for λI= 3 meV, λR= 4 meV, µ= 0, and for a FMR\nenergy/planckover2pi1Ω = 0.06 meV.\nprojection /an}bracketle{tsz/an}bracketri}htwhich we have numerically calculated\nfor different twist angles in Sec. II of the Supplemental\nMaterial66.\nAway from the MA, the band dispersion gets larger\nasθdecreases and, in particular, the separation between\nbands with opposite /an}bracketle{tsz/an}bracketri}ht, involved in the SP process, in-\ncreases. This behavior is due to the angle dependence\nof the effective Fermi velocity v∗of TBG/WSe 2, which\nreduces, in the first order in the SOC, to that of TBG,\nnamely (see Secs. I and II of the supplemental Mate-\nrial66)\nv∗∼vF1−3α2\n1+6α2(12)\nThe expression of the GD [Eq. ( 11)] includes transitions\nbetweenbandswithopposite /an}bracketle{tsz/an}bracketri}ht(seeSec. IVoftheSup-\nplemental Material66). These transitions depend on the\nstatistical weight ∆ f(E) =f(E/angbracketleftsz/angbracketright)−f(E−/angbracketleftsz/angbracketright) where\nf(x) is the Fermi-Dirac function and E/angbracketleftsz/angbracketrightis the energy\nband with a spin orientation /an}bracketle{tsz/an}bracketri}ht.\nIn Fig.4, we plot a pictorial representationof the band\nstructureofthecontinuummodel[Eq.( 6)]andtheFermi-\nDirac distribution f(E) at a given temperature T. The\nband dispersiongets largeras θmovesawayfrom the MA\n(Fig.3) andthe separationbetween thebandswith oppo-\nsite/an}bracketle{tSz/an}bracketri}htincreases. As a consequence, the corresponding\nstatistical weight ∆ f(E) is enhanced compared to the\ncase around the MA. This behavior explains the drop of\nthe GD at the MA.\nAround the MA ( θ+\nMandθ−\nM), the statistical weight\n∆f(E) is reduced compared to that at the MA since the\nbandsE+,−andE−,−get closer (Fig. 3).\nThis behavior gives rise to the small peak at the MA\n(Fig.2), which disappearsat low temperature ( kBT <λ)\nwhere bands around the MA have the same statistical\nweight ∆f(E) = 1 (see Sec. IV of the Supplemental Ma-\nterial66). In this case, the GD is basically dependent on\nthe effective Fermi velocity v∗[Eq. (12)] which vanishesat exactly the MA. Such dependence is responsible for\nthe cancellation of several terms contributing to the self-\nenergy [Eq. ( 9)], as they are proportional to v∗[Eq. (12)]\n(see Sec. IV of the Supplemental Material66).\nLet us nowturn to thecaseofadirtyinterfacewherethe\nspin transfer should now also include the non-conserved\nmomentum processes. The corresponding self-energy\n[Eq. (9)] can also be expressed in terms of the thermal\nweight ∆f(E) governing the interband transitions (see\nSec. IV of the Supplemental Material66).\nRegarding the flatness of the bands, the dirty processes\nat the MA acquire, as in the clean limit, small thermal\nweights compared to the twist angles away from the MA,\nwhere the band are dispersive. In the dirty limit, the\nGilbert damping correction is, then, expected to drop at\nthe MA as found in the case of a clean interface.\nIt comes out that the twist angle dependence of δαG\nis a direct probe of the emergence of the flat bands in\nTBG. On the other hand, the temperature dependence\nof the fine structure around the MA provides an accurate\nmeasurement of the MA, with a precision below 0 .005◦\n(see Fig. S.4of the Supplemental Material66). It also\ngives an estimation of the SOC induced in TBG adjacent\nto a monolayer of TMD.\nIt is worth stressing that in our model we did not take\ninto account the electron-electron interactions which\nsignificantly distort the electronic band structure of\nTBG80–83. Near the MA, the dominant electron-electron\ninteraction is found to be the Coulomb interaction with\nan amplitude estimated to be 10-15 meV82, which is\nlarger than the width of the flat bands ∼2−5 meV\nand the SOC considered in the present work. How\nare the results of Fig.2 modified in the presence of\nCoulomb interaction? Treating this interaction within\nthe Hartree-Fock approximation revealed that the\nHartree term considerably widens the bands while the\nexchange term leads, basically, to broken-symmetry\nphases. At the charge neutrality, the Hartree term\nvanishes and the exchange potential, which concerns\nbands with identical spins, opens a gap of 4 meV80,82,83,\nwhich is of the order of the SOC amplitudes. As a\nconsequence, the statistical weight ∆ f(E) of the bands\nwith opposite /an}bracketle{tSz/an}bracketri}htis expected to increase, but keeping\nlarger values at small angles compared to the MA.\nMoreover, the bandwidth, around the MA, is found to\nrelatively increase under the exchange term80,82,83, but\nremains smaller than 3 meV, which preserve the flatness\nof the bands. It comes out, that our results hold in\nundoped TBG under Coulomb interaction, and can be\nused to extract the value of the MA at which the Gilbert\ndamping correction drops. Away from the neutrality,\nthe bands are substantially distorted by the Coulomb\ninteraction80,82,83and our results should be taken with a\ngrain of salt since they account for filling νfactors away\nfrom−0.5<ν <0.5, where the bandwidth, at the MA,\nis less than 4 meV.5\nFigure 3. Band structure of TBG/WSe 2in the continuum limit [Eq. ( 6)] atθ= 0.5◦(a),θ=θ−\nM= 1.043◦(b),θ=θ+\nM= 1.058◦\n(c) andθ= 1.2◦(d). The dashed lines represent the bands at the MA ( θM= 1.05◦). The red (blue) arrows correspond to the\nout-of-plane electronic spin projection /angbracketleftsz/angbracketright= +1 (/angbracketleftsz/angbracketright=−1)56. Calculations are done for λI= 3 meV and λR= 4 meV.\nFigure 4. Schematic representation of the band structure\nEσ,±(Eq.6) and the Fermi-Dirac distribution f(E). The\nbands in dashed and green lines correspond, respectively, t o\nthe MA and to a twist angle θfar from the MA. The red\n(blue) arrows represent the projection of the out-of-plane\nspin projection /angbracketleftSz/angbracketright= +1 (/angbracketleftSz/angbracketright=−1). Around the MA,\nthe bands are almost flat and the statistical weights ∆ f(E),\ncorresponding to the transitions between E−,+→E+,+and\nE−,−→E+,−, are small compared to the case of a twist angle\naway from the MA, where the band dispersion is larger.\nBesides interactions, strain is found to be a key pa-\nrameter in the emergence of flat bands in TBG68,70. The\neffect of strain can be included in our model by deriving\nthe strain induced correction to the Hamiltonian given\nby Eq. (3), taking into account the strain dependence of\nthe vectors qjconnecting the Dirac points70. The twist\nangle, at which δαGdrops, can then provide a way to\nmeasure the strain in TBG.\nExperimental realization. – Our proposed setup\nconsists of an interface between a FI and a fully hBN\nencapsulated TBG/WSe 2heterostructure (Fig. 1). The\nhBN layer acts as a tunnel barrier which prevents the\ndiffusion of the FI atoms into the graphene layer84.\nOn the other hand, the encapsulation provides a clean\ninterface and prevents the graphene degradation84\nwhich is a challenging issue in the STM and ARPES\nexperiments24–26, carried out on non-encapsulated TBG\nsamples.\nIt should be stressed that the hBN encapsulated\nTBG/WSe 2heterostructure has been already realizedin Refs. [ 56and57]. Furthermore, the spin transport\nthrough a clean interface between a FI and 2D material\nhas been experimentally achieved84,85. The 2D materials\nwere fully encapsulated by hBN84or covered by a thin\nlayer of an oxide insulator (as MgO)85to avoid the\ninterdiffusion with the FI.\nOurproposedtechniquetomeasurethe MAcan, then, be\nimplemented experimentally with a clean interface and\nat room temperature. Moreover, an insitumanipulation\nof the twist angle can be realized as in Refs. [ 86–89].\nConclusion. – To conclude, we have proposed an\nexperiment to probe the flat bands of TBG and to\nmeasure its MA accurately. The experiment is based\non a spin pumping measurement through a junction\nseparating a FI and a TBG adjacent to a monolayer of\nWSe2. We first derived the continuum model of TBG\nwith SOC, which constitutes a first step to develop an\nanalytical understanding of the emergence of a stable\nsuperconducting state at small twist angles observed\nin TBG in proximity to WSe 256. We then determined\nanalytically the Gilbert damping correction δαGinduced\nby the presence of the TBG/WSe 2heterostructure. Our\nresults show that the twist angle dependence of δαG\nexhibits a drop at the MA with a temperature-dependent\nfine structure. This feature provides an accurate de-\ntermination of the MA and an estimation of the SOC\ninduced in TBG by its proximity to the TMD layer. Our\nproposed set-up can be readily implemented regarding\nthe state-of-the art of the experimental realizations of\nSP in 2D materials and TBG-based heterostructure.\nOur work opens the gate to a twist tunable spintronics\nin twisted layered heterostructures.\nAcknowledgments. – We thank Mamoru Matsuo and\nShu Zhang for stimulating discussions. We are indebted\nto Jean-No¨ el Fuchs and Daniel Varjas for a critical read-\ning of the manuscript. S. H. acknowledges the kind hos-\npitality of the Institute for Solid State Physics (ISSP)\nwhere this work was carried out. S. H. also thanks the\nhospitality of the Max Planck Institute for the Physics of\nComplex Systems (MPI-PKS). S. H. acknowledges finan-\ncial support from the ISSP International visiting profes-6\nsors program and the MPI-PKS visitors program.\n∗sonia.haddad@fst.utm.tn\n1Y.Cao, V.Fatemi, S. Fang, K.Watanabe, T. Taniguchi, E.\nKaxiras, and P. 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We consider a TBG where the two layers l= 1,2 are rotated oppositely θ2=−θ1=θ\n2.\nThe Hamiltonian of a graphene layer lrotated at an angle θlis\nhl(k) =eiθl\n2σzh(0)\nl(k)e−iθl\n2σz(S.1)\nwhereh(0)\nl(k) is the Hamiltonian of the unrotated layer ( l) given, in the continuum limit, by\nh(0)\nl(k) =−/planckover2pi1vFk·σ∗, (S.2)\nwhere the momentum kis written relatively to the Dirac point Kl,ξ,vFis the Fermi velocity, ξis the valley index,\nσ∗= (ξσx,σy) andσi(i=x,y,z) are the sublattice-Pauli matrices.\nThe leading contributions of the interlayer tunneling can be limited to t hree nearest hopping processes in the\nmomentum space connecting states |k/an}bracketri}ht1, around the Dirac point K1,ξof layer (1), to the states |k+qjξ/an}bracketri}ht2around\nK2,ξ, the Dirac point of layer (2). The qjξvectors are given by65\nq1ξ=ξkθ(0,1),q2ξ=q1ξ+ξGM\n1=ξkθ/parenleftigg\n−√\n3\n2,−1\n2/parenrightigg\n,\nq3=q1ξ+ξ/parenleftbig\nGM\n1+GM\n2/parenrightbig\n=ξkθ/parenleftigg√\n3\n2,−1\n2/parenrightigg\n, (S.3)\nwherekθ= 2kDsinθ\n2∼θkDandkD=|K1,ξ|=|K2,ξ|=4π\n3a,abeing the graphene lattice parameter. The/parenleftbig\nGM\n1,GM\n2/parenrightbig\nis the moir´ e BZ basis given by GM\ni=RT\ntGi,Giare the lattice basis vectors of the monolayer reciprocal lattice\nG1=2π\na/parenleftbig\n1,−1/√\n3/parenrightbig\nandG2=2π\na/parenleftbig\n0,2/√\n3/parenrightbig\n.Rtis the rotation tensor written, in the sublattice basis, at a small\ntwist angle as\nR(θ) =/parenleftbigg\n0−θ\nθ0/parenrightbigg\n. (S.4)\nIn the basis {|k/an}bracketri}ht1,|k+qj,ξ/an}bracketri}ht2}, the Hamiltonian, at the valley ξ, reads as65\nH(k) =\nh1(k)T1T2T3\nT†\n1h2,1(k) 0 0\nT†\n20h2,2(k) 0\nT†\n30 0 h2,3(k)\n, (S.5)\nFor the relaxed TBG the Tjmatrices are given by69\nT1=/parenleftbigg\nw w′\nw′w′′/parenrightbigg\n,T2=eiξGM\n1·r/parenleftbigg\nw w′e−iξΦ\nw′eiξΦw′′/parenrightbigg\n,T3=eiξ(GM\n1+GM\n2)·r/parenleftbigg\nw w′eiξΦ\nw′e−iξΦw′′/parenrightbigg\n, (S.6)\nHereh1(k) is given by Eq. S.1andh2,j(k)≡h2(k+qjξ) =h1(k+qjξ), where the momentum kis written relatively\ntoK1,ξ. Φ =2π\n3andris the shortest inplane shifts between carbon atoms of the two laye rs8,67. Hereafter, we neglect\nthe relative sliding between the layers which is not relevant in the phys ics of TBG8,65. Choosing the A sublattice in10\nboth layers ( i= 1,2) as the origin of the unit cell, turns out to take r=0in Eq.S.68. The parameters w,w′andw′′\nare the tunneling amplitudes which take the same value w=w′=w′′∼110 meV in the rigid TBG68.\nIn the relaxed lattice, these amplitudes are no more equal w∼w′′∼90 meV and w′= 117 meV69. In the present\nwork, we do not consider the lattice relaxation effect, since the SOC parameters λI, λR∼4meV are small compared\nto the difference between the interlayer amplitudes ∆ w=w−w′∼20meV. In the unrelaxed lattice, the Tjmatrices\ncan be written as\nT1=w(Iσ+σx),T2=w/parenleftigg\nIσ−1\n2σx+ξ√\n3\n2σy/parenrightigg\n,T3=w/parenleftigg\nIσ−1\n2σx−ξ√\n3\n2σy/parenrightigg\n. (S.7)\nHere,Iσis the identity matrix acting on the sublattice indices.\nConsidering, in Eq. S.5, thekdependent term as a perturbation, the effective Hamiltonian can be written, to the\nleading order in k, as\nH(1)(k) =/an}bracketle{tΨ|H(k)|Ψ/an}bracketri}ht\n/an}bracketle{tΨ|Ψ/an}bracketri}ht, (S.8)\nwhere Ψ = ( ψ0(k),ψ1(k),ψ2(k),ψ3(k)) is the zero energy eigenstate of H(k=0). Ψ is constructed on the two-\ncomponent sublattice spinor ψ0(k) (ψj(k)) of layer 1 (layer 2) taken at the momentum k(k+qjξ) around the Dirac\npointK1,ξ(K2,ξ) at the valley ξ.ψ0is the zero energy eigenstate of h1. The Ψ components satisfy\nh1ψ0+/summationdisplay\njTjψj= 0,andT†\njψ0+hjψj= 0,withh1ψ0= 0, (S.9)\nwherehj≡h2(qj,ξ). Then\nψj=−h−1\njT†\njψ0,and/summationdisplay\njTjh−1\njT†\nj= 0. (S.10)\nTo the leading order in k,H(1)(k), takes the following form\nH(1)(k) =/an}bracketle{tΨ|H(k)|Ψ/an}bracketri}ht\n/an}bracketle{tΨ|Ψ/an}bracketri}ht=1\n/an}bracketle{tΨ|Ψ/an}bracketri}ht\nψ†\n0h1(k)ψ0+ψ†\n0/summationdisplay\njTjh−1\njhj(k)h−1\njT†\njψ0\n, (S.11)\nH(1)(k) =−/planckover2pi1v∗ψ†\n0k·σ∗ψ0, (S.12)\nEq.S.12is obtained by neglecting θinhj(k), which turns out to take qj,ξ=0inhj(k)65.\nv∗is the effective velocity of the energy band of TBG around the zero e nergy which vanishes at the MA θm, and is\ngiven by65\nv∗=vF1−3α2\n1+6α2(S.13)\nwhereα=w\n/planckover2pi1vFq0,q0=|qjξ| ∼4π\n3aθ.\nIn our numerical calculations (Fig. 3of the main text), we take w= 118 meV and /planckover2pi1vF/a∼2.68 eV which corresponds\ntoθm= 1.05◦for the first MA65.\nS.2. II. DERIVATION OF THE LOW-ENERGY HAMILTONIAN OF TBG WIT H SOC\nWe now consider the heterostructure consisting of TBG adjacent to a monolayer of WSe 2as shown in Fig.1 of the\nmain text, where we denote the graphene layer in contact with the T MD by layer (2). This layer is subject to a SOC\ninduced by proximity effect by the TMD, and the corresponding Hamilt onian can be written as56\nh2(k) =h1(k)+hSOC+m\n2σz (S.14)\nwherehSOCis given by Eq. 1of the main text.11\nTo derive the continuum model of TBG/WSe 2, we follow the perturbative approach of Ref.65presented in the\nprevioussection. Now, thebasisΨ = ( ψ0(k),ψ1(k),ψ2(k),ψ3(k)) isconstructedonthefour-componentspin-sublattice\nspinorψ0(k) andψj(k), (j= 1,2,3) corresponding, respectively, to layer (1) and layer (2). ψ0(k) is written as\nψ0(k)T= (ψ0,A↑,ψ0,A↓,ψ0,B↑,ψ0,B↓). In this basis, the Hamiltonian of TBG/WSe 2takes the form\nHξ,SOC(k) =\nh1(k)T1T2T3\nT†\n1h2,1(k) 0 0\nT†\n20h2,2(k) 0\nT†\n30 0 h2,3(k)\n. (S.15)\nThe momentum kis measured relatively to the Dirac point K1ξof layer (1), h2,j(k) =h2(k+qjξ), (j= 1,2,3) and\nh2(k) includes now the SOC terms (Eq. S.14).\nWe take the sublattice A as the origin of the unit cell in each layer. The Tjmatrices are written as the tensor product\nof those given by Eq. S.7, with the 2 ×2 identity spin-matrix Is.\nRegarding the small values of the SOC, we assume that Hξ,SOC(k) has a zero eigenenergy and the corresponding\neigenstate Ψ satisfies the condition given by Eq. S.10.\nFollowingthesameprocedureasintheprevioussection, wederivefr omEq.S.11theeffectivelowenergyHamiltonian\nH(1)\nξ,SOC(k) of TBG/WSe 2by substituting hj(k) by the Hamiltonian of layer (2), rotated at θ/2 and including SOC\nas\nh2,rot(k) =−/planckover2pi1k·σ∗Is+λI\n2ξsz+λR\n2(ξσxsy−σysx)+λKM\n2ξσzsz−λR\n2θ(ξσysy+σxsx), (S.16)\nHereafter, we neglect the Kane and Mele term whose contribution, to the leading order in k, is found to vanish. We\nalso disregard the last term in Eq. S.16, which results into a higher order correction in θ.\nTo the first orderin the SOC coupling, we obtain the continuum model of TBG/WSe 2described by the Hamiltonian\nH(1)\nξ,SOC(k) given by Eq. 3in the main text. This Hamiltonian contain a SOC term ( hSOC\neff) with renormalized Ising\nand Rashba interactions\n˜λI=6α2\n/an}bracketle{tΨ|Ψ/an}bracketri}htλI,˜λR=3α2\n/an}bracketle{tΨ|Ψ/an}bracketri}htλR,and/an}bracketle{tΨ|Ψ/an}bracketri}ht ∼1+6α2(S.17)\n˜λIand˜λRare enhanced by decreasing the twist angle from the MA.\nFigure S.1. Energy bands of TBG/WSe 2around zero energy as function of the dimensionless momentu m amplitude k/q0at\ndifferent twist angles. The bands are represented up to the cu toffkc=q0/2. Calculations are done for λI= 3 meV and\nλR= 4 meV56. The MA is θM= 1.05◦.12\nTo the leading order in k, the four eigenergies of the Hamiltonian H(1)\nξ,SOC(Eq.3of the main text), denoted E(k)σ,±,\nare given by\nE(k)σ,±=σ\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/radicalig\nf1(k)±6α2/radicalbig\nf2(k), (S.18)\nf1(k) = (/planckover2pi1vF)2/parenleftbig\n1−3α2/parenrightbig2||k||2+9\n2α4/parenleftbig\n2λ2\nI+λ2\nR/parenrightbig\n,\nf2(k) = (/planckover2pi1vF)2/parenleftbig\n1−3α2/parenrightbig2||k||2/parenleftbigg\nλ2\nI+1\n4λ2\nR/parenrightbigg\n+9\n16α4λ4\nR, (S.19)\nwhereσ=±is the band index. E(k)σ,±are depicted in Fig. S.1at different twist angles.\nAt the Dirac point, the eigenergies reduce to\nE±,+=±3α2\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/radicalig\nλ2\nI+λ2\nR,E±,−=±3α2\n/an}bracketle{tΨ|Ψ/an}bracketri}htλI. (S.20)\nIt is worth to note that in TBG, the flatness of the bands around th e charge neutrality point strongly depends on\nthe heterostrain which may emerge in the graphene layers during th e fabrication procedure70. The interplay between\nstrain and SOC in TBG/WSe 2goes beyond the scope of the present work.\nIn figures S.2andS.3, we plot the electronic band structure of TBG/WSe 2at different twist angles. The solid\nlines correspond to the numerical results obtained within the contin uum model taking into account 148 bands for\neach spin and valley. The calculations are done for the relaxed TBG wit h interlayer momentum hopping amplitudes\nw= 55meV and w= 105meV (Eq. S.6). We considered these values to reproduce the numerical band st ructure\nobtained in Ref.[ 56] using the continuum model. We have taken into account the lattice r elaxation, as in Ref. 56,\nsince it is expected to be important at small angles. The numerical re sults are compared to the eigenergies of the\neffective Hamiltonian H(1)\nξ,SOC, given by Eq. 3of the main text (dashed line), and to the approximated expression s\n(Eq.S.18) represented by dotted gray lines. It should be stressed that H(1)\nξ,SOCis derived for a rigid TBG, for which\nwe have taken an interlayer hopping amplitude w= 105meV, which gives rise to a MA θM∼1.1◦as in Ref [ 56].\nFor the sake of simplicity, we did not consider the relaxation effect, w hich is not significant around the MA56, where\nwe consider the spin pumping effect. The derivation of the effective H amiltonian of the relaxed TBG adjacent to\nWSe2is left to a future work.\nAs shownby Fig. S.2andS.3, the effective Hamiltonian H(1)\nξ,SOC(Eq.3ofthe main text) providesa good description\nof the band structure of TBG/WSe 2. It can be taken as a framework to unveil the origin of the observe d stable\nsuperconducting state in this heterostructure56. However, at relatively small angles ( ∼0.5◦), the Fermi velocities of\nH(1)\nξ,SOCare overestimated (Fig. S.2). This discrepancy is due to the assumption of a rigid TBG lattice which is not\njustified at small angles56.\nS.3. III. ELECTRONIC GREEN FUNCTION\nThe Matsubara Green function associated to the effective Hamilton ianH(1)\nξ,SOC(Eq.5) is\nˆg(k,iωn) = [iωnISIσ−Hξ,SOC(k)]−1, (S.21)\nwhereISandIσare the 2 ×2 spin and band identity matrices, respectively. ˆ g(k,iωn) can be expressed as\nˆg(k,iωn) = ˆg0(k,iωn)Is+ˆ g·s. (S.22)\ns= (sx,sy,sz) are spin-Pauli matrices, ˆ g0(k,iωn) and the components ˆ gi(i=x,y,z) ofˆ gare written as\nˆg0(k,iωn) =A0(k,iωn)+C0(k,iωn),ˆgx(k,iωn) =Bx(k,iωn)+Dx(k,iωn),\ngy(k,iωn) =By(k,iωn)+Dy(k,iωn),ˆgz(k,iωn) =Az(k,iωn)+Cz(k,iωn). (S.23)13\n(a) (b) (c)\nFigure S.2. Electronic band structure of TBG/WSe 2calculated, at a twist angle (a) θ= 0.5◦, (b)θ= 0.79◦, and (c) θ= 0.87◦.\nCalculations are based on the continuum model and including 148 bands for each moir´ e valley and spin. The line color deno tes\nthe value of the out-of-plane spin projection /angbracketleftSz/angbracketright. The dashed black lines represent the eigenergies of the fou r-band effective\nHamiltonian given by Eq. 3of the main text, and the gray dotted lines denote the approxi mated eigenergies given by Eq. S.18.\nCalculations are done for λR= 4meV and λI= 3meV. The band structure is represented in the moir´ e Brill ouin zone where κ\nandκ′correspond respectively to the Dirac point K1of layer (1) and K2layer (2) at the valley ξ= +. The bottom panel is a\nzoomed-in representation around the high symmetry point κ.\n(a) (b) (c)\nFigure S.3. Electronic band structure of TBG/WSe 2calculated around the MA θMat (a)θ= 1.05◦, (b)θM= 1.1◦, (c)\nθ= 1.2◦. The bottom panel is a zoomed-in representation around the h igh symmetry point κ. The data are the same as in\nFig.S.2.14\nTheA,B,CandDoperators are written in terms of the band-Pauli matrices σx,y,zand the corresponding identity\nmatrixIσ\nAi(k,iωn) =Ai1(k,iωn)Iσ+Aiz(k,iωn)σz\nCi(k,iωn) =Cix(k,iωn)σx+Ciy(k,iωn)σy\nBj(k,iωn) =Bj1(k,iωn)Iσ+Bjz(k,iωn)σz\nDj(k,iωn) =Djx(k,iωn)σx+Djy(k,iωn)σy (S.24)\nherej=x,yandi= 0,z.\nIn the limit of small SOC couplings λR,λI≪/planckover2pi1vFq0, A, B, C, and D become\nA01(k,iωm) =Y00−E2\n1\n2(E2\n3−E2\n1)/bracketleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2/bracketrightbigg\n+E2\n3−Y00\n2(E2\n3−E2\n1)/bracketleftbigg1\ni/planckover2pi1ωm−E3+1\ni/planckover2pi1ωm−E4/bracketrightbigg\n(S.25)\nY00=(/planckover2pi1vF)2(1−3α2)2\n/an}bracketle{tΨ|Ψ/an}bracketri}ht2||k||2+1\n2/parenleftbigg3α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2\n+/parenleftbigg3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2\nA0z(k,iωm) =Y0\n2(E2\n1−E2\n3)/braceleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracerightbigg\n(S.26)\nY0=−3\n2/parenleftbigg3α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht\nC0x(k,iωm) =1\n2(E2\n1−E2\n3)/braceleftbigg\nY2/bracketleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg\n−Y1/bracketleftbigg\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−E3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg/bracerightbigg\nξkx (S.27)\nY1=/planckover2pi1vF(1−3α2)\n/an}bracketle{tΨ|Ψ/an}bracketri}ht, Y2=−Y1/parenleftbigg3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2\nC0y(k,iωm) =1\n2(E2\n1−E2\n3)/braceleftbigg\nY2/bracketleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg\n−Y1/bracketleftbigg/parenleftbigg\nE11\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−E3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg/bracerightbigg\nky (S.28)\nAz1(k,iωm) =ξ1\n2(E2\n1−E2\n3)/braceleftbigg\nY4/bracketleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg\n−Y3/bracketleftbigg\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−E3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg/bracerightbigg\n(S.29)\nY3=−3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht, Y4=−Y3/bracketleftigg\n(/planckover2pi1vF)2\n/an}bracketle{tΨ|Ψ/an}bracketri}ht2(1−3α2)2||k||2−1\n2/parenleftbigg3α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2\n−/parenleftbigg3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2/bracketrightigg\nAzz(k,iωm) =ξY5\n2(E2\n3−E2\n1)/braceleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2−1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracerightbigg\n(S.30)\nY5=1\n2/parenleftbiggλR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2\nCzx(k,iωm) =Y6\n2(E2\n3−E2\n1)/braceleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2−1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracerightbigg\nkx (S.31)\nY6=/planckover2pi1vF(1−3α2)6α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht215\nCzy(k,iωm) =Y6\n2(E2\n3−E2\n1)/braceleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2−1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracerightbigg\nξky\n(S.32)\nBx1(k,iωm) =Y7\n2(E2\n3−E2\n1)/braceleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2−1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracerightbigg\nky (S.33)\nY7=/planckover2pi1vF(1−3α2)3α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht2\nBxz(k,iωm) =Y81\n2(E2\n1−E2\n3)ky×\n/braceleftbigg1\nE1/bracketleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/bracketrightbigg\n+1\nE3/bracketleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracketrightbigg/bracerightbigg\n(S.34)\nY8=−/planckover2pi1vF(1−3α2)3α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht23α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht\nDxx(k,iωm) =Y91\n2(E2\n1−E2\n3)ξkxky×\n/braceleftbigg1\nE1/bracketleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/bracketrightbigg\n−1\nE3/bracketleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracketrightbigg/bracerightbigg\n(S.35)\nY9= (/planckover2pi1vF)2(1−3α2)23α2λR\n/an}bracketle{tΨ|Ψ/an}bracketri}ht\nDxy(k,iωm) =1\n2(E2\n1−E2\n3)/braceleftbigg\nY11/bracketleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg\n−Y10/bracketleftbigg\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−E3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg/bracerightbigg\n(S.36)\nY10=−3α2λR\n2/an}bracketle{tΨ|Ψ/an}bracketri}ht, Y11=Y10/bracketleftigg\n(/planckover2pi1vF)2(1−3α2)2(k2\nx−k2\ny)+/parenleftbigg3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2/bracketrightigg\nBy1(k,iωm) =−Y71\n2(E3\n3−E2\n1)/braceleftbigg1\ni/planckover2pi1ωm−E1+1\ni/planckover2pi1ωm−E2−1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracerightbigg\nkx(S.37)\nByz(k,iωm) =−Y81\n2(E2\n1−E2\n3)kx/braceleftbigg1\nE1/bracketleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/bracketrightbigg\n−1\nE3/bracketleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracketrightbigg/bracerightbigg\n(S.38)\nDyx(k,iωm) =ξ1\n2(E2\n1−E2\n3)/braceleftbigg\nY12/bracketleftbigg1\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−1\nE3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg\n+Y10/bracketleftbigg\nE1/parenleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm−E2/parenrightbigg\n−E3/parenleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/parenrightbigg/bracketrightbigg/bracerightbigg\n(S.39)\nY12=3α2λR\n3/an}bracketle{tΨ|Ψ/an}bracketri}ht/bracketleftigg\n−(/planckover2pi1vF)2(1−3α2)2(k2\nx−k2\ny)+/parenleftbigg3α2λI\n/an}bracketle{tΨ|Ψ/an}bracketri}ht/parenrightbigg2/bracketrightigg\nDyy(k,iωm) =−Y91\n(E2\n1−E2\n3)kxky/braceleftbigg1\nE1/bracketleftbigg1\ni/planckover2pi1ωm−E1−1\ni/planckover2pi1ωm+E1/bracketrightbigg\n−1\nE3/bracketleftbigg1\ni/planckover2pi1ωm−E3−1\ni/planckover2pi1ωm−E4/bracketrightbigg/bracerightbigg\n,(S.40)\nwhereE1=E+,+(k),E2=E−,+(k),E3=E+,−(k) andE4=E−,−(k) (Eq.S.18).16\nS.4. IV. MAGNON GREEN FUNCTION AND GILBERT DAMPING\nA. Interfacial exchange coupling between a ferro- magnetic insulator (FI) and a TBG\nWe consider the Hamiltonian of the ferromagnetic insulator (FI) in th e independent magnon approximation as\nHFI=/summationdisplay\nk/planckover2pi1ωkb†\nkbk, (S.41)\nwhere/planckover2pi1ωk≃ D|k|2+/planckover2pi1γhdcis a dispersion of magnons, Dis a spin stiffness, γis the gyromagnetic ratio, hdcis a\nstatic magnetic field. In the spin pumping setup, only the static part associated to k=0is relevant. Considering\nonly the uniform spin precession, the Hamiltonian of the FI can be simp ly written as\nHFI=/planckover2pi1ω0b†\n0b0, (S.42)\nwherebqis the Fourier transformation of the site representation bidefined as\nbi=1√NFI/summationdisplay\nqeiq·ribq≃1√NFIb0, (S.43)\nb†\ni=1√NFI/summationdisplay\nqe−iq·rib†\nq≃1√NFIb†\n0, (S.44)\nwhereNFIis the number of unit cells in the FI.\nWe consider the (retarded) magnon Green function as\nGR(q,ω) =/integraldisplay\ndtGR(q,t)eiωt, (S.45)\nGR(q,t) =−i\n/planckover2pi1θ(t)/an}bracketle{t[S+\nq(t),S−\nq(0)]/an}bracketri}ht=−2iS0\n/planckover2pi1θ(t)/an}bracketle{t[bq(t),b†\nq(0)]/an}bracketri}ht. (S.46)\nIn the absence of the junction, the magnon Green function is\nGR\n0(q,ω) =2S0//planckover2pi1\nω−ωq+iδ. (S.47)\nWe introduce spin relaxation of the bulk FI phenomenologically as\nGR\n0(q,ω) =2S0//planckover2pi1\nω−ωq+iαGω, (S.48)\nwhereαGis a dimensionless strength of the Gilbert damping, which is of order of 10−4–10−3. We note that a line\nshape of the ferromagnetic resonance is proportional to Im GR\n0(q=0,ω)79.\nIn the presence of the interfacial coupling and for a uniform spin pr ecession, the magnon Green function is given\nby the Dyson equation as\nGR\n0(q= 0,ω) =2S0//planckover2pi1\nω−ωq=0+iαGω−(2S0//planckover2pi1)ΣR(q= 0,ω), (S.49)\nwhere ΣR(ω) is the self-energy. Although the real part of ΣR(ω) is related to the shift of the ferromagnetic resonance,\nwe neglect it for simplicity. Then, the magnon Green function is rewrit ten as\nGR\n0(0,ω) =2S0//planckover2pi1\nω−ωq=0+i(αG+δαG)ω, (S.50)\nδαG(ω) =−2S0\n/planckover2pi1ωImΣR(q=0,ω). (S.51)\nWe note that δαG(ω) depends on ωin general. However, since the ferromagnetic resonance peak is sh arp enough\n(αG+δαG≪1), we can replace ωwithω0= Ω (the peak position of the ferromagnetic resonance):\nδαG≃ −2S0\n/planckover2pi1ΩΣR(q= 0,Ω). (S.52)17\nThe Hamiltonian of the interfacial coupling is given as\nHint=/summationdisplay\n/angbracketlefti,j/angbracketrightTij(S+\nis−\nj+h.c.). (S.53)\nHere,S±\niis a spin ladder operator of the FI and is described by magnon annihilat ion/creation operators ( biandb†\ni)\nas\nS+\ni=/radicalbig\n2S0bi, S−\ni=/radicalbig\n2S0b†\ni, (S.54)\nwhereS0is an amplitude of the localized spin in the FI. s±\njis a spin ladder operator of electrons in two-dimensional\nelectron systems (twisted bilayer graphene) and is described by th e electron annihilation/creation operators ( cjσand\nc†\njσ) as\ns+\nj=c†\nj↑cj↓, s−\nj=c†\nj↓cj↑. (S.55)\nWe define the Fourier transformation as\ncjσ=1√\nN/summationdisplay\nkeik·rjckσ, (S.56)\nc†\njσ=1√\nN/summationdisplay\nke−ik·rjc†\nkσ, (S.57)\nwhereNis the number of unit cells and rjis the position of the site jin TBG. Then, we obtain\ns+\nj=1\nN/summationdisplay\nk,k′e−ik·rj+ik′·rjc†\nk↑ck′↓, (S.58)\ns−\nj=1\nN/summationdisplay\nk,k′e−ik·rj+ik′·rjc†\nk↓ck′↑. (S.59)\nWe define the Fourier transformation of s±\njas\ns+\nj=1\nN/summationdisplay\nqeiq·rjs+\nq, (S.60)\ns−\nj=1\nN/summationdisplay\nqeiq·rjs−\nq. (S.61)\nFroms−\nj= (s+\nj)†, we obtain the relation s−\nq= (s+\n−q)†. The inverse Fourier transformation is given as\ns+\nq=/summationdisplay\nje−iq·rjs+\nj, (S.62)\ns−\nq= (s+\n−q) =/summationdisplay\nje−iq·rjs−\nj. (S.63)\nEspecially for q=0, we obtain\ns+\n0=/summationdisplay\njs+\nj, s−\n0=/summationdisplay\njs−\nj, (S.64)\nUsing Eqs. ( S.56) and (S.57), we obtain\ns+\nq=1\nN/summationdisplay\nje−iq·rj/summationdisplay\nk,k′e−ik·rj+ik′·rjc†\nk↑ck′↓=/summationdisplay\nkc†\nk↑ck+q↓, (S.65)\ns−\nq=1\nN/summationdisplay\nje−iq·rj/summationdisplay\nk,k′e−ik·rj+ik′·rjc†\nk↓ck′↑=/summationdisplay\nkc†\nk↓ck+q↑. (S.66)18\nFor a clean interface, we can set Tij=T. Then, using Eqs. ( S.43) and (S.44), the Hamiltonian of the interface is\nwritten as\nHint=T√2S0√NFI/summationdisplay\n/angbracketlefti,j/angbracketright(b0s−\nj+b†\n0s+\nj)≃T√2S0√NFI\nb0/parenleftig/summationdisplay\njs−\nj/parenrightig\n+b†\n0/parenleftig/summationdisplay\njs+\nj/parenrightig\n\n=/radicalbig\n2S0b0˜s−+/radicalbig\n2S0b0˜s+, (S.67)\nwhere ˜s±is defined as ˜ s±= (T/√NFI)s±\n0.\nBy the second-order perturbation, the self-energy of the magn on atq=0is calculated as\nΣR(ω) =/integraldisplay\ndtΣR(t)eiωt, (S.68)\nΣR(t) =−i\n/planckover2pi1θ(t)/an}bracketle{t[˜s+(t),˜s−(0)]/an}bracketri}ht. (S.69)\nThe self-energy can be related to a retarded component of the dy namic spin susceptibility per unit cell as\nΣR(ω) =−T2N\nNFIχ(0,ω), (S.70)\nχ(q,ω) =/integraldisplay\ndtχ(q,t)eiωt, (S.71)\nχ(q,t) =i\nN/planckover2pi1θ(t)/an}bracketle{t[s+\n−q(t),s−\nq(0)]/an}bracketri}ht. (S.72)\nχ(q,t) is calculated for one-band of TBG without spin-orbit interaction as\nχ(q,t) =1\nN/summationdisplay\nqf(ξk)−f(ξk+q)\n/planckover2pi1ω+iδ+ξk+q−ξk, (S.73)\nwhereξk=ǫk−µ,ǫkis a dispersion of electrons, µis a chemical potential. This is just a Lindhard function. We\nnote thatχ(q,t) is independent of the system size (area). For systems with spin-o rbit interaction, we have to extend\nthe Lindhard function into the spin-dependent one.\nThen, the enhancement of the Gilbert damping is written as\nδαG=−2S0\n/planckover2pi1ΩImΣR(q=0,Ω)\n=2S0T2N\nNFI/planckover2pi1ω0χ(q=0,Ω). (S.74)\nWe note that the number of the unit cell of twisted bilayer graphene is written as N=S/AwhereSis a area of the\njunction and Ais an area of a unit cell of twisted bilayer graphene. We also note that the number of the unit cell\nof the FI is written as NFI=Sd/a3wheredis a thickness of the FI, ais a lattice constant of the FI. Using these\nparameters, we obtain\nδαG=2S0T2a3\nAd/planckover2pi1ω0χ(q=0,Ω). (S.75)\nWe note that δαGis proportional to 1 /din consistent with experimental results. If YIG (Yttrium Iron Garn et) is\nchosen as the ferromagnet insulator, the parameter is given in the Table.\nB. Electronic spins in the FI magnetization frame\nRegarding the dependence on szof the electronic Hamiltonian (Eq. 3of the main text), one should consider a 3 D\nFI magnetization as in Ref.79. The average spin vector is along the orthoradial spherical vecto r/an}bracketle{tSFI/an}bracketri}ht=/an}bracketle{tSFI/an}bracketri}htux′.\nThe radial vector uz′forms an angle θmwith thezaxis perpendicular to the interface. The third axis y′is in the\n(xoy) plane and its unit vector is the orthoradial inplane vector uy′=−sinΦmux+cosΦ muyas shown in Fig. S.4.19\nTable I. Experimental parameters.\nMicrowave frequency ω01GHz\nAmplitude of spins of FI S010\nLattice constant of FI a12.376˚A\nThickness of FI d≥10nm\nInterfacial exchange coupling J∼1K (not known)\nFigure S.4. Magnetization-fixed coordinate frame ( x′,y′,z′) with respect to the Laboratory frame ( x,y,z).\nIn the FI spin frame ( x′,y′,z′), the components of the electronic spin operators are given by:\nsx′\nk=sk·ux′= cosθmcosΦmsx\nk+cosθmsinΦmsy\nk−sinθmsz\nk\nsy′\nk=sk·uy′=−sinΦmsx\nk+cosΦ msy\nk\nsz′\nk=sk·uz′= sinθmcosΦmsx\nk+sinθmsinΦmsy\nk+cosθmsz\nk\n(S.76)\nWe define the ladder electronic spin operators as\nsx′,±\nk=sy′\n±k±isz′\n±k=1\n2/summationdisplay\nσ,σ′,k′c†\nk′,σ/parenleftig\nσx′,±\ns/parenrightig\nσ,σ′c†\nk′±k,σ′ (S.77)\nwheresx′,±=sx(−sinΦm±isinθmcosΦm)+sy(cosΦm±isinθmsinΦm)±icosθmsz.\nC. Magnon self-energy\nIn the second order perturbation with respect to the interfacial exchange interaction Tq, the interfacial self-energy\nis given by78\nΣ(q,iωn) =|Tq|2\n4β/summationdisplay\nk,iωmTr/bracketleftig\nσx′,−\nsˆg(k,iωm)σx′,+\nsˆg(k+q,iωm+iωn)/bracketrightig\n(S.78)\nwhere ˆg(k,iωm) is the electronic Green function given by Eq. S.22.\nThe trace term is of the form:\nTr[a∗·σs(ˆg0+ˆg·σs)a·σs(ˆg′\n0+ˆg′·σs)] (S.79)\nwhere the vector a= (−sinΦm+isinθmcosΦm,cosΦm+isinθmsinΦm,icosθm) is written in the laboratory frame\n(x,y,z).\nWe set ˆg= ˆg(k,iωm) and ˆg′= ˆg(k,iωm+iωn). Taking into account the operator character of ˆ gone could use the\nidentity\n(a·σs)(b·σs) = (a·b)I+i(a×b)·σs (S.80)20\nGiven the expressions of ˆ gand ˆg′in Eq.S.23, the trace term (Eq. S.79) reduces to:\nTr[a∗·σs(ˆg0+ˆg·σs)a·σs(ˆg′\n0+ˆg′·σs)] =/summationdisplay\ni=0,1,2Fi(k,iωm,iωn) (S.81)\nwhere\nF0(k,iωm,iωn) = 4/parenleftbig\nA01A′\n01+A0zA′\n0z+C0xC′\n0x+C0yC′\n0y/parenrightbig\nF1(k,iωm,iωn) =−2{cosθmcosΦm/parenleftbig\nA01B′\nx1−Bx1A′\n01+A0zB′\nxz−A′\n0zBxz+C0xD′\nxx−C′\n0xDxx+C0yD′\nxy−C′\n0yDxy/parenrightbig\n+cosθmsinΦm/parenleftbig\nA01B′\ny1−By1A′\n01+A0zB′\nyz−A′\n0zByz+C0xD′\nyx−C′\n0xDyx+C0yD′\nyy−C′\n0yDyy/parenrightbig\n−sinθm/parenleftbig\nA01A′\nz1−A′\n01Az1+A0zA′\nzz−A′\n0zAzz+C0xC′\nzx−C′\n0xCzx+C0yC′\nzy−C′\nzyC0y/parenrightbig/bracerightbig\nF2(k,iωm,iωn) =−2cos2θmcos2Φm/parenleftbig\nBx1B′\nx1+BxzB′\nxz+DxxD′\nxx+DxyD′\nxy/parenrightbig\n−2cos2θmsin2Φm/parenleftbig\nBy1B′\ny1+ByzB′\nyz+DyxD′\nyx+DyyD′\nyy/parenrightbig\n−2sin2θm/parenleftbig\nAz1A′\nz1+AzzA′\nzz+CzxC′\nzx+CzyC′\nzy/parenrightbig\n−cos2θmsin2Φ m/parenleftbig\nBx1B′\ny1+BxzB′\nyz+DxxD′\nyx+DxyD′\nyy+B′\nx1By1+B′\nxzByz+D′\nxxDyx+D′\nxyDyy/parenrightbig\n+cosΦ msin2θm/parenleftbig\nBx1A′\nz1+BxzA′\nzz+DxxC′\nzx+DxyC′\nzyB′\nx1Az1+B′\nxzAzz+D′\nxxCzx+D′\nxyCzy/parenrightbig\n+sinΦ msin2θm/parenleftbig\nBy1A′\nz1+ByzA′\nzz+DyxC′\nzx+DyyC′\nzyB′\ny1Az1+B′\nyzAzz+D′\nyxCzx+D′\nyyCzy/parenrightbig\n(S.82)\nThe terms with a prime are expressed in terms of iω′\nn=iωn+iωm.\nRegarding the kdependence of the A,B,CandDoperators (Eqs. S.25-S.40), onlyF0, the last term in F1and the\nthree first terms in F2give non-vanishing contributions after summing over kin Eq.12.\nOn the other hand, the terms between parentheses in the first an d second line in F2expression give the same\ncontribution. As a result, the GD is found to be independent of the a zimuthal angle Φ m, which expresses isotropy of\nthe electronic band structure Eσ,±(Eq.S.18). However, the GD depends on the out-of-plane orientation of th e FI\nmagnetization via the angle θm.\nAccording to Eq. S.78, the terms to calculate are of the form\n/summationdisplay\nkF(k)/summationdisplay\nωm1\ni/planckover2pi1ωm−Ei1\ni/planckover2pi1ω′m−Ej, (S.83)\nwhereω′\nm=ωm+ωnandF(k) is a function of k= (k,ϕk).\nThe summation over ωmin Eq.S.83can be written as\n/summationdisplay\nωm1\ni/planckover2pi1ωm−Ei1\ni/planckover2pi1ω′m−Ej=1\ni/planckover2pi1ωn−(Ej−Ei)/summationdisplay\nωm/bracketleftbigg1\ni/planckover2pi1ωm−Ei−1\ni/planckover2pi1ω′m−Ej/bracketrightbigg\n=−1\ni/planckover2pi1ωn−(Ej−Ei)1\nkBT/integraldisplay\ncdz\n2πih(z)f(z) (S.84)\nwhereh(z) =1\nz−Ei−1\nz+i/planckover2pi1ωn−Ej,f(z) is the Fermi-Dirac function, Cis clockwise contour around the poles z=i/planckover2pi1ωm.\nEquation S.83reduces, then, to\n/summationdisplay\nωm1\ni/planckover2pi1ωm−Ei1\ni/planckover2pi1ω′m−Ej=f(Ej)−f(Ei)\ni/planckover2pi1ωn−(Ej−Ei)(S.85)\nTaking the analytic continuation i/planckover2pi1ωn=/planckover2pi1ω+iη, Eq.S.83becomes\nlim\nη→0/summationdisplay\nkF(k)ηf(Ej)−f(Ei)\n(/planckover2pi1ω−Ej+Ei)2+η2= lim\nη→0/summationdisplay\nkF(k)(f(Ej)−f(Ei))L(/planckover2pi1ω−(Ej−Ei)), (S.86)\nL(x) =η\nx2+η2being the Lorentzian function. The sum over k= (k,ϕk) in Eq.S.86reduces toA\n(2π)2/integraltextkc\n0kdk/integraltext2π\n0dϕk,\nwhereAis the moir´ e superlattice area, kcis a cutoff on the momentum amplitude k, below which the continuum\nmodel for the monolayer is justified. We take kc=q0/2, whereq0=4π\n3aθis the separation between the Dirac points\nK1,ξandK2,ξof, respectively, layer (1) and layer (2) at a given monolayer valley ξ.21\nD. Gilbert damping correction\nFor a uniform spin precession, the Gilbert damping correction δαG, at the FMR frequency Ω, can be expressed as78\nδαG=−2S0\n/planckover2pi1ΩImΣ(q=0,Ω) (S.87)\nThe imaginary part of the self-energy is of the form ImΣ( q=0,Ω) =T2\n0\n/planckover2pi1Ω˜Σ(q=0,Ω), where ˜Σ(q=0,Ω) is a dimen-\nsionless integral. Introducing the average SOC λ=1\n2(λI+λR),δαGcan be written as\nδαG=−α0\nG/parenleftbiggλ\n/planckover2pi1Ω/parenrightbigg2\n˜Σ(q=0,Ω) (S.88)\nwhereα0\nG= 2S0T2\n0\nλ2.\nIn Fig.S.5, we plot the normalized Gilbert damping correction δαG/α0\nGas a function of the twist angle θand\nthe FMR energy /planckover2pi1Ω at different temperatures. The bottom panels are a zoomed repr esentation around the MA.\nFig.S.5shows that, the GD increases by decreasing the twist angle and sha rply drops at the MA, regardless of the\ntemperature range and the FMR frequency .\nAt high temperature ( kBT >λ), the GD exhibits, around the MA, a fine structure characterized by a peak which\ndisappears at low temperature. The origin of this peak is, as discuss ed in the main text, due to the dispersion of the\nenergy bands of TBG/WSe 2and their corresponding thermal weights ∆ f(E) =f(E/angbracketleftSz/angbracketright)−f(E−/angbracketleftSz/angbracketright) (Eq.S.86).\nFigure S.5. Color plot of the normalized Gilbert damping cor rectionδαG/α0\nGas a function of the twist angle θand the FMR\nenergy/planckover2pi1Ω atkBT= 0.1 meV ((a) and (d)), kBT= 1 meV ((b) and (e)) and kBT= 25 meV ((c) and (f)). The bottom panels\nshow the behavior of the GD around the MA. Calculations are do ne forµ= 0,λI= 3 meV and λR= 4meV.\nIn Fig.S.6we plot ∆f(E) corresponding to the transitions between E−,+→E+,+andE−,−→E+,−in the case\nof the undoped system.\nFiguresS.6(a) and (b) show that, at high temperature ( kBT > λ I,λR), ∆f(E) increases as the band dispersion\ngets larger and reaches its minimal value at the MA. This behavior exp lains the drop of the GD at the MA and its\nenhancement at small twist angles.\nIn figure S.6(c), we plot ∆ f(E) around the MA, at relatively high thermal energy compared to the SOC, where the\nGD exhibits a peak at the MA (Fig. 2of the main text). In this case, ∆ f(E) is maximal at the MA and decreases at\nthe anglesθ+\nMandθ−\nMclose to the MA. This feature results from the decrease of the ene rgy separation between E−,−\nandE+,−atθ+\nMandθ−\nM, compared to that at θM(Fig.3of the main text). At low temperature, and around the\nMA, one gets ∆ f(E) = 1 for the transitions E−,−→E+,−andE−,+→E+,+. As a consequence, the GD behavior is\nnow only dependent on the effective Fermi velocity v∗which vanishes at the MA. As a consequence, the small peak22\nFigure S.6. Statistical weight ∆ f(E) corresponding to the transitions between E−,+→E+,+(a) andE−,−→E+,−((b) and\n(c)) at different temperatures. The dots represent the energ yE+,+(a) andE+,−((b) and (c)) at the MA and the arrows mark\nthe limit of the band E+,+(a) andE+,−((b) and (c)) at the indicated twist angle. In (c), ∆ f(E) is shown around the MA\nfor the transition between E−,−→E+,−. Calculations are done for the SOC λI= 3 meV, λR= 4 meV56and in the undoped\nTBG (µ= 0).\nof the GD, emerging at the MA at relatively high temperature, disapp ears.\nFigureS.7shows the behavior of the normalized GD correction δαG/α0\nGas function of the chemical potential µ\natkBT= 25meV and for the FMR energy /planckover2pi1Ω = 0.06meV. The decrease of δαGis a consequence of the thermal\nweight. The results shown in Fig. S.7are expected to hold in the presence of Coulomb interaction if the wid th of the\nbands at the MA remains less than 4 meV, which is the case of the filling f actorνsatisfying −0.5<ν <0.580.\nFigure S.7. Normalized GD correction δαG/α0\nGas function of the chemical potential µatkBT= 25meV and for different twist\nangles. The upper limit of µisµc=/planckover2pi1vFkccorresponding to the momentum cutoff kc=q0\n2. Calculations are done for the SOC\nλI= 3 meV, λR= 4 meV56,kBT= 25meV and for the FMR energy /planckover2pi1Ω = 0.06meV.\nIn Fig.S.8, we plot the normalized GD correction δαG/α0\nGas function of the SOC parameters, λIandλR, for\ndifferent twist angles, at kBT= 25meV, /planckover2pi1Ω = 0.06meV and in the case of the undoped system. The drop of δαG\nat the MA is a robust feature regardless of the amplitude of the SOC . However, there is a relative increase of δαG,\nat the MA, if the bands E−,+andE+,+(orE−,−andE+,−) are in resonance with the FMR energy, as shown in\nFig.S.8(c). This resonance can be only reached for relatively small values o fλR.\nAs shown in Fig. S.2, the energy spectrum of the effective model (dashed lines) are slig htly more dispersive, at\nsmall twist angles ( θ∼0.5◦), than those obtained by including higher bands (solid lines). This disc repancy should be\ntaken into account when fixing the value of the cutoff kcup to which the sum in Eq. S.86is evaluated. To determine\nthe role of the cutoff on the SP effect, we plot, in Fig. S.9, the GD correction δαGas a function of the twist angle at\ndifferent cutoffs kc≤q0\n2, whereq0=|K1,ξK2,ξ|is the momentum separation between the Dirac points K1,ξandK2,ξ\nof respectively layer (1) and layer (2) at a given valley ξ.\nFig.S.9shows that the GD correction drops at the MA regardless of the cu toff values. The larger the cutoff, the\nsharper the drop.23\nFigure S.8. Normalized GD correction δαG/α0\nGas function of the SOC λRandλIat a twist angle θ= 0.5◦(a),θ= 0.8◦(b)\nand at the MA θ= 1.05◦(c). Calculations are done for µ= 0,kBT= 25meV and for the FMR energy /planckover2pi1Ω = 0.06meV.\nFigure S.9. Normalized GD correction δαG/α0\nGas function of the twist angle for different values of the cuto ff parameter kc.\nCalculations are done for µ= 0,kBT= 25meV, λI= 3meV, λR= 4meV, and for the FMR energy /planckover2pi1Ω = 0.06meV."    },    {        "title": "1605.01694v2.Theory_of_magnon_motive_force_in_chiral_ferromagnets.pdf",        "content": "Theory of magnon motive force in chiral ferromagnets\nUtkan G ung ord u\u0003and Alexey A. Kovalev\nDepartment of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska, Lincoln, Nebraska 68588, USA\nWe predict that magnon motive force can lead to temperature dependent, nonlinear chiral damping\nin both conducting and insulating ferromagnets. We estimate that this damping can signi\fcantly\nin\ruence the motion of skyrmions and domain walls at \fnite temperatures. We also \fnd that in\nsystems with low Gilbert damping moving chiral magnetic textures and resulting magnon motive\nforces can induce large spin and energy currents in the transverse direction.\nPACS numbers: 85.75.-d, 72.20.Pa, 75.30.Ds, 75.78.-n\nEmergent electromagnetism in the context of spintron-\nics [1] brings about interpretations of the spin-transfer\ntorque [2, 3] and spin-motive force (SMF) [4{10] in terms\nof \fctitious electromagnetic \felds. In addition to pro-\nviding beautiful interpretations, these concepts are also\nvery useful in developing the fundamental understanding\nof magnetization dynamics. A time-dependent magnetic\ntexture is known to induce an emergent gauge \feld on\nelectrons [5]. As it turns out, the spin current gener-\nated by the resulting \fctitious Lorentz force (which can\nalso be interpreted as dynamics of Berry-phase leading\nto SMF) in\ruences the magnetization dynamics in a dis-\nsipative way [5, 6, 11{16], a\u000becting the phenomenologi-\ncal Gilbert damping term in the Landau-Lifshitz-Gilbert\n(LLG) [17] equation. Inadequacy of the simple Gilbert\ndamping term has recently been seen experimentally in\ndomain wall creep motion [18]. Potential applications\nof such studies include control of magnetic solitons such\nas domain walls and skyrmions [19{31], which may lead\nto faster magnetic memory and data storage devices\nwith lower power requirements [32{34]. Recently, phe-\nnomena related to spin currents and magnetization dy-\nnamics have also been studied in the context of energy\nharvesting and cooling applications within the \feld of\nspincaloritronics [35{39].\nMagnons, the quantized spin-waves in a magnet, are\npresent in both conducting magnets and insulating mag-\nnets. Treatment of spin-waves with short wavelengths as\nquasiparticles allows us to draw analogies from systems\nwith charge carriers. For instance, the \row of thermal\nmagnons generates a spin transfer torque (STT) [40{42]\nand a time-dependent magnetic texture exerts a magnon\nmotive force. According to the Schr odinger-like equa-\ntion which governs the dynamics of magnons in the adia-\nbatic limit [41], the emergent \\electric\" \feld induced by\nthe time-dependent background magnetic texture exerts\na \\Lorentz force\" on magnons, which in turn generates\na current by \\Ohm's law\" (see Fig. 1). Despite the sim-\nilarities, however, the strength of this feedback current\nhas important di\u000berences from its electronic analog: it is\ninversely proportional to the Gilbert damping and grows\n\u0003ugungordu@unl.eduwith temperature.\nIn this paper, we formulate a theory of magnon feed-\nback damping induced by the magnon motive force. We\n\fnd that this additional damping strongly a\u000bects the dy-\nnamics of magnetic solitons, such as domain walls and\nskyrmions, in systems with strong Dzyaloshinskii-Moriya\ninteractions (DMI). We also \fnd that the magnon mo-\ntive force can lead to magnon accumulation (see Fig. 1),\nnon-vanishing magnon chemical potential, and large spin\nand energy currents in systems with low Gilbert damp-\ning. To demonstrate this, we assume di\u000busive transport\nof magnons in which the magnon non-conserving relax-\nation time, \u001c\u000b, is larger compared to the magnon conserv-\ning one,\u001cm(\u001c\u000b\u001d\u001cm). For the four-magnon thermal-\nization,\u001cm=~=(kBT)(Tc=T)3, and for LLG damping,\n\u001c\u000b=~=\u000bkBT, this leads to the constraint \u000b(Tc=T)3\u001c1\n[43, 44].\nEmergent electromagnetism for magnons. We initially\nassume that the magnon chemical potential is zero. The\nvalidity of this assumption is con\frmed in the last sec-\ntion. E\u000bects related to emergent electromagnetism for\nmagnons can be captured by considering a ferromagnet\nwell below the Curie temperature. We use the stochastic\nLLG equation:\ns(1 +\u000bn\u0002)_n=n\u0002(He\u000b+h); (1)\nwheresis the spin density along n,He\u000b=\u0000\u000enF[n]\nis the e\u000bective magnetic \feld, F[n] =R\nd3rF(n) is the\nfree energy and his the random Langevin \feld. It is\nconvenient to consider the free energy density F(n) =\nJ(@in)2=2 +^Dei\u0001(n\u0002@in) +H\u0001n+Kun2\nzwhereJis\nthe exchange coupling, ^Dis a tensor which describes the\nDMI [47],H=Ms\u00160Haezdescribes the magnetic \feld,\nKudenotes the strength of uniaxial anisotropy, Msis\nthe saturation magnetization, Hais the applied magnetic\n\feld, and summation over repeated indices is implied. At\nsu\u000eciently high temperatures, the form of anisotropies is\nunimportant for the discussion of thermal magnons and\ncan include additional magnetostatic and magnetocrys-\ntalline contributions.\nLinearized dynamics of magnons can be captured by\nthe following equation [48]:\ns(i@t+ns\u0001At) =\u0002\nJ(@i=i\u0000ns\u0001[Ai\u0000Di=J])2+'\u0003\n ;\n(2)arXiv:1605.01694v2  [cond-mat.mes-hall]  13 Jul 20162\nμ/μ0\n-0.3-0.2-0.100.10.20.3\nμ/μ0\n-0.75-0.50-0.2500.250.500.75\nFIG. 1. (Color online) A moving magnetic texture, such as\na domain wall (top) or an isolated skyrmion (bottom), gen-\nerates an emergent electric \feld and accumulates a cloud of\nmagnons around it. In-plane component of nsand electric\n\feldEare represented by small colored arrows and large black\narrows, respectively. Magnon chemical potential \u0016is mea-\nsured in\u00160=\u0018~vD=J \u0001 for domain wall and \u00160=\u0018~vD=JR\nfor skyrmion, where \u0018is the magnon di\u000busion length. Soli-\ntons are moving along + xaxis with velocity v,nz=\u00061 at\nx=\u00071 for domain wall and nz= 1 at the center for the\nisolated hedgehog skyrmion. Material parameters for Co/Pt\n(J= 16pJ/m, D= 4mJ/m2,Ms= 1:1MA/m,\u000b= 0:03,\nat room temperature [45]) were used for domain wall lead-\ning to \u0001\u00197nm, and Cu 2OSeO 3parameters ( J= 1:4pJ/m,\nD= 0:17mJ/m2,s= 0:5~=a3,a= 0:5nm with\u000b= 0:01, at\nT\u001850K [46]) for skyrmion leading to R\u001950nm. System\nsize is taken to be 6\u0001 \u00022\u0001 for domain wall and 6 R\u00026Rfor\nskyrmion.\nwhere'absorbs e\u000bect of anisotropies, DMI and the\nmagnetic \feld,  =nf\u0001(e0\nx+ie0\ny) describes \ructu-\nationsnf=n\u0000nsq\n1\u0000n2\nfaround slow component\nns(jnj=jnsj= 1,ns?nf) in a rotated frame in\nwhiche0\nz=ns,Di=^Dei, andA\u0016\u0002\u0011 ^R@\u0016^RTcorre-\nsponds to the gauge potential with \u0016=x;y;z;t . Note\nthat in the rotated frame, we have n!n0=^Rnand\n@\u0016!(@\u0016\u0000A0\n\u0016\u0002) withA0\n\u0016\u0002= (@\u0016^R)^RT. In deriv-\ning Eq. 2, we assumed that the exchange interaction is\nthe dominant contribution and neglected the coupling\nbetween the circular components of  and ydue to\nanisotropies [41, 49].\nThe gauge potential in Eq. (2) leads to a reactive\ntorque in the LLG equation for the slow dynamics [50].\nAlternatively, one can simply average Eq. (1) over the\nfast oscillations arriving at the LLG equation with the\nmagnon torque term [40]:\ns(1 +\u000bns\u0002)_ns\u0000ns\u0002Hs\ne\u000b=~(j\u0001D)ns;(3)\nwhereHs\ne\u000b=\u0000\u000ensF[ns] is the e\u000bective \feld for theslow magnetization calculated at zero temperature [51],\nji= (J=~)hns\u0001(nf\u0002@inf)iis the magnon current and\nDi=@i+ (^Dei=J)\u0002is the chiral derivative [48, 52, 53].\nMagnon feedback damping. The magnon current jis\ninduced in response to the emergent electromagnetic po-\ntential, and can be related to the driving electric \feld\nEi=~ns\u0001(@tns\u0002Dins) by local Ohm's law j=\u001bE\nwhere\u001bis magnon conductivity. The induced elec-\ntric \feld Ecan be interpreted as a magnon generaliza-\ntion of the spin motive force [6]. The magnon feed-\nback torque\u001c=~\u001b(E\u0001D)nshas dissipative e\u000bect on\nmagnetization dynamics and leads to a damping tensor\n^\u000bemf=\u0011(ns\u0002Dins)\n(ns\u0002Dins) in the LLG equation\nwith\u0011=~2\u001b=s[54].\nA general form of the feedback damping should also\ninclude the contribution from the dissipative torque\n[40]. Here we introduce such \f-terms phenomenologically\nwhich leads to the LLG equation:\ns(1 +ns\u0002[^\u000b+^\u0000])_ns\u0000ns\u0002Hs\ne\u000b=\u001c; (4)\nwhere\u001cis the magnon torque term and we separated the\ndissipative ^ \u000band reactive ^\u0000 contributions:\n^\u000b=\u000b+^\u000bemf\u0000\u0011\f2Dins\nDins; (5)\n^\u0000 =\u0011\f[(ns\u0002Dins)\nDins\u0000Dins\n(ns\u0002Dins)];\nwhere in general the form of chiral derivatives in the \f-\nterms can be di\u000berent. Given that \fand\u000bare typically\nsmall for magnon systems, the term ^ \u000bemfwill dominate\nthe feedback damping tensor. An unusual feature of the\nchiral part of the damping is that it will be present even\nfor a uniform texture. While the DMI prefers twisted\nmagnetic structures, this can be relevant in the presence\nof an external magnetic \feld strong enough to drive the\nsystem into the ferromagnetic phase.\nIn conducting ferromagnets, charge currents also lead\nto a damping tensor of the same form where the strength\nof the damping is characterized by \u0011e=~2\u001be=4e2swith\n\u001beas the electronic conductivity [11, 13, 15], which\nshould be compared to \u0011in conducting ferromagnets\nwhere both e\u000bects are present. Since the magnon feed-\nback damping \u0011grows as/1=\u000b, the overall strength of\nmagnon contribution can quickly become dominant con-\ntribution in ferromagnets with small Gilbert damping.\nUnder the assumption that magnon scattering is domi-\nnated by the Gilbert damping such that the relaxation\ntime is given by \u001c\u000b= 1=2\u000b!, magnon conductivity is\ngiven by\u001b3D\u00181=6\u00192\u0015~\u000bin three-dimensions and \u001b2D\u0018\n1=4\u0019~\u000bin two-dimensions [40] where \u0015=p\n~J=skBTis\nthe wavelength of the thermal magnons. For Cu 2OSeO 3\nin ferromagnetic phase, we \fnd \u0011\u00192nm2. Similarly,\nfor a Pt/Co/AlO xthin \flm of thickness t= 0:6nm yield\n\u0011\u00191nm2at room temperature. This shows that the\nmagnon feedback damping can become signi\fcant in fer-\nromagnets with sharp textures and strong DMI.\nDomain wall dynamics. We describe the domain wall\npro\fle in a ferromagnet with DMI by Walker ansatz3\ntan(\u0012(x;t)=2) = exp(\u0006[x\u0000X(t)]=\u0001) whereX(t) and\n\u001e(t) denote the center position and tilting angle of the\ndomain wall [55], \u0001 =p\nJ=K 0is the domain wall width,\nK0=Ku\u0000\u00160M2\ns=2 includes the contributions from\nuniaxial anisotropy as well as the demagnetizing \feld\nand ^D=\u0000D(sin\r11 + cos\rez\u0002) contains DMI due to\nbulk and structure inversion asymmetries whose relative\nstrength is determined by \r. After integrating the LLG\nequation, we obtain the equations of motion for a domain\nwall driven by external perpendicular \feld [56]:\n\u0000XX_X=\u0001 + _\u001e=FX;\u0000\u001e\u001e_\u001e\u0000_X=\u0001 =F\u001e;(6)\nwhere \u0000XX=\u000b+\u0011(D=J)2sin2(\r+\u001e)=3 and \u0000\u001e\u001e=\n\u000b+\u0011[2=3\u00012+ (\u0019D=2J\u0001) cos(\r+\u001e) + (D=J)2cos2(\r+\n\u001e)] are dimensionless angle-dependent drag coe\u000ecients,\nFX=H=s andF\u001e= [Ksin 2\u001e+ sin(\r+\u001e)D\u0019=2\u0001]=sare\ngeneralized \\forces\" associated with the collective coordi-\nnatesXand\u001e,Kis the strength of an added anisotropy\ncorresponding, e.g., to magnetostatic anisotropy K=\nNx\u00160M2\ns=2 whereNxis the demagnetization coe\u000ecient.\nIn deriving these equations, we have neglected higher or-\nder terms in \u000band\f[57].\nTime-averaged domain wall velocity obtained from\nnumerical integration of the equations of motion for\na Co/Pt interface with Rashba-like DMI is shown in\nFig. 2. Thermal magnon wavelength at room tempera-\nture (\u00190:3nm) is much shorter than the domain wall size\n\u0001 =p\nJ=K 0\u00197nm, so the quasiparticle treatment of\nmagnons is justi\fed. We observe that damping reduces\nthe speed at \fxed magnetic \feld, and this e\u000bect is en-\nhanced with increasing DMI strength Dand diminishing\nthe Gilbert damping \u000b(see Fig. 2).\nAnother important observation is that in the presence\nof the feedback damping, the relation between applied\n\feld and average domain wall velocity becomes nonlin-\near. This is readily seen from steady state solution of the\nequations of motion before the Walker breakdown with\n\u001e=\u001e0which solves sin( \r+\u001e0)D\u0019=2s\u0001 =\u0000[H=s][\u000b+\n\u0011(D=J)2sin2(\r+\u001e0)=3]\u00001(noting that D=\u0001\u001dK,\nimplying a N\u0013 eel domain wall [56, 58]) and X=vt,\nleading to the cubic velocity-\feld relation ( sv=\u0001)(\u000b+\n\u0011[2sv=J\u0019 ]2=3) =Hfor \feld-driven domain wall motion.\nThe angle\u001e0also determines the tilting of Eas seen in\nFig. 1.\nSkyrmion dynamics. Under the assumption that the\nskyrmion retains its internal structure as it moves, we\ntreat it as a magnetic texture ns=ns(r\u0000q(t)) with\nq(t) being the time-dependent position (collective coor-\ndinate [61]) of the skyrmion. We consider the motion\nof a skyrmion under the temperature gradient r\u001f=\n\u0000rT=T, which exerts a magnon torque:\n\u001c= (1 +\fTns\u0002)(Lr\u001f\u0001D)ns; (7)\nwhereLis the spin Seebeck coe\u000ecient and \fTis the \\\f-\ntype\" correction. These are given by L3D\u0018kBT=6\u00192\u0015\u000b\nand\fT\u00193\u000b=2 in three-dimensions and L2D\u0018kBT=4\u0019\u000b\nand\fT\u0019\u000bin two-dimensions within the relaxation time\nD=4mJ/m2\nD=2mJ/m2\nD=1mJ/m2\n0 10 20 30 4002004006008001000\nμ0Ha(mT)〈X〉(m/s)Co/Pt\nD=1.5mJ/m2D=1mJ/m2D=0.5mJ/m2\n0 1 2 3 402004006008001000\nμ0Ha(mT)〈X〉(m/s)Pt/CoFeB/MgOFIG. 2. (Color online) Domain wall velocity as a func-\ntion of the magnetic \feld and varying strength of DMI for\nCo/Pt and Pt/CoFeB/MgO \flms. Solid (dashed) lines cor-\nrespond to dynamics at zero (room) temperature. We used\nmaterial parameters Ms= 1:1MA/m,J= 16pJ/m, K0=\n0:34MJ/m3,\u000b= 0:03 [45] for Co/Pt, and Ms= 0:43MA/m,\nJ= 31pJ/m, K0= 0:38MJ/m3,\u000b= 4\u000210\u00003[31, 59, 60] for\nPt/CoFeB/MgO.\napproximation [41, 42]. Multiplying the LLG equation\nEq. (4) withR\nd2r@qjns\u0001ns\u0002and substituting _ns=\n\u0000_qi@qins, we obtain the equation of motion for v=_q:\ns(W\u0000Qz\u0002)v+ (\fT\u0011D\u0000Qz\u0002)Lr\u001f=F:(8)\nAbove,W=\u00110\u000b+\u0011\u000b0can be interpreted as the con-\ntribution of the renormalized Gilbert damping, Q=R\nd2rns\u0001(@xns\u0002@yns)=4\u0019is the topological charge of the\nskyrmion,\u00110is the dyadic tensor, \u0011Dis the chiral dyadic\ntensor which is\u0018\u00110for isolated skyrmions and vanishes\nfor skyrmions in SkX lattice [48] (detailed de\fnitions of\nthese coe\u000ecients are given in the Supplemental Material\n[62]). The \\force\" term F=\u0000rU(q) due to the e\u000bec-\ntive skyrmion potential U(q) is relevant for systems with\nspatially-dependent anisotropies [63], DMI [64], or mag-\nnetic \felds. In deriving this equation, we only considered\nthe dominant feedback damping contribution ^ \u000bemfwhich\nis justi\fed for small \u000band\f. For temperature gradients\nand forces along the x-axis we obtain velocities:\nvx=\u0000L@x\u001f(Q2+W\fT\u0011D) +FxW\ns(Q2+W2);\nvy=\u0000L@x\u001fQ(\fT\u0011D\u0000W) +FxQ\ns(Q2+W2): (9)\nThe Hall angle de\fned as tan \u0012H=vy=vxis strongly af-\nfected by the renormalization of Wsince tan\u0012H=Q=W\nfor a \\force\" driven skyrmion and tan \u0012H\u0019(\fT\u00110\u0000\nW)=Qfor a temperature gradient driven skyrmion. Sim-\nilar to the domain wall velocity in Fig. 2, the Hall e\u000bect\nwill depend on the overall temperature of the system.\nWe \fnd that for a skyrmion driven by @x\u001f, the Hall an-\ngle\u0012Hmay \rip the sign in magnets with strong DMI\nas the temperature increases. We estimate this should\nhappen in Cu 2OSeO 3atT\u001850K using a typical radial\npro\fle for a rotationally symmetric skyrmion given by\nUsov ansatz cos( \u0012=2) = (R2\u0000r2)=(R2+r2) forr\u0014R\nandR\u00192\u0019J=D\u001952nm.\nMagnon pumping and accumulation. The motion of\nskyrmions induces a transverse magnon current across4\nthe sample. This e\u000bect can be quanti\fed by the av-\nerage magnon current due to magnon motive force per\nskyrmion:\nj=\u001bZ\nd2rE=\u0019R2= (v\u0002ez)4\u001b~2Q=R2: (10)\nThe current can only propagate over the magnon di\u000bu-\nsion length; thus, it can be observed in materials with\nlarge magnon di\u000busion length or small Gilbert damping.\nμ/μ0\n-1.5-1.0-0.500.51.01.5\nFIG. 3. (Color online) An array of moving skyrmions (only 3\nshown in the \fgure) induces a transverse current and accumu-\nlation of magnons along the edges . \u0016is obtained by numer-\nically solving the di\u000busion equation using material parame-\nters for Pt/CoFeB/MgO given in the caption of Fig. 2 with\nR= 35nm. System height and distance between skyrmion\ncenters are taken to be 3 R.\nSo far, we have assumed a highly compressible limit in\nwhich we disregard any build up of the magnon chem-\nical potential \u0016. In a more realistic situation the build\nup of the chemical potential will lead to magnon di\u000bu-\nsion. To illustrate the essential physics, we consider a\nsituation in which the temperature is uniform. For slow\nmagnetization dynamics in which magnons quickly estab-\nlish a stationary state (i.e. R=v\u001d\u001c\u000bfor skyrmions and\n\u0001=_X\u001d\u001c\u000bfor domain walls, which is satis\fed at high\nenough temperatures) we write a stationary magnon dif-\nfusion equation:\nr2\u0016=\u0016\n\u00182+r\u0001E; (11)\nwhere\u0018=\u0015=2\u0019\u000bis the magnon di\u000busion length and\nwe used the local Ohm's law \u0000r\u0016=j=\u001b\u0000E. Renor-\nmalization of magnon current in Eq. (3) then follows\nfrom solution of the screened Poisson equation j=\u001bE+(\u001b=4\u0019)rR\nd3r0(r0\u0001E)e\u0000jr\u0000r0j=\u0018=jr\u0000r0jin three dimen-\nsions andj=\u001bE+ (\u001b=2\u0019)rR\nd2r0(r0\u0001E)K0(jr\u0000r0j=\u0018)\nin two dimensions where K0is the modi\fed Bessel func-\ntion for an in\fnitely large system [65]. By analyzing\nthe magnon current due to magnon accumulation ana-\nlytically and numerically, we \fnd that renormalization\nbecomes important when the length associated with the\nmagnetic texture is much smaller than the magnon dif-\nfusion length.\nFinally, we numerically solve Eq. (11) for isolated soli-\ntons (see Fig. 1) and for an array of moving skyrmions\n(see Fig. 3). Given that the width of the strip in Fig. 3\nis comparable to the magnon di\u000busion length one can\nhave substantial accumulation of magnons close to the\nboundary. Spin currents comparable to the estimate in\nEq. (10) can be generated in this setup and further de-\ntected by the inverse spin Hall e\u000bect [66]. From Eq. (10),\nfor a skyrmion with R= 35nm moving at 10m/s in\nPt/CoFeB/MgO with D= 1:5mJ/m2[31], we obtain an\nestimate for spin current js=j~\u001810\u00007J/m2which\nroughly agrees with the numerical results. This spin cur-\nrent will also carry energy and as a result will lead to a\ntemperature drop between the edges.\nConclusion. We have developed a theory of magnon\nmotive force in chiral conducting and insulating ferro-\nmagnets. The magnon motive force leads to tempera-\nture dependent, chiral feedback damping. The e\u000bect of\nthis damping can be seen in the non-linear, temperature\ndependent behavior of the domain wall velocity. In ad-\ndition, observation of the temperature dependent Hall\nangle of skyrmion motion can also reveal this additional\ndamping contribution. 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Kovalev\nDepartment of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska, Lincoln, Nebraska 68588, USA\nTHIELE’S EQUATION OF MOTION FOR SKYRMION\nWe consider the motion of a rotationally symmetric skyrmion under the influence of applied temperature gradient.\nAssuming that skyrmion drifts without any changes to its internal structure, ns(r,t) =ns(r−q(t)) where qis the\nposition of the skyrmion, we multiply the LLG equation with the operator/integraltext\nd2r∂qjns·ns×and integrate over the\nregion containing the skyrmion and obtain the following equation motion:\ns(W−Q/primez×)v+ (βTηD−Qz×)L∂χ=F. (1)\nwhere v=˙qis the skyrmion velocity, W=η0α+η(α0−β2α2),Lis the spin Seebeck coefficient, χ= 1/T,F=−∇U,\nQis the topological charge defined in the main text and Q/prime=Q−ηβα 1. In terms of the polar coordinates ( θ,φ) of\nns, the dyadic tensor η0and the chiral dyadic tensor ηDare given by\nη0=π/integraldisplayR\n0dr/parenleftbigg(r∂rθ)2+ sin2θ\nr/parenrightbigg\n, ηD=η0+D\nJπ/integraldisplayR\n0dr(sinθcosθ+r∂rθ), (2)\nThe damping terms αican be expanded in powers of D/J asαi=/summationtext\njαβi,Dj(D/J)j, whereαβi,Djis given by\nαβ0,D2=π/integraldisplayR\n0dr(r∂rθ)2cos2θ+ sin2θ\nr\nαβ0,D=π/integraldisplayR\n0dr(r∂rθ)r∂rθtanθcos2θ+ sin2θ\nr2\nαβ0,D0=π/integraldisplayR\n0dr(r∂rθ)22 sin2θ\nr3(3)\nαβ,D2=2π/integraldisplayR\n0dr(∂rθ) sinθ(cos2θ+ 1)\nαβ,D=4π/integraldisplayR\n0dr(∂rθ) sinθcos2θtanθ+r∂rθ\nr\nαβ,D0=2π/integraldisplayR\n0dr(∂rθ) sinθcos2θtan2θ+ (r∂rθ)2\nr2(4)\nαβ2,D2=π/integraldisplayR\n0dr/parenleftbiggcos2θsin2θ+ (r∂rθ)2\nr/parenrightbigg\nαβ2,D=2π/integraldisplayR\n0dr/parenleftbiggcos2θsin2θtanθ+ (r∂rθ)3\nr2/parenrightbigg\nαβ2,D0=π/integraldisplayR\n0dr/parenleftbiggcos2θsin2θtan2θ+ (r∂rθ)4\nr3/parenrightbigg\n(5)\nThe integrals can be evaluated by using an approximate radial profile for θ(r). Using Usov ansatz yields the values\nenumerated in Table I for αβi,Dj. Only the dominant terms are kept in the main text.2\n1 D/J (D/J)2\n1496π\n15R252π\n5R16π\n5\nβC\nR2472π\n15R16π\n3\nβ2 5056\n105R22804π\n105R464π\n105\nTABLE I. List of feedback damping coefficients αβi,Djfor a rotationally symmetric skyrmion using Usov ansatz cos( θ/2) =\n(R2−r2)/(R2+r2) forr≤Rand 0 forr > R . Rows correspond to α0,α1andα2, expanded in powers of D/J. Above,\nC≈449. Remaining parameters are given as η0= 16π/3,ηD=η0+ (4πR/3)(D/J).\nTRANSPORT COEFFICIENTS\nTexture-independent part of the transport coefficients can be obtained using the Boltzmann equation within the\nrelaxation-time approximation in terms of the integral [1, 2]\nJij\nn=1\n(2π)3/planckover2pi1/integraldisplay\nd/epsilon1τ(/epsilon1)(/epsilon1−µ)n(−∂/epsilon1f0)/integraldisplay\ndS/epsilon1vivj\n|v|(6)\nasσ=J0and Π =−J1/J0. Above,τ(/epsilon1) is the relaxation time, /epsilon1(k) =/planckover2pi1ωk,vi=∂ωk/∂ki,dS/epsilon1is the area d2k\ncorresponding to a constant energy surface with /epsilon1(k) =/epsilon1,f0is the Bose-Einstein equilibrium distribution. Under the\nassumption that the scattering processes are dominated by Gilbert damping, we set τ(/epsilon1)≈1/2αω. By evaluating the\nintegral after these substitutions, we obtain σ2D≈F−1/6π2λ/planckover2pi1αin three dimensions ( d= 3), where λ=/radicalbig\n/planckover2pi1J/skBT\nis the wavelength of the thermal magnons, F−1=/integraltext∞\n0d/epsilon1/epsilon1d/2e/epsilon1+x/(/epsilon1+x)(e/epsilon1+x−1)2∼1 evaluated at the magnon gap\nx=/planckover2pi1ω0/kBT. Similarly for d= 2, we obtain σ2D≈F−1/4π/planckover2pi1α.\nThe spin Seebeck coefficient Lis given by−/planckover2pi1σΠ = /planckover2pi1J1, for which we obtain L3D≈F0kBT/6π2λαin 3D and\nL2D≈F0kBT/4παin 2D, where F0=/integraltext∞\n0d/epsilon1/epsilon1d/2/(e/epsilon1+x−1)2∼1. Ford >2 and small x, the numerical factor F0\ncan be expressed in terms of Riemann zeta function and Euler gamma function as ζ(d/2)Γ(d/2 + 1) [3]. In the main\ntext, the numerical factors F−1andF0are omitted.\n[1] N. Ashcroft and N. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).\n[2] A. A. Kovalev and Y. Tserkovnyak, EPL (Europhysics Lett. 97, 67002 (2012).\n[3] R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, 1996), 2nd ed."    },    {        "title": "2303.07025v2.Experimental_investigation_of_the_effect_of_topological_insulator_on_the_magnetization_dynamics_of_ferromagnetic_metal___BiSbTe__1_5_Se__1_5___and__Ni__80_Fe__20___heterostructure.pdf",        "content": "Experimental investigation of the effect of topological insulator on the magnetization\ndynamics of ferromagnetic metal: BiSbTe 1.5Se1.5andNi80Fe20heterostructure\nSayani Pal, Soumik Aon, Subhadip Manna, Sambhu G Nath, Kanav Sharma & Chiranjib Mitra∗\nIndian Institute of Science Education and Research Kolkata,\nMohanpur 741246, West Bengal, India\n(Dated: November 27, 2023)\nWe have studied the spin-pumping phenomenon in ferromagnetic metal( Ni80Fe20)/topological\ninsulator( BiSbTe 1.5Se1.5) bilayer system to understand magnetization dynamics of ferromagnetic\nmetal (FM) in contact with a topological insulator (TI). TIs embody a spin-momentum-locked\nsurface state that spans the bulk band gap. Due to this special spin texture of the topological surface\nstate, the spin-charge interconversion efficiency of TI is even higher than that of heavy metals. We\nevaluated the parameters like effective damping coefficient ( αeff), spin-mixing conductance ( g↑↓\neff)\nand spin current density ( j0\nS) to demonstrate an efficient spin transfer in Ni80Fe20/BiSbTe 1.5Se1.5\nheterostructure. To probe the effect of the topological surface state, a systematic low-temperature\nstudy is crucial as the surface state of TI dominates at lower temperatures. The exponential increase\nof ∆Hfor all different thickness combinations of FM/TI bilayers and the enhancement of effective\ndamping coefficient ( αeff) with lowering temperature confirms that the spin chemical potential bias\ngenerated from spin-pumping induces spin current into the TI surface state. Furthermore, low-\ntemperature measurements of effective magnetization (4 πMeff) and magnetic anisotropy field ( Hk)\nshowed anomaly around the same temperature region where the resistivity of TI starts showing\nmetallic behavior due to the dominance of conducting TI surface state. The anomaly in Hkcan\nresult from the emerging exchange coupling between the TI surface state and the local moments of\nthe FM layer at the interface without any long-range ferromagnetic order in TI at the interface.\nINTRODUCTION\nSpintronics is one of the emerging fields that has\nwitnessed remarkable progress on both fundamen-\ntal and technological fronts over the past couple of\ndecades. Phenomena like spin-orbit torque [1], spin\nHall effect[2], giant magnetoresistance [3], tunnelling\nmagnetoresistance [4], domain wall motion [5] provide\nbasics for applications in memory devices[6], storage\ntechnology[7], logic gates [8] and magnetic sensors [9].\nThese devices utilize the spin degrees of freedom of\nelectrons and their interaction with orbital moments\nthrough spin-orbit coupling. Complete knowledge of the\nprocess of generation, manipulation, and detection of\nspin degrees of freedom or the spin current is essential for\nwidespread applications in this field. If one focuses on\nthe currently available spin current generation processes,\nspin-pumping [10, 11] is one of the most efficient methods\nwhere the precessing magnetization in the ferromagnet\n(FM) injects spin current into the adjacent layer by\ntransferring spin angular momentum. This raises a\nneed to study the effect of spin pumping with special\nemphasis on exploring new materials which can give\nrise to significant spin-charge interconversion efficiency.\nTopological insulators (TI) are a new class of materials\nthat have an interesting spin texture of the surface\nstate, owing to spin-momentum locking[14–17]. The\nmomentum direction of the electron in the surface state\nof TI is perpendicularly locked to its spin polarization\n∗Corresponding author:chiranjib@iiserkol.ac.indirection. Thus the spin-charge interconversion for TI\nis even higher than the heavy metals which makes TIs\nsuitable for spintronics application[12]. As surface states\nare robust against deposition of FM layers on top of TI\n[18], the topological surface states remain intact and\ngapless [19]. TI/FM bilayers have been successfully\nused for the spin current generation in spin-pumping\nexperiments[20, 34–37]. The effect of spin pumping can\nbe witnessed in the enhanced damping coefficient ( αeff)\nvalue of the ferromagnet upon excitations of ferromag-\nnetic resonance (FMR) because, in the spin pumping\nprocess, the net transfer of spin angular momentum into\nTI layer brings about an additional damping torque on\nthe precessing magnetization in the FM. It is difficult\nto fabricate a perfect TI thin film where the bulk state\nof TI is completely insulating. Thus for a complete\nunderstanding of the effect of TI surface state on FM\nmagnetization dynamics, the low-temperature study is\nnecessary where the surface states of TI dominate.\nIn this paper, we present the study of the spin-pumping\nphenomenon in ferromagnetic metal (FM)/ topological\ninsulator (TI) bilayer system. We chose Ni80Fe20\nas the FM layer and BiSbTe 1.5Se1.5as the TI layer.\nCurrently, BiSbTe 1.5Se1.5is one of the best 3D TI\nmaterials in which bulk conduction in thin films is\nnegligible even at room temperature and the dominance\nof surface state is very prominent at lower temperatures\n[21–23]. In our low-temperature measurements, we have\nwitnessed exponential enhancement of FMR linewidth\n(∆H) and effective damping coefficient ( αeff) at lower\ntemperatures. It supports the proposal of the spin\nchemical potential bias induced spin current injectionarXiv:2303.07025v2  [cond-mat.mes-hall]  24 Nov 20232\ninto the surface state of TI given by Abdulahad et al. [50].\nFor further investigation of the effect of the TI surface\nstate on the FM magnetization, we have also studied\nlow-temperature variations of effective magnetization\nand anisotropy field. We calculated the interfacial\nmagnetic anisotropy of the bilayer to be in-plane of the\ninterface. At low temperatures, this magnetic anisotropy\nfield shows a hump-like feature concomitant with the\nresistivity behavior of BiSbTe 1.5Se1.5with temperature.\nIt predicts the existence of exchange coupling between\nthe surface states of TI and the local moments of the FM\nlayer which acts perpendicular to the TI/FM interface.\nWe have also evaluated the values of spin-transport\nparameters like spin-mixing conductance, g↑↓\neffand spin\ncurrent density, j0\nsat room temperature to ensure a suc-\ncessful spin injection into the TI layer from the FM layer.\nSAMPLE PREPARATION AND\nCHARACTERIZATION\nFor this particular work, we have prepared dif-\nferent thickness combinations of topological insu-\nlator(TI)/ferromagnet(FM) bilayer heterostructure.\nBiSbTe 1.5Se1.5(BSTS ) has been taken as the TI\nmaterial and Permalloy( Ni80Fe20) has been used as\nthe ferromagnetic material. BSTS thin films were\ngrown on silicon (Si 111) substrate using pulsed laser\ndeposition(PLD) technique [24, 25]. The target material\nwas prepared using 99 .999% pure Bi, Sb, Te, and Se in a\n1:1:1.5:1.5 stoichiometric ratio. The films were deposited\nthrough ablation of the target by a KrF excimer laser\n(248 nm, 25 ns pulse width) at a low repetition rate of\n1Hz and 1 .2Jcm−2laser fluence keeping the substrate\ntemperature fixed at 2500Cand the chamber partial\npressure at 0.5 mbar (base pressure 2 ×10−5mbar)\nwith a continuous flow of Ar gas. After deposition,\nTI films were immediately transferred into the thermal\nevaporation chamber for the deposition of the FM\nlayer. Commercially available 99 .995% pure permalloy\n(Ni80Fe20) pallets were used for deposition. The Py\nfilm was deposited [26] on top of TI film at a rate of\n1.2˚A(crystal monitor: Inficon SQM 160) keeping the\nchamber pressure fixed at 1 ×10−6torr (base pressure\n1×10−7torr). For the characterization of the films\nX-ray diffraction analysis (XRD), field emission scanning\nelectron microscope (FE-SEM) imaging, and atomic\nforce microscopy (AFM) facilities have been used. X-ray\nreflectometry technique has been used for thickness\nmeasurements here. For convenience we are defining the\nBSTS of different thicknesses as follows: 10nm BSTS as\nBSTS1, 21nm BSTS as BSTS2, 28nm BSTS as BSTS3,\nand 37nm BSTS as BSTS4.RESULTS AND DISCUSSION\nFor a systematic study of the FM/TI bilayer system,\nwe have done in-plane FMR measurements in reflection\nmode geometry using a short-circuited CPW as shown\nin fig.1a. We obtained typical FMR signal at different\nmicrowave frequencies for Py(15nm)/BSTS2 sample in\nfig.1b. From the Lorentz formula fitting [53] of the FMR\nsignal we extracted the frequency dependence of the field\nlinewidth (∆ Hvs.f) and the resonance frequency vs.\nresonance field ( fvsH) data as shown in fig.2a and fig.2b\nrespectively. These give us valuable information about\nthe magnetization dynamics in ferromagnet which can\nbe described within the framework proposed by Landau,\nLifshitz, and Gilbert [30],\nd⃗M\ndt=−γ⃗M×⃗Heff+αeff\nMS⃗M×d⃗M\ndt(1)\nwhere, γis the gyromagnetic ratio, ⃗Mis the magneti-\nzation vector, MSis the saturation magnetization, Heff\nis the effective magnetic field which includes the exter-\nnal field, demagnetization and crystalline anisotropy field\nandαeffis the effective damping coefficient of the sys-\ntem.\nFor a given magnetic material at ferromagnetic res-\nonance, the resonance field and frequency follow Kittel\nequation[27] given by,\nf=γ\n2πq\n(H+Hk)(H+Hk+ 4πMeff) (2)\nwhere H,Hk, and 4 πMeffare the externally ap-\nplied field, magnetic anisotropy field, and effective mag-\nnetization respectively. We have obtained Hkand\n4πMefffor different FM/TI bilayer systems by fitting\nthe Kittel equation to the fvs. Hcurve as shown\nin fig.2b. The obtained 4 πMeffvalue contains satura-\ntion magnetization(4 πMs) and other anisotropic contri-\nbutions. We can evaluate 4 πMsvalue by analyzing the\nthickness dependent measurement of 4 πMeffof the FM\nlayer. In the lower thickness region of the ferromagnetic\nthin films, 4 πMeffis inversely proportional to the film\nthickness and follows the equation[28],\n4πMeff= 4πMs−2Ks\nMsd(3)\nwhere Ksis the surface anisotropy constant and dis the\nthickness of the FM film. The slope of the linear fit\ngives the anisotropy field contribution to 4 πMeffand\nthe intercept gives the 4 πMsvalue as shown in fig.2c.\nThe 4 πMeffdoes not depend on the thickness varia-\ntion of BSTS at room temperature but 4 πMefffor Py(t)\nmonolayer samples and for Py(t)/BSTS2 bilayer sam-\nples vary linearly with the inverse Py thickness as shown\nin Fig.2c. From the linear fitting (Eq.3) of 4 πMeff3\n(a)\n (b)\nFIG. 1. (a) In the left diagram, a schematic illustration of the experimental set-up has shown where the FM/TI bilayer is\nplaced upside down on top of a CPW, and in the right diagram, net injected spin current ( Ipump\nS ) due to spin-pumping into\nthe TI layer (BSTS) from the FM layer (Py) has shown, it results faster magnetization relaxation in FM; (b) Ferromagnetic\nResonance spectra of absorption at different frequencies for Py/BSTS bilayer system at room temperature after background\nsubtraction.\n(a)\n (b)\n (c)\nFIG. 2. (a) Field linewidth (∆ H) variation with resonance frequencies ( f) at 300K for Py/BSTS bilayer samples with different\nPy thicknesses. Eq.4 has been used for fitting the curve and to determine the damping coefficient ;(b) Resonance field ( H) vs.\nresonance frequency ( f) for Py(20nm)/BSTS2 system at different temperatures . Eq.2 has been used for fitting the curve and\nto determine the effective magnetization; (c) Effective magnetization (4 πMeff) variation with thickness of Py(t), Py(t)/BSTS2\nand Py(15nm/BSTS(t) at room temperature. Eq3 has been used for fitting the curve and to evaluate saturation magnetization\n(4πMS) and magnetic anisotropy field( Hk).\n(a)\n (b)\nFIG. 3. (a) αeffvariation with Py thickness for Py(t)/BSTS2 heterostructure at room temperature which fits in Eq.5; (b)\nαeffas a function of BSTS thickness for Py(15nm)/BSTS(t) heterostructure at room temperature.4\n(a)\nFIG. 4. Temperature dependence of resistivity of the BSTS\nsample of thickness 21nm deposited on Si(111) substrate.\nvs. 1 /tPydata for the Py(t) and Py(t)/BSTS2 sam-\nples, we evaluated the saturation magnetization, Msof\nthe Py/BSTS bilayer that has been decreased from that\nof the bare Py sample by an amount of 183 emu/cc3.\nIt is a result of the loss of ferromagnetic order in the\nPermalloy layer. Due to the intermixing of the Py and\nBSTS at the interface, a magnetic dead layer could have\nformed at the interface which resulted in the reduction\nof saturation magnetization value as suggested by some\nprevious studies [42–44] also. The Ksvalue has de-\ncreased from 0 .092±0.008erg/cm2in bare Py film to\n0.091±0.015erg/cm2in Py/BSTS2 bilayer. So interfa-\ncial anisotropy constant, Ki(=KPy/TI\ns −KPy\ns) for the\nPy/BSTS2 sample is −0.001erg/cm2. From the nega-\ntive value of Ki, we can ensure an in-plane magnetic\nanisotropy in the Py/BSTS interface at room temper-\nature. A detailed discussion of magnetic anisotropy has\nbeen provided in the last section where the temperature\nvariation of Hkis discussed.\nαeffcan be determined by analysing ∆ Hat different\nfrequencies. ∆ Hcontains both the intrinsic and extrin-\nsic contributions to the damping. Linewidth due to in-\ntrinsic damping is directly proportional to the resonance\nfrequency( f) and follows the equation[29],\n∆H= ∆H0+ (2παeff\nγ)f (4)\nwhere ∆ H0describes inhomogeneous linewidth broad-\nening [38, 39] due to different extrinsic contributions\nlike magnetic inhomogeneities [40, 41], surface roughness,\nand defects in the sample. We have evaluated the αeff\nvalues by fitting the ∆ Hvsfcurve for FM/TI bilayers\nas shown in fig.2a. This αeffconsists of Gilbert damp-\ning in the bulk ferromagnet( αFM) and the enhanced\ndamping( αSP) resulting from spin pumping into the ad-\njacent TI layer [31–33], αeff=αFM+αSP. The αFM\nvalue for bare Py film of thickness 15nm was calculated\nto be 0.0074 and for the FM/TI bilayer system there has\nbeen significant enhancement in the αeffvalue over thebare Py value due to spin pumping, αSP. In this het-\nerostructure, αeffincreases gradually as the thickness of\nPy decreases both for Py(t) and Py(t)/BSTS2 samples as\nshown in fig.3a. From the linear fit of αeffvs. 1/tPydata\nwe have obtained the spin-mixing coefficient, g↑↓\nefffor the\nBSTS/Py interface to be 5 .26×1018±0.71×1018m−2\nby using the equation[34, 37],\nαeff−αFM=gµB\n4πMstFMg↑↓\neff(5)\nwhere, gandµBare the g-factor and Bohr magneton\nrespectively. We have also calculated the spin current\ndensity( j0\ns) for the FM/TI heterostructure using the g↑↓\neff\nvalue in the following equation[20, 36],\nj0\ns=g↑↓\neffγ2h2\nmℏ[4πMsγ+p\n(4πMs)2γ2+ 4ω2]\n8πα2[(4πMs)2γ2+ 4γ2](6)\nwhere γ,hm,ℏ,ω, and αare the gyromagnetic ratio,\nmicrowave magnetic field, Planck’s constant, Larmour\nprecession frequency, and effective damping parameter\nrespectively. Using Eq.6 the j0\nsvalue for Py/BSTS2\nwas obtained to be 0 .901×10−10±0.122×10−10Jm−2\nin our experiment. The g↑↓\neffandj0\nsvalues obtained\nfrom Py thickness-dependent study of αeffare in a\ncomparable range of the previously reported values\nfor other combinations of ferromagnet and TI bilayer\nstructures [34, 35, 37]. This gives evidence of successful\nspin injection into the BSTS layer from the Py layer\ndue to spin pumping [31–33, 50]. We also report the TI\nthickness-dependent study of αeffas shown in fig.3b.\nFor bilayer structures of Py(15nm)/BSTS2(t) there is\na sudden jump in the αeffvalue from that of the bare\nFM film ( αFM = 0.0074) because of spin pumping.\nThen with the thickness variation of TI layer in the\nrange of 10nm to 37nm, αeffincreases slowly from\n0.015 to 0.02. The TI thickness dependence of αeff\nfor Py(15nm)/BSTS(t) bilayer is almost linear which\ncertainly can not be described by the conventional\nspin diffusion theory [48] for FM/NM proposed by\nTserkovnyak et al. [47]. For Py/BSTS heterostructure,\nαeffvs. tBSTS study suggests an efficient spin-sink\nnature of the TI bulk with increasing thickness at\nroom temperature [49]. From the room temperature\nstudy we certainly can not distinguish the TI surface\nstate contribution from the TI bulk state contribution\nbecause growing a BSTS thin film with a perfectly\ninsulating bulk state is still very challenging. Thus it\nwas imperative to study the effect of topological surface\nstate at low-temperature where bulk states of TI get\nsuppressed and surface states of TI starts to dominate.\nIn this section, we have focused on low-temperature\nmeasurements specifically to understand the effect of\ntopological surface states (TSS) on the magnetization\nrelaxation of FM. At higher temperatures, a significant\namount of bulk carriers are available to participate\nin the transport but with the reduction of phonon5\n(a)\n (b)\nFIG. 5. (a)Temperature dependence of the field linewidth (∆ H) for different thickness combinations of Py/BSTS bilayer\nsystems and for a bare Py thin film. The solid lines are the fits in the expression exp(−T/T 0); (b)Temperature dependence of\neffective damping coefficient, αeffof Py(20nm)/BSTS2 and bare Py(20nm) film.\n(a)\n (b)\nFIG. 6. (a)Temperature dependence of effective magnetization of Py(20nm/BSTS2); (b)Temperature dependence of the\nanisotropy field of Py(20nm)/BSTS2.\nscattering, surface carriers dominate at a lower tem-\nperature. From the resistivity vs. temperature data of\nBSTS2 in fig.4, we can see an insulating behavior of\nresistivity due to the enhanced insulating nature of the\nbulk state of TI at higher temperatures and a metallic\nbehavior of resistivity below a certain temperature\nwhere the topological surface states dominate. We\nmeasured temperature variation of FMR linewidth\n(∆H), enhanced damping coefficient ( αeff), anisotropy\nfield ( Hk) and effective magnetization (4 πMeff). For\ndifferent thickness combinations of Py/BSTS bilayer,\nwe obtained the ∆ Hvariation with temperature. It\nincreases exponentially with decreasing temperature\nthat fits the expression, exp(−T/T 0) as shown in fig.5a.\nFor bare Py(15nm) film, we can note that there is no\nsignificant variation in ∆ Hat low temperatures as can\nbe seen from the curve at the bottom of fig.5a. To\ngain further understanding, the temperature variation\nofαeffhas also been studied for Py(20nm)/BSTS2\nas shown in fig.5b and compared with αefffor barePy film. From the enhancement of αeffvalue for\nPy(20nm)/ BSTS2 at room temperature we can ensure a\nsuccessful spin injection due to the spin pumping effect.\nBut the exponential increase of αeffwith decreasing\ntemperature for the bilayer implies a huge increment in\nthe amount of spin angular momentum transfer into the\nTI layer at lower temperatures. We attribute the origin\nof the exponential increase of αeffand ∆ Hat lower\ntemperatures to the spin chemical potential bias induced\nspin current into the surface state of TI as proposed by\nAbdulahad et al. [50]. The induced spin current into\nthe TI surface state at lower temperatures corresponds\nto the rapid relaxation of magnetization precession of\nFM which is reflected in the exponential increase of ∆ H\nandαeffof the ferromagnet.\nTo further investigate the effect of TI surface state on\nthe magnetization of FM, we studied the temperature\nvariation of 4 πMeffandHkfor Py(20nm)/BSTS2. In\nour previous study [26] with bare Py thin films, we have6\nseen that 4 πMeffincreases monotonically as saturation\nmagnetization increases with lowering the temperature.\nBut from fig.6a, we can see that the low-temperature\ndependence of 4 πMefffor Py/BSTS2 bilayer deviates\nfrom the single layer Py film [Supplementary fig.S11(a)].\nThis anomaly in 4 πMeffis related to the change of mag-\nnetic anisotropy energy of the system as well as the other\neffects like spin chemical potential induced current and\nexchange coupling between TSS and FM as mentioned\nby Abdulahad et al. [50]. In a previous section, we\nevaluated the interfacial magnetic anisotropy coefficient\n(Ki=−0.001erg/cm2) to be in-plane of the interface of\nthe Py/BSTS2 bilayer. The anisotropy field associated\nwith the system anisotropy energy shows an interesting\nnature as we lower the temperature. We can see from\nfig.6b that Hkincreases initially with decreasing tem-\nperature until a certain value is reached and then the\nanisotropy field weakens against a further decrease in\ntemperature. Thus we get a hump-like feature of HK\nfor the same temperature region where 4 πMeffshows\nthe anomaly and it is concomitant with the resistivity vs\ntemperature behavior of the BSTS2 sample. The low-\ntemperature behavior of Hkand 4 πMeffcan be justified\nby the argument proposed by Abdulahad et al. [50]. In\ntheir phenomenological model, they propose an existence\nof exchange interaction between the surface states of TI\nand local moments of the ferromagnetic layer. Several\ntheoretical as well as experimental predictions confirm\nthe existence of gapless topological surface states even\nafter transition metal deposition on TI [51, 52]. These\nsurface states can couple with the local moments of the\nFM through exchange interaction without any long-range\nferromagnetic order. This exchange coupling acts per-\npendicular to the TI surface and weakens the in-plane\nanisotropy at lower temperatures where the surface states\nof TI dominate.\nCONCLUSIONS\nIn summary, we have carried out spin-pumping ex-\nperiment in BiSbTe 1.5Se1.5(TI)/ Ni80Fe20(FM) bilayer\nsystem. From the thickness-dependent measurements of\nFM/TI bilayers, we obtained the spin-transport param-\neters like damping coefficient due to spin-pumping, spin\nmixing conductance, and spin current density at room\ntemperature. These results demonstrate a successful spin\ntransfer from the FM layer to the TI layer due to spin-\npumping. We have performed low-temperature measure-\nments to specifically understand the surface state con-\ntribution of TI on the FM magnetization because the\nsurface states of TI are more pronounced at lower tem-\nperatures. We have confirmed the suppression of the\ninsulating bulk state of TI at lower temperatures from\nthe resistivity vs. temperature data of TI. In our low-\ntemperature measurements of FMR linewidth and ef-fective damping coefficient, we have witnessed an expo-\nnential increase in both parameters with the decrease in\ntemperature. It suggests a spin chemical potential bias-\ninduced spin current injection into the surface states of\nTI that gets enhanced at low temperatures [50]. We have\nalso studied temperature variations of the effective mag-\nnetization of the system. It showed a deviation from\nthe bare Py film [26] in the temperature regime where\nTI surface states dominate. This deviation of effective\nmagnetization results from the change in the anisotropy\nenergy of the system. At room temperature, we eval-\nuated the magnetic anisotropy energy coefficient which\nis found to be in-plane of the interface. This in-plane\nanisotropy weakens when conducting surface state of TI\nstarts to dominate. It reflects from the hump-like feature\nin the magnetic anisotropy field vs. temperature data of\nthe bilayer system. The decrease in in-plane magnetic\nanisotropy below a certain temperature can result from\nthe exchange coupling between the surface states of TI\nand the local moments of the FM layer which act per-\npendicular to the interface [50]. Combining the results of\nour low-temperature measurements we can conclude that\nthere exists an exchange coupling between the TI surface\nstate and FM which does not create any long-range ferro-\nmagnetic order in the TI and is unable to alter the overall\nspin texture of the TI surface state at the interface[18].\nHowever, it affects the magnetization dynamics of the\nferromagnetic metal quite significantly. These added fea-\ntures of enhancing the damping coefficients enables an-\nother fast control of magnetization dynamics in the FM\nlayer.\nACKNOWLEDGEMENTS\nThe authors sincerely acknowledge the Ministry\nof Education, Government of India and Science\nand Engineering Research Board (SERB) (grant no:\nEMR/2016/007950), and Department of Science and\nTechnology (grant no. DST/ICPS/Quest/2019/22) for\nfinancial support. S.P. acknowledges the Department\nof Science and Technology(DST)-INSPIRE fellowship In-\ndia, S. A. acknowledges the Ministry of Education of the\nGovernment of India, S.M. acknowledges the Council Of\nScientific and Industrial Research(CSIR), India, S.G.N\nand K.S acknowledges the University Grant Commis-\nsion, India for research fellowship. The authors would\nlike to thank Dr. Partha Mitra of the Department of\nPhysics, Indian Institute of Science Education and Re-\nsearch Kolkata, for providing the lab facilities for sample\ndeposition. 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Journal of Magnetism and Mag-\nnetic Materials, 166(1-2), pp.6-26."    },    {        "title": "0806.4656v2.Theory_of_spin_magnetohydrodynamics.pdf",        "content": "Theory of Spin Magnetohydrodynamics\nYaroslav Tserkovnyak and Clement H. Wong\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n(Dated: June 10, 2021)\nWe develop a phenomenological hydrodynamic theory of coherent magnetic precession coupled\nto electric currents. Exchange interaction between electron spin and collective magnetic texture\nproduces two reciprocal e\u000bects: spin-transfer torque on the magnetic order parameter and the Berry-\nphase gauge \feld experienced by the itinerant electrons. The dissipative processes are governed\nby three coe\u000ecients: the ohmic resistance, Gilbert damping of the magnetization, and the \\ \f\ncoe\u000ecient\" describing viscous coupling between magnetic dynamics and electric current, which stems\nfrom spin mistracking of the magnetic order. We develop general magnetohydrodynamic equations\nand discuss the net dissipation produced by the coupled dynamics. The latter in particular allows\nus to determine a lower bound on the magnetic-texture resistivity.\nPACS numbers: 72.15.Gd,72.25.-b,75.75.+a\nI. INTRODUCTION\nConduction electrons moving in a ferromagnet interact\nwith the magnetization through the exchange interaction.\nIf the exchange \feld is strong and slowly varying in space\nand time, the electron spin will adiabatically follow the\ndirection of the magnetization. We may then consider\nelectrons with spins up and down along the magnetiza-\ntion direction as two distinct species of particles, and for\nconvenience call them spin up/down electrons. As is well\nknown, a spin up/down electron wave packet acquires a\nBerry phase1that in\ruences their orbital motion. In ef-\nfect, the electrons experience a Lorentz force due to \\\fc-\ntitious\" electromagnetic \felds which are local functions\nof the magnetization.2\nIn this \fctitious electrodynamics, spin up/down elec-\ntrons have opposite charges and di\u000berent conductivities.\nTheir motion and associated currents interact with the\nmagnetization through what is commonly called current-\ndriven spin-transfer torques. We call this interplay be-\ntween spin currents and magnetization spin magneto-\nhydrodynamics , in analogy to the classical theory of\nmagnetohydrodynamics,3where the magnetic \felds cou-\nple to electric currents in conducting \ruids, and the cur-\nrents in turn generate magnetic \felds. In our spin magne-\ntohydrodynamics, the Maxwell's equations for the mag-\nnetic \feld are replaced by the Landau-Lifshitz-Gilbert\n(LLG) equation for the magnetization. In this paper, we\nneglect full dynamics of the real electromagnetic \felds,\nfocusing on the spin-related phenomena.\nThe electron spin follows the magnetization direction\nperfectly only in the limit of an in\fnitely large exchange\n\feld. In reality, there will be some misalignment and\nassociated spin relaxation. This is usually described\nphenomenologically as a dissipative spin torque with a\ncoe\u000ecient\fin the Landau-Lifshitz equation.4,5,6In a\none-dimensional ring geometry, we will derive the com-\nplete set of coupled spin-magnetohydrodynamical equa-\ntions, starting from the semi-phenomenological dynami-\ncal equations for nonequilibrium currents and magneti-\nzation. We recast the reactive spin torque mediated bythe Berry phase in this thermodynamic context. In our\ntheory, we take an alternative view that the \fterm arises\nfrom a correction to the Berry-phase electromotive force\n(EMF) in the equation of motion for the charge current,\nwith the appropriate dissipative spin torque established\nby the Onsager reciprocity.\nThis physics is presently vigorously studied (exper-\nimentally as well as theoretically) in the contexts of\ncurrent-driven magnetic excitations and domain-wall\nmotion4,5,6,7,8,9,10and the reciprocal spin accumula-\ntions and voltages generated by the \fctitious gauge\n\felds.11,12,13,14,15,16Since the mesoscopic regime (mainly\ndealing with variants of magnetic spin valves, tunnel\njunctions, and magnetic multilayers) is at present well\nexplored,17we will limit our attention here to the case of\ncontinuous magnetic systems.\nII. NONDISSIPATIVE SPIN TORQUE\nSince the underlying physics is rich and complex in\nthe most general setting, we will limit our discussion to\na simple setting, which we believe captures all the es-\nsential ingredients of the spin magnetohydrodynamics.\nConsider a uniform current in a ferromagnetic ring, as-\nsuming for simplicity incompressible electric \rows (the\ncontinuity equation prohibits current inhomogeneities for\nan incompressible electron \ruid). The electric current is\nthen the only dynamical variable describing the electron\n\ruid. The magnetic texture here could be a domain wall\nor magnetic spiral, for example (in higher dimensions we\ncould have topological twists and kinks such as vortices,\nhedgehogs, or skyrmions). See Fig. 1 for a schematic\nof the setup. In the Landau-Lifshitz phenomenology of\nferromagnetic dynamics well below the Curie tempera-\nture, only the instantaneous direction of the magnetiza-\ntionm(x;t) (or, equivalently, spin density) is assumed\nto be a dynamic variable. The magnitude of the spin\ndensitySalong mis assumed to be uniform and con-\nstant in time. We will separately drive the current with\na time-dependent external magnetic \rux \b( t) inside thearXiv:0806.4656v2  [cond-mat.mes-hall]  6 Jan 20092\n!!J(t)m!H(x,t)!(t)e\"\nFIG. 1: (color online). Schematics of our principal \\study\ncase:\" Uniform electric current J(t) carried by itinerant elec-\ntrons can be driven by the external magnetic \rux \b( t) gen-\nerating the EMF E=\u0000@t\b=c. The magnetic texture m(x;t)\nresponds to the e\u000bective \feld H(x;t), which may have an ex-\nternal contribution applied to the wire independently of \b.\nThe reactive magnetohydrodynamic coupling stems from the\nBerry phase \b0, which is acquired by the electron spin (shown\nin blue) following the instantaneous magnetic pro\fle (shown\nin red) around the loop. \b0corresponds geometrically to the\nsolid angle enclosed by the electron spin. Coupled dissipative\nprocesses arise once we relax the projection approximation,\nallowing for some orientational spin mistracking and dephas-\ning as electrons propagate through the magnetic texture.\nring, and the magnetic dynamics with a magnetic \feld\nh(x;t) applied directly to the wire.\nThe \frst step in our phenomenology is to identify\nthe free energyFas a function of the thermodynamic\nvariablesJandm(x;t) (or their thermodynamic con-\njugates), which completely determine the macroscopic\nstate of our system, assuming local thermal equilibrium.\nNeglecting spin, the gauge-invariant free energy associ-\nated with an electric current in the ring is given by\nF(J;\b) = (J\u0000 \b=c)2=2L, where we de\fne LJto be\nthe current corresponding to the canonical momentum\nof the electrons. Lis the self-inductance of the ring and\ncis the speed of light. However, spin up/down electrons\npropagating through a quasistatic magnetic texture18ac-\ncumulate also a Berry phase,1which gives a \fctitious\ncontribution to the vector potential associated with a\n\fctitious EMF.11This vector potential is given (in some\nconvenient gauge) by14A0\nx= (~c=e) sin2(\u0012=2)@x\u001e, pro-\nducing gauge-invariant \fctitious \rux,\n\b0=I\ndxA0\nx=~c\n2eI\ndx(1\u0000cos\u0012)@x\u001e: (1)\n(\u0012;\u001e) are the spherical angles parametrizing m(x).e>0\nis minus the electron charge. Eq. (1) is the \rux associated\nwith spin-up electrons adiabatically following magnetic\ntexture, with the opposite result for spin-down electrons.\nThe free energy accounting for the Berry phase be-\ncomes\nF0(J;\b;\b0[m(x;t)]) = [J\u0000 (\b +p\b0)=c]2=2L; (2)wherepis the polarization of the spin s-dependent con-\nductivity\u001bs:p= (\u001b\"\u0000\u001b#)=(\u001b\"+\u001b#) (assuming fast\nspin relaxation or halfmetallic ferromagnets). The elec-\ntric current is given by\nJ\u0011\u0000c@\bF0= [J\u0000 (\b +p\b0)=c]=L=@JF0;(3)\nwhich is thus the thermodynamic conjugate of J. The\nequation of motion for current in our simple electric cir-\ncuit is given by Ohm's law,\n@tJ\u0011L@tJ+@t(\b +p\b0)=c=\u0000RJ: (4)\nwhereRis the resistance of the wire. Naturally, the\ndynamic Berry phase is seen to give a contribution to\nthe EMF:11\nE0\u0011\u0000p@t\b0=c=PI\ndxm\u0001(@xm\u0002@tm);(5)\nwhich is a well-known result.2(We de\fnedP=p~=2e.)\nNow that the free energy of the current is coupled to\nthe magnetization of the ring through the Berry-phase\n\rux, there will be a corresponding reactive coupling of\nthe magnetization to the current. We describe magnetic\ndynamics by the Landau-Lifshitz-Gilbert equation19\n@tm=H\u0002m=S\u0000\u000bm\u0002@tm; (6)\nwhere the e\u000bective \feld His de\fned by the functional\nderivative, H\u0011@mF(so that locally H?m), and\n\u000bis the dimensionless Gilbert damping20parameter.\nThe total free energy of our magnetoelectric system is\nF(m;J;\b) =F(m)+F0(J;\b;\b0[m(x;t)]), whereF(m)\nis a standard free energy of the ferromagnet. Variation of\ntheF0with respect to mgives current-driven spin torque\napplied to the magnetic dynamics:21\u001c0\u0011@mF0\u0002m,\nwhere@mF0\u0011@m\b0@\b0F0=\u0000pJ@ m\b0=c. Di\u000berentiat-\ning Berry phase (1) with respect to m, we \fnd\n\u001c0=PJ@xm: (7)\nSince ~=2eis the electron spin-charge conversion factor,\nwe can give another interpretation of this term. It is sim-\nply the rate of change of the angular momentum of the\nconducting electrons with spins locked to the magnetic\npro\fle. The spins of the up/down electrons rotate in the\nopposite directions so that, if the spin up/down conduc-\ntivities are the same (and thus P= 0), the net change in\ntheir angular momentum vanishes. Putting this term on\nthe left-hand side, we get\n@tm\u0000PJ@xm=S=@mF(m)\u0002m=S\u0000\u000bm\u0002@tm:(8)\nThe left-hand side of this equation is the rate of change\nof the total angular-momentum density of the magneto-\nelectric system,2while the right-hand side gives the usual\nLLG torque on the system.3\nIII. DISSIPATIVE SPIN TORQUE\nLLG equation (6) with torque (7) and Ohm's law (4)\nwith the \fctitious EMF (5) now constitute coupled equa-\ntions of our spin magnetohydrodynamic theory, with the\nreactive coupling mediated by Berry phase (1). We re-\nproduce them here for clarity (after putting the magne-\ntization equation in the Landau-Lifshitz form):\n@tJ=\u0000RJ; @ tm=H\u0002m\u0000\u000bH\n(1 +\u000b2)S: (9)\nThese are the equations of motion for a quasistationary,\nthermodynamic system near equilibrium.22In equilib-\nrium, the current Jis zero and magnetization is static.\nOut of equilibrium, the \frst-order time derivatives of\n(J;m) are completely speci\fed by the instantaneous val-\nues of their thermodynamic conjugates ( J;H). The right-\nhand side is a linear expansion in these conjugates with\ndissipative coe\u000ecients Rand\u000bthat cause the system to\nrelax back to equilibrium. So far, the dissipation in the\ncurrent and magnetization is separate and physically un-\nrelated. We now add the dissipative couplings which will\nbe key results of this paper.\nWe proceed phenomenologically by adding to the cur-\nrent equation (4) correction \u0001 E0to the Berry-phase EMF\nand correction R0to resistance, due to coupling with the\nmagnetic texture m(x;t). The modi\fed Ohm's law then\nbecomes:\n@tJ=\u0000(R+R0)J+ \u0001E0(10)\nTo avoid a slew of uninteresting coe\u000ecients and\nanisotropies, we will constrain the phenomenology by as-\nsuming spin-rotational symmetry of the magnetic texture\nand the inversion symmetry of the wire. Under the lat-\nter,m!m,J!\u0000J,@x!\u0000@x, andE0!\u0000E0. In the\nspirit of the standard quasistationary description,22we\nexpand only up to the linear order in the nonequilibrium\nquantitiesJand@tm, so that terms of the form, e.g.,\nJ2@tm\u0001@xmare excluded. To the second order in @xm,\nthe only possible terms satisfying these requirements are:\n\u0001E0\u0000R0J=\n\fPI\ndx@xm\u0001@tm\u0000\u0011\f2P2\n\u000bSJI\ndx(@xm)2:(11)\nThe \frst term stems physically from a spin mistracking\nof electrons propagating through the magnetic texture.14\nSince the mistracking should scale as 1 =\u0001xc(vanishing in\nthe limit of in\fnite exchange \u0001 xc), we may anticipate the\ndissipative coupling to be governed by a small parame-\nter\f\u0018~=\u001cs\u0001xc, where\u001csis a characteristic (transverse)\nspin-dephasing time. The \u0011term in Eq. (10) describes the\nresistance associated with magnetic texture, which is of-\nten discussed in the context of magnetic domain walls.23\nBoth terms in Eq. (11) are odd under time reversal, like\nohmic resistance and Gilbert damping. Finally, we notethat including in Eq. (11) a reactive term of the form\n(5) would not add anything new to the following consid-\nerations, as long as we treat Pas a phenomenological\ncoe\u000ecient.\nOur modi\fcation of Ohm's law must respect the\nOnsager reciprocity principle.22Substituting @tmfrom\nEqs. (9) into Eq. (11), we see how the e\u000bective \feld H\n(which is conjugate to m) a\u000bects the dynamics of J. The\nOnsager theorem is now readily applied to determine how\nthe electric current J(which is conjugate to J) should\nmodify the dynamics of m. We write the \fnal result as\na correction to the spin torque (7):\n\u0001\u001c0=\fPJm\u0002@xm: (12)\nThe complete equation of motion of the magnetic texture\nin the LLG form thus becomes\n@tm=H\u0002m=S\u0000\u000bm\u0002@tm+ \u0001\u001c0=S; (13)\nwith \u001c0implicitly included in H.\nEqs. (10) and (13) are our \fnal coupled deterministic\nequations. We can rewrite them in a more explicit form\nas\nL@tJ+ (R+R0)J+@t\b=c=\nPI\ndx@xm\u0001(\f\u0000m\u0002)@tm;\nS(1 +\u000bm\u0002)@tm+m\u0002H=PJ(1 +\fm\u0002)@xm:(14)\nHere, the deterministic spin-torque contribution (7) is for\nclarity separated out of the e\u000bective \feld H, which here\nconsists of the usual purely magnetic contributions. The\nleft-hand sides in these equations contain the ordinary\nOhm's law (corrected for the magnetic-texture resistance\nR0) and the LLG terms, respectively, while the right-hand\nsides describe the reactive Berry-phase coupling and its\ndissipative \fcorrection.\nEq. (12) was derived microscopically in Refs. 4,6,24,25,\nrelating\fto electron spin dephasing: \f\u0018~=\u001cs\u0001xc(con-\nsistent with our anticipation above). Its Onsager coun-\nterpart in Eq. (11) was \frst obtained phenomenologi-\ncally in Ref. 14 and microscopically in Ref. 13. These\n\\\fterms\" are now accepted to be crucial in understand-\ning current-driven magnetic dynamics and the reciprocal\ngauge \felds.\nIV. DISSIPATION POWER\nSuppose we perturb our system with some nonequi-\nlibrium current and magnetic texture, after which the\nsystem evolves back toward equilibrium according to the\nequations of motion, producing entropy. If the system is\nsteadily driven, the heat will be dissipated to the envi-\nronment at some \fnite rate. From standard thermody-\nnamics, the dissipation power is4\nP[m(x;t);J(t)]\u0011\u0000J@tJ\u0000I\ndxH\u0001@tm=RJ2+I\ndx\u0014\n\u000bS(@tm)2\u00002\fPJ@xm\u0001@tm+\u0011\f2P2\n\u000bSJ2(@xm)2\u0015\n:(15)\nAccording to the second law of thermodynamics, the dis-\nsipation (15) must always be positive, which means that\n\u0011\u00151. This gives us the lower bound on the resistivity\nof the magnetic texture:\n\u001a=\u0011\f2P2\n\u000bS(@xm)2\u0015\f2P2\n\u000bS(@xm)2: (16)\nIn models where \u000bcomes solely from the coupling of\nthe magnetization to the conducting electrons (which is\nin fact believed to be the dominant cause for Gilbert\ndamping in metallic ferromagnets), we may expect the\nlower bound (16) to give an estimate for the texture re-\nsistivity. For a mean-\feld Stoner-model treatment of\nGilbert damping, we found \u000b=\f, while for an s\u0000d\nmodel we had \u000b= (s=S)\f, wheresis the portion of\nspin density carried by the selectrons,Sis the total spin\ndensity, and \f=~=\u001cs\u0001xcin both cases (with the spin-\ndephasing time \u001csgoverned by the magnetic and spin-\norbit impurities).6In both models, therefore, \u000bS=s\f,\ngiving for the resistivity estimate (up to the second order\nin spatial derivative)\n\u001a&(\fP2=s)(@xm)2; (17)\nwhich involves only quantities related to conducting elec-\ntrons. Taking parameters relevant to Permalloy wires:7\np\u00181,\f\u001810\u00002, domain-wall width of 20 nm, and\nthe magnetization of 103emu=cm3, we \fnd the resistiv-\nity (17) to be \u001a\u001810\u00004\u0016\n\u0001cm. This is smaller than\nthe domain-wall resistivity calculated to the (1 =\u0001xc)2\norder in spin mistracking of the magnetic pro\fle (but\nstill quadratic order in texture), in the absence of spinrelaxation,23whose overall prefactor appears to be larger\nthan in our Eq. (17) for transition metals. We thus con-\nclude that our \u0011may in practice be much larger than\nunity (which is the lower bound necessary for the consis-\ntency of our phenomenology).\nLet us also note in the passing that in the special case\nof\u000b=\fand\u0011= 1, the magnetic dissipation (15) ac-\nquires a very simple form:\nP[m(x;t)]!\u000bSI\ndx\u0012\n@tm\u0000PJ\nS@xm\u00132\n; (18)\nwhich is nothing but the Gilbert dissipation with the ad-\nvective time derivative Dt=@t+v@x(v=\u0000PJ=S).\nIt is clear that this limit describes dissipative magnetic\ndynamics that are simply carried by the electric \row at\nspeedv. In this case, the spin torques disappear if we\nwrite the LLG equation (6) with Dtin the place of @t.8\nV. THERMAL NOISE\nAt \fnite temperatures, thermal agitation causes \ructu-\nations of the current and magnetization, which are cor-\nrelated due to their coupling. A complete description\nrequires that we supplement the stochastic equations of\nmotion with the correlators of these \ructuations. It is\nconvenient to regard these \ructuations as being due to\na stochastic external magnetic \feld \u000ehand a stochastic\ncurrent source \u000eJ: their noise correlators are then related\nto the dissipative coe\u000ecients of the theory according to\nthe \ructuation-dissipation theorem (FDT). Constructing\nthe noise sources by following the standard procedure,22\nour \fnal coupled stochastic equations become:\nL@tJ+~R(J+\u000eJ) +@t\b=c=PI\ndx@xm\u0001(\f\u0000m\u0002)@tm; (19)\nS(1 +\u000bm\u0002)@tm+m\u0002(H+\u000eh) =PJ@xm+P(J+\u000eJ)\fm\u0002@xm; (20)\nwhere we have explicitly separated the deterministic spin-torque contribution PJ@xmout of the e\u000bective \feld H,\nwhich here consists of the usual purely magnetic contributions. The left-hand sides in these equations contain the\nordinary Ohm's law (corrected for the magnetic-texture resistance: ~R=R+R0) and the LLG terms, respectively,\nwhile the right-hand sides describe the reactive Berry-phase coupling and its dissipative \fcorrection.\nWritingfJ;Hg=\u0000^\r\nf@tJ;@tmg, we read out the \\matrix\" ^ \rfrom Eqs. (19) and (20):\n^\rJ;J=1\nR0;^\rJ;h(x)=\u0000\fP\nR0@xm;^\rh(x);J=\fP\nR0@xm;\n^\rhi(x);hi0(x0)=S\u000fii0jmj(x)\u000e(x\u0000x0) +\u000bS\u000eii0\u000e(x\u0000x0)\u0000\f2P2\nR0@xmi(x)@xmi0(x0) (21)\nwhere\u000fijkis the antisymmetric Levi-Civita tensor. Symmetrizing matrix ^ \rimmediately produces Langevin sources5\nsatisfying the FDT,22in the limit that ~!\u001ckBT:\nh\u000eJ(t)\u000eJ(t0)i= 2kBT\u000e(t\u0000t0)=~R;h\u000eJ(t)\u000eh(t0)i= 0;\nh\u000ehi(x)\u000ehi0(x0)i= 2kBTh\n\u000bS\u000eii0\u000e(x\u0000x0)\u0000(\f2P2=~R)@xmi@x0mi0i\n\u000e(t\u0000t0): (22)\nApart from the obvious contributions, we have a magnetic \feld noise proportional to \f2, in the form of a nonlocal\ntensor Gilbert damping. The nonlocal Gilbert damping is apparent, if the electrons are not externally driven, @t\b = 0,\nin the limit L!0 of a large ring, in which case the magnetic equation decouples to give\nS(1 +\u000bm\u0002)@tm+m\u0002(H+\u000eh+\u000eh0) =P2\n~R(1 +\fm\u0002)@xmI\ndx0@x0m\u0001(\f\u0000m\u0002)@tm: (23)\nHere, we moved the spin torque driven by the Nyquist noise to the left as\n\u000eh0=\u0000P\u000eJm\u0002@xm: (24)\n\u000eh0thus enters the equation as a statistically independent current-driven noise source. Writing the right-hand side of\nEq. (23) as\n\u0000m\u0002I\ndx0$K(x;x0)@tm(x0); (25)\nwhere\nKii0(x;x0) =P2\nR0(m\u0002@xm\u0000\f@xm)i(m\u0002@x0m+\f@x0m)i0; (26)\nand extracting the symmetric part of the tensor Kii0(x;x0), we arrive at the total Gilbert damping tensor\nGii0(x;x0) =\u000b\u000eii0\u000e(x\u0000x0) +P2\nS~R\u0002\n(m\u0002@xm)i(m\u0002@x0m)i0\u0000\f2@xmi@x0mi0\u0003\n: (27)\nThis is exactly the form required by the FDT, consistent\nwith the correlator for \u000eh+\u000eh0. The e\u000bective Gilbert\ndamping can thus appear both negative and positive in\ndi\u000berent regions. The minimal texture resistivity (16),\nhowever, insures that we have a nonnegative damping\nglobally. This Gilbert damping originates physically in\nthe spin torques that are generated by the magnetically-\ndriven \fctitious EMF. Nonlocal @x@x0magnetic noise was\nrecently constructed in Ref. 26 (neglecting spin relax-\nation and\f) by heuristically converting Nyquist current\nnoise into magnetic \ructuations via adiabatic spin trans-\nfer. Although the DFT-required nonlocal @x@x0Gilbert\ntensor (27) was established in that paper (apart from the\n\f2piece), only here we are able to derive it directly from\nthe fundamental Langevin sources of the coupled mag-\nnetohydrodynamic theory, dictated by the FDT. As es-\ntimated in Ref. 26, this nonlocal contribution to Gilbert\ndamping is in practice important (in comparison to \u000b) in\nnanoscale magnetic structures.\nVI. SUMMARY\nWe developed a general phenomenological theory of\nmagnetohydrodynamic coupling in isotropic metallic fer-romagnets. The reactive coupling between magnetic tex-\nture dynamics on the one hand and electric \rows on the\nother stems from the Berry phase accumulated by elec-\ntron spin following the quasistationary magnetic texture.\nDissipative terms of the coupled dynamic equations orig-\ninate in the electron spin mistracking of the magnetic or-\nder parameter and the associated spin dephasing. Apart\nfrom the usual Gilbert damping, the latter leads to a\nviscous coupling between electric currents and magnetic\ntexture dynamics, parametrized by a single parameter \f.\nWe also obtain a small correction to the texture resistiv-\nity at order \f2. Finally, our thermodynamic description\nof the magnetohydrodynamic coupling allows us to de-\nrive the stochastic Langevin contributions to the e\u000bective\n\feld and electric current, according to the \ructuation-\ndissipation theorem.\nAcknowledgments\nWe acknowledge stimulating discussions with Gerrit E.\nW. Bauer, Arne Brataas, and Mark D. Stiles. This work\nwas supported in part by the Alfred P. Sloan Foundation.\n1M. V. Berry, Proc. R. Soc. London A 392, 45 (1984).2G. E. Volovik, J. Phys. C: Sol. State Phys. 20, L83 (1987).6\n3L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electro-\ndynamics of Continuous Media , vol. 8 of Course of Theo-\nretical Physics (Pergamon, Oxford, 1984), 2nd ed.\n4S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n5A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Euro-\nphys. Lett. 69, 990 (2005).\n6Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W.\nBauer, Phys. Rev. B 74, 144405 (2006); H. J. Skadsem,\nY. Tserkovnyak, A. Brataas, and G. E. W. Bauer, ibid.\n75, 094416 (2007); Y. Tserkovnyak, A. Brataas, and G. E.\nBauer, J. Magn. Magn. Mater. 320, 1282 (2008).\n7A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and\nT. Shinjo, Phys. Rev. Lett. 92, 077205 (2004); M. Hayashi,\nL. Thomas, Y. B. Bazaliy, C. Rettner, R. Moriya, X. Jiang,\nand S. S. P. 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Brataas,\nG. E. W. Bauer, and P. J. Kelly, Phys. Rep. 427, 157\n(2006); D. C. Ralph and M. D. Stiles, J. Magn. Magn.\nMater. 320, 1190 (2007).\n18A. Stern, Phys. Rev. Lett. 68, 1022 (1992); Y. Aharonov\nand A. Stern, Phys. Rev. Lett. 69, 3593 (1992).\n19E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part\n2, vol. 9 of Course of Theoretical Physics (Pergamon, Ox-\nford, 1980), 3rd ed.\n20T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n21Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B\n57, R3213 (1998).\n22L. D. Landau and E. M. Lifshitz, Statistical Physics, Part\n1, vol. 5 of Course of Theoretical Physics (Pergamon, Ox-\nford, 1980), 3rd ed.\n23M. Viret, D. Vignoles, D. Cole, J. M. D. Coey, W. Allen,\nD. S. Daniel, and J. F. Gregg, Phys. Rev. B 53, 8464\n(1996); C. H. Marrows, Adv. Phys. 54, 585 (2005).\n24H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn.\n75, 113706 (2006).\n25R. A. Duine, A. S. N\u0013 u~ nez, J. Sinova, and A. H. MacDonald,\nPhys. Rev. B 75, 214420 (2007).\n26J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. B 78, 140402(R) (2008);"    },    {        "title": "1406.2491v2.Influence_of_Ta_insertions_on_the_magnetic_properties_of_MgO_CoFeB_MgO_films_probed_by_ferromagnetic_resonance.pdf",        "content": "arXiv:1406.2491v2  [cond-mat.mtrl-sci]  13 Aug 2014Influence ofTa insertions onthe magneticproperties ofMgO /CoFeB/MgOfilms probed by\nferromagnetic resonance\nMariaPatriciaRouelliSabino,Sze TerLim,andMichaelTran\nDataStorage Institute,Agency for Science, Technology and Research, 5Engineering Drive 1,117608 Singapore\n(Dated: April 3,2018)\nAbstract We show by vector network analyzer ferromagnetic resonance measurements that low Gilbert damping α,\ndownto0.006,canbeachievedinperpendicularlymagnetize dMgO/CoFeB/MgOthinfilmswithultrathininsertionsofTa\nintheCoFeBlayer. AlthoughincreasingthenumberofTainse rtionsallowsthickerCoFeBlayerstoremainperpendicular ,\nthe effective areal magnetic anisotropy does not improve withmore insertions, whichcome withan increase in α.\nPerpendicularmagnetic anisotropy (PMA) is the key to furth er downscaling of spin transfer torque magnetoresistive\nrandommemorydevices,asitallowstwokeyrequirementstob esatisfied: lowcriticalcurrent Ic0andhighthermalstability\n∆, the latter of which is proportional to the energy barrier Ebbetween the two stable magnetic states. The spin torque\nswitching efficiency, defined as Eb/Ic0, is commonly used as a metric to account for both requirement s. For a Stoner-\nWohlfarthmodel,itisgivenby[1]( /planckover2pi1/4e)·(η/α),whereαistheGilbertdampingparameter,and ηisthespinpolarization\nfactor, which is related to the tunnel magnetoresistance ra tio (TMR) byη=[TMR(TMR+2)]1/2/[2(TMR+1)]. It thus\nbecomes evident that for high switching e fficiency, one has to decrease αwhile keeping TMR high. Magnetic tunnel\njunctions(MTJs)basedonCoFeB /MgOsystemsarewell knownto providehighTMR[2] andhaverec entlybeenshown\nto possess PMA, which is attributed to the CoFeB /MgO interface.[3] A Ta layer is usually placed adjacent to th e CoFeB\nto induce the proper crystallization necessary for PMA and h igh TMR [4]. In Ta /CoFeB/MgO systems, however, spin\npumping to the Ta increases α. [5] Moreover, the CoFeB layer also needs to be ultrathin (ty pically less than 1.5nm) in\nordertoexhibitPMA.[3]Toimprovethethermalstabilityas devicesarescaleddowntosmallerdiameters,increasingth e\neffectivearealanisotropyenergydensity Kef ftisdesired.\nOne approach to address these issues is the use of double-MgO structures, i.e., those in which both the barrier layer\nand cappinglayer straddlingthe free layerare made of MgO. I mproved Ic0and/or∆have beenreportedin devicesusing\ndoubleMgOfreelayers. [6,7,8,9]Theimprovementintherma lstabilityisattributedtotheadditionalCoFeB /MgOinter-\nface,whereaslower Ic0isassociatedwithlow α. Indeed,αdownto0.005hasbeenmeasuredinin-planeMgO /FeB/MgO\nfilms,[10] which agrees with device measurements.[11] The s tacks investigated in these damping studies, however, did\nnot have the Ta layer used in practical free layers with perpe ndicular anisotropy.[9, 12] In addition, although the inte r-\nfacial anisotropy in the out-of-plane devices measured by T sunegi et al.[11] can be as high as 3.3 mJ /m2, the effective\nperpendicularanisotropywasratherlow ( Kef ft≈0.04mJ/m2) relativetothat ofa Ta /CoFeB/MgOstack[3]. In thiswork,\nweexploretheinfluenceofTainsertionswithintheCoFeBlay erofMgO/CoFeB/MgOfilmsbymagnetometryandvector\nnetwork analyzer ferromagnetic resonance (VNA-FMR) measu rements. The insertion of extremely thin Ta layers (0.3\nnm) inside the CoFeB layer aids crystallization, allowing a larger total CoFeB thickness to remain perpendicular, [13]\nwith aneffectivearealanistropycomparableto thatofTa /CoFeB/MgO.\nTwo sample series were deposited by magnetron sputtering on SiO2substrates with seed layers of Ta 5 /TaN 20/Ta 5\nin an ultrahigh vacuum environment (all thicknesses in nm). The stack configurations of the two sample series are: (1)\nMgO 3/CoFeB 1.0/Ta 0.3/CoFeB 0.5 - 1.5/MgO 3 (“single-insertion”) and (2) MgO 3 /CoFeB 1.0/Ta 0.3/CoFeB 0.5 -\n1.5/Ta0.3/CoFeB1.0/MgO3(“double-insertion”),wheretheCoFeBcompositionis Co40Fe40B20(at%). TheTainsertion\nlayer thickness is in the regime allowing strong ferromagne tic coupling between the CoFeB layers. [16] Two other\nsample serieswere grownasreferences: (a)MgO3 /CoFeB1.0-2.5/MgO3(“zero-insertion”),and(b)seed /CoFeB1.0-\n2.5/MgO3(“single-MgO”).Forallthedouble-MgOsamples,anult rathinCoFeBlayerbelowthebottomMgOlayerwas\nalsodepositedforgoodMgOgrowth. Weconfirmedfromseparat emeasurementsthatthislayerdoesnotcontributetothe\nmagneticsignal. All sampleswerecappedwith15nmofTa forp rotectionandwereannealedpost-growthat 300◦Cfor1\nh in vacuum. Although3 nm MgO is too thick for practical use in MTJs, it was chosen to ensure continuityof the MgO\nlayers and lessen the influence of the layersbeyondit.[10, 1 4, 15] (Measurementsof similar sampleswith 1 nm of MgO\nonbothsidesofthe magneticlayeryieldedthesametrends.)\nMagnetization measurements were performed using an altern ating gradient magnetometer (AGM). The PMA im-\nproveswith doublingof the CoFeB /MgO interface and with increasingnumber of Ta insertions, n, as shown in Fig. 1(a)\nfor samples with a similar total nominal CoFeB thickness tnom≈2.5 nm. We also confirmed that we cannot obtain a\n1perpendicular easy axis in double-MgO structures without T a insertions.[17] The double-insertion sample, on the othe r\nhand,exhibitslargeout-of-planeremanenceas shownin the inset of Fig. 1(a). A coercivefield less than0.01T (inset) is\ntypicalofCoFeBfilmswithPMA[18,16].\n/s49 /s50 /s51 /s52/s49/s50/s51/s52/s53\n/s45/s48/s46/s48/s49 /s48/s46/s48/s49\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s32/s115/s105/s110/s103/s108/s101/s32/s77/s103/s79\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s48\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s49\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s50/s77/s47/s65/s32/s40/s65 /s109/s50\n/s47/s109/s50\n/s41/s32/s120/s32/s49/s48/s45/s54\n/s116\n/s110/s111/s109/s32/s40/s110/s109/s41/s48/s40/s98/s41\n/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s97/s46/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s70/s105/s101/s108/s100/s32/s40/s84/s41/s116\n/s110/s111 /s109/s32 /s32/s50/s46/s53/s110/s109/s40/s97/s41\nFigure 1: (Color online) (a) Out-of-planeAGM loops for samp les with single-MgO (red squares), zero-insertion(purple\ninvertedtriangles),single-insertion(bluecircles),an ddouble-insertion(greentriangles),withtotalnominalC oFeBthick-\nnesstnom≈2.5nm. Insetshowsalow-fieldout-of-planeloopforthesame doubleinsertionsample. (b)Magneticmoment\nperunitarea( M/A)asafunctionofthetotalnominalCoFeBthicknessforallsa mpleseries. Linearfitsareshownassolid\nlines.MSandtMDLcanbeextractedfromtheslopeand xintercept,respectively,andaresummarizedinTable 1.\nIt is known that Ta can create a magnetically dead layer (MDL) when it is near a magnetic layer.[19] We plot the\nmagneticmomentperarea against tnom[Fig.1(b)]to obtainthethicknessoftheMDL foreachseries fromthexintercept\nofalinearfit. TheresultsaresummarizedinTable1,alongwi ththeMSvaluesobtainedfromtheslope. WefindanMDL\nthickness tMDLof 0.24±0.09 nm for n=1 and 0.7±0.1 nm for n=2. These thicknesses are similar to the total Ta\ninsertionthicknessintherespectiveseries,andareconsi stentwiththepictureofCoFeBintermixingwithTatoproduc ea\nmagneticallydeadvolume. The tMDLvalueforthesingle-MgOsamples(0.26 ±0.08nm)agreeswithvaluesfoundinthe\nliterature.[14, 20, 21] On the other hand, no dead layer was f ound for the zero-insertionsamples, which is similar to the\nresultsinRef. [19].\nTable1: SummaryofMagneticProperties\nSeries tMDL(nm) MS(MA/m) Ki(mJ/m2) Kv(MJ/m3)\nsingle MgO 0 .26±0.08 1.51±0.08 1.61±0.07−0.29±0.08\ndouble MgO, n=0 0.04±0.06 1.29±0.05 0.91±0.09 0.37±0.05\ndouble MgO, n=1 0.24±0.09 1.12±0.05 2.18±0.08−0.34±0.04\ndouble MgO, n=2 0.7±0.1 1.10±0.04 2.4±0.1−0.25±0.03\nVNA-FMR was used to measure the e ffective anisotropy field and damping parameter of the samples . In the VNA-\nFMR setup, the samples were placed face down on a coplanar wav eguide and situated in a dc magnetic field of up to\n1.2 T applied perpendicular to the film plane. The transmission scattering parameter S21was measured at a specific\nfrequency while the dc field was swept. For each sweep, the rea l and imaginary parts of the resonance response were\nfitted simultaneouslyusingthecomplexsusceptibilityequ ation\nχ(H)=Mef f(H−Mef f+i∆H\n2)\n(H−Mef f)2−/parenleftBig2π/planckover2pi1f\ngµB/parenrightBig2+i∆H(H−Mef f)(1)\nwherefis the frequencyofthe ac field, Mef f=MS−H⊥\nK,∆His the fullwidth at half-maximum, H⊥\nKis the anisotropy\nfield perpendicular to the plane, gis the spectroscopic splitting factor, µBis the Bohr magneton, and /planckover2pi1is the reduced\nPlanck’s constant. Nonmagnetic contributions to the S21parameter and a linear time-dependentdrift of the instrume nts\nwere taken into account during the fit. We note that only one re sonance peak is observed within the range studied. A\nrepresentativefitofthesusceptibilitydataisshowninFig .2(a)foradouble-insertionsamplewith tnom=2.5nm. Inusing\nEq.1,a valueof g=2isfirst assumedtoobtainvaluesfor Mef fand∆H, whichdoesnotaffectthe finalresult.\nForeachfrequency,a resonancefield\nµ0Hres(f)=2π/planckover2pi1\ngµBf+µ0Mef f (2)\naccording to Kittel’s equation is calculated and plotted ag ainst the frequency, as shown in Fig. 2(c). A linear fit, now\nwithgandMef fas fitting parameters, is then performed. The e ffective anisotropyenergydensity Kef fcan be calculated\n2from the effective anisotropy field HKef f(=−Mef f) asKef f=HKef fMS/2, noting that a positive anisotropy constant\ncorrespondsto aperpendiculareasyaxis.\nToobtainα,we performa linearfit ofthe measuredFMRlinewidthasafunc tionofthefrequencyto\nµ0∆H(f)=4π/planckover2pi1α\ngµBf+µ0∆H0 (3)\nwhere∆H0is the inhomogeneous linewidth broadening, and the value of gused is the fitted value from Eq. 2. We\nnote that two-magnon scattering contributions to the linew idth are eliminated owing to the perpendicular measurement\nconfiguration[22]. Such a fit is shown in Fig. 2(c). Only data p oints taken well beyond the saturation field for each\nsamplewereusedinthefit, andasymptoticanalysisasdescri bedinRef. [23]fortheaccessiblefrequencyrangewasalso\nperformed.\nWe define an effective thickness tef f=tnom−tMDLand show the calculated Kef ftef f(to which Ebis proportional)\nfor both sample series in Fig. 3(a). The x-axis error bars originate from the fitting error in obtainin gtMDL. We find that\nfortef f>2 nm,double-insertionsampleshavehigher Kef ftef fthan single-insertionsamplesforthe same tef f. However,\nthe maximum Kef ftef fachieved for both the single- and double-insertion series d oes not significantly exceed Kef ftef f\nmeasuredin ourthinnestsingle-MgOsample( tef f=1.0nm),similar tothatobservedin MTJmeasurements.[7]\nTounderstandthisfurther,we considerthedi fferentcontributionsto Kef ftef f,whichisgivenby\nKef ftef f=Ki+(Kv−µ0M2\nS\n2)tef f (4)\nwhereKiis the total interfacial anisotropy constant, including al l CoFeB/MgO interfaces; Kvis the volume anisotropy\nconstant; and the demagnetizing energy is given by the M2\nSterm. We assume that any interfacial anisotropy from the\nTa/CoFeB interface is negligible.[24] Kiis commonly derived from the yintercept of a linear Kef ftef fversustef ffit,\nwhereasKvcan be calculated from the slope if MSis known. Because it is possible that for CoFeB thicknesses b elow\n1.0 nm,Kiis degraded because of Ta reaching the CoFeB /MgO interface,[25] we consider only the linear region of the\ncurve during the fit. The calculated values, given in Table 1, demonstrate that the absence of a Ta insertion leads to the\nlowest value of Ki(0.91±0.09mJ/m2), explaining why n=0 samples did not exhibit a perpendiculareasy axis. On the\nother hand, Kiforn=1 (2.18±0.08mJ/m2) andn=2 (2.4±0.1mJ/m2) are both larger than the single-MgO series\n(1.61±0.07mJ/m2), as would be expected from the additional PMA from the secon d CoFeB/MgO interface. However,\nthe anisotropy per interface did not double with the additional CoFeB /MgO, which may be attributed to the di fferent\ndegrees of crystallization for single- and double-MgO samp les. An indication of better crystallization into CoFe in th e\nsingle-MgO series is its higher MS.Kvis negative and does not vary appreciably in samples where Ta is present, in\ncontrast to the positive value found for zero-insertion sam ples. The role of Ta with regard to Kvis not yet understood,\nas previouslypointed out by Sinha et al. [20], and a detailed study of the amount, proximity,and profile of Ta would be\nnecessarytoclarifythesee ffects.\nTurningourattentionto α,weidentifyasingle-MgOsample( tef f≈0.8nm)andasingle-insertionsample( tef f≈1.3\nnm)withacomparable Kef ftef f≈0.2mJ/m2. Weimmediatelynoticethat αforthesingle-insertionsampleisaroundtwo\ntimeslowerthanthatforthe single-MgOsample.\nThis dramaticdecrease in αmay be attributedto the suppressionof spin pumpingby the Mg O layers straddlingboth\nsides of the precessing magnet.[10, 26] Indeed, measuremen ts of zero-insertion samples show no thickness dependence\n[purple dashed line in Fig. 3(b)] and a low mean value of α=0.0035±0.0002 comparable to the bulk damping of\nCo40Fe40B20[5,27]. However,adecreasein αwithincreasing tef fcanstillbeseeninboththesingle-anddouble-insertion\nseries. One possible reason is the alloying of CoFeB and Ta, a s Ta is known to readily intermix with CoFeB [21], and\nhigher damping may be expected from CoFeBTa alloys [28]. The relative percentage of CoFeBTa alloy decreases with\nincreasing CoFeB thickness, coinciding with the αdecrease. This picture is also consistent with the jump in αfrom\nsingle-to double-insertionsamples, i.e.,thereismoreCo FeBTa alloybecausetherearemoreTa insertions. [11, 10]\nItmayalsobepossiblethatspinpumpingtotheTainsertionl ayeroccurs,asinthecaseofthePdinterlayerinCoFe /Pd\nmultilayers[29]. The complexityof oursystem, however,pr eventsusfromusinga simple multilayermodel. One reason\nisthatthemiddleCoFeBlayer(inthedouble-insertioncase )mayhavedifferentpropertiesfromtheCoFeBlayersadjacent\nto MgO, because CoFeB crystallizes from the MgO interface, [ 30] with which the middle CoFeB has no contact. The\ndegree of Ta intermixing also depends on the deposition orde r and will vary across the structure.[19] At this point, we\ncannot discriminate the mechanism behind the damping behav ior. It may be worthwhile to study the use of CoFeBTa\nalloysasinterlayerstopossiblyhavemorecontroloverthe amountanddistributionofTa inthestack.[31]\nInconclusion,wehavedemonstratedPMAandlowdampingindo uble-MgOstructures. AthinTainsertionlayerwas\nfound to significantly increase the PMA - no perpendicular ea sy axis was realized in our MgO /Co40Fe40B20/MgO films\nwithout Ta - and adding more insertionsallowed thicker CoFe B layers to remain perpendicular. However, the maximum\nKef ftef findouble-MgOsamplesiscomparableonlywiththatofthesin gle-MgOsampleforthisCoFeBcomposition.[9]\nOn the other hand, αfor double MgO films increases with the number of insertions b ut is still lower than that of single\n3MgO films for the entire range. Considering both trends with n, we find that the optimal stack in the range of samples\nwe studiedis a double-MgO, n=1 sample with tnom=1.75nm,which exhibits Kef ftef f=0.27mJ/m2at a low damping\nvalueof0.006.\nAcknowledgement\nWe expressgratitudeforsupportfromtheA*STARGraduateAc ademySINGAProgram.\nReferences\n[1] J. Z. Sun, S. L. Brown, W. Chen, E. A. Delenia, M. C. Gaidis, J. Harms, G. Hu, Xin Jiang, R. Kilaru, W. Kula,\nG.Lauer,L.Q.Liu,S.Murthy,J.Nowak,E.J.Oâ ˘A´ZSullivan,S.S.P.Parkin,R.P.Robertazzi,P.M.Rice,G.Sa ndhu,\nT.Topuria,andD.C. Worledge: Phys.Rev.B88,104426(2013) .\n[2] S. Yuasa: J. Phys.Soc.Jpn.77,031001(2008).\n[3] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and\nH. Ohno: Nat.Mater. 9,721(2010).\n[4] D. C. Worledge, G. Hu, David W. Abraham, J. Z. Sun, P. L. Tro uilloud, J. Nowak, S. Brown, M. C. Gaidis, E. J.\nO’Sullivan,andR. P.Robertazzi: Appl.Phys.Lett.98,0225 01(2011).\n[5] X. Liu,W.Zhang,M.J. Carter,andG. 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Noshiro,C. Yoshida, Y. Yamazaki, A. Taka hashi, Y. Iba, A. Hatada, M. Nakabayashi,T. Takenaga,\nM. Aoki,andT.Sugii.IEDM,29.1.1(2012).\n5/s50/s48/s48 /s50/s50/s48 /s50/s52/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s40/s99/s41/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s70/s105/s101/s108/s100/s32/s40/s109 /s84/s41/s32\n/s32\n/s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s109 /s84/s41/s32/s68/s97/s116/s97 /s32\n/s32/s70/s105/s116\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s71/s72/s122/s41/s32/s68/s97/s116/s97\n/s32/s70/s105/s116/s83\n/s50/s49/s32/s82/s101/s97/s108/s32/s40/s97/s46/s117/s46/s41\n/s40/s97/s41\n/s40/s98/s41/s83\n/s50/s49/s32/s73/s109/s97/s103/s46/s32/s40/s97/s46/s117/s46/s41\n/s65/s112/s112/s108/s105/s101/s100/s32/s70/s105/s101/s108/s100/s32/s40/s109/s84/s41\nFigure 2: (a) Real and (b) imaginary parts of the S21parameter obtained from VNA-FMR measurements for a double-\ninsertion sample with tnom=2.5 nm at 12 GHz while a perpendiculardc magnetic field is swept. The lines are fits to an\nexpressionusingEq.1, takingnonmagneticcontributionst oS21anda lineardriftintoaccount. (c)Field-sweptlinewidth\nandresonancefieldsforthesamesampleasafunctionoffrequ ency. Thelinearfitsdescribedinthetextareusedtoextract\nHKef f(=−Mef f)andα.\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s48 /s49 /s50 /s51/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s115/s105/s110/s103/s108/s101/s32/s77/s103/s79\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s48\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s49\n/s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s50/s75\n/s101/s102/s102/s116\n/s101/s102/s102/s32/s40/s109/s74/s47/s109/s50\n/s41\n/s40/s97/s41/s40/s49/s48/s45/s51\n/s41\n/s116\n/s101/s102/s102/s32/s40/s110/s109/s41/s40/s98/s41\nFigure3: (a) Kef ftef fand(b)αversuseffectiveCoFeBthickness tef fobtainedfromfield-sweptVNA-FMRmeasurements\nfor all sample series. Solidlines in (a)are linear fits. Purp ledashedline in (b) correspondsto the mean αvalueaveraged\noverall zero-insertionsamples,whichwasfoundtobeconst antwithinerroracrosstheentirethicknessrangestudied.\n6"    },    {        "title": "2211.08048v2.Nonlinear_sub_switching_regime_of_magnetization_dynamics_in_photo_magnetic_garnets.pdf",        "content": "1 \n Nonlinear s ub-switching regime of magnetization  dynamics in photo -magnetic  garnets  \nA. Frej, I. Razdolski, A. Maziewski, and A. Stupakiewicz  \nFaculty of Physics, University of Bialystok, 1L Ciolkowskiego, 1 5-245 Bialystok, Poland  \nAbstract.  We analyze, both experimentally and numerically, the nonlinear regime of the \nphoto -induced coherent magnetization dynamics in cobalt -doped yttrium iron garnet films. \nPhoto -magnetic excitation with femtosecond laser pulses reveals a strongly nonlinear \nrespo nse of the spin subsystem with a significant increase of the effective Gilbert damping. By \nvarying both laser fluence and the external magnetic field, we show that this nonlinearity \noriginates in the anharmonicity of the magnetic energy landscape. We numer ically map the \nparameter workspace for the nonlinear photo -induced spin dynamics below the photo -\nmagnetic switching threshold. Corroborated by numerical simulations of the Landau -Lifshitz -\nGilbert equation, our results highlight the key role of the cubic sy mmetry of the magnetic \nsubsystem in reaching the nonlinear spin precession regime. These findings expand the \nfundamental understanding of laser -induced nonlinear spin dynamics as well as facilitate the \ndevelopment of applied photo -magnetism.  \n1. INTRODUCTION  \nRecently,  a plethora of fundamental mechanisms for magnetization dynamics induced by \nexternal stimul i at ultrashort time scale s has been actively d iscussed  [1-5]. The main interest \nis not only in the excit ation of  spin precession but in the switching of ma gnetization between \nmultiple stable states, as it open s up rich possibilities for non-volatile magnetic  data  storage \ntechnology . One of t he most intriguing example s is the phenomenon  of ultrafast switching of \nmagnetization with laser pulses. Energy -efficie nt, non -thermal mechanisms of laser -induced \nmagnetization switching require a theoretical understanding of coherent magnetization \ndynamics in a strongly non -equilibrium environment  [6]. This quasiperiodic motion  of \nmagnetization  is often mode led as  an oscillator where the key parameters , such as frequency \nand damping , are considered within the framework of  the Landau -Lifshit z-Gilbert  (LLG)  \nequation  [1, 7] . Although it is inhe rently  designed to describe small -angle spin precession \nwith in the  linear approximation, there are attempts to extend this formalism into the \nnonlinear regime where the precession parameters become angle -dependent  [8]. This is \nparticularly important in light of the discovery of the so -called p recessional switching , where  \nmagnetization , having been  impulsively driven  out of equilibrium, ends its precessional \nmotion in a different minimum of the potential energy  [6, 9 -11]. Obviously,  such \nmagnetization trajectories are characterized by very large precession angles  (usually on the \norder of tens of  degrees ). It is, however, generally believed that the magnetization excursion \nfrom the equilibrium of about 10 -20 degrees is already sufficient for the violat ion of the linear \nLLG approach  [12, 13] . Thus, an intermediate regime under the switching stimulus threshold \nexists, taking a large area in the phase space and presenting an intriguing c hallenge in \nunderstanding  fundamental spin dynamics.  \nAn impulsive optical stimulus often results in a thermal excitation mechanism, inducing \nconcomitant temperature variations , which can impact the parameters of spin precession  [14-\n16]. This highlights the  special role of the non -thermal optical mechanisms  of switching  [17-2 \n 19]. Among those , we outline photo -magnetic excitation , which has been recently \ndemonstrated in dielectric Co -doped YIG  (YIG:Co)  films  [6, 11] . There, laser photons at a \nwavelength of 1300 nm resonantly excite the 5E → 5T2 electron transition s in Co -ions, resulting \nin an emerging photo -induced magnetic anisotropy  and thus in a highly efficient excitation of \nthe magnetic subsystem  [6]. This photo -induced  effective  anisotropy field features  a near ly \ninstant aneous  rise time (within the femtosecond pump laser pulse  duration), shifting the \nequilibrium direction for the magnetization and thus triggering its l arge -amplitude precession.  \nIn the sub -switching regime  (at excitation strengths just below the switching threshold ), the \nfrequency of the photo -induced magnetization precession has been shown to depend on the \nexcit ation wavelength  [20]. However, nonlinearities in magnetization dynamics in the sub-\nswitching regime have  not yet been described in detail, and the underlying mechanism  for the \nfrequency variations  is not understood.  \nIn this work , we systematically examine the intermediate sub -switching regime characterized \nby large angles of magnetization precession and the nonlinear response of the spin system to \nphoto -magnetic excitations. We show a strong increase of the effective Gilbert damping at \nelevated lase r-induced excitation levels and quantify its nonlinearity within the existing \nphenomenological formalism  [8]. We further map the nonlinear regime in the phase space \nformed by the  effective  photo -induced anisotropy field  and the external magnetic field.   \n \nFig. 1. Sketch of m agnetization dynamics at various  stimulus  levels . Owing to the highly nonlinear \nmagnetization dynamics in the switching regime, the nonlinearity onset manifests in the sub -switching \nregime too . \nThis paper is organized in the following order: in the first part, we describe the details of the \nexperiment for laser -induced large -amplitude magnetization precession. Next, we present the \nexperimental results, followed  by the fitting analysis . Then, we complement our findings with \nthe results of  numerical simulation of the photo -magnetic spin dynamics. Afterward , we \ndiscuss the workspace of parameters for the sub-switching regime of laser -induced \nmagnetization precession. The paper ends with c onclusions.  \n \n3 \n 2. EXPERIMENTAL DETAILS  \nThe experiments were done on  a 7.5 μm -thick YIG:Co  film with a composition of \nY2CaFe 3.9Co0.1GeO 12. The Fe ions at the tetrahedral and octahedral sites are replaced by Co -\nions [21]. The sample was grown by liquid -phase epitaxy on a 400  μm-thick  gadolinium gallium \ngarnet (GGG) substrate. It exhibits eight possible magnetization states along the garnet’s cubic \ncell diagonals due to its cubic magnetocrystalline anisotropy ( 𝐾1=−8.4×103 𝑒𝑟𝑔/𝑐𝑚3) \ndominating the energy landscape over the uniaxial anisotropy ( 𝐾𝑢=−2.5×103 𝑒𝑟𝑔/𝑐𝑚3). \nOwing to the 4 ° miscut, additional in -plane anisotropy is introduced, tilting the magnetization \naxes and  resulting in slightly lower energy of half of the magnetiz ation states in comparison \nto the others. In the absence of the external magnetic field, the equilibrium magnetic state \ncorresponds to the magnetization in the domains close to the <111> -type directions in YIG:Co \nfilm.  Measurements of the Gilbert damping  𝛼 using the fe rromagnetic resonance technique \nresulted in  𝛼≈0.2. This relatively high damping is inextricably linked to the C o dopants  [22-\n24]. \nThe n onlinearity  of an oscillator is usually addressed by varying the intensity of the stimulus \nand comparing the response of the system under study. Here , we investigated the nonlinear \nmagnetization dynamics by varying the optical pump fluence and , thus , the strength of the \nphoto -magnetic effective field driving the magnetization out of the equilibrium. We \nperfor med systematic studies in various magnetic states of YIG:Co governed by the magnitude \nof the  external magnetic field s. The magnetic field  𝐻⊥ was applied perpendicular to the sample \nplane and in -plane magnetic field  𝐻 was applied along the [110] direction of the YIG:Co crystal \nby means of an electromagnet. Owing to the introduced miscut, the studied YIG:Co exhibits \nfour magnetic domains at 𝐻=0 [25]. The large jump at an in-plane magnetic f ield close to \nzero shows the magnetization switching in the domain structures between four magnetic \nphases. The optical spot size in this experiment was around 100 ��m while the size of smaller \ndomains was around 5 μm, resulting in the spatial averaging of the domains in the \nmeasurements. This behavior of magnetic domains was dis cussed and visualized in detail  by \nmagneto -optical Faraday effect in our previous papers  [6, 25] . With an increase of the \nmagnetic field up to a round 𝐻=0.4 kOe, larger and smaller domains are formed due to the \ndomain wall motion, eventually resulting in a formation of a single domain in a noncollinear \nstate.  Upon further increase, the magnetization rotates towards the direction of the applied \nfield until a collinear state  with in -plane magnetization orientation  is reached  at about 2 kOe \n(see Fig. 2) . 4 \n  \nFig. 2. Magnetization reversal using static magneto -optical Faraday effect under perpendicular H (a) \nand in -plane H (b) magnetic fields. The grey area indicates the magnetization switching in magnetic \ndomain structure  [25]. The green area shows the saturation range with a collinear state of \nmagnetization.  \nDynamic nonlinearities in the magne tic response were studied employing the pump -probe \ntechnique relying on the optical excitation of the spin precession in YIG :Co film. The pumping \nlaser pulse  at 1300 nm , with  a duration of 50 fs and a repetition rate of 500 Hz , induce d spin \ndynamics through  the photo -magnetic mechanism  [6]. The transient Faraday rotation of the  \nweak probe beam  at 625 nm was used to monitor  the dynamics of the  out-of-plane  \nmagnetization  component Mz. The diameter  of the pump spot was around 1 40 μm , while  the \nprobe beam was focused within the pump spot  with a size of around 50 μm . The fluence of \nthe pump beam was varied in the range  of 0.2  6.5 mJ/cm2, below the switching threshold of \nabout 39 mJ/cm2 [20]. At 1300 nm pump wavelength, the optical absorption in our garnet is \nabout 12%. An estimation of the temperature increase ΔT due to the heat load for the laser \nfluence of 6.5 mJ/cm2 results in ΔT <1 K (see Methods of Ref. 6). The polarization of both \nbeams was linear and set along  the [100] crystallographic direction in YIG:Co for the pump and \nthe [010]  direction  for the probe  pulse . The experiments  were done  at room temperature . At \neach magnetic field, we performed a series of laser fluence -dependent pump -probe \nexperiments measuring the transients of an oscillating magnetization component  normal to \nthe sample plane . We then used a phenomenological damped oscillator response function to \n5 \n fit the experimental data and retrieve the fit  parameters such as a mplitude, frequency, \nlifetime and effective damping. In what follows, we analyze the obtained nonlinearities in the \nresponse of the magnetic system and employ numerical simulations to reproduce the \nexperimental findings.  \n \n3. RESULTS  \nA. Time -resolved photo -magnetic dynamics  \nIn order to determine the characteristics of the photo -magnetic precession , we carried out \ntime -resolved measurements  of a transient Faraday rotation  ∆𝜃𝐹 in YIG:Co  film. Fig. 3(a-d) \nexemplifies a few typical datasets obtained for four v arious pump fluences (between 1.7 and \n6.5 mJ/cm2) in magnetic fields of various strength s. A general trend demonstrating a decrease \nof the precession amplitude and an increase of  its frequency is seen  upon the magnetic field \nincrease . To get further insights into the magnetization dynamics, these datasets were fitted \nwith a damped sine function on top of a non -oscillatory, exponentially decaying background : \n∆𝜃𝐹(∆𝑡)=𝐴𝐹sin(2𝜋𝑓∆𝑡+𝜙)exp (−∆𝑡\n𝜏1)+𝐵exp (−∆𝑡\n𝜏2),   (1) \nwhere 𝛥𝑡 is pump and probe time difference, 𝐴𝐹 is the amplitude, 𝑓 is the frequency , 𝜙 is the \nphase, 𝜏1 is the decay time of precession, and 𝜏2 is the decay time of the background with an \namplitude 𝐵. \n \nFig. 3. Time -resolved Faraday rotation  at different  magnetic fields  H (a-d) and laser fluence s (I1-I4 \ncorrespond  to 1.7, 3.2, 5.0,  and 6.5 mJ/cm2, respectively) . The normalized MZ on the vertical axis is \ndefined as  ΔF/max, where max is obtained for saturation magnetization rotation at H (see. Fig. 2a). \nThe curves are offset  vertically without rescaling.  The s olid lines are fittings with the  damped sine \nfunction  (Eq. 1) .  \n6 \n  \nFig. 4. Photo -magnetic  precession parameters as a  function of p ump fluence  in different  external \nmagnetic field  H: a) amplitude  of the Faraday rotation AF, b) frequency  of the precession , and c) \neffective damping. Different colors correspond to different external magnetic fields. The s olid lines are \nthe linear fits  where applicable , while the dashed lines are the visual guides.  Some of the error bars \nare smaller than the data point symbols.  \nAt low applied fields  𝐻<1 kOe, where the photo -magnetic  anisotropy field ( 𝐻𝐿) contribution \nto the total effective magnetic field is the strongest, the largest magnetization precession \namplitude is observed. Figure 4 show s the most important parameters of the magnetiza tion \nprecession, that is, amplitude, frequency and effective damping  (Fig. 4a -c). The latter is \nobtained from the frequency and the lifetime as  (2𝜋𝑓𝜏1)−1. Although the amplitude \ndependence on the pump fluence is mostly linear, the other two parameters exhibit a more \ncomplicated dependence, which is indicative of the noticeable nonlinearity in the magnetic \nsystem. In particular,  at 𝐻=0.4 and 0.5 kOe, we observe d an increase in the effective \ndamping with laser fluence, resulting in a faster decay of the magnetic precession. This is \nfurther corroborated by the frequency decrease seen in Fig. 4b. It is seen that the behavior of \nthe magnetic subsystem is noticeably dissimilar at low ( below 1  kOe) and high  (above 2 kOe)  \nmagnetic fields. At higher magnetic fie lds 𝐻>1 kOe we were unable to observe nonlinear \nmagnetization response at pump fluences up to 10 mJ/cm2. This is indicative of a significant \ndifference in the dynamic response in the collinear and noncollinear states of the magnetic \nsubsystem.  \n \n4. Nonlinear precession of magnetization in anisotropic cubic crystal s \nThe data shown in Fig. 4c clearly indicates the nonlinearity in the magnetic response \nmanifesting in the increase of the effective damping with the excitation (laser) fluence. \nPreviously, similar behavior was found in a number of metallic systems  [26-29] and quickly \nattributed to laser heating. Interestingly, Chen et al . [30] found a decrease of the effective \ndamping with laser fluence in FePt, while invoking the temper ature dependence of magnetic \ninhomogeneities to explain the results. There, the impact of magnetic inhomogeneity -driven \ndamping contribution exhibits a similar response to laser heating and an increase in the static \nmagnetic field. A more complicated mecha nism relying on the temperature -dependent \n7 \n competition between the surface and bulk anisotropy contributions and resulting in the \nmodification of the effective anisotropy field has been demonstrated in ultrathin Co/Pt \nbilayers  [31, 32] . \nNonlinear spin dynamics is a rapidly developing subfield enjoying rich prospects for ultrafast \nspintronics  [33]. Importantly, all those works featured thermal excitation of magnetization \ndynamics in metallic, strongly absorptive systems. In  stark contrast, we argue that the \nmechanism in the Co-doped YIG studied here is essentially non -thermal. This negligible \ntemperature change  ΔT is unable to induce significant variations of the parameters in the \nmagnetic syst em of YIG:Co (T N=450 K), thus ruling out the nonlinearity mechanism discussed \nabove. Rather, we note the work by M üller et al.  [34], where the non -thermal nonlinear \nregime of magnetization dynamics in CrO 2 at high laser fluences was ascribed to the spin -wave \ninstabilities at large precession amplitudes  [35]. We also note the recently debated and \nphysically rich mechanisms of magnetic nonlinearities, such as spin inertia  [36-39] and \nrelativistic effects  [40, 41] . Yet, we argue that in our case of a cubic magnetic anisotropy -\ndominated energy landscape, a much simpler explanation for the nonlinear spin dynamics can \nbe suggested. In particular, we  attribute the amplitude -dependent effective dampin g to the \nanharmonicity of the p otential well for magnetization . \n \nFig. 5. Energy landscape  as a function of the polar angle  𝜃𝜑=45°in the linear (𝐻=2.5 kOe, green) \nand nonlinear (𝐻=0.4 kOe, red) precession regime s. The d ashed lines are the parabolic fits in the \nvicinity of the minima . 𝜃 is the polar angle of magnetization  orientation  measured from the normal to \nthe sample plane  along  the [001] axis in YIG:Co .  \nWe performed  numerical calculations of the energy  density  landscape 𝑊(𝜃,𝜑):  \n𝑊(𝜃,𝜑)=𝑊𝑐+𝑊𝑢+𝑊𝑑+𝑊𝑧    (2) \ntaking into account the following terms in the free energy of the system:  the Zeeman energy  \n𝑊𝑧=−𝑴∙𝑯, demagnetizing field  term 𝑊𝑑=−2𝜋𝑀𝑠2sin2𝜃, cubic 𝑊𝑐=𝐾1∙\n(sin4𝜃sin2𝜑cos2𝜃+sin2𝜃cos2𝜃cos2𝜑+sin2𝜃cos2𝜃sin2𝜑) and uniaxial anisotropy \n𝑊𝑢=𝐾𝑢sin2𝜃 (𝜃 and 𝜑 are the polar and azimuthal angles, respectively ). In the calculations, \nwe assume 𝐾1=−9∙ 103 erg/cm3, 𝐾𝑢=−3∙103 erg/cm3, and 𝑀𝑠 is the saturation \n8 \n magnetization of 7.2 Oe [25]. Then, following  [8] and [42], we calculate the precession \nfrequency 𝑓 and the effective damping 𝛼𝑒𝑓𝑓: \n𝑓=𝛾\n2𝜋𝑀𝑠sin𝜃√𝛿2𝑊\n𝛿𝜃2𝛿2𝑊\n𝛿𝜑2−(𝛿2𝑊\n𝛿𝜃𝛿𝜑)2\n,    (3) \n𝛼𝑒𝑓𝑓=𝛼0𝛾(𝛿2𝑊\n𝛿𝜃2+𝛿2𝑊\n𝛿𝜑2sin−2𝜃)\n8𝜋2𝑓𝑀𝑠,     (4) \nwhere  the 𝛾 is gyromagnetic ratio , and 𝛼0 is the Gilbert damping in YIG:Co  [23, 24] . In Fig. 5, \nwe only show the total energy as a function  of the polar angle 𝜃, to illustrate the \nanharmonicity of the potential at small external in -plane magnetic fields. Experimental data \nand calculations of the energy  𝑊(𝜃,𝜑) have been published in Refs.  [25, 43] . There, it is seen \nthat at relative ly small external magnetic fields  canting  the magnetic state , the proximity of a \nneighboring energy minimum (to the right) effectively modifies the potential well for the \ncorresponding o scillator (on the left) , introducing an anharmonicity . On the other hand, at \nsufficiently large magnetic fields, whic h, owing to the Zeeman energy term, modify the \npotential such that a single minimum emerges (shown in Fig. 5 in green), no nonlinearity is \nexpected. This is also in line with the decreas ing impact of the cubic symmetry in the magnetic \nsystem, which is res ponsible for the anharmonicity of the energy potential.  \nTo get yet another calculated quantity that can be compare d to the experiment, we \nintroduced the photo -magnetically in duced effective anisotropy term  𝐾𝐿. This contribution \ndepends on the  laser fluence  I through the effective light -induced field 𝐻𝐿∝𝐼 as: \n𝐾𝐿=−2𝐻𝐿𝑀𝑠cos2𝜃     (5) \nThe presence of this term displaces the equilibrium for net magnetization. The equilibrium \ndirection s can be obtained by minimizing the total energy with and without the photo -\nmagn etic anisotropy term. Then, k nowing the angle between the perturbed and unperturbed \nequilibrium directions for the magnetization, we calculate d the precession amplitude 𝐴. We \nnote the difference between the amplitudes 𝐴𝐹, which refers to the Faraday rotation of the \nprobe beam, and 𝐴 standing for the opening angle of magnetization precession. Alth ough both \nare measured in degrees, their meaning is different.  \nHaving repeated this for a few levels of optical excitation, we obtain ed a linear slope of the \namplitude vs  excitation  strength  dependence. Figure 6 (a-c) illustrates the amplitude, \nfrequency , and (linear ) effective  damping as a function of the external magnetic field. The \nagreement between the calculated parameters and those obtained from fitting the \nexperimental data is an impressive indication of the validity of our total energy approach. \nFurther, the linear effective damping value of 𝛼≈0.2 obtained in the limit of  strong field s, is \nin good agreement with the values known for our Co -doped YIG from previous works  [6, 24] . \nIn principle, the effective damping  in garnets can increase  towards lower magnetic fields.  \nConventionally attributed to the extrinsic damping contributions, this behavior has been \nobserved in rare -earth iron garnets before as well and ascribed to the generation of the \nbackward volume spin w ave mode by ultrashort laser pulses  [44]. It is worth noting that there \nis no nonlinearity phenomenologically embedded in the approach given above.   9 \n  \nFig. 6. Photo -magnetic precession parameters  at various magnetic fields: amplitude (a), frequency (b) , \nand (linear) effective damping (c). The points are from the experimental data, the solid lines are \ncalculated as described in the text. The dark rectangular points are obtained in the FMR experimen ts. \nThe g rey shaded  area indicates the presence of a  domain st ate (DS). The g reen  shaded  area show s the \nmagnetization saturation state . \nYet, the data presented in Fig. 4c indicates the persistent nonlinear behavior of the effective \ndamping. To clarify the role of the potential anharmonicity, we fitted the potentials 𝑊(𝜃,𝜑) \nusing a parabolic function with an anharmonic term : \n𝑊(𝑥)=𝑊0+𝑘[(𝑥−𝑥0 )2+𝛽𝑥(𝑥−𝑥0)4]    (6) \nHere 𝑥=𝜃 or 𝜑, and 𝛽𝑥 is the anharmonicity parameter. We calculated it independently for \n𝜃 and 𝜑 for each dataset of 𝑊(𝜃,𝜑) obtained at different values of the external magnetic \nfield 𝐻 by fitting the total energy with Eq. (6) in the vicinity of the energy minimum  (Fig. 5) . \nThis anharmonicity should be examined on equal footing with the no nlinear damping \ncontribution. To quantify the latter, we follow the approach by Tiberkevich & Slavin  [8] and \nanalyze the effective damping dependencies on the precession amplitude by means of fitting \na second -order polynomial to them : \n𝛼=𝛼0+𝛼2𝐴2.      (7) \nThe examples of th e fit curves are shown in Fig. 7 a, demonstrating a good quality of the fit \nwithin a  certain range of the amplitudes  𝐴 (below 45 ). It should, however, be noted that the \nmodel in Ref.  [8] has been developed for the in -plane magnetic anisotropy, and thus its \napplicability for our case is limited. This is the re ason why we do not go beyond the amplitude \ndependence of the effective damping and do not analy ze the frequency dependence on 𝐴 in \n10 \n the limit of strong effective fields.  We note that the amplitude  𝐴, the opening angle of the \nprecession, should be understood as a mathematical parameter only, and not as a true \nexcursion angle of magnetization obtained in the real experimental conditions. There, large \neffective Gilbert damping values and a short decay t ime of the photo -magnetic anisotropy \npreclude the excursion of magnetization from its equilibrium to reach these 𝐴 values.  \n \nFig. 7. a) Effective damping in the linear and nonlinear precession regimes  of the precession amplitude \n𝐴. The lines are the second -order polynomial fits with Eq. (7). b ) Magnetic field dependence of the \nnonlinearity parameters: n onlinear damping coefficient 𝛼2 (points, obtained from experiments)  and \nthe 𝑊(𝜃) potential  anharmonicity  normalized  𝛽𝜃 (red line , calculated ).  \nWe note that the anharmonicity parameter 𝛽𝑥 calculated for the  W(θ) profiles was found to \nbe a few orders of magnitude larger than that obtained for W(𝜑). This difference in the \nanharmonicity justifies our earlier decision to focus on the shape of W(θ) potential only (cf. \nFig. 5). This means that the potential for magnetization in the azimuthal plane is muc h closer \nto the parabolic shape  and much larger amplit udes of the magnetization precession are \nrequired for it to start manifesting nonlinearities in dynamics. As such, we only consider the \nanharmonicity 𝛽𝑥 originating in the W(θ) potential energy. I n Fig. 7b, we compare the 𝛽𝜃 (red \nline) and 𝛼2 (points) dependencies on the external in -plane magnetic field. It is seen that its \ngeneral shape is very similar, corroborating our assumption that the potential anharmonicity \nis the main driving force behind the obse rved nonlinearity. We argue that thanks to the c ubic \nmagnetic anisotropy in YIG:Co  film, the potential anharmonicity -related mechanism of \nnonlinearity allows for reaching the nonlinear regime at moderate excitation levels.  \n \n5. Simulation s of laser -induced magnetization dynamics  \n11 \n To further prove that the ob served nonlinearities in magnetization dynamics do not require \nintroducing additional inertial or relativistic terms  [33], we complemented our experimental \nfindings with numerical simulations  of the LLG equation:  \n𝑑𝐌\n𝑑𝑡=−𝛾[𝐌×𝐇eff(𝑡)]+𝛼\n𝑀𝑠(𝐌×𝑑𝐌\n𝑑𝑡),    (8) \nwhere 𝐻𝑒𝑓𝑓 is the effective field derived from Eq. (2) as : \n𝐇eff(𝑡)=−∂𝑊𝐴\n∂𝑴+𝐇L(𝑡),     (9) \nWe employ ed the simulation model  from Ref.  [11] and added a term corresponding to the \nexternal magnetic field  𝐻. Calculations performed for a broad range of laser fluence s and \nexternal field values allowed us to obtain a set of  traces  of the magnetization dynamics . Figure \n8 show s a great deal of similarity  between simulations and experimen tal data (cf.  Fig. 3). It is \nseen that t he frequency increases with  increasing external  field  𝐻 while  the amplitude \ndecreases  (see Fig.  8a). The simulations for various stimulus strengths show the expected \ngrowth of the precession amplitude  (see Fig.  8b). \n \nFig. 8. Photo -magnetic  precession obtained in numerical simulations of the LLG equation  for: a) field \ndependence  at moderate excitation level  and b) power dependence  (I=4, 10, 16, and 22 arb. units ) at \n𝐻=0.4 kOe. \nWe further repeated our fit procedure with Eq.(1) to obtain the precession parameters from \nthese data. Figure 9 show s the values of the amplitude  and frequency of the precession in the \npower regime.  At a low field  𝐻=0.4 kOe (red) , the nonlinearity is clearly visible and \ncomparable with experimental data, as seen in Fig. 4. Similarly, at high field s (green) , the \nbehavior is mostly linear.  Figure 9a shows a great deal of similarity between simulations  \n(amplitude parameter)  and experi mental data  (normalize d value AF/max) (cf. Fig. 4a). The \nanalysis of the damping parameter (Fig. 9c) also confirms the exp erimental findings (as in Fig. \n7a), revealing the existence of two regimes, linear and nonlinear . The results of the s imulations \nconfirm that the observation of  the nonlinear response of the magnetic system  can be \nattributed  to the anharmonicity of the energy landscape.  \n12 \n Notably, in the simulations , as well as in the experimental data, we not only observe a second -\norder co rrection to the effective damping 𝛼2, but also a deviation from Eq.(7)  at even larger \namplitudes  (cf. Fig. 7 a and Fig. 9c). The latter manifests as a reduction of the effective damping \ncompared to the expected 𝛼0+𝛼2𝐴2 dependence shown with dashed lines. This higher -order \neffect is unlikely to originate in the multi -magnon scattering contribution since the latter \nwould only further increase the effective damping  [8]. We rather believe that th is is likely an \nartifact of the used damped oscillator model  where  in the range of  𝛼𝑒𝑓𝑓≈1 the quasiperiodic \ndescription of magnetization precession ceases to be physically justified.    \n \nFig. 9. Power dependence of  the a) amplitude  and b) frequency as obtained in the simulations  for low \n(red dataset) and high (green dataset) external magnetic field s. c) Effective damping  in the linear and \nnonlinear  precession regimes . \n \n6. Photo -induced phase diagram of sub -switching regime  \nIt is seen from both experimental and numerical results above that the cubic symmetry of the \nmagnetic system is key for the observed nonlinear magnetization dynamics. To quantify the \nparameter space for the nonlinearity, we first estimate the realistic values of the effective \nlight -induced magnetic field 𝐻𝐿. Throughout a number of works on photo -magnetism in Co -\ndoped garnets, a single -ion approach to magnetic anisotropy is consistent ly utilized. We note \n13 \n that in YIG:Co, it is the Co ions at tetrahedral sites that are predominantly responsible for the \ncubic anisotropy of  the magnetic energy landscape  [22]. In the near -IR range, these ions are \nresonantly excited at the 1300 nm wavelength, resulting in improved efficiency of the photo -\nmagnetic stimulus , as compared to previous works  [45]. Further, we note that at the \nmagnetization switching threshold, about 90% of the Co3+ ions with a concentration on the \norder of 1020 cm-3 are excited with incident photons  [11, 46] . Taking into account the single -\nion contribution to the anisotropy 𝛥𝐾1~105 erg/cm3 [47], and assuming a linear relation \nbetween the absorbed laser power (or fluence) and the effective photo -magnetic field 𝐻𝐿, for \nthe latter we find that 𝐻𝐿~1 kOe is sufficient for the magnetization switching. This means that \nthe sub -switching regime of magnetiz ation dynamics (cf. Fig. 1) refers to the laser fluences (as \nwell as wavelengths) , resulting in smaller effective fields.  \nWe reiterate that in previous works, the impact of the external magnetic field on the photo -\nmagnetically driven magnetization precess ion has not been given detailed attention.  To \naddress this gap , we plotted  the amplitude of the precession  𝐴 calculated in the same way as \nabove in the  sub-switching regime (Fig. 10) . As expected, the amplitude generally increases  \nwith 𝐻𝐿. However, we n ote a critical external field of about 0.5 kOe at which the desired \namplitudes can be reached at smaller light -induced effective fields 𝐻𝐿. At this field, where the \nsystem enters a single domain state, the potential curvature around the energy minimum \ndecreases, thus facilitating the large -angle precession. In other words, external magnetic field s \ncan a ct as leverage for the effective field of the photo -induced anisotropy, thus reducing the \nmagnetization switching threshold.  An exhaustive study of magneti zation switching across the \nparameter space shown in Fig. 10 remains an attractive perspective for future studies.  \n \nFig. 10. Calculated amplitude m ap of the photo -induced magnetization  precession in YIG:Co film. \nIn our analysis, we only considered a truly photo -magnetic excitation and neglected the laser -\ninduced effects of thermal origi n. It is, however, known that laser -driven heating can introduce \nan additional, long -lasting modification of magnetic anisotropy in iron garnets  [48, 49] . The \n14 \n relatively long relaxation times associated with cooling are responsible for the concomitant \nmodulation of the precession parameters and thus facilitate nonlinearities in the response of \nthe magnetic system. Yet, 1300 nm laser excitation of magnetization dynamics in YIG :Co film  \nwas shown to be highly polarization -dependent  [6], thus indicating the dominant role of the \nnon-thermal excitation mechanism. On the other hand, the unavoid able laser -induced heating \nwith experi mental values of laser fluence in YIG:Co film  has been estimated to not exceed 1 K  \n[6]. As such, we do not expect modification of the Gilbert damping associated with the \nproximity of the ma gnetization compensation or N éel temperature in the ferromagnetic \ngarnet  [50]. However, a detailed investigation of the temperature -dependent nonlinear \nmagnetization dynamics in the vicinity of the compensation point or a magnetic phase \ntransition  [51, 52]  represents another promising research direction.  Further , exploring the \nnonlinear  regime  in the response of the magnetic system to intense THz stimul i along the lines \ndiscussed in  [33] enjoys a rich potentia l for spintronic applications.  \n  \n7. CONCLUSIONS  \nIn summary, we studied, both experimentally and numerically, the nonlinear regime of \nmagnetization dynamics in photo -magn etic Co -doped YIG film. After  excitation with \nfemtosecond laser pulses at fluences below the magnetization switching threshold, there is a \nrange of external magnetic field where the magnetic system demonstrates strongly non linear \nprecession characterized by a significant increase of t he effective Gilbert damping. We \nattribute this nonlinearity to the anharmonicity of the potential for the magnetic oscillator \nenhanced by the dominant role of the cubic magnetocrystalline anisotropy. The effective \ndamping and its nonlinear contribution, a s obtained from numerical simulations, both \ndemonstrate a very good agreement with the experimental findings. Simulations of the \nmagnetization dynamics by means of  the LLG equation further confirm the nonlinearity in the \nmagnetic response below the switchi ng limit. Finally, we provide estimations for the realistic , \neffective photo -magnetic fields 𝐻𝐿 and map the workspace of the parameters in the sub -\nswitching, nonlinear regime of photo -induced magnetization dynamics.  \n \nACKNOWLEDGMENTS  \nThis work has been fu nded by the  Foundation for Polish Science ( Grant No. POIR.04.04.00 -00-\n413C/17)  and the National Science Centre Poland (Grant No. DEC -2017/25/B/ST3/01305) . \n \nREFERENCES  \n \n[1]  A. Kirilyuk, A. V. Kimel and T. Rasing, \"Ultrafast optical manipulation of magnetic order,\" \nRev. Mod. Phys., 82 (3), 2731 (2010).  15 \n [2]  J. Walowski and M. 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Pearson,1J. B. Ketterson,2V. Novosad,1M. Wu,3and A. Hoffmann1\n1Materials Science Division, Argonne National Laboratory, Argonne IL 60439, USA\n2Department of Physics and Astronomy, Northwestern Univers ity, Evanston IL 60208, USA\n3Department of Physics, Colorado State University, Fort Col lins CO 80523, USA\n(Dated: June 25, 2021)\nDue to its transverse nature, spin Hall effects (SHE) provide the possibility to excite and detect\nspin currents and magnetization dynamics even in magnetic i nsulators. Magnetic insulators are out-\nstanding materials for the investigation of nonlinear phen omena and for novel low power spintronics\napplications because of their extremely low Gilbert dampin g. Here, we report on the direct imaging\nof electrically driven spin-torque ferromagnetic resonan ce (ST-FMR) in the ferrimagnetic insulator\nY3Fe5O12based on the excitation and detection by SHEs. The driven spi n dynamics in Y 3Fe5O12\nis directly imaged by spatially-resolved microfocused Bri llouin light scattering (BLS) spectroscopy.\nPreviously, ST-FMR experiments assumed a uniform precessi on across the sample, which is not\nvalid in our measurements. A strong spin-wave localization in the center of the sample is observed\nindicating the formation of a nonlinear, self-localized sp in-wave ‘bullet’.\nMagneticmemoryandlogicdevicesrelyontheefficient\nmanipulation of the orientation of their magnetization\nusing low power1,2. Recently, there has been revitalized\ninterest in the ferrimagnetic insulator yttrium iron gar-\nnet (YIG, Y 3Fe5O12) motivated by the discovery of spin-\ntronic effects by combining this material and heavy met-\nals such as Pt3–7. Its extremely small magnetic damping\nenables low power data transmission and processing on\nthe basis of magnons, the elementary quanta of magnetic\nexcitations.3–5,8–12. In addition the low damping YIG\nalso enables nonlinear phenomena where the superposi-\ntion principle breaks down11. Previous work reported on\nthe formation of spin-wave caustics13, Bose Einstein con-\ndensationofmagnons14andnonlinearmode conversion15\nto name only a few. Recently, it has become possible to\ngrow nanometer-thick YIG films, which allow the prepa-\nration of micro- and nanostructured devices5,8,9,12,16.\nTherefore,the studyofnonlinearspin dynamicsin minia-\nturized YIG systems has only just begun.\nIndependent of the progress of the YIG film growth,\nthe development in employing spin-orbit interaction in\nheavy metals17,18and their alloys19in contact with a\nferromagnet (FM) has flourished. The SHE20,21can be\nused for the generation of strong current-driven torques\non the magnetization in the FM layer. The resultant\nspin current can drive spin-torque ferromagnetic reso-\nnance (ST-FMR) in bilayers consisting of ferromagnetic\nand nonmagnetic metals and be detected by a hom-\ndyne mixing of the microwavesignal with the anisotropic\nmagnetoresistance22. Recent theories propose that ST-\nFMR can be extended to insulating FM/normal metal\nbilayers. Here, the detection of magnetization precession\noccurs by spin pumping and a rectification of the spin\nHall magnetoresistance23,24. We showed recently that\nthis rectification process is indeed possible in YIG/Pt\nbilayers25. All previousanalysisof electric measurements\nassume uniform precession across the sample22,26. In or-\nder to validate this assumption it is highly desirable to\nimageaccurrent-driven spin dynamics spatially-resolvedand frequency-resolved. These investigations provide\ninteresting insights in the underlying physics, such as\nwhether bulk or edge modes are preferably excited by\nST-FMR or nonlinear spin dynamics may occur.\nIn this letter, we show experimentally the excitation of\nspin dynamics in microstructured magnetic insulators by\nthe SHE of an adjacent heavy metal and observe the for-\nmation of a nonlinear, self-localized spin-wave intensity\nin the center of the sample27–29. The magnetization dy-\nnamics in a nanometer-thick YIG layer is driven simulta-\nneouslybythe Oerstedfield andaspin torqueoriginating\nfrom a spin current generated by the SHE of an attached\nPt layer. The dynamics is detected in two complemen-\ntary ways: (1) Electrically, by a rectification mechanism\n(a) (b)\n(c) (d)\n-2.0-1.5-1.0-0.50.00.5DC voltage VDC (µV) \n12001000800600\nMagnetic field H (Oe) Data\n Total fit\n SMR\n Spin pumping1.5\n1.0\n0.5\n0.0DC voltage VDC (µV) \n12001000800600\nMagnetic field H (Oe)\nFIG. 1. (a) Schematic of the ST-FMR experimental setup (b)\nST-FMR mechanism in the YIG/Pt bilayer. The alternat-\ningrfcurrent drives an Oersted field hrfexerting a field-like\ntorqueτHon the magnetization M. At the same time a oscil-\nlatorytransverse spinaccumulationat theYIG/Ptinterfac e is\ngenerated by the SHE which results in a damping-like torque\nτSTT. (c) and (d) Typical dcvoltage spectra recorded at in-\nplane angles of φ= 30◦andφ= 240◦andP= +10 dBm.2\n-4-2024 VDC (µV)\n-1000 -500 0500 1000\n Magnetic field H (Oe)+15 dBm\n+12 dBm\n+10 dBmf = 4 GHz(a)\n(b)\n-3-2-10123SMR voltage VSMR (µV)\n250 200 150 10050 0\nIn-plane angle φ (°)-400-2000200400Spin-pumping\nvoltage VSP (nV)-3-2-1VDC (µV)subsidiary \n   mode main\nmode\nFIG. 2. (a) Typical VDCspectra at a constant frequency\nf= 4 GHz for various applied microwave powers. The in-\nset shows the resonance peak at P= +15 dBm. Two modes\nare detected. (b) In-plane angular dependence of the SMR,\nVSMR, and of the spin-pumping contribution, VSP, to the dc\nvoltage. The solid lines represent fits ∝cosφsin 2φ.\nof the spin Hall magnetoresistance (SMR)30–32as well as\nby spin pumping3–5,33–35and (2) Optically, by spatially-\nresolved Brillouin light scattering (BLS) microscopy36.\nThe experimental findings are further validated by mi-\ncromagnetic simulations37.\nYIG(40 nm)/Pt bilayers were fabricated by in-situ\nmagnetron sputtering under high-purity argon atmo-\nsphere on single crystal gadolinium gallium garnet\n(GGG, Gd 3Ga5O12) substrates of 500 µm thickness with\n(111) orientation16. For the electrical measurements a\nPt thickness of 2 nm was used, while for the optical in-\nvestigations the thickness was 5 nm in order to minimize\nthe influence of additional heating effects by the laser.\nIn a subsequent fabrication process, stripes in the shape\nof 30×5µm2(electrical measurements) and 5 ×5µm2\n(optical measurements) were patterned by photolithog-\nraphy and ion milling5. A coplanar waveguide (CPW)\nmade of Ti/Au (3 nm/120 nm) was structured on top of\nthe bar allowing the signal line to serve as a lead for the\nYIG/Pt bar as illustrated in Fig. 1(a). In this ST-FMR\nconfiguration a bias-T is utilized to allow for simultane-\nous transmission of a microwave signal with dcvoltage\ndetection via lock-in technique across the Pt. For this\npurpose the amplitude of the rfcurrent is modulated at\n3 kHz. We use a BLS microscope with a spatial reso-\nlution of 250 nm, where the laser spot is focused onto\nthe sample and the frequency shift of the back reflected\nlight is analyzed by a multi-pass tandem Fabry P´ erot\ninterferometer36. The detected BLS intensity is propor-\ntional to the square of the dynamic magnetization, i.e.,\nthe spin-wave intensity.In order to excite a dynamic response by ST-FMR in\ntheYIGsystema rfsignalispassedthroughthePtlayer.\nThe magnetization dynamics is governed by a modified\nLandau-Lifshitz-Gilbert equation23,24:\ndM\ndt=−|γ|M×Heff+αM×dM\ndt+|γ|/planckover2pi1\n2eMsdFJs,(1)\nwhereγis the gyromagnetic ratio, Heff=hrf+HD+H\nis the effective magnetic field including the microwave\nmagnetic field hrf, demagnetization fields HD, and the\nbias magnetic field H.αis the Gilbert damping param-\neter [the second term describes the damping torque τα,\nFig.1(b)] and Jsis a transverse spin current at the inter-\nface generated by the SHE from the alternating charge\ncurrent in the Pt layer23,24:\nJs=Re(g↑↓\neff)\neM×(M×µs)+Im(g↑↓\neff)\neM×µs.(2)\nHere,g↑↓\neffis the effective spin-mixing conductance and µs\nis the spin accumulation at the YIG/Pt interface. The\nfirstterminEq.( 2)describesananti-damping-liketorque\nτSTTand the second term is a field-like torque τH. As\nillustrated in Fig. 1(b) and described by Eq. ( 1) the mag-\nnetizationis drivenbythe independent torquetermscon-\ntaininghrfandJs.\nFirst, we describe the electrical characterization of the\nYIG/Pt bars by means of ST-FMR. Figure 1(c) and\n(d) illustrate typical dcvoltage spectra; exemplarily, we\nshow spectra recorded at in-plane angles of φ= 30◦and\nφ= 240◦, with applied rfpowerP= +10 dBm. A sig-\nnalisobservedwhen thesystemisdrivenresonantly. The\ndata is analyzed using the model proposed by Chiba et\nal. (supplementary information)23,24. According to the\nmodel, two signals contribute to the dcvoltage: (1) Spin\npumping which manifests in a symmetric contribution to\nthe Lorentzian lineshape. (2) Spin Hall magnetoresis-\ntance which is a superimposed symmetric and antisym-\nmetric Lorentzian curve [Fig. 1(c,d)].\nFIG. 3. Color-coded dispersion relation measured by BLS\nmicroscopy. The laser spot was focused onto the center of\nthe sample while the rffrequency as well as magnetic field\nwere varied. As for the electrical measurements two modes\nare detected by BLS. The inset shows corresponding field de-\npendence of the resonance measured by electrical means.3\nFIG. 4. Spatially-resolved BLS map of the 5 ×5µm2large\nYIG/Pt sample. The magnetic field H= 665 Oe is applied\natφ∼45◦. (a) - (d) Driving microwave frequency increases\nfrom 3.7 GHz to 3.85 GHz, microwave power P= +17 dBm.\nFigure2(a) illustrates dcvoltage spectra at a fixed\nmicrowave frequency f= 4 GHz for three different ap-\nplied powers. The offset is due to the longitudinal spin\nSeebeck effect6,7(see supplementary information) and\ndoes not affect the conclusions drawn from the resonance\nsignal7. The inset in Fig. 2(a) shows the resonance peak\natP= +15 dBm. Clearly, a less intense, secondary\nmode in addition to the main mode is detected. Accord-\ning to the Chiba model23,24thedcvoltage signal can be\ndeconvoluted into a spin-pumping and a SMR contribu-\ntion as also shown in Fig. 1(c) and (d). To analyze the\ndata employing the model we use a spin-mixing conduc-\ntance ofg↑↓\neff= 3.36×1014Ω−1m−2and a spin-Hall angle\nofγSHE= 0.0938. A fit to the angular-dependent data\nyields a phasedifference between Oersted field and the ac\ncurrent of δ= 64±5◦[see Fig. 1(c,d)]. Figure 2(b) shows\nthe angular dependences of the fitted spin-pumping and\nthe SMR signals. The model predicts the same angular\ndependent behavior ∝cosφsin2φfor spin pumping and\nSMR. As seen in Fig. 2(b), we find a good agreement be-\ntweentheory(solidlines)andexperimentforbothcurves.\nPlease note that we observe a small, non-vanishing volt-\nage at angles φ=n·90◦,n∈N, where the model sug-\ngests zero voltage23–25. In this angular range the model\nbreaks down and the experimental data cannot be fitted\n(see supplementary information).\nIn the following we compare the electrical measure-\nmentswiththeresultsobtainedbyBLSimaging. Theop-\ntical measurements wereperformed on YIG(40 nm)/Pt(5\nnm) bars having a lateral size of 5 ×5µm2. The ex-\nternal magnetic field is applied at an angle of φ∼45◦\nwhere the dcvoltage detection is maximized [Fig. 2(b)].\nFigure3shows the dispersion relation measured by BLS\nin a false color-coded image where red indicates a highspin-wave intensity and the blue area shows the ab-\nsence of spin waves. The measured dispersion is in\nagreement with the electrical measurements as shown in\nthe inset: As the field increases the resonance shifts to\nhigher frequencies as is expected from the Kittel equa-\ntion,f=|γ|\n2π/radicalbig\nH(H+4πMeff), where Meffis the effec-\ntive magnetization.\nAs is apparent from Fig. 2(a) and Fig. 3magnetization\ndynamics can be excited in a certain bandwidth around\nthe resonance which is determined by the specific de-\nvice characteristics. Furthermore, both figures (electrical\nand optical detection) suggest that there is an additional\nmode below the main mode. At first, one might identify\nthis mode as an edge mode39,40. However, this is not the\ncase as it will be discussed below.\nIn order to spatially map the spin-wave intensity, the\napplied magnetic field is kept fixed at H= 665 Oe. Fig-\nure4illustrates the experimental observations in false\ncolor-coded images. At an excitation frequency below\nthe resonance frequency, e.g., f= 3.7 GHz no magneti-\nzation dynamics is detected [Fig. 4(a)]. As the frequency\nincreases the system is driven resonantly and a strong\nspin-wave intensity is observed from f= 3.725 GHz\ntof= 3.8 GHz, Fig. 4(b,c). Increasing the frequency\neven further results in a diminished signal, Fig. 4(d) for\nf= 3.85 GHz. At even larger frequencies no magnetiza-\ntion dynamics is detected as it is also apparent from the\ndispersion illustrated in Fig. 3. In conventional electrical\nST-FMR measurements, a uniform spin-wave intensity\ndistribution across the lateral sample dimensions is as-\nsumed. However, as our experimental results show, this\nassumption is not fulfilled: A strong spin-wave signal is\nlocalized in the center of the YIG/Pt bar. It is desir-\nable to experimentally investigate at what minimum ex-\ncitation power the formation of the localization occurs.\nHowever, in the investigated range of powers we always\nobserve a localization in the center of the sample (see\nsupplementary information). For rfpowers of less than\n+11 dBm the signal is below our noise-floor.\nIn spite of this experimental limitation, we also car-\nried out micromagnetic simulations in order to gain fur-\nther insight into the underlying magnetization dynamics.\nThe simulations confirmed qualitatively the experimen-\ntal observations as is depicted in Fig. 5: Two modes can\nbe identified in the simulations, Fig. 5(a). In the low\npower regime, which is not accessible experimentally, we\nfind that the spatial magnetization distribution of the\nmain mode is almost uniform and the less intense sub-\nsidiary mode is localized at the edges ( hrf= 0.25 Oe,\nnot shown). With increasing rfpower, the spatial dis-\ntributions of both modes transform and at a threshold\nofhrf≈1 Oe a localization of both modes in the center\nof the sample is observed. Figure 5(b,c) show the corre-\nsponding spatial dynamic magnetization distributions at\nhrf= 5 Oe and agreewell with the experimental findings,\nFig.4.\nThis spatial profile can be understood asthe formation\nof a nonlinear, self-localized ‘ bullet’-like spin-wave inten-4\n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 Intensity  (a.u.) \n3.8 3.7 3.6 3.5 3.4 3.3 \nFrequency f (GHz)(a) \nmain mode subsidiary mode \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 Normalized integrated \n BLS intensity  (a.u.) 50 40 30 20 \nrf  power P (mW) 1.0 \n0.8 \n0.6 \n0.4 \n0.2 Normalized integrated \nintensity simulation (a.u.) \n20 15 10 5 0rf magnetic field h rf (Oe) (d) \n1.0 \n0.8 \n0.6 \n0.4 \n0.2 \n0.0 (b) \n (c) subsidiary mode main mode \n1 µm \nFIG. 5. Micromagnetic simulations: (a) The spectrum reveal s\ntwo modes. Spatially-resolved magnetization distributio n of\nthe main mode, (b), and the less intense, subsidiary mode,\n(c). (d) The normalized integrated BLS intensity saturates\nat high excitation powers P, which is validated by micromag-\nnetic simulations at large driving rfmagnetic fields hrf.\nsity caused by nonlinear cross coupling between eigen-\nmodes in the system15. This process is mainly deter-\nmined by nonlinear spin-wave damping which transfers\nenergy from the initially excited ferromagnetic resonance\ninto other spin-wave modes rather than into the crys-\ntalline lattice15. To check this assumption, we plot-\nted in Fig. 5(d) the normalized integrated BLS intensity\nas well as the integrated spatial magnetization distribu-\ntion as a function of the applied microwave power and\ntherfmagnetic field, respectively. Both integrated sig-\nnals demonstrate a nonlinear behavior and saturate at\nhigh powers/microwave magnetic fields. This observa-\ntion is a direct manifestation of nonlinear damping: en-\nergy is absorbed by the ferromagnetic resonance and re-\ndistributedtosecondaryspin-wavemodesmoreandmore\neffectively15.\nUntil now, ST-FMR experiments assumed a uni-\nform magnetization precession22–24,26. However, as our\nspatially-resolved BLS results demonstrate and con-\nfirmed by micromagnetic simulations, the driven lateral\nspin-wave intensity distribution in insulating FMs devi-ates from this simple model at higher excitation powers\nwhich are common in ST-FMR measurements. The for-\nmation of a localized spin-wave mode was not considered\nin previous ST-FMR experiments neither in metals nor\nin insulators. Our findings have direct consequences on\nthe analysis and interpretation of ST-FMR experiments.\nThe precession amplitude is not uniform across the sam-\nple implying that the effective spin-mixing conductance\ng↑↓\neffis actually an average over the sample cross sec-\ntion. In areas where the precession amplitude is large,\ng↑↓\neffis underestimated, whereas it is over estimated in\nlow-intensityareas. This also complicates the determina-\ntion of the spin-Hall angle from ST-FMR measurements.\nMicromagnetic simulations show phase inhomogeneity,\nspecifically aroundthe perimeter ofthe mode. The phase\ninhomogeneity tends to equally lag and lead the main\nuniform phase of the center mode; effectively the phase\ninhomogeneity then leads to no significant change to the\nlineshape. However,assuming the phaseat the perimeter\nto be uniformly leading the bulk phase results in a cor-\nrectionto the lineshape that is still negligiblebecause the\neffective areaand amplitude where the phase is deviating\nis significantly smaller than the bulk area. Nevertheless,\nin general the issue of inhomogeneous phase distribution\nmay complicate the analysis of electrical ST-FMR spec-\ntra, especially in smaller samples.\nInconclusion,wedemonstratedthattheconceptofST-\nFMR can be extended to magnetic insulators where the\nformation of a nonlinear, self-localized spin-wave inten-\nsity driven by an accurrent was observed. We adopted\nan electrically-driven ST-FMR excitation and detection\nscheme in magnetic insulator (YIG)/heavy normal metal\n(Pt)bilayersthatwasoriginallydevelopedforall-metallic\nsystems. A dcvoltage in YIG/Pt bilayers was observed\nunder resonance condition by a SMR-mediated spin-\ntorque diode effect in agreement with theoretical pre-\ndictions. Spatially-resolved BLS microscopy revealed a\nstrong ‘bullet’-like spin-wave localization in the center\nof the sample due to nonlinear cross coupling of eigen-\nmodes in the system. Since the observed electrical signal\nis sufficiently large and the signal-to-noise ratio is rea-\nsonably good, down-scaling of sample dimensions to the\nnanometer-scale is feasible.\nACKNOWLEDGMENTS\nWe thank Stephen Wu for assistance with ion milling.\nThe work at Argonne was supported by the U.S. De-\npartment of Energy, Office of Science, Materials Science\nand Engineering Division. Lithography was carried out\nat the Center for Nanoscale Materials, an Office of Sci-\nence user facility, which is supported by DOE, Office of\nScience, Basic Energy Science under Contract No. DE-\nAC02-06CH11357. The work at Colorado State Univer-\nsity was supported by the U. S. Army Research Office\n(W911NF-14-1-0501),theU.S.NationalScienceFounda-\ntion (ECCS-1231598),C-SPIN(one ofthe SRCSTARnet5\nCenters sponsored by MARCO and DARPA), and the U. S. Departme nt of Energy (DE-SC0012670).\n∗jungfleisch@anl.gov\n1D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008).\n2A. 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Grundler, Advanced Materials 21, 2927\n(2009)."    },    {        "title": "1502.01420v2.Nonlinear_analysis_of_magnetization_dynamics_excited_by_spin_Hall_effect.pdf",        "content": "arXiv:1502.01420v2  [cond-mat.mes-hall]  12 Mar 2015Nonlinear analysis of magnetization dynamics excited by sp in Hall effect\nTomohiro Taniguchi\nNational Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan.\n(Dated: October 6, 2018)\nWe investigate the possibility of exciting self-oscillati on in a perpendicular ferromagnet by the\nspin Hall effect on the basis of a nonlinear analysis of the Lan dau-Lifshitz-Gilbert (LLG) equation.\nIn the self-oscillation state, the energy supplied by the sp in torque during a precession on a constant\nenergy curve should equal the dissipation due to damping. Al so, the current to balance the spin\ntorque and the damping torque in the self-oscillation state should be larger than the critical current\nto destabilize the initial state. We find that these conditio ns in the spin Hall system are not satisfied\nby deriving analytical solutions of the energy supplied by t he spin transfer effect and the dissipation\ndue to the damping from the nonlinear LLG equation. This indi cates that the self-oscillation of a\nperpendicular ferromagnet cannot be excited solely by the s pin Hall torque.\nPACS numbers: 75.78.-n, 05.45.-a, 75.78.Jp, 75.76.+j\nI. INTRODUCTION\nNonlinear dynamics such as fast switching and self-\noscillation (limit cycle) has been a fascinating topic in\nphysics1,2. Magnetization dynamics excited by the spin\ntransfer effect3,4in a nanostructured ferromagnet5–12\nprovide fundamentally important examples of such non-\nlinear dynamics. The magnetization switching was first\nobserved in Co/Cu metallic multilayer in 20005. Three\nyears later, self-oscillation was reported in a similar\nsystem6. In these early experiments on the spin transfer\neffect, linear analysis was used to estimate, for exam-\nple, the critical current destabilizing the magnetization\nin equilibrium13,14. However, recently it became clear\nthat nonlinear analysis is necessary to quantitatively an-\nalyze the magnetization dynamics2,15–26. For example,\ncurrent density to excite self-oscillation can be evaluated\nby solvinga nonlinearvectorequation calledthe Landau-\nLifshitz-Gilbert (LLG) equation23,24.\nOriginally, the spin transfer effect was studied by ap-\nplyinganelectriccurrentdirectlytoaferromagneticmul-\ntilayer. Recently, however, an alternative method em-\nploying the spin Hall effect has been used to observe the\nspin transfer effect27–40. The spin-orbit interaction in a\nnonmagnetic heavy metal scatters the spin-up and spin-\ndown electrons to the opposite directions, producing a\npure spin current flowing in the direction perpendicular\nto an applied current. The pure spin current excites the\nspin torque, called spin Hall torque, on a magnetization\nin a ferromagnet attached to a nonmagnet. The direc-\ntionofthespinHalltorqueisgeometricallydetermined27,\nand its magnitude shows a different angular dependence\nthan the spin torque in the ferromagnetic multilayer3.\nTherefore, it is fundamentally unclear whether the phys-\nical phenomena observed in the multilayer5–12can be re-\nproduced in the spin Hall system, and thus, new phys-\nical analysis is necessary. The magnetization switching\nof both in-plane magnetized and perpendicularly mag-\nnetized ferromagnets by spin Hall torque was recently\nreported28–31,36,37. Accordingly, it might be reasonableto expect reports on self-oscillation by spin Hall torque.\nHowever, whereas self-oscillation has been observed in\nthe in-plane magnetized system32, it has not been re-\nported yet in the perpendicularly magnetized system.\nThe purpose of this paper is to investigate the possibil-\nityofexcitingself-oscillationbyspinHalltorquebasedon\na nonlinear analysis of the LLG equation. We argue that\ntwo physical conditions should be satisfied to excite self-\noscillation. The first condition isthat the energythat the\nspintorquesuppliesduringaprecessiononaconstanten-\nergy curve should equal the dissipation due to damping.\nThe second condition is that the current to balance the\nspin torque and the damping torquein the self-oscillation\nstate should be larger than the critical current to desta-\nbilize the initial state. This is because the magnetization\ninitially stays at the minimum energy state, whereas the\nself-oscillation corresponds to a higher energy state. We\nderive exact solutions of the energy supplied by the spin\ntransfer effect and the dissipation due to damping in the\nspin Hall system by solving the nonlinear LLG equation,\nandfindthat theseconditionsarenotsatisfied. Thus, the\nself-oscillation of a perpendicular ferromagnet cannot be\nexcited solely by the spin Hall torque.\nThe paper is organized as follows. The physical condi-\ntions to excite a self-oscillation is summarized in Sec. II.\nThese conditions are applied to the spin Hall system in\nSec. III. Section IV is devoted to the conclusions.\nII. PHYSICAL CONDITIONS TO EXCITE\nSELF-OSCILLATION\nLet us first summarize the physical conditions neces-\nsaryto excite self-oscillation. The magnetization dynam-\nics are described by the LLG equation\ndm\ndt=−γm×H−γHsm×(p×m)+αm×dm\ndt,(1)\nwheremandpare the unit vectors pointing in the\ndirections of the magnetization and the spin polariza-2\ntion of the spin current, respectively. The gyromag-\nnetic ratio and the Gilbert damping constant are de-\nnoted as γandα, respectively. The magnetic field H\nrelates to the energy density of the ferromagnet Evia\nH=−∂E/∂(Mm), where Mis the saturation magneti-\nzation. The strength of the spin torque, Hs, is propor-\ntional to the current density j. Since the LLG equation\nconserves the norm of the magnetization, the magnetiza-\ntion dynamics can be described as a trajectory on a unit\nsphere. The energy density Eshows constant energy\ncurves on this sphere. For example, when the system has\nuniaxial anisotropy, the constant energy curves are lati-\ntudelines. Theself-oscillationisasteadyprecessionstate\non a constant energy curve excited by the field torque,\nthe first term on the right-hand side of Eq. (1). This\nmeans that the second and third terms of Eq. (1), aver-\naged over the constant energy curve, cancel each other.\nIn other words, the energy supplied by the spin trans-\nfer effect during the precession on the constant energy\ncurve equals the dissipation due to the damping. This\ncondition can be expressed as2,24\n/contintegraldisplay\ndtdE\ndt=Ws+Wα= 0, (2)\nwhere the energy supplied by the spin transfer effect and\nthe dissipation due to the damping during the precession\non the constant energy curve of Eare given by2,15–26\nWs(E) =γM/contintegraldisplay\ndtHs[p·H−(m·p)(m·H)],(3)\nWα(E) =−αγM/contintegraldisplay\ndt/bracketleftBig\nH2−(m·H)2/bracketrightBig\n.(4)\nThe time integral is over a precession period on a con-\nstant energy curve. We emphasize that Eqs. (3) and (4)\nare functions of the energy density E. We denote the\nminimum and maximum values of EasEminandEmax,\nrespectively. When the energy density also has saddle\npointsEsaddle,Emaxin the following discussion can be\nreplaced by Esaddle. To excite the self-oscillation, there\nshould be a certain value of the electric current density\nthat satisfies Eq. (2) for Emin< E < E maxin a set of\nreal numbers. Therefore, Eq. (2) can be rewritten as\n∃j∈R,Ws+Wα= 0. (5)\nWe denote the current satisfying the first condition, Eq.\n(2), or equivalently Eq. (5), as j(E).\nAnother condition necessary to excite self-oscillation\nrelatestothefactthatthemagnetizationinitiallystaysat\nthe minimum energy state. To excite any kind of magne-\ntization dynamics, the spin torque should destabilize the\ninitial state, which means that a current density larger\nthan the critical current density, jc=j(Emin), should be\ninjected. Then, the condition\nj(E)> j(Emin), (6)z\nxy\njmspin Hall torque\nHt(a)\n(b)\nspin Hall torquespin Hall torquedamping\ndampingz xyHt // xHt // y\nFIG. 1: (a) Schematic view of system. The current density\njflows in the nonmagnet along the x-axis, exciting the spin\nHall torque pointing in the y-direction on the magnetization\nmin the ferromagnet. The applied magnetic field is denoted\nasHt. (b) Schematic view of the precession trajectory of\nthe magnetization on the constant energy curve. The solid\ncircle is the trajectory in the absence of the magnetic field\nor in the presence of the field along the z-axis, whereas the\ndashed elliptical lines are those in the presence of the field in\nthexandy-axes. The solid and dotted arrows represent the\ndirections of the spin Hall torque and the damping torque,\nrespectively.\nshould be satisfied to excite the self-oscillation. If this\ncondition is not satisfied, the magnetization directly\nmoves to a constant energy curve including the saddle\npoint without showing a stable steady precession, and\nstops dynamics because the spin torque does not balance\nthe damping torquefor Emin< E < E saddle. An example\nof such dynamics is shown below; see Fig. 3. We empha-\nsize that Eqs. (5) and (6) are applicable to any kind of\nphysical system showing a self-oscillation.\nIII. SPIN HALL SYSTEM\nLet us apply the abovediscussions to the spin Hall sys-\ntem schematically shown in Fig. 1 (a), where the electric\ncurrent flows in the nonmagnet along the xdirection,\nwhereas the ferromagnet is attached along the zdirec-\ntion. The spin polarization of the spin current is geomet-\nrically determined as p=ey. In the spin Hall system,\nthe spin torque strength Hsis given by\nHs=/planckover2pi1ϑj\n2eMd, (7)3\nwhereϑanddare the spin Hall angle and the thickness\nof the ferromagnet, respectively. The magnetic field H\nconsists of the applied field Htand the perpendicular\nanisotropy field HKmzez. We can assume that Ht>0\nwithout losing generality because the sign of Htonly af-\nfects the sign of j(E) derived below. Since we are in-\nterested in a perpendicular ferromagnet, we assume that\nHK> Ht>0. Figure 1 (b) schematically shows the\nprecession trajectory of the magnetization on a constant\nenergycurve, where the directions ofthe spin Hall torque\nand the damping torque are represented by the solid and\ndotted arrows, respectively. The spin Hall torque is par-\nallel to the damping torque for my>0, whereas it is\nanti-parallel to the damping torque for my<0. This\nmeans that the spin Hall torque dissipates energy from\nthe ferromagnetwhen my>0, andsuppliesthe energyto\nthe ferromagnet when my<0. Then, due to the symme-\ntry of the trajectory, the net energy supplied by the spin\nHall torque, Ws, is zero when the applied magnetic field\npoints to the x- orz-direction. This means that Eq. (2)\ncannot be satisfied, and thus, self-oscillation cannot be\nexcited in the spin Hall system in the absence of the ap-\nplied magnetic field, or in the presence of the field point-\ning in the x- orz-direction. Therefore, in the following\nwe focus on the applied magnetic field pointing in the\ny-direction. The magnetic field and the energy density\nare given by\nH=Htey+HKmzez, (8)\nE=−MHtmy−MHK\n2m2\nz. (9)\nThe minimum energy of Eq. (9) is\nEmin=−MHK\n2/bracketleftBigg\n1+/parenleftbiggHt\nHK/parenrightbigg2/bracketrightBigg\n,(10)\nwhich corresponds to a point mstable =\n(0,Ht/HK,/radicalbig\n1−(Ht/HK)2). On the other hand,\nEq. (9) has a saddle point at msaddle= (0,1,0),\ncorresponding to the energy density\nEsaddle=−MHt. (11)\nSince the magnetization initially stays at the minimum\nenergy state, and the magnetization dynamics stops\nwhenmreaches the saddle point msaddle, we consider\nthe energy region of Emin< E < E saddle. To calculate\nEqs. (3) and (4), it isnecessarytosolveanonlinearequa-\ntiondm/dt=−γm×H, whichdetermines the precession\ntrajectory of mon the constant energy curve. Since the\nconstant energy curve of Eq. (9) is symmetric with re-\nspect to the yz-plane, it is sufficient for the calculation of\nEqs. (3) and (4) to derive the solutions of mfor half of\nthe trajectory in the region of mx>0, which are exactly\ngiven by\nmx(E) = (r2−r3)sn(u,k)cn(u,k),(12)my(E) =r3+(r2−r3)sn2(u,k),(13)\nmz(E) =/radicalBig\n1−r2\n3−(r2\n2−r2\n3)sn2(u,k),(14)\nwhereu=γ/radicalbig\nHtHK/2√r1−r3t, andrℓare given by\nr1(E) =−E\nMHt, (15)\nr2(E) =Ht\nHK+/radicalBigg\n1+/parenleftbiggHt\nHK/parenrightbigg2\n+2E\nMHK,(16)\nr3(E) =Ht\nHK−/radicalBigg\n1+/parenleftbiggHt\nHK/parenrightbigg2\n+2E\nMHK.(17)\nThe modulus of Jacobi elliptic functions, sn( u,k) and\ncn(u,k), is\nk=/radicalbiggr2−r3\nr1−r3. (18)\nThe derivations of Eqs. (12), (13), and (14) are shown in\nAppendix A. The precession period is\nτ(E) =2K(k)\nγ/radicalbig\nHtHK/2√r1−r3, (19)\nwhereK(k) is the first kind of complete elliptic integral.\nThe work done by spin torque and the dissipation due to\ndamping, WsandWα, are obtained by substituting Eqs.\n(12), (13), and (14) into Eqs. (3) and (4), integrating\nover [0,τ/2], and multiplying a numerical factor 2 be-\ncause Eqs. (12), (13), and (14) are the solution of the\nprecession trajectory for a half period. Then, WsandWα\nforEmin< E < E saddleare exactly given by\nWs=8MHs√r1−r3\n3Ht/radicalbig\nHK/(2Ht)Hs, (20)\nWα=−4αM√r1−r3\n3/radicalbig\nHK/(2Ht)Hα, (21)\nwhereHsandHαare given by\nHs=Ht/parenleftbigg1−r2\n1\nr1−r3/parenrightbigg\nK(k)−/parenleftbiggE\nM+H2\nt\nHK/parenrightbigg\nE(k),(22)\nHα=Ht/parenleftbigg1−r2\n1\nr1−r3/parenrightbigg\nK(k)+/parenleftbigg5E\nM+3HK+2H2\nt\nHK/parenrightbigg\nE(k).\n(23)\nHere,E(k) is the second kind of complete elliptic inte-\ngral. The derivations of Eqs. (20) and (21) are shown in4\nHt/H K=0.1, 0.3, 0.5, 0.7, 0.9current, j(E) \nenergy, E0 0.2 0.4 0.6 0.8 1.000.20.40.60.81.0\nFIG. 2: The dependence of the current j(E), Eq. (24),\nfor several values of Ht/HKon the energy density E. For\nsimplicity, the horizontal and vertical axes are normalize d as\nj(E)/jcandE/(Esaddle−Emin)−[Emin/(Esaddle−Emin)] to\nmakej(Emin) = 1,Emin= 0, and Esaddle= 1.\ntime (μs)0 0.2 0.4 0.6 0.8 1.001.0\n-1.0magnetization mz\nmymx\nFIG. 3: Typical magnetization dynamics excited by the\nspin Hall effect. The parameter values are taken from\nexperiments36–38,42asM= 1500 emu/c.c., HK= 540 Oe,\nα= 0.005,γ= 1.764×107rad/(Oe·s),d= 1 nm, ϑ= 0.1,\nandHt= 50 Oe. The current magnitude is 14 ×106A/cm2,\nwhile the critical current, Eq. (25), is 13 ×106A/cm2.\nAppendix B. The current j(E) forEmin< E < E saddle\nis given by\nj(E) =2αeMd\n/planckover2pi1ϑHtHα\n2Hs. (24)\nThe currents for E→EminandE→Esaddleare41\nj(Emin) =2αeMd\n/planckover2pi1ϑHK\nHt/HK/bracketleftBigg\n1−1\n2/parenleftbiggHt\nHK/parenrightbigg2/bracketrightBigg\n,(25)\nj(Esaddle) =2αeMd\n/planckover2pi1ϑ/parenleftbigg3HK−2Ht\n2/parenrightbigg\n.(26)\nEquation (24) is the current density satisfying Eq. (2),\nor equivalently Eq. (5). Then, let us investigate whetherEq. (24) satisfies Eq. (6). It is mathematically difficult\nto calculate the derivative of Eq. (24) with respect to\nEfor an arbitrary value of E, although we can confirm\nthatj(Emin)> j(Esaddle) forHt< HK. We note that\na parameter determining whether Eq. (6) is satisfied is\nonlyHt/HKbecause the otherparameters, such as αand\nM, are just common prefactors for any j(E). As shown\nin Fig. 2, j(E) is a monotonically decreasing function of\nEfor a wide range of Ht/HK, i.e., Eq. (6) is not satis-\nfied. This result indicates that the magnetization stays\nin the equilibrium state when j < jc=j(Emin), whereas\nit moves to the constant energy curve of Esaddlewithout\nshowing stable self-oscillation when j > jcbecause the\nspin Hall torque does not balance the damping torque\non any constant energy curve between EminandEsaddle.\nThe magnetizationfinally stopsits dynamics at ±msaddle\nbecause all torques become zero at these points. Figure\n3 shows a typical example of such dynamics, in which the\ntime evolution of each component is shown. Therefore,\nself-oscillation solely by the spin Hall torque cannot be\nexcitedin the perpendicularferromagnet. Thisis apossi-\nble reason why the self-oscillation has not been reported\nyet.\nRecently, many kinds of other torques pointing in\ndifferent directions or having different angular depen-\ndencies, such as field-like and Rashba torques, have\nbeen proposed28,29,36,37,40,43–45. These effects might\nchange the above conclusions. Adding an in-plane\nanisotropy21,22, tilting the perpendicular anisotropy40,\nor using higher order anisotropy might be another\ncandidate. Spin pumping is also an interesting phe-\nnomenon because it modifies the Gilbert damping\nconstant46–49. It was shown in Refs.48,50that the en-\nhancement of the Gilbert damping constant in a fer-\nromagnetic/nonmagnetic/ferromagnetic trilayer system\ndepends on the relative angle of the magnetization. This\nmeans that the Gilbert damping constant has an angular\ndependence. In a such case, it might be possible to sat-\nisfy Eqs. (5) and (6) by attaching another ferromagnet\nto the spin Hall system and by choosing an appropriate\nalignment of the magnetizations. The above formulas\nalso apply to these studies. In Appendix C, we briefly\ndiscuss a technical difficulty to include the effect of the\nfield-like torque or Rashba torque.\nIV. CONCLUSION\nIn conclusion, wedevelopedamethod forthe nonlinear\nanalysisofthe LLGequationinthe spinHall systemwith\na perpendicular ferromagnet. We summarized physical\nconditions to excite self-oscillation by the spin transfer\neffect. The first condition, Eq. (2), or equivalently Eq.\n(5), implies that the energy supplied by the spin torque\nduring a precession on a constant energy curve should\nequal the dissipation due to damping. The second con-\ndition, Eq. (6), implies that the current to balance the\nspin torque and the damping torquein the self-oscillation5\nstate should be larger than the critical current to desta-\nbilize the initial state. By solving the nonlinear LLG\nequation, we derived exact solutions of the energy sup-\nplied bythe spintransfereffect andthe dissipationdue to\ndamping, and showed that these conditions are not sat-\nisfied. These results indicate that self-oscillation cannot\nbe excited solely by the spin Hall torque.\nThe author would like to acknowledge T. Yorozu for\nhis great constructive help on this work. The author also\nthanks M. Hayashi, H. Kubota, and A. Emura for their\nkind supports. This work was supported by JSPS KAK-\nENHI Grant-in-Aid for Young Scientists (B) 25790044.\nAppendix A: Precession trajectory on a constant\nenergy curve\nHere, we show the derivation of Eqs. (12), (13), and\n(14). The precession trajectory on a constant energy\ncurve is determined by dm/dt=−γm×H. They-\ncomponent of this equation is dmy/dt=γHKmxmz.\nThus, we find\n/integraldisplay\ndt=1\nγHK/integraldisplaydmy\nmxmz. (A1)\nAs mentioned in Sec. III, since the constant energycurve\nof Eq. (9) is symmetric with respect to the yz-plane, it\nis sufficient to derive the solutions of mfor half of the\ntrajectory in the region of mx>0. Using Eandmy,mx\nandmzare expressed as\nmx=/radicalbigg\n1−m2y+2E\nMHK+2Ht\nHKmy,(A2)\nmz=/radicalbigg\n−2E\nMHK−2Ht\nHKmy. (A3)\nThe initial state of myis chosen as my(0) =r3, where\nr3is given by Eq. (17). Then, myat a certain time tis\ndetermined from Eq. (A1) as\nγ/radicalbig\n2HtHK/integraldisplayt\n0dt\n=/integraldisplaymy\nr3dm′\ny/radicalBig\n(m′y−r1)(m′y−r2)(m′y−r3).(A4)\nWe introduce a new parameter sasmy=r3+(r2−r3)s2.\nThen, we find\nγ/radicalbigg\nHtHK\n2√r1−r3t=/integraldisplays\n0ds′\n/radicalbig\n(1−s′2)(1−k2s′2),(A5)\nwhere the modulus kis given by Eq. (18). The solution\nofsiss= sn(u,k). Therefore, myis given by Eq. (13).\nEquations (12) and (14) are obtained by substituting Eq.\n(13) into Eqs. (A2) and (A3).We note that Eqs. (12), (13), and (14) are periodic\nfunctions with the period given by Eq. (19). On the\nother hand, when E=Esaddle, the magnetization stops\nitsdynamicsfinallyatthesaddlepoint m= (0,1,0). The\nsolution of the constant energy curve of Esaddlewith the\ninitial condition my(0) =r3can be obtained by similar\ncalculations, and are given by\nmx= 2/parenleftbigg\n1−Ht\nHK/parenrightbiggtanh(νt)\ncosh(νt), (A6)\nmy=−1+2Ht\nHK+2/parenleftbigg\n1−Ht\nHK/parenrightbigg\ntanh2(νt),(A7)\nmz= 2/radicalBigg\nHt\nHK/parenleftbigg\n1−Ht\nHK/parenrightbigg1\ncosh(νt),(A8)\nwhereν=γ/radicalbig\nHt(HK−Ht).\nAppendix B: Derivation of Eqs. (20) and (21)\nUsing Eqs. (12), (13), and (14), the explicit form\nof Eq. (3) for the spin Hall system is given by Ws=\nγMHs/integraltext\ndtws, wherewsis given by\nws= (Ht−HKr3)(1−r2\n3)\n+/braceleftbig\n−2Htr3+HK/bracketleftbig\nr3(r2+r3)−(1−r2\n3)/bracketrightbig/bracerightbig\n(r2−r3)sn2(u,k)\n+{−Ht+HK(r2+r3)}(r2−r3)2sn4(u,k).\n(B1)\nSimilarly, Eq. (21) for the spin Hall system is given by\nWα=−αγM/integraltext\ndtwα, wherewαis given by\nwα= (1−r2\n3)(Ht−HKr3)2\n−/bracketleftbig\n2H2\ntr3−H2\nK(r2+r3)(1−2r2\n3)+2HtHK(1−r2r3−2r2\n3)/bracketrightbig\n×(r2−r3)sn2(u,k)\n−[Ht−HK(r2+r3)]2(r2−r3)2sn4(u,k).\n(B2)\nThen, WsandWαare obtained by integrating over\n[0,τ/2], and multiplying a numerical factor 2. The fol-\nlowing integral formulas are useful,\n/integraldisplayu\ndu′sn2(u′,k) =u−E[am(u,k),k]\nk2,(B3)\n/integraldisplayu\ndu′sn4(u′,k) =sn(u,k)cn(u,k)dn(u,k)\n3k2\n+2+k2\n3k4u\n−2(1+k2)\n3k4E[am(u,k),k],(B4)\nwhereE(u,k), am(u,k), and dn( u,k) are the second kind\nofincomplete elliptic integral,Jacobiamplitude function,\nand Jacobi elliptic function, respectively.6\nAppendix C: The effect of the field-like torque or\nRashba torque\nThe direction of the field-like torque or the Rashba\ntorque is given by m×p, wherepis the direction of the\nspin polarization. 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B 76, 092402\n(2007)."    },    {        "title": "1702.08408v2.Current_Induced_Damping_of_Nanosized_Quantum_Moments_in_the_Presence_of_Spin_Orbit_Interaction.pdf",        "content": "Current Induced Damping of Nanosized Quantum Moments in the Presence of\nSpin-Orbit Interaction\nFarzad Mahfouzi\u0003and Nicholas Kioussisy\nDepartment of Physics and Astronomy, California State University, Northridge, CA, USA\n(Dated: November 10, 2021)\nMotivated by the need to understand current-induced magnetization dynamics at the nanoscale,\nwe have developed a formalism, within the framework of Keldysh Green function approach, to study\nthe current-induced dynamics of a ferromagnetic (FM) nanoisland overlayer on a spin-orbit-coupling\n(SOC) Rashba plane. In contrast to the commonly employed classical micromagnetic LLG simula-\ntions the magnetic moments of the FM are treated quantum mechanically . We obtain the density\nmatrix of the whole system consisting of conduction electrons entangled with the local magnetic\nmoments and calculate the e\u000bective damping rate of the FM. We investigate two opposite limiting\nregimes of FM dynamics: (1) The precessional regime where the magnetic anisotropy energy (MAE)\nand precessional frequency are smaller than the exchange interactions, and (2) The local spin-\rip\nregime where the MAE and precessional frequency are comparable to the exchange interactions. In\nthe former case, we show that due to the \fnite size of the FM domain, the \\Gilbert damping\"does\nnot diverge in the ballistic electron transport regime, in sharp contrast to Kambersky's breathing\nFermi surface theory for damping in metallic FMs. In the latter case, we show that above a critical\nbias the excited conduction electrons can switch the local spin moments resulting in demagnetization\nand reversal of the magnetization. Furthermore, our calculations show that the bias-induced anti-\ndamping e\u000eciency in the local spin-\rip regime is much higher than that in the rotational excitation\nregime.\nPACS numbers: 72.25.Mk, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nUnderstanding the current-induced magnetization\nswitching (CIMS) at the nanoscale is mandatory for the\nscalability of non-volatile magnetic random access mem-\nory (MRAM) of the next-generation miniaturized spin-\ntronic devices. However, the local magnetic moments of a\nnanoisland require quantum mechanical treatment rather\nthan the classical treatment of magnetization commonly\nemployed in micromagnetic simulations, which is the cen-\ntral theme of this work.\nThe \frst approach of CIMS employs the spin transfer\ntorque (STT)1,2in magnetic tunnel junctions (MTJ) con-\nsisting of two ferromagnetic (FM) layers (i.e., a switch-\nable free layer and a \fxed layer) separated by an insulat-\ning layer, which involves spin-angular-momentum trans-\nfer from conduction electrons to local magnetization3,4.\nAlthough STT has proven very successful and brings the\nprecious bene\ft of improved scalability, it requires high\ncurrent densities ( \u00151010A/cm2) that are uncomfort-\nably high for the MTJ's involved and hence high power\nconsumption. The second approach involves an in-plane\ncurrent in a ferromagnet-heavy-metal bilayer where the\nmagnetization switching is through the so-called spin-\norbit torque (SOT) for both out-of-plane and in-plane\nmagnetized layers.5{8The most attractive feature of the\nSO-STT method is that the current does not \row through\nthe tunnel barrier, thus o\u000bering potentially faster and\nmore e\u000ecient magnetization switching compared to the\nMTJs counterparts.\nAs in the case of STT, the SO-STT has two compo-\nnents: a \feld-like and an antidamping component. Whilethe \feld-like component reorients the equilibrium direc-\ntion of the FM, the antidamping component provides the\nenergy necessary for the FM dynamics by either enhanc-\ning or decreasing the damping rate of the FM depending\non the direction of the current relative to the magneti-\nzation orientation as well as the structural asymmetry\nof the material. For su\u000eciently large bias the SOT can\novercome the intrinsic damping of the FM leading to ex-\ncitation of the magnetization precession.8The underlying\nmechanism of the SOT for both out-of-plane and in-plane\nmagnetized layers remains elusive and is still under de-\nbate. It results from either the bulk Spin Hall E\u000bect\n(SHE)9{12, or the interfacial Rashba-type spin-orbit cou-\npling,13{16or both17{19.\nMotivated by the necessity of scaling down the size\nof magnetic bits and increasing the switching speed, the\nobjective of this work is to develop a fully quantum me-\nchanical formalism, based on the Keldysh Green function\n(GF) approach, to study the current-induced local mo-\nment dynamics of a bilayer consisting of a FM overlayer\non a SOC Rashba plane, shown in Fig. 1.\nUnlike the commonly used approaches to investigate\nthe magnetization dynamics of quantum FMs, such as the\nmaster equation20, the scattering21or quasi-classical22\nmethods, our formalism allows the study of magnetiza-\ntion dynamics in the presence of nonequilibrium \row of\nelectrons.\nWe consider two di\u000berent regimes of FM dynamics: In\nthe \frst case, which we refer to as the single domain\ndynamics, the MAE and the precession frequency are\nsmaller than the exchange interactions, and the FM can\nbe described by a single quantum magnetic moment, of\na typically large spin, S, whose dynamics are governedarXiv:1702.08408v2  [cond-mat.mes-hall]  27 Apr 20172\nFIG. 1: (Color online) Schematic view of the FM/Rashba\nplane bilayer where the FM overlayer has length Lxand is\nin\fnite (\fnite) along the y-direction for the case of a single\ndomain (nano-island) discussed in Sec. III (IV). The magne-\ntization,~ m, of the FM precesses around the direction denoted\nby the unit vector, ~ nM, with frequency !and cone angle, \u0012.\nThe Rashba layer is attached to two normal (N) leads which\nare semi-in\fnite along the x-direction, across which an exter-\nnal bias voltage, V, is applied.\nmainly by the quantized rotational modes of the magne-\ntization. We show that the magnetic degrees of freedom\nentering the density matrix of the conduction electron-\nlocal moment entagled system simply shift the chemical\npotential of the Fermi-Dirac distribution function by the\nrotational excitations energies of the FM from its ground\nstate. We also demonstrate that the e\u000bective damping\nrate is simply the netcurrent along the the auxiliarym-\ndirection, where m= -S, -S+1, :::, +S, are the eigenval-\nues of the total Szof the FM. Our results for the change\nof the damping rate due to the presence of a bias volt-\nage are consistent with the anti-damping SOT of clas-\nsical magnetic moments,16,23, where due to the Rashba\nspin momentum locking, the anti-damping SOT, to low-\nest order in magnetic exchange coupling, is of the form,\n~ m\u0002(~ m\u0002^y), where ^yis an in-plane unit vector normal\nto the transport direction.\nIn the adiabatic and ballistic transport regimes due to\nthe \fnite S value of the nanosize ferromagnet our formal-\nism yields a \fnite \\Gilbert damping\", in sharp contrast to\nKambersky's breathing Fermi surface theory for damp-\ning in metallic FMs.24On the other hand, Costa and\nMuniz25and Edwards26demonstrated that the prob-\nlem of divergent Gilbert damping is removed by takinginto account the collective excitations. Furthermore, Ed-\nwards points out26the necessity of including the e\u000bect of\nlong-range Coulomb interaction in calculating damping\nfor large SOC.\nIn the second case, which corresponds to an indepen-\ndent local moment dynamics, the FM has a large MAE\nand hence the rotational excitation energy is compara-\nble to the local spin-\rip excitation (exchange energy).\nWe investigate the e\u000bect of bias on the damping rate of\nthe local spin moments. We show that above a criti-\ncal bias voltage the \rowing conduction electrons can ex-\ncite (switch) the local spin moments resulting in demag-\nnetization and reversal of the magnetization. Further-\nmore, we \fnd that, in sharp contrast to the single do-\nmain precessional dynamic, the current-induced damping\nis nonzero for in-plane and out-of-plane directions of the\nequilibrium magnetization. The bias-induced antidamp-\ning e\u000eciency in the local moment switching regime is\nmuch higher than that in the single domain precessional\ndynamics.\nThe paper is organized as follows. In Sec. II we present\nthe Keldysh formalism for the density matrix of the en-\ntagled quantum moment-conduction electron system and\nthe e\u000bective dampin/antdamping torque. In Sec. III we\npresent results for the current-induced damping rate in\nthe single domain regime. In Sec. IV we present results\nfor the current-induced damping rate in the independent\nlocal regime. We conclude in Sec. V.\nII. THEORETICAL FORMALISM\nFig. 1 shows a schematic view of the ferromagnetic\nheterostructure under investigation consisting of a 2D\nferromagnet-Rashba plane bilayer attached to two semi-\nin\fnite normal (N) leads whose chemical potentials are\nshifted by the external bias, Vbias. The magnetization\nof the FM precesses around the axis speci\fed by the\nunit vector, ~ nM, with frequency !and cone angle \u0012.\nThe FM has length, LFM\nx, along the transport direction.\nThe total Hamiltonian describing the coupled conduc-\ntion electron-localized spin moment system in the het-\nerostructure in Fig. 1 can be written as,\nHtot=X\nrr0;\u001b\u001b0Trfsdgh\u0010\n1s^H\u001b\u001b0\nrr0+\u000err0\u000e\u001b\u001b01s\u0016r+\u000err0Jsd~ \u001b\u001b\u001b0\u0001~sd(r) +\u000e\u001b\u001b0\u000err0HM\u0011\n \u0003\nfs0\ndgr0\u001b0 fsdgr\u001bi\n: (1)\nHere,~sd(r) is the local spin moment at atomic position\nr, the trace is over the di\u000berent con\fgurations of the lo-\ncal spin moments, fsdg, fsdgr\u001b=jfsdgi\n e\nr\u001bis the\nquasi-particle wave-function associated with the conduc-tion electron (  e) entangled to the FM states ( jfsdgi),\nJsdis thes\u0000dexchange interaction, 1sis the identity ma-\ntrix in spin con\fguration space, and ^ \u001bx;y;z are the Pauli\nmatrices. We use the convention that, except for r, bold3\nsymbols represent operators in the magnetic con\fgura-\ntion space and symbols with hat represent operators in\nthe single particle Hilbert space of the conduction elec-\ntrons. The magnetic Hamiltonian HMis given by\nHM=\u0000g\u0016BX\nr~Bext(r)\u0001~sd(r) (2)\n\u0000X\nhr;r0iJdd\nrr0\ns2\nd~sd(r0)\u0001~sd(r)\u0000X\nrJsd\nsd~sc(r)\u0001~sd(r);\nwhere, the \frst term is the Zeeman energy due to the\nexternal magnetic \feld, the second term is the magnetic\ncoupling between the local moments and the third term\nis the energy associated with the intrinsic magnetic \feld\nacting on the local moment, ~sd(r), induced by the local\nspin of the conduction electrons, ~sc(r).\nThe Rashba model of a two-dimensional electron gas\nwith spin orbit coupling interacting with a system of\nlocalized magnetic moments has been extensively em-\nployed14,27,28to describe the e\u000bect of enhanced spin-orbit\ncoupling solely at the interface on the current-induced\ntorques in ultrathin ferromagnetic (FM)/heavy metal\n(HM) bilayers. The e\u000bects of (i) the ferromagnet induc-\ning a moment in the HM and (ii) the HM with strong\nspin-orbit coupling inducing a large spin-orbit e\u000bect in\nthe ferromagnet (Rashba spin-orbit coupling) lead to a\nthin layer where the magnetism and the spin-orbit cou-\npling coexist.27\nThe single-electron tight-binding Hamiltonian29for\nthe conduction electrons of the 2D Rashba plane, H\u001b\u001b0\nrr0\nwhich is \fnite along the transport direction xand in\fnite\nalong theydirection is of the form,\n^H\u001b\u001b0\nxx0(kya) = [tcos(kya)\u000e\u001b\u001b0\u0000tsosin(kya)\u001bx\n\u001b\u001b0]\u000exx0(3)\n+t(\u000ex;x0+1+\u000ex+1;x0)\u000e\u001b\u001b0+itso(\u000ex;x0+1\u0000\u000ex+1;x0)\u001by\n\u001b\u001b0:\nHere,x;x0denote atomic coordinates along the trans-\nport direction, ais the in-plane lattice constant, and tso\nis the Rashba SOI strength. The values of the local e\u000bec-\ntive exchange interaction, Jsd= 1eV, and of the nearest-\nneighbor hopping matrix element, t=1 eV, represent a\nrealistic choice for simulating the exchange interaction of\n3dferromagnetic transition metals and their alloys (Fe,\nCo).30{32The Fermi energy, EF=3.1 eV, is about 1 eV\nbelow the upper band edge at 4 eV consistent with the\nab initio calculations of the (111) Pt surface33. Further-\nmore, we have used tso=0.5 eV which yields a Rashba\nparameter, \u000bR=tsoa\u00191.4 eV \u0017A (a=2.77 \u0017A is the in-\nplane lattice constant of the (111) Pt surface) consis-\ntent with the experimental value of about 1-1.5 eV \u0017A34\nand the ab initio value of 1 eV \u0017A28. However, because\nother experimental measurements for Pt/Co/Pt stacks\nreport35a Rashba parameter which is an order of mag-\nnitude smaller, in Fig.3 we show the damping rate for\ndi\u000berent values of the Rashba SOI.. For the results in\nSec. IV, we assume a real space tight binding for propa-\ngation along y-axis.The single particle propagator of the coupled electron-\nspin system is determined from the equation of motion\nof the retarded Green function,\n\u0012\nE\u0000i\u0011\u0000^\u0016\u0000^H\u0000HM\u0000Jsd\n2^~ \u001b\u0001^~sd\u0013\n^Gr(E) =^1;(4)\nwhere,\u0011is the broadening of the conduction electron\nstates due to inelastic scattering from defects and/or\nphonons, and for simplicity we ignore writing the identity\nmatrices ^1 and 1in the expression. The density matrix\nof the entire system consisting of the noninteracting elec-\ntrons (fermionic quasi-particles) and the local magnetic\nspins is determined (see Appendix A for details of the\nderivation for a single FM domain) from the expression,\n^\u001a=ZdE\n\u0019^Gr(E)\u0011f(E\u0000^\u0016\u0000HM)^Ga(E): (5)\nIt is important to emphasize that Eq. (5) is the central\nresult of this formalism which demonstrates that the ef-\nfect of the local magnetic degrees of freedom is to shift the\nchemical potential of the Fermi-Dirac distribution func-\ntion by the eigenvalues, \"m, ofHMjmi=\"mjmi, i.e.,\nthe excitation energies of the FM from its ground state.\nHere,jmiare the eigenstates of the Heisenberg model\ndescribing the FM. The density matrix can then be used\nto calculate the local spin density operator of the con-\nduction electrons, [ ~sc(r)]mm0=P\nss0\u001amm0\nss0;rr~ \u001bss0=2, which\nalong with Eqs. (2), (4), and (5) form a closed set\nof equations that can be solved self consistently. Since,\nthe objective of this work is the damping/anti-damping\n(transitional) behavior of the FM in the presence of bias\nvoltage, we only present results for the \frst iteration.\nEq. (5) shows that the underlying mechanism of the\ndamping phenomenon is the \row of conduction electrons\nfrom states of higher chemical potential to those of lower\none where the FM state relaxes to its ground state by\ntransferring energy to the conduction electrons. There-\nfore, the FM dynamical properties in this formalism is\ncompletely governed by its coupling to the conduction\nelectrons, where conservation of energy and angular mo-\nmentum dictates the excitations as well as the \ructua-\ntions of the FM sate through the Fermi distribution func-\ntion of the electrons coupled to the reservoirs. This is\ndi\u000berent from the conventional Boltzmann distribution\nfunction which is commonly used to investigate the ther-\nmal and quantum \ructuations of the magnetization.\nDue to the fact that the number of magnetic con\fgura-\ntions (i.e. size of the FM Hilbert space) grows exponen-\ntially with the dimension of the system it becomes pro-\nhibitively expensive to consider all possible eigenstates\nof theHMoperator. Thus, in the following sections we\nconsider two opposite limiting cases of magnetic con\fg-\nurations. In the \frst case we assume a single magnetic\nmoment for the whole FM which is valid for small FMs\nwith strong exchange coupling between local moments\nand small MAE. In this case the dynamics is mainly gov-\nerned by the FM rotational modes and local spin \rips can4\nFIG. 2: (Color online) Schematic representation of the quasi-\nparticles of the FM and conduction electron entangled states.\nThe horizontal planes denote the eigenstates, jS;miof the\ntotalSzof the FM with eigenvalues m=\u0000S;\u0000S+ 1;:::; +S\nalong the auxiliarym-direction. Excitation of magnetic state\ninduces a shift of the chemical potential of the Fermi-Dirac\ndistribution function leading to \row of quisiparticles along the\nm-direction which corresponds to the damping rate of the FM.\nThe FM damping involves two processes: (1) An intra-plane\nprocess involving spin reversal of the conduction electron via\nthe SOC; and (2) An inter-plane process involving quasiparti-\ncle \row of majority (minority) spin along the ascending (de-\nscending)m-direction due to conservation of total angular\nmomentum, where the interlayer hopping is accompanied by\na spin \rip of conduction electrons.\nbe ignored. In the second case we ignore the correlation\nbetween di\u000berent local moments and employ a mean \feld\napproximation such that at each step we focus on an indi-\nvidual atom by considering the local moment under con-\nsideration as a quantum mechanical object while the rest\nof the moments are treated classically. We should men-\ntion that a more accurate modeling of the system should\ncontain both single domain rotation of the FM as well\nas the local spin \ripping but also the e\u000bect of nonlocal\ncorrelations between the local moments and conduction\nelectrons, which are ignored in this work.\nIII. SINGLE DOMAIN ROTATIONAL\nSWITCHING\nIn the regime where the energy required for the excita-\ntion of a single local spin moment ( \u0019meV ) is much larger\nthan the MAE (\u0019\u0016eV) the low-energy excited states cor-\nrespond to rotation of the total angular momentum of the\nFM acting as a single domain and the e\u000bects of local spin\n\rips described as the second term in Eq 2, can be ignored.\nIn this regime all of the local moments behave collectively\nand the local moment operators can be replaced by the\naverage spin operator, ~sd(r) =P\nr0~sd(r0)=Nd=sd~S=S,\nwhereNdis the number of local moments and ~Sis the\ntotal angular momentum with amplitude S. The mag-\nnetic energy operator is given by HM=\u0000~B\u0001S, where,\n~B=g\u0016B~Bext+Jsd~ sc. Here, for simplicity we assume\n~ scto be scalar and independent of the FM state. The\neigenstates of HMoperator are then simply the eigen-states,jS;mi, of the total angular momentum Sz, with\neigenvalues m!=\u0000S!;:::; +S!, where!=Bzis the\nLarmor frequency. Thus, the wave function of the cou-\npled electron-spin con\fguration system, shown schemat-\nically in Fig. 2 is of the form,  ms0r(t) =jS;mi\n s0r(t).\nOne can see that the magnetic degrees of freedom corre-\nsponding to the di\u000berent eigenstates of the Szoperator,\nenters as an additional auxiliary dimension for the elec-\ntronic system where the variation of the magnetic energy,\nhS;mjHMjS;mi=m!, shifts the chemical potentials of\nthe electrons along this dimension. The gradient of the\nchemical potential along the auxiliary direction, is the\nLarmor frequency ( \u0016eV\u0019GHz ) which appears as an\ne\u000bective \\electric \feld\"in that direction.\nSubstituting Eq (5) in Eq (A1)(b) and averaging over\none precession period we \fnd that the average rate of\nangular momentum loss/gain, which we refer to as the\ne\u000bective \\ damping rate \"per magnetic moment, can be\nwritten as\nTm=1\n2=(T\u0000\nm\u0000T+\nm); (6)\nwhere,\nT\u0006\nm=Jsd\n2SNdTrel[^\u001b\u0007S\u0006\nm^\u001am;m\u00061]: (7)\nis the current along the auxiliarym-direction in Fig. 2\nfrom them$m+ 1 (\u0006sign) state of the total Szof the\nFM. Here,Trel, is the trace over the conduction electron\ndegrees of freedom, and S\u0006\nm=p\nS(S+ 1)\u0000m(m\u00061)\nare the ladder operators. It is important to note that\nwithin this formalism the damping rate is simply the net\ncurrent across the mth-layer along the auxiliary direction\nassociated with the transition rate of the FM from state\nmto its nearest-neighbor states ( m\u00061).\nFig. 3 shows the damping rate as a function of the pre-\ncession cone angle, \u0012= cos\u00001(m\nS), for di\u000berent values of\nbias and for an in-plane e\u000bective magnetic \feld (a) along\nand (b) normal to the transport direction, and (c) an out-\nof-plane magnetic \feld. For cases (a) and (c) the damp-\ning rate is negative and relatively independent of bias for\nlow bias values. A negative damping rate implies that the\nFM relaxes towards the magnetic \feld by losing its angu-\nlar momentum, similar to the Gilbert damping rate term\nin the classical LLG equation, where its average value\nover the azimuthal precession angle, '=!t, is of the\nform,T=\u0000\u000bsdRd'\n2\u0019~ m\u0002(~ m\u0002~B)\u0001~ nM, which is nonzero\n(zero) when the unit vector ~ nMis along (perpendicular\nto) the e\u000bective magnetic \feld. The dependence of the\ndamping rate on the bias voltage when the e\u000bective mag-\nnetic \feld~Bis inplane and normal to the transport direc-\ntion can be understood by the spin-\rip re\rection mech-\nanism accompanied by Rashba spin-momentum locking\ndescribed in Ref.16. One can see that a large enough bias\ncan result in a sign reversal of the damping rate and hence\na magnetization reversal of the FM. It's worth mention-\ning that due to the zero-point quantum \ructuations of5\nthe magnetization, at \u0012= 0;\u0019(i.e.m=\u0006S) we have\nT 6=0 which is inversely proportional to the size of the\nmagnetic moment, S.\nIn Fig. 4(a) we present the e\u000bective damping rate ver-\nsus bias for di\u000berent values of the Rashba SOC. The re-\nsults show a linear response regime with respect to the\nbias voltage where both the zero-bias damping rate and\nthe slope,dT=dV increases with the Rashba SOC. This\nis consistent with Kambersky's mechanism of Gilbert\ndamping due to the SOC of itinerant electrons,24and the\nSOT mechanism16. Fig. 4(b) shows that in the absence\nof bias voltage the damping rate is proportional to t2\nsoand\nthe e\u000bect of the spin current pumped into the left and\nright reservoirs is negligible. This result of the t2\nsodepen-\ndence of the zero-bias damping rate is in agreement with\nrecent calculations of Costa and Muniz25and Edwards26\nwhich took into account the collective excitations. In the\npresence of an external bias, Tvaries linearly with the\nSOC, suggesting that to the lowest order it can be \ftted\nto\nT= sin2(\u0012)tso(c1tso~!+c2eVbias); (8)\nwherec1andc2are \ftting parameters.\nThe bias-induced e\u000eciency of the anti-damping SOT,\n\u0002\u0011~!(T(Vbias)\u0000T(0))=eVbiasT(0), describes how e\u000e-\ncient is the energy conversion between the magnetization\ndynamics and the conduction electrons. Accordingly, for\na given bias-induced e\u000eciency, \u0002, one needs to apply an\nexternal bias equal to ~!=e\u0002 to overcome the zero-bias\ndamping of the FM. Fig. 5 displays the anti-damping ef-\n\fciency versus the position of the Fermi energy of the FM\nfrom the bottom (-4 t=-4 eV) to the top (4 t=4eV) of the\nconduction electron band for the two-dimensional square\nlattice. The result is independent of the bias voltage and\nthe Larmor frequency in the linear response regime ( i.e.\nVbias;!\u001ct). We \fnd that the e\u000eciency peaks when the\nFermi level is in the vicinity of the bottom or top of the\nenergy band where the transport is driven by electron- or\nhole-like carriers and the Gilbert damping is minimum.\nThe sign reversal of the antidamping SOT is due to the\nelectron- or hole-like driven transport similar to the Hall\ne\u000bect.36\nClassical Regime of the Zero Bias Damping rate |\nIn the following we show that in the case of classical\nmagnetic moments ( S!1 ) and the adiabatic regime\n(!!0), the formalism developed in this paper leads to\nthe conventional expressions for the damping rate. In this\nlimit the system becomes locally periodic and one can\ncarry out a Fourier transformation from m\u0011Szspace\nto azimuthal angle of the magnetization orientation, ',\nspace. Conservation of the angular momentum suggests\nthat the majority- (minority-) spin electrons can propa-\ngate only along the ascending (descending) m-direction,\nwhere the hopping between two nearest-neighbor m-\nlayers is accompanied by a spin-\rip. As shown in Fig.\n2 the existence of spin-\rip hopping requires the presence\nof intralayer SOC-induced noncollinear spin terms which\nrotate the spin direction of the conduction electrons as\n04590135180−10−505B=[|B|,0,0]Damping Rate ( µeV)\n  \n4590135180B=[0,|B|,0]\nCone Angle, θ (deg)4590135180B=[0,0,|B|]\nVbias=−5 mV\nVbias=0\nVbias=5 mV(a) (b) (c)\nStudent Version of MATLABFIG. 3: (Color online) E\u000bective damping rate for a single\nFM domain as a function of the precession cone angle, \u0012, for\nvarious bias values under an e\u000bective magnetic \feld which is\nin-plane (a) along and (b) normal to the transport direction\nand (c) out-of-plane. The length of the FM along the xdi-\nrection isLx= 25awhile it is assumed to be in\fnite in the\ny-direction, ~!= 10\u0016eV, the broadening parameter \u0011= 0,\nkBT= 10meV and the domain magnetic moment S= 200.\nThe results are robust with larger values of Sin either the\nballistic,\u0011\u001c~!, or dirty,\u0011\u001d~!, regimes.\n−2−1012−3−2−101\nBias Voltage (mV)Damping Rate ( µeV)\n  \n−0.5 0 0.5−15−10−505\nSpin Orbit Coupling, tso (eV)  \ntso=0\ntso=0.1 eV\ntso=0.2 eVVbias=−5 mV\nVbias=0\nVbias=5 mV(a) (b)\nStudent Version of MATLAB\nFIG. 4: (Color online) Damping torque versus (a) bias voltage\nand (b) spin-orbit coupling strength, for m= 0 corresponding\nto the precession cone angle of 90o. The precession axis of the\nFM is along the y-direction and the rest of the parameters are\nthe same as in Fig. 3. The zero-bias damping rate versus SOC\nshows at2\nsodependence while the damping rate under non-\nzero bias exhibits nearly linear SOC dependence.\nthey propagate in each m-layer. This is necessary for\nthe persistent \row of electrons along the 'auxiliary di-\nrection and therefore damping of the magnetization dy-\nnamics. Using the Drude expression of the longitudinal\nconductivity along the '-direction for the damping rate,\nwe \fnd that, within the relaxation time approximation,\n\u0011=!!1 , where the relaxation time of the excited con-\nduction electrons is much shorter than the time scale of6\n−4−3−2−101234−3\n−2\n−1\n0\n1\n2\n3\nFermi Energy (eV)Antidamping Efficiency (%)\n−4−2024−40−200Damping Rate ( µeV)\n  \nVbias=5 mV\nVbias=0\nVbias=−5 mV\nStudent Version of MATLAB\nFIG. 5: (Color online) Bias-induced precessional anti-\ndamping e\u000eciency, \u0002 = ~!(T(Vbias)\u0000T(0))=eVbiasT(0), ver-\nsus the Fermi energy of the 2D Rashba plane in Fig.1, where\nthe energy band ranges from -4 eV to +4 eV. The magnetiza-\ntion precesses around the in-plane direction ( y\u0000axis) normal\nto the transport direction and the rest of the parameters of\nthe system are the same as in Fig. 3. Note, for magnetization\nprecession around the xandzaxis,T(Vbias) =T(0) for all\nprecession cone angles and hence \u0002=0. Inset shows the damp-\ning rate versus the Fermi energy for di\u000berent bias values used\nto calculate precessional anti-damping e\u000eciency.\nthe FM dynamics, Tis given by\nT=\u0000!\n\u0011X\nnZdkxdkyd'\n(2\u0019)3(v'\nn~k)2f0(\"n~k(')):(9)\nHere,v'\nn~k=@\"n~k(')=@' is the group velocity along the\n'-direction in Fig. 2, and \"n;~k=\"0(j~kj)\u0006j~h(~k)jfor the\n2D-Rashba plane, where \"0(j~kj) is the spin independent\ndispersion of the conduction electrons and ~h=atso^ez\u0002\n~k+1\n2Jsd~ m, is the spin texture of the electrons due to\nthe SOC and the s\u0000dexchange interaction. For small\nprecession cone angle, \u0012, the Gilbert damping constant\ncan be determined from \u000b=\u0000T=sd!sin2(\u0012), where the\nzero-temperature Tis evaluated by Eq. (9). We \fnd\nthat\n\u000b\u00191\n\u0011t2\nso\u0002\n(k+\nFa)2D+(EF) + (k\u0000\nFa)2D\u0000(EF)\u0003\n(1+cos2(\r));\n(10)\nwhereD+(\u0000)(E) is the density of states of the majority\n(minority) band, \ris the angle between the precession\naxis and the normal to the Rashba plane, and the Fermi\nwave-vectors ( k\u0006\nF) are obtained from, \"0(k\u0006\nF) =EF\u0007\nJsd=2. Eq. (10) shows that the Gilbert damping increases\nas the precession axis changes from in-plane ( \r=\u0019=2) to\nout of plane ( \r= 0),37which can also be seen in Fig. 3.\nIt is important to emphasize that in contrast to Eq. (9)\nthe results shown in Fig. 4 correspond to the ballistic\nregime with \u0011= 0 in the central region and the relaxation\nof the excited electrons occurs solely inside the metallic\nreservoirs. To clarify how the damping rate changes from\n10−810−610−410−2100−80−60−40−20020\nBroadening (eV)Damping Rate ( µeV)\n  \nS=200, Vbias=0\nS=200, Vbias=3 mV\nS=300, Vbias=0\nS=300, Vbias=3 mV\nStudent Version of MATLABFIG. 6: (Color online) Precessional damping rate versus\nbroadening of the states in the presence (solid lines) and ab-\nsence (dashed lines) of bias voltage for two values of the do-\nmain sizeS= 200 and S= 300. In both ballistic, \u0011=!\u00190,\nand di\u000busive, \u0011=!\u001d1, regimes the precessional damping rate\nis independent of the domain size, while in the intermediate\ncase, the amplitude of the minimum of damping rate shows a\nlinear dependence versus S. Note that the value of the broad-\nening at which the damping rate is minimum varies inversely\nproportional to the domain size, S.\nthe ballistic to the di\u000busive regime we present in Fig.\n6 the damping rate versus the broadening, \u0011, of states\nin the presence (solid line) and absence (dashed line) of\nbias voltage. We \fnd that in both ballistic ( \u0011=!\u00190)\nand di\u000busive ( \u0011=!\u001d1) regimes the damping rate is\nindependent of the size of the FM domain, S. On the\nother hand, in the intermediate regime the FM dynamics\nbecome strongly dependent on the e\u000bective domain size\nwhere the minimum of the damping rate varies linearly\nwithS. This can be understood by the fact that the\ne\u000bective chemical potential di\u000berence between the \frst,\nm=\u0000Sand last,m=Slayers in Fig.3 is proportional\ntoSand for a coherent electron transport the conduc-\ntance is independent of the length of the system along\nthe transport direction. Therefore, in this case the FM\nmotion is driven by a coherent dynamics.\nIV. DEMAGNETIZATION MECHANISM OF\nSWITCHING\nIn Sec. III we considered the case of a single FM do-\nmain where its low-energy excitations, involving the pre-\ncession of the total angular momentum, can be described\nby the eigenstates jmiofSzand local spin \rip processes\nwere neglected. However, for ultrathin FM \flms or FM\nnanoclusters, where the MAE per atom ( \u0019meV ) is com-\nparable to the exchange energy between the local mo-\nments (Curie temperature), the low-energy excitations\ninvolve both magnetization rotation and local moments\nspin-\rips due to conduction electron scattering which can\nin turn change also S. In this case the switching is ac-7\nFIG. 7: (Color online) Spatial dependence of the local damp-\ning rate for the spin-1 =2 local moments of a FM island under\ndi\u000berent bias voltages ( \u00060:4V) and magnetization directions.\nFor the parameters we chose the size of the FM island to be\n25\u000225a2, the e\u000bective magnetic \feld, jBj= 20meV , the\nbroadening, \u0011= 0, andkBT= 10 meV.\ncompanied by the excitation of local collective modes that\ne\u000bectively lowers the amplitude of the magnetic ordering\nparameter. For simplicity we employ the mean \feld ap-\nproximation for the 2D FM nanocluster where the spin\nunder consideration at position ris treated quantum me-\nchanically interacting with all remaining spins through\nan e\u000bective magnetic \feld, ~B. The spatial matrix ele-\nments of the local spin operator are\n[^~sd;r]r1r2=~ sd(r1)\u000er1r2(1\u0000\u000er1r)1s+1\n2\u000er1r2\u000er1r~ \u001c;(11)\nwhere,~\u001cs are the Pauli matrices. The magnetic energy\ncan be expressed as, HM(r) =\u0000~B(r)\u0001~\u001c=2, where, the\ne\u000bective local magnetic \feld is given by,\n~B(r) =g\u0016B~Bext+ 4X\nr0Jdd\nrr0~ sd(r0) + 2Jsd~ sc(r):(12)\nThe equation of motion for the single particle propa-\ngator of the electronic wavefunction entangled with the\nlocal spin moment under consideration can then be ob-\ntained from,\n\u0012\nE\u0000^\u0016\u0000HM(r)\u0000^H\u0000Jsd\n2^~ \u001b\u0001^~sd;r\u0013\n^Gr\nr(E) =^1:(13)\nThe density matrix is determined from Eq. (5) which\ncan in turn be used to calculate the spin density of the\nconduction electrons, ~ sc(r) =Tr(^~ \u001b^\u001arr)=2, and the di-\nrection and amplitude of the local magnetic moments,\n~ sd(r) =Tr(~\u001c^\u001arr)=2.\nFig. 7 shows the spatial dependence of the spin-1\n2local\nmoment switching rate for a FM/Rashba bilayer (Fig.\n−101−30−20−100102030\n  Damping Torque (meV)B=[|B|,0,0]\n−101\nBias Voltage (V)B=[0,|B|,0]\n−101B=[0,0,|B|]\n|B|=1 meV\n|B|=20 meV(c) (a) (b)\nStudent Version of MATLABFIG. 8: (Color online) Bias dependence of the average (over\nall sites) damping rate of the FM island for in-plane e\u000bective\nmagnetic \feld (or equilibrium magnetization) (a) along and\n(b) normal to the transport direction and (c) out-of-plane\nmagnetic \feld for two values of jBj.\n1) for two bias values ( Vbias=\u00060:4V) and for an in-\nplane e\u000bective magnetic \feld (a) along and (b) normal\nto the transport direction, and (c) an out-of-plane mag-\nnetic \feld. The size of the FM island is 25 a\u000225a, where\nais the lattice constant. Negative local moment switch-\ning rate (blue) denotes that, once excited, the local mo-\nment relaxes to its ground state pointing along the di-\nrection of the e\u000bective magnetic \feld; however positive\nlocal damping rate (red) denotes that the local moments\nremain in the excited state during the bias pulse dura-\ntion. Therefore, the damping rate of the local moments\nunder bias voltage can be either enhanced or reduced\nand even change sign depending on the sign of the bias\nvoltage and the direction of the magnetization. We \fnd\nthat the bias-induced change of the damping rate is high-\nest when the FM magnetization is in-plane and normal to\nthe transport directions similar to the single domain case.\nFurthermore, the voltage-induced damping rate is peaked\nclose to either the left or right edge of the FM (where the\nreservoirs are attached) depending on the sign of the bias.\nNote that there is also a \fnite voltage-induced damping\nrate when the magnetization is in-plane and and along\nthe transport direction ( x) or out-of-the-plane ( z).\nFig. 8 shows the bias dependence of the average (over\nall sites) damping rate for in- (a and b) and out-of-plane\n(c) directions of the e\u000bective magnetic \feld (direction\nof the equilibrium magnetization) and for two values of\njBj. This quantity describes the damping rate of the\namplitude of the magnetic order parameter. For an in-\nplane magnetization and normal to the transport direc-\ntion (Fig. 8) the bias behavior of the damping rate is lin-\near and \fnite in contrast to the single domain [Fig. 3(a)]\nwhere the damping rate was found to have a negligible\nresponse under bias. On the other hand, the bias behav-\nior of the current induced damping rate shows similar\nbehavior to the single domain case when the equilibrium8\n−4 −2 0 2 4−30−20−100102030\nFermi Energy (eV)Antidamping Efficiency (%)\n  \nB=[20,0,0] meV\nB=[0,20,0] meV\nB=[0,0,20] meV\nStudent Version of MATLAB\nFIG. 9: (Color online) Bias-induced local anti-damping\ne\u000eciency due to local spin-\rip, \u0002 = jBj(T(Vbias)\u0000\nT(0))=eVbiasT(0), versus Fermi energy for di\u000berent equilib-\nrium magnetization orientations. For the calculation we chose\nVbias= 0:2 V and the rest of the Hamiltonian parameters are\nthe same as in Fig. 7.\nmagnetization direction is in-plane and normal to the\ntransport direction (Fig. 8(b)). For an out-of-plane ef-\nfective magnetic \feld [Fig. 8(c)] the damping torque has\nan even dependence on the voltage bias.\nIn order to quantify the e\u000eciency of the voltage in-\nduced excitations of the local moments, we calculate the\nrelative change of the average of the damping rate in the\npresence of a bias voltage and present the result versus\nthe Fermi energy for di\u000berent orientations of the magne-\ntization in Fig 9. We \fnd that the e\u000eciency is maximum\nfor an in-plane equilibrium magnetization normal to the\ntransport direction and it exhibits an electron-hole asym-\nmetry. The bias-induced antidamping e\u000eciency due to\nspin-\rip can reach a peak around 20% which is much\nhigher than the peak e\u000eciency of about 2% in the sin-\ngle domain precession mechanism in Fig. 5 for the same\nsystem parameters.\nFuture work will be aimed in determining the switch-\ning phase diagram16by calculating the local antidamping\nand \feld-like torques self consistently for di\u000berent FM\ncon\fgurations.\nV. CONCLUDING REMARKS\nIn conclusion, we have developed a formalism to in-\nvestigate the current-induced damping rate of nanoscale\nFM/SOC 2D Rashba plane bilayer in the quantum\nregime within the framework of the Kyldysh Green func-\ntion method. We considered two di\u000berent regimes of FM\ndynamics, namely, the single domain FM and indepen-\ndent local moments regimes. In the \frst regime we as-\nsume the rotation of the FM as the only degree of free-\ndom, while the second regime takes into account only\nthe local spin-\rip mechanism and ignores the rotation ofthe FM. When the magnetization (precession axis) is in-\nplane and normal to the transport direction, similar to\nthe conventional SOT for classical FMs, we show that the\nbias voltage can change the damping rate of the FM and\nfor large enough voltage it can lead to a sign reversal. In\nthe case of independent spin-1 =2 local moments we show\nthat the bias-induced damping rate of the local quantum\nmoments can lead to demagnetization of the FM and has\nstrong spatial dependence. Finally, in both regimes we\nhave calculated the bias-induced damping e\u000eciency as a\nfunction of the position of the Fermi energy of the 2D\nRashba plane.\nAppendix A: Derivation of Electronic Density\nMatrix\nUsing the Heisenberg equation of motion for the an-\ngular momentum operator, ~S(t), and the commutation\nrelations for the angular momentum, we obtain the fol-\nlowing Landau-Lifshitz equations of motion,\n\u0007i@\n@tS\u0006(t) =hzS\u0006(t)\u0000h\u0006(t)Sz(t) (A1a)\n\u0000i@\n@tSz(t) =1\n2\u0000\nh+(t)S\u0000(t)\u0000h\u0000(t)S+(t)\u0001\n(A1b)\n~hmm0(t) =1\n~X\nrJsd~smm0\nc(r) +g\u0016B\u000emm0~B(t);(A1c)\nwhere,S\u0006=Sx\u0006Sy(\u001b\u0006=\u001bx\u0006\u001by), is the angu-\nlar momentum (spin) ladder operators, ~smm0\nc(r) =\n1\n2P\n\u001b\u001b0~ \u001b\u001b\u001b0\u001amm0\n\u001b\u001b0;rris the local spin density of the con-\nduction electrons which is an operator in magnetic con-\n\fguration space. Here, \u001ais the density matrix of the\nsystem, and the subscripts, r;m;\u001b refer to the atomic\ncite index, magnetic state and spin of the conduction\nelectrons, respectively. In the following we assume a pre-\ncessing solution for Eq (A1)(a) with a \fxed cone angle\nand Larmor frequency !=hz. Extending the Hilbert\nspace of the electrons to include the angular momentum\ndegree of freedom we de\fne  ms0i(t) =jS;mi\n s0i(t).\nThe equation of motion for the Green function (GF) is\nthen given\n\u0010\nE\u0000i\u0011\u0000^H(k) +n!\u0000n\n2SJsd(k)\u001bz\u0011\n^Gr\nnm(E;k) (A2)\n\u0000p\nS(S+ 1)\u0000n(n+ 1)\n2SJsd(k)\u001b\u0000^Gr\nn+1m(E;k)\n\u0000p\nS(S+ 1)\u0000n(n\u00001)\n2SJsd(k)\u001b+^Gr\nn\u00001m(E;k) =^1\u000enm\nwhere,n= (\u0000S;\u0000S+1;:::;S ) and the gauge transforma-\ntion n\u001bi(t)!ein!t n\u001bi(t) has been employed to remove\nthe time dependence. The density matrix of the system\nis of the form\n^\u001anm=e\u0000i(n\u0000m)!tSX\np=\u0000SZdE\n2\u0019^Gr\nnp2\u0011fp^\u0016^Ga\npm (A3)9\nwhere,fp^\u0016(E) =f(E\u0000p!\u0000^\u0016) is the equilibrium Fermi\ndistribution function of the electrons. Due to the fact\nthatp!are the eigenvalues of HM=\u0000g\u0016B~B\u0001S, one\ncan generalize this expression by transforming into a ba-\nsis where the magnetic energy is not diagonal which in\nturn leads to Eq (5) for the density matrix of the con-\nduction electron-local moment entagled system.\nAppendix B: Recursive Relation for GFs\nSince in this work we are interested in diagonal blocks\nof the GFs and in general for FMs at low temperaturewe haveS\u001d1, we need a recursive algorithm to be able\nto solve the system numerically. The surface Keldysh\nGFs corresponding to ascending ^ gu;r=<, and descending\n^gd;r=<, recursion scheme read,\n^gu;r\nn(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^H(k)\u0000^\u0006rn(E;k)\u0000n\n2SJsd(k)\u001bz\u0000(S\u0000\nn)2\n4S2Jsd(k)\u001b+^gu;r\nn\u00001(E;k)\u001b\u0000Jsd(k)(B1)\n^\u0006u;<\nn(E;k) =\u0000X\n\u000b\u0010\n2i\u0011n+^\u0006r\nn;\u000b(E;k)\u0000^\u0006a\nn;\u000b(E;k)\u0011\nfn\u000b+(S\u0000\nn)2\n4S2Jsd\u001b+^gu;r\nn\u00001^\u0006u;<\nn\u00001^gu;a\nn\u00001\u001b\u0000Jsd (B2)\n^gd;r\nn(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^\u0006rn(E;k)\u0000^H(k)\u0000n\n2SJsd(k)\u001bz\u0000(S+\nn)2\n4S2Jsd(k)\u001b\u0000^gu;r\nn+1(E;k)\u001b+Jsd(k)(B3)\n^\u0006d;<\nn(E;k) =\u0000X\n\u000b\u0010\n2i\u0011n+^\u0006r\nn;\u000b(E;k)\u0000^\u0006a\nn;\u000b(E;k)\u0011\nfn\u000b+(S+\nn)2\n4S2Jsd\u001b\u0000^gd;r\nn+1^\u0006d;<\nn+1^gd;a\nn+1\u001b+Jsd (B4)\nwhere, ^\u0006r\nn(E;k) =P\n\u000b^\u0006r\n\u000b(E\u0000!n;k) corresponds to the\nself energy of the leads, \u000b=L;R refers to the left and\nright leads in the two terminal device in Fig. 3 and S\u0006\nm=p\nS(S+ 1)\u0000m(m\u00061). Using the surface GFs we can\ncalculate the GFs as follows,\n^Gr\nn;m(E;k) =1\nE\u0000!n\u0000i\u0011n\u0000^H(k)\u0000^\u0006rn\u0000n\n2SJsd(k)\u001bz\u0000^\u0006r;u\nn\u0000^\u0006r;d\nn; n =m (B5)\n=S+\nn\n2S^gu;r\nn(E;k)Jsd(k)\u001b\u0000^Gr\nn+1;m(E;k); n6=m (B6)\n=S\u0000\nn\n2S^gd;r\nn(E;k)Jsd(k)\u001b+^Gr\nn\u00001;m(E;k); n6=m (B7)\nwhere the ascending and descending self energies are given by,\n^\u0006r;u\nn=(S\u0000\nn)2\n4S2Jsd(k)\u001b+^gu;r\nn\u00001(E;k)\u001b\u0000Jsd(k) (B8)\n^\u0006r;d\nn=(S+\nn)2\n4S2Jsd(k)\u001b\u0000^gd;r\nn+1(E;k)\u001b+Jsd(k) (B9)\nThe average rate of angular momentum loss/gain can be obtained from the real part of the loss of angular momentum\nin one period of precession,\nT0\nn=1\n2(T0\u0000\nn\u0000T0+\nn) =1\n2= X\nkTr[S\u0000\nn\n2S\u001b+Jsd(k)^\u001ann+1(k)\u0000S+\nn\n2S\u001b\u0000Jsd(k)^\u001ann\u00001(k)]!\n(B10)10\nwhich can be interpreted as the current \rowing across the layer n.\nT0\u0000=+\nn =X\nkZdE\n2\u0019iTrnh\n^\u0006d=u;r\nn(E;k)\u0000^\u0006d=u;a\nn(E;k)i\n^G<\nnn(E;k) +^\u0006d=u;<\nn (E)h\n^Gr\nnn(E;k)\u0000^Ga\nnn(E;k)io\n;(B11)\nAcknowledgments\nThe work at CSUN is supported by NSF-Partnership\nin Research and Education in Materials (PREM) GrantDMR-1205734, NSF Grant No. ERC-Translational Ap-\nplications of Nanoscale Multiferroic Systems (TANMS)-\n1160504, and US Army of Defense Grant No. W911NF-\n16-1-0487.\n\u0003Electronic address: Farzad.Mahfouzi@gmail.com\nyElectronic address: nick.kioussis@csun.edu\n1J. 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Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China  \nEmail: kkmeng@ustb.edu.cn ; yjiang@ustb.edu.cn  \n \nAbstract: The Y 3Fe5O12 (YIG) films with perpendicular magnetic anisotropy (PMA) \nhave recently attracted a great deal of attention for spintronics  applications. Here, w e \nreport the induced PMA in the ultrathin YIG films grown on \n(Gd 2.6Ca0.4)(Ga 4.1Mg 0.25Zr0.65)O12 (SGGG) substrate s by epitaxial strain without \npreprocessing. Reciprocal space mapping shows  that the film s are  lattice -matched to \nthe substrate s without strain relaxation. Through ferromagnetic resonance and \npolarized neutron reflectometry measurements, we find that these YIG films have \nultra-low Gilbert damping constant  (α < 1×10-5) with a magnetic dead layer as thin as \nabout 0.3 nm  at the YIG/SGGG interfaces. Moreover, the transport behavior of the \nPt/YIG/SGGG films reveals an enhancement of spin mixing conductance and a large \nnon-monotonic magnetic field dependence of anomalous Hall effect as compared with \nthe Pt/YIG/Gd 3Ga5O12 (GGG) films. The non- monotonic anomalous Hall signal is \nextracted in the temperature range from 150 to 350 K, which has been ascribed to the possible non -collinear magnetic order at the Pt/YIG interface induced by uniaxial \nstrain.  \n \nThe spin transport in ferrim agnetic insulator (FMI)  based devices has received \nconsiderable interest due to its free of current -induced Joule heating and beneficial for \nlow-power spintronic s applications  [1, 2]. Especially, the high-quality Y3Fe5O12 (YIG)  \nfilm as a widely  studied FMI  has low damping  constant,  low magnetostriction and \nsmall magnetocrystalline anisotropy,  making it a key material for magnonics  and spin \ncaloritronics . Though the magnon s can carr y information over distances as long as \nmillimeters  in YIG film , there remain s a challenge to control its magnetic anisotropy \nwhile maintaining the low damping constant [3] , especially for  the thin film with \nperpendicular  magnetic anisotropy (PMA) , which  is very useful for spin polarizers,  \nspin-torque oscillators, magneto -optical d evices  and m agnon valve s [4-7]. In addition, \nthe spin- orbit torque (SOT) induced magnetization switching with low current \ndensities has been realized in non -magnetic heavy metal  (HM)/FMI heterostructures , \npaving the road towards  ultralow -dissipation SOT de vices based on FMI s [8-10]. \nFurthermore, p revious theoretical studies have pointed that  the current density will \nbecome much smaller if the domain structures were topologically protected (chiral) [11].  However,  most FMI  films  favor in-plane easy axis dominated by shape \nanisotropy , and the investigation is eclipsed as compared with ferromagnetic materials \nwhich show abundant and interesting domain structures such as chiral domain walls and magnetic skyrmions et al.  [12-17]. Recently, the interface- induc ed chiral domain walls  have been observed in centrosymmetric oxides Tm 3Fe5O12 (TmIG) thin films, \nand the  domain walls  can be propelled by spin current from an adjacent platinum \nlayer  [18]. Similar with the TmIG films, the possible chiral  magnetic structures are \nalso expected in the YIG films with lower damping  constan t, which  would further \nimprove the chiral domain walls’ motion speed.  \nRecently, several ways  have been reported to attain the perpendicular ly \nmagnetized YIG films , one of which is utiliz ing the lattice distortion and \nmagnetoelastic effect induced by  epitaxial strain [1 9-22]. It is noted that  the strain \ncontrol can not only enable the field -free magnetization switching but also assist the \nstabilization of the non- collinear  magnetic textures  in a broad range of magnetic field \nand temperature. Therefore, abundant and interesting physical phenomena  would \nemerge in epitaxial grown YIG films with PMA. However, either varying the buffer \nlayer or doping would increase the Gilbert damping constant of YIG, which will  \naffect the efficiency of the SOT induced magnetization switching  [20, 21]. On the \nother hand, these preprocessing would lead to a more complicate magnetic structures  \nand impede  the further discussion of spin transport properties such as possible \ntopological Hall effect  (THE).  \nIn this work, we realized the PMA of ultrathin YIG films deposited on SGGG \nsubstrates  due to epitaxial strain . Through ferromagnetic resonance (FMR) and \npolarized neutron reflectometry (PNR) measurements, we have found that the YIG \nfilms had small Gilbert damping constant with a magnetic dead layer as thin as about \n0.3 nm  at the YIG/SGGG interfaces. Moreover, we have carried out the transport measurements of the Pt/YIG/SGGG films and observed a large non -monotonic \nmagnet ic field dependence of the anomalous Hall resistivity, which did not exis t in \nthe compared Pt/YIG/GGG films. The non -monotonic anomalous Hall signal was \nextracted in the temperature range from 150 to 350 K, and we ascribed it  to the \npossible non -collinear magnetic order at the Pt/YIG interfaces induced by uniaxial \nstrain.  \n \nResults  \nStructural and magnetic characterization.  The epitaxial YIG films with varying \nthickness from 3 to 90 nm  were grown on the [111] -oriented GGG substrate s (lattice \nparameter a = 1.237 nm) and SGGG substrates  (lattice parameter a = 1.248 nm)  \nrespectively by pulsed laser deposition technique  (see methods). After the deposition, \nwe have investigated the surface morphology of the two kinds of films using atomic \nforce microscopy (AFM) as shown in Fig. 1 ( a), and the two films have a  similar and \nsmall surface roughness ~0.1 nm. Fig. 1 ( b) shows the  enlarged  XRD ω-2θ scan \nspectra of the YIG (40 nm) thin film s grow n on the two different substrates (more \ndetails are shown in the Supplementary Note 1 ), and they  all show predominant (444) \ndiffraction peaks without any other diffraction peaks, excluding impurity phases or other crystallographic orientation s and indicat ing the single -phase  nature. According \nto the (444) diffraction pe ak position and the reciprocal space map of the (642) \nreflection of a 40 -nm-thick YIG film grown on SGGG as shown in Fig. 1(c), we have \nfound that the lattice constant of SGGG  (~1.248 nm) substrate was larger than the YIG layer  (~1.236 nm). We quantify thi s biaxial strain as ξ  = (aOP - aIP)/aIP, where a OP \nand aIP represent the pseudo  cubic lattice constant calculated from the ou t-of-plane  \nlattice constant d(4 4 4) OP and in-plane lattice constant d(1 1 0) IP, respectively, \nfollowing the equation of  \n2 2 2lkhad\n++= , with h, k, and l  standing for the Miller \nindices of the crystal planes . It indicates that the SGGG substrate provides a tensile \nstress ( ξ ~ 0.84%) [21]. At the same time, the magnetic properties of the YIG films \ngrown on the two different substrates were measured via VSM magnetometry at room \ntemperature. According to the magnetic field ( H) dependence of the magnetization  (M) \nas shown in Fig. 1 (d), the magnetic anisotropy of the YIG film grown on SGGG \nsubstrate has been modulated by strain, while the two films have similar in -plane \nM-H curves.  \nTo further investigate the quality of the YIG films  grown on SGGG substrates  \nand exclude  the possibility of the strain induced large stoichiometry  and lattice \nmismatch, compositional analyse s were carried out using x -ray photoelectron \nspectroscopy (XPS) and PNR. As shown in Fig. 2 (a), the difference of binding \nenergy between the 2p 3/2 peak and the satellite peak is about 8.0 eV, and the Fe ions \nare determined to be in the 3+ valence state.  It is found that there is  no obvious \ndifference for Fe elements in the YIG films grown on GGG and SGGG substrates. \nThe Y 3 d spectrums show a small energy shift as shown in Fig. 2 (b) and the binding \nenergy shift may be related to the lattice strain and the variation of bond length  [21]. \nTherefore, the stoichiometry of the YIG  surface has not been dramatically modified \nwith the strain control.  Furthermore, we have performed the PNR meas urement  to probe the depth dependent struc ture and magnetic information of YIG films grown on \nSGGG substrates. The PNR  signals and scattering length density (SLD) profiles  for \nYIG (12.8 nm)/SGGG films by applying an in- plane magnetic field of 900 mT at \nroom temperature are shown in Fig. 2 ( c) and ( d), respectively.  In Fig. 2(c), R++ and \nR-- are the nonspin -flip reflectivities, where the spin polarizations are the same for the \nincoming and reflected neutrons. The inset of Fig. 2(c) shows the experimental and  \nsimulated spin -asymmetry (SA), defined as SA = ( R++ – R--)/(R++ + R--), as a function \nof scattering vector Q. A reasonable fitting was obtained with a three- layer model for \nthe single YIG film, containing the interface layer , main YIG layer and surface layer. \nThe nuclear SLD and  magnetic SLD are directly proportional to the nuclear scattering \npotential and the magnetization , respectively . Then, the depth- resolved structural and \nmagnetic SLD profiles delivered by fitting are s hown in Fig. 2(d) . The Z -axis \nrepresents the distance for the vertical direction of the film, where Z  = 0 indicates the \nposition at the YIG/SGGG interface. It is obvious that  there is few Gd diffusion into \nthe YIG film, and the dead layer  (0.3 nm ) is much thinner than the reported values \n(5-10 nm) between YIG  (or T mIG) and substrates [23 -25]. The net magnetization of \nYIG is 3.36 μB (～140  emu/cm3), which is similar with that of bulk YIG [2 6]. The \nPNR results also showed that besides the YIG/ SGGG interface region, there is also  \n1.51- nm-thick  nonmagnetic surface layer, which may be  Y2O3 and is likely to be \nextremely important in magnetic proximity effect [ 23]. \n Dynamical characterization and spin transport properties.  To quantitatively \ndetermine  the magnetic anisotropy and dynamic properties of the YIG films, the FMR \nspectra were measured at room temperature using an electron paramagnetic resonance \nspectrometer with rotating the films. Fig. 3(a) shows the geometric configuration of the angle reso lved FMR measurements. We use the FMR absorption line shape to \nextract the resonance field  (H\nres) and peak -to-peak linewidth ( ΔHpp) at different θ  for \nthe 40 -nm-thick YIG fil ms grown on GGG and SGGG substrates, respectively.  The \ndetails for 3 -nm-thick  YIG film are show n in the Supp lementary  Note 2 . According to \nthe angle  dependence of H res as shown in Fig. 3(b), one can find that as compared \nwith the YIG films grown on GGG substrate s, the minimum Hres of the 40- nm-thick \nYIG film grown on SGGG substrate increases with varying θ from 0° to 90° .On the \nother hand,  according to the frequency dependence of Hres for the YIG (40 nm)  films \nwith applying H  in the XY  plane as shown in Fig. 3(c), in contrast to the YIG/GGG \nfilms, the H res in YIG/SGGG films could not be fitted by the in-plane magnetic \nanisotropy Kittel formula  21)] 4 ( )[2(/\neff res res πM H Hπγ/ f + = . All these results \nindicate that the easy axis of YIG  (40 nm) /SGGG films  lies out -of-plane. The angle  \ndependent  ΔHpp for the  two films are  also compared as shown in Fig. 3(d) , the \n40-nm-thick YIG film grown on SGGG substrate has  an optimal value of Δ Hpp as low \nas 0.4 mT at θ =64°, and the corresponding FMR absorption line and Lorentz fitting \ncurve are shown in Fig. 3(e).  Generally , the ΔHpp is expected to be minimum \n(maximum) along  magnetic easy (hard) axis, which is basically coincident with the \nangle dependent  ΔHpp for the YIG films grown on GGG substrates.  However, as shown in Fig. 3(d), the  ΔHpp for the YIG/SGGG films  shows an anomalous variation. \nThe lowest  ΔHpp at θ=64° could be ascribed to the high YIG film quality and  ultrathin  \nmagnetic dead layer at the YIG/SGGG interface.  It should be noted that , as compared \nwith YIG/GGG films , the Δ Hpp is independent on the frequency from 5 GHz to 14 \nGHz as shown in Fig. 3(f). Then, w e have calculate d the Gilbert damping constant α  \nof the  YIG (40 nm)/SGGG films by extracting the Δ Hpp at each frequency as shown in \nFig. 3(f). The obtained α  is smaller tha n 1 × 10−5, which  is one order of magnitude \nlower than t he report in  Ref. [20]  and would  open new perspectives for the \nmagnetization dynamics.  According to the theor etical theme, the ΔHpp consists of \nthree parts: Gilbert damping, two magnons scattering relaxation process and \ninhomogeneities, in which both the Gilbert damping and the two magnons scattering \nrelaxation process depend on frequency. Therefore, the large Δ Hpp in the YIG/SGGG \nfilms  mainly  stems from the inhomogeneities, w hich will be discussed next with the \nhelp of the transport measurements. All of the above results have proven that the  \nultrathin YIG films grown on SGGG substrate s have not only evident  PMA  but also \nultra-low Gilbert damping constant.  \nFurthermore, we have also investigated the spin transport properties for  the high \nquality YIG film s grown on SGGG substrate s, which  are basically sensitive to  the \nmagnetic details of YIG. The magnetoresistance (MR) has been proved as a powerful \ntool to effectively explore magnetic information originating from the interfaces [ 27]. \nThe temperature dependent spin Hall magnetoresistance (SMR) of the Pt (5 nm)/YIG \n(3 nm) films grown on the two different substrates were measured using a small and non-perturbative current densit y (~ 106 A/cm2), and the s ketches of the measurement  \nis shown in Fig. 4 (a). The β scan of the longitudinal  MR, which is defined as \nMR=ΔρXX/ρXX(0)=[ρXX(β) -ρXX(0)]/ρXX(0) in the YZ  plane  for the two films  under a 3 T \nfield (enough to saturate the magnetization  of YIG ), shows cos2β behavior s with \nvarying temperature  for the Pt/YIG/GGG and Pt/YIG/SGGG films as shown in Fig. 4 \n(b) and (c), respectively. T he SMR of  the Pt/YIG /SGGG films  is larger than that of \nthe Pt/YIG /GGG films with the same thickness of YIG  at room temperature,  \nindica ting an enhanced  spin mixing conductance ( G↑↓) in the Pt/YIG /SGGG films. \nHere, it should be noted that the spin transport properties for the Pt layers ar e \nexpected to be the same because of the similar resistivity and film s quality . Therefore, \nthe SGGG substrate not only induces the PMA but also enhances G ↑↓ at the Pt/YIG \ninterface. Then, we have also investigated the field dependent Hall resistivities  in the \nPt/YIG/SGGG films at the temperature range from 260 to 350 K  as shown in Fig. 4(d).  \nThough the conduction electrons cannot penetrate into the FMI layer, the possible \nanomalous Hall effect (AHE) at the HM/FMI interface is proposed to emerge, and the \ntotal Hall resistivity can usually be expressed as the sum of various contributions  [28, \n29]: \nS-A S H ρ  ρ H R ρ + + =0 ,         (1)  \nwhere R0 is the normal Hall coefficient, ρ S the transverse manifestation of SMR, and \nρS-A the spin Hall anomalous Hall effect (SAHE) resistivity. Notably, the external field \nis applied out -of-plane, and  ρs (~Δρ1mxmy) can be neglected [ 29]. Interestingly, the \nfilm grown on SGGG substrate shows a bump and dip feature during the hysteretic measurements in the temperature range from 260 to 350 K. In the following \ndiscussion, we term the part of extra anomalous signals as the anomalous SAHE resistivity ( ρ\nA-S-A). The ρ A-S-A signals clearly coexist with the large background of \nnormal Hall effect. Notably, the broken (space) inversion symmetry with strong \nspin-orbit coupling (SOC)  will induce the Dzyaloshinskii -Moriya interaction (DMI) . \nIf the DMI could be compared with the Heisenberg exchange interaction and the \nmagnetic anisotropy  that were controlled by st rain, it c ould stabilize non-collinear \nmagnetic textures  such as skyrmions, producing  a fictitious magnetic field and the \nTHE . The ρA-S-A signals  indicate that a chiral spin texture may exist, which  is similar \nwith B20-type compounds Mn 3Si and Mn 3Ge [ 30,31]. To more clearly demonstrate \nthe origin of the anomalous  signals, we have subtracted the normal Hall term , and the \ntemperature dependence of ( ρS-A + ρ A-S-A) has been shown in Fig. 4 (e). Then, we can \nfurther discern the peak and hump structure s in the temperature range from 260 to 350 \nK. The SAHE  contribution ρS-A can be expressed  as 𝜌𝑆−𝐴=𝛥𝜌2𝑚𝑍 [32, 33],\n where  \n𝛥𝜌2 is the coefficient depending on the imaginary part of G ↑↓, and mz is the unit \nvector of the magnetization  orientation along  the Z direction . The extracted Hall \nresist ivity ρA-S-A has been shown in Fig. 4 (f), and the temperature dependence of the \nlargest ρA-S-A (𝜌𝐴−𝑆−𝐴Max) in all the films have been shown in Fig. 4 (g). Finite values of \n𝜌𝐴−𝑆−𝐴Max exist  in the temperature range from 150 to 350 K , which is much d ifferen t \nfrom that in B20 -type bulk chiral magnets which are subjected to low temperature and \nlarge magnetic field [34]. The large non -monotonic magnetic field dependence of anomalous Hall resistivity  could not stem from the We yl points, and the more detailed \ndiscussion was shown in the Supplementary Note 3.  \nTo further discuss the origin of the anomalous transport signals, we have \ninvestigated the small field dependence of the Hall resistances for Pt  (5 nm) /YIG (40 \nnm)/SGGG films as shown in Fig. 5(a). The out-of-plane hysteresis loop  of \nPt/YIG/SGGG is not central symmetry, which indicates the existence of an  internal \nfield leading to opposite  velocities  of up to down and down to domain walls in the  \npresence of current along the +X  direction. The large field dependences of the Hall \nresistances are shown in Fig. 5(b), which could not be described by Equation (1). \nThere are large variations for the Hall signals when the external magnetic field  is \nlower than the saturation field ( Bs) of YIG film (~50 mT at 300 K and ~150 mT at 50 \nK). More interestingly, we have firstly applied a large out -of-plane external magnetic \nfield of +0.8 T ( -0.8 T) above Bs to saturate the out -of-plane magnetization \ncomp onent MZ > 0 ( MZ < 0), then decreased the field to zero, finally the Hall \nresistances were measured in the small field range ( ± 400 Oe), from which  we could \nfind that the shape was reversed as shown in Fig. 5(c). Here, we infer that the magnetic structures  at the Pt/YIG interface grown on SGGG substrate could not be a \nsimple linear magnetic order.  Theoretically , an additional chirality -driven Hall effect \nmight be present in the ferromagnetic regime due to spin canting [3 5-38]. It has been \nfound that  the str ain from an insulating substrate could produce a tetragonal distortion, \nwhich would drive an orbital selection, modifying  the electronic properties and the \nmagnetic ordering of manganites. For A\n1-xBxMnO 3 perovskites, a compressive strain makes the ferromagnetic configuration relatively more stable than the \nantiferromagnetic state [3 9]. On the other hand, the strain would induce the spin \ncanting [ 40]. A variety of experiments and theories have reported that the ion \nsubstitute, defect and magnetoelast ic interaction would cant the magnetization of YIG \n[41-43]. Therefore, if we could modify the magnetic order by epitaxial strain, the \nnon-collinear magnetic structure is expected to emerge in the YIG film.  For YIG \ncrystalline structure, the two Fe sites ar e located on the octahedrally coordinated 16(a) \nsite and the tetrahedrally coordinated 24(d) site, align ing antiparallel with each other  \n[44]. According to the XRD and RSM results, the tensile strain due to SGGG \nsubstrate would result in the distortion ang le of the facets of the YIG unit cell smaller \nthan 90 ° [45]. Therefore, the magneti zations  of Fe at two sublattice s should  be \ndiscussed separately rather than as a whole. Then, t he anomalous  signals  of \nPt/YIG/SGGG films  could be  ascribed to the emergence o f four different Fe3+ \nmagnetic orientation s in strained Pt/YIG films, which are shown in Fig. 5(d).  For \nbetter to understand our results, w e assume that, in analogy with ρ S, the ρA-S-A is larger \nthan  ρA-S and scales linearly with m ymz and mxmz. With applying a large external field \nH along  Z axis, the uncompensated magnetic moment at the tetrahedrally coordinated \n24(d) is along with the external fields H  direction for |H | > Bs, and the magnetic \nmoment tends to be along A (-A) axis  when the external fields is swept  from 0.8 T \n(-0.8 T) to 0 T. Then,  if the Hall resistance  was measured at small out -of-plane field , \nthe uncompensated magnetic moment  would switch from A (-A) axis to B (-B) axis. In \nthis case, the ρ A-S-A that scales with Δ ρ3(mymz+mxmz) would change the sign because the mz is switched from the Z  axis to - Z axis as shown in Fig. 5(c). However, there is \nstill some problem that needs to be further clarified. There are no anomalous  signals  \nin Pt/YIG/GGG films that could be ascribed to the weak strength of Δρ3 or the strong \nmagnetic anisotropy . It is still valued for further discussion of the origin of Δ ρ3 that \nwhether it could stem from the skrymions et al ., but until now we have not observed \nany chiral domain structures in Pt/YIG/SGGG films through the Lorentz transmission \nelectron microscopy. Therefore, we hope that future work would  involve more \ndetailed magnetic microscopy imaging and microstructure analysis, which can further elucidate the real microscopic origin of the large non -monotonic magnetic field \ndependence of anomalous Hall resistivity.   \n \nConclusion  \nIn conclusion, the YIG film with PMA could be realized using both epitaxial strain \nand growth -induced anisotropies. These YIG films grown on SGGG substrates had \nlow G ilbert damping constants (<1 ×10\n-5) with a magnetic dead layer as thin as about \n0.3 nm  at the YIG/SGGG interface. Moreover, we observe d a large non -monotonic \nmagnetic field dependence of anomalous Hall resistivity in Pt/YIG/SGGG films, \nwhich did not exist in Pt/YIG/GGG films. The non -monotonic anomalous portion of \nthe Hall signal was extracted in the temperature range from 150 to 350 K and w e \nascribed it to the possible non -collinear magnetic order at the Pt/YIG interface \ninduced by uniaxial strain. The present work not only demonstrate that the strain \ncontrol can effectively tune the electromagnetic properties of FMI but also open up the exp loration of non -collinear spin texture for fundamental physics and magnetic \nstorage technologies based on FMI.  \n \nMethods  \nSample preparation.  The epitaxial YIG films with varying thickness from 3 to 90 \nnm were grown on the [111] -oriented GGG substrate s (lattice parameter a =1.237 nm) \nand SGGG substrates  (lattice parameter a  =1.248 nm)  respectively by pulsed laser \ndeposition technique . The growth  temperature was TS =780 ℃ and the oxyg  \npressure  was varied from  10 to 50 Pa . Then,  the films were annealed at 780℃ for 30 \nmin at the oxygen pressure of  200 Pa . The Pt (5nm) layer was deposited on  the top of  \nYIG films  at room temperature by magnetron sputtering. After the deposition, the \nelectron beam lithography and Ar ion milling were used to pattern Hall bars, and a lift-off process was used to form contact electrodes . The size of all the Hall bars is 20 \nμm×120 μm. \nStructural and magnetic characterization. The s urface morphology was measured \nby AFM (Bruke Dimension Icon).  Magnetization measurements were carried out \nusing a Physical Property Measurement System (PPMS) VSM.  A detailed \ninvestigation of the magnetic information  of Y IG was investigated by  PNR at the \nSpallation Neutron Source of China.   \nFerromagnetic resonance measurements. The measurement setup is depicted in Fig. \n3(a). For FMR measurements, the DC magnetic field was modulated with an AC field. \nThe transmitted signal was detected by a lock -in amplifier. We observed the FMR spectrum of the sample by sweeping the external magnetic field. 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(c) High -resolution XRD \nreciprocal space map of t he YIG film deposited on the SGGG substrate. (d) Field \ndependence of the normalized magnetization of the YIG films  grown  on the two \ndifferent substrates . \n \n \nFig. 2 Structural  and magnetic  properties of YIG films.  Room temperature XPS \nspectra of (a) Fe 2p and (b) Y 3d for YIG films grown  on the two substrates . (c) P NR \nsignals (with a 900 mT in -plane field) for the spin -polarized R++ and R-- channels. \nInset: The experimental and simulated SA  as a function of scattering vector Q.  (d) \nSLD profiles of the  YIG/SGGG films.  The nuclear SLD and  magnetic SLD is directly \nproportional to the nuclear scattering potential and the magnetization , respectively.  \n \n \n \n \nFig. 3 Dynamical properties of YIG films . (a) The geometric configuration of the \nangle dependent FMR measurement. (b) The angle  dependence of the H res for the YIG \nfilms on GGG and SGGG substrates. (c) The frequency dependence of the H res for \nYIG films grown on GGG and S GGG substrates. (d) The ang le dependence of Δ Hpp \nfor the YIG films on GGG and SGGG substrates. (e) FMR spectrum of the \n40-nm-thick YIG film grown on SGGG substrate with 9.46 GHz at θ =64°. (f) The \nfrequency dependence of Δ Hpp for the 40 -nm-thick YIG films grown on GGG and \nSGGG substr ates.  \n \nFig. 4 Spin transport properties  of Pt/YIG (3nm) films . (a) The definition of the \nangle, the axes and the measurement configurations. ( b) and ( c) Longitudinal MR at \ndifferent temperatures in  Pt/YIG/GGG and Pt/YIG/SGGG films respectively (The \napplied magnetic field is 3 T). (d) Total Hall resistivities vs  H for Pt/YIG/SGGG films \nin the temperature range from 260 to 300 K. (e)  (ρS-A+ρA-S-A) vs H for two films in the \ntemperature range from 260 to 300 K. (f) ρ A-S-A vs H for Pt/YIG/SGGG films at 300K. \nInset: ρS-A and ρS-A + ρ A-S-A vs H for Pt/YIG/SGGG films at 300K. (g) Temperature \ndependence of the 𝜌𝐴−𝑆−𝐴𝑀𝑎𝑥. \n \n \n \nFigure 5 S pin transport properties  of Pt/YIG ( 40 nm) films . (a) and (b) The Hall \nresistances vs H for the Pt/YIG/SGGG films in the temperature range from 50 to 300 \nK in small  and large magnetic field range, respectively. (c) The Hall resistances vs  H \nat small magnetic field range after sweeping a large out -of-plane  magnetic field +0.8 \nT (black line) and - 0.8 T (red line) to zero.  (d) An illustration of the orientations of the \nmagnetizations Fe ( a) and Fe ( d) in YIG films with the normal in -plane magnetic \nanisotropy (IMA), the ideal  strain induced PMA and the actual magnetic anisotropy \ngrown on SGGG in our work.  \n"    },    {        "title": "1903.08395v2.Nonlinear_magnetization_dynamics_driven_by_strong_terahertz_fields.pdf",        "content": "Nonlinear magnetization dynamics driven by strong terahertz \felds\nMatthias Hudl,1Massimiliano d'Aquino,2Matteo Pancaldi,1See-Hun Yang,3Mahesh G. Samant,3Stuart\nS. P. Parkin,3, 4Hermann A. D urr,5Claudio Serpico,6Matthias C. Ho\u000bmann,7and Stefano Bonetti1, 8,\u0003\n1Department of Physics, Stockholm University, 106 91 Stockholm, Sweden\n2Department of Engineering, University of Naples \\Parthenope\", 80143 Naples, Italy\n3IBM Almaden Research Center, San Jose CA 95120, USA\n4Max-Planck Institut f ur Mikrostrukturphysik, 06120 Halle, Germany\n5Department of Physics and Astronomy, Uppsala University, 751 20 Uppsala, Sweden\n6DIETI, University of Naples Federico II, 80125 Naples, Italy\n7SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA\n8Department of Molecular Sciences and Nanosystems,\nCa' Foscari University of Venice, 30172 Venezia Mestre, Italy\n(Dated: September 26, 2019)\nWe present a comprehensive experimental and numerical study of magnetization dynamics in a\nthin metallic \flm triggered by single-cycle terahertz pulses of \u001820 MV/m electric \feld amplitude\nand\u00181 ps duration. The experimental dynamics is probed using the femtosecond magneto-optical\nKerr e\u000bect, and it is reproduced numerically using macrospin simulations. The magnetization\ndynamics can be decomposed in three distinct processes: a coherent precession of the magnetiza-\ntion around the terahertz magnetic \feld, an ultrafast demagnetization that suddenly changes the\nanisotropy of the \flm, and a uniform precession around the equilibrium e\u000bective \feld that is relaxed\non the nanosecond time scale, consistent with a Gilbert damping process. Macrospin simulations\nquantitatively reproduce the observed dynamics, and allow us to predict that novel nonlinear mag-\nnetization dynamics regimes can be attained with existing table-top terahertz sources.\nSince Faraday's original experiment [1] and until two\ndecades ago, the interaction between magnetism and\nlight has been mostly considered in a unidirectional way,\nin which changes to the magnetic properties of a mate-\nrial cause a modi\fcation in some macroscopic observable\nof the electromagnetic radiation, such as polarization\nstate or intensity. However, the pioneering experiment of\nBeaurepaire et al. [2], where femtosecond optical pulses\nwere shown to quench the magnetization of a thin-\flm\nferromagnet on the sub-picoseconds time scales, demon-\nstrated that intense laser \felds can conversely be used\nto control magnetic properties, and the \feld of ultrafast\nmagnetism was born. Large research e\u000borts are nowadays\ndevoted to the attempt of achieving full and deterministic\ncontrol of magnetism using ultrafast laser pulses [3{10], a\nfundamentally di\u000ecult problem that could greatly a\u000bect\nthe speed and e\u000eciency of data storage [11].\nRecently, it has been shown that not only femtosecond\nlaser pulses, but also intense single-cycle terahertz (THz)\npulses [12] can be used to manipulate the magnetic order\nat ultrafast time scales in di\u000berent classes of materials\n[13{18]. The main peculiarity of this type of radiation,\ncompared with more conventional femtosecond infrared\npulses, is that the interaction with the spins occurs not\nonly through the overall energy deposited by the radia-\ntion in the electronic system, but also through the Zee-\nman torque caused by the magnetic \feld component of\nthe intense THz pulse. This is a more direct and e\u000ecient\nway of controlling the magnetization, and to achieve the\nfastest possible reversal [19, 20]. However, an accurate\ndescription of the magnetization dynamics triggered by\nstrong THz pulses is still missing.In this Letter, we present a combined experimental and\nnumerical study of the magnetization dynamics triggered\nby linearly polarized single-cycle THz pulses with peak\nelectric (magnetic) \felds up to 20 MV/m (67 mT). We\ninvestigate not only the fast time scales that are com-\nparable to the THz pulse duration ( \u00181 ps), but also\nthe nanosecond regime, where ferromagnetic resonance\n(FMR) oscillations are observed. Moreover, we write\nan explicit form of the Landau-Lifshitz-Gilbert (LLG)\nequation [21, 22] suitable to analyze terahertz-driven dy-\nnamics, that we use to predict yet-unexplored nonlin-\near magnetization dynamics regimes uniquely achievable\nwith this type of excitation mechanism.\nExperimental details. Room temperature experi-\nmental data is obtained from a time-resolved pump-\nprobe method utilizing the magneto-optical Kerr e\u000bect\n(MOKE), Refs. [23, 24]. A sketch of the experimental\nsetup is presented in Fig. 1 (a). Strong THz radiation is\ngenerated via optical recti\fcation of 4 mJ, 800 nm, 100\nfs pulses from a 1 kHz regenerative ampli\fer in a lithium\nniobate (LiNbO 3) crystal, utilizing the tilted-pulse-front\nmethod [25]. In contrary to the indirect (thermal) cou-\npling present in visible- and near-infrared light-matter in-\nteraction, THz radiation can directly couple to the spin\nsystem via magnetic dipole interaction (Zeeman inter-\naction) [26]. In this respect, a fundamental aspect is\nthe orientation of the THz polarization, which is con-\ntrolled using a set of two wire grid polarizers, one vari-\nably oriented at \u000645\u000eand a second one \fxed to +90\u000e\n(or -90\u000e) with respect to the original polarization direc-\ntion of ETHz. As depicted in Fig. 1 (b), the magnetic\n\feld component of the THz pulse HTHzis \fxed alongarXiv:1903.08395v2  [cond-mat.mes-hall]  25 Sep 20192\n(a)\n(e) (f)(b)\n+HTHz\n-HExtOUTWPLNO OAPM\nEM+SP P\nMPump\nProbe±45° ±90°\n0 2.5 5.0 7.5 10−100−50050100MOKE signal (a.u.)\n0 250 500 750−100−50050100\n+HExt\n-HExt\n0 2.5 5.0 7.5 10\nTime (ps)−100−50050MOKE signal (a.u.)\n0 250 500 750\nTime (ps)−100−50050\n+HTHz\n-HTHz\n(a)\n(c)\n(e)(d)\n(f)(b)\n+HTHz\n-HExtOUTWPLNO OAPM\nEM+SP P\nMPump\nProbe±45° ±90°\nFIG. 1. (Color online) (a) Schematic drawing of the exper-\nimental setup: BD - Balanced detection using two photodi-\nodes and a lock-in ampli\fer, WP - Wollaston prism, EM + S\n- Electromagnet with out-of-plane \feld and sample, OAPM\n- O\u000b-axis parabolic mirror, P - Wire grid polarizer, LNO -\nLithium niobate, and M - Mirror. (b) Sample geometry: The\nelectric \feld component of the THz pulse at the sample po-\nsition is oriented parallel to the y-axis direction, ETHzky,\nand the magnetic \feld component parallel to the x-axis direc-\ntion,HTHzkx. A static magnetic \feld HExtis applied along\nthez-axis direction. (c-f) Experimental MOKE data showing\nthe in\ruence of reversing the external magnetic \feld ( \u0006HExt,\nHTHz= const.) on the THz-induced demagnetization (c) and\non the FMR oscillations (d). The in\ruence of reversing the\nTHz magnetic \feld pulse ( \u0006HTHz,HExt.= const.) on the\nTHz-induced demagnetization and on the FMR oscillations is\nshown in (e) and (f), respectively. (The shaded green area is\na guide to the eye.)\nthex-axis direction and is therefore \ripped by 180\u000eby\nrotating the \frst polarizer. An amorphous CoFeB sample\n(Al2O3(1.8nm)/Co 40Fe40B20(5nm)/Al 2O3(10nm)/Si\nsubstrate) is placed either in the gap of a \u0006200 mT elec-\ntromagnet or on top of a 0.45 T permanent magnet. In\nboth cases, the orientation of the externally applied \feld\nHExtis along the z-axis direction, i.e. out of plane with\nrespect to the sample surface. However, a small compo-\nnent of this external bias \feld lying in the sample plane\nparallel to the y-axis direction has to be taken into ac-count, due to a systematic (but reproducible) small mis-\nalignment in positioning the sample. The THz pump\nbeam, with a spot size \u001f\u00191 mm (FWHM), and the 800\nnm probe beam, with a spot size \u001f\u0019200\u0016m (FWHM),\noverlap spatially on the sample surface in the center of\nthe electromagnet gap. Being close to normal incidence,\nthe MOKE signal is proportional to the out-of-plane com-\nponent M zof the magnetization, i.e. polar MOKE geom-\netry [27]. The probe beam re\rected from the sample sur-\nface is then analyzed using a Wollaston prism and two\nbalanced photo-diodes, following an all-optical detection\nscheme [4].\nResults and discussion. The experimental data demon-\nstrating THz-induced demagnetization and the magnetic\n\feld response of the spin dynamics is shown in Fig. 1\n(c-f) when \u00160HExt= 185 mT. For short timescales on\nthe order of the THz pump pulse, \u001c\u00181 ps, the polar\nMOKE is sensitive to the coherent response of the mag-\nnetization, i.e. its precession around the THz magnetic\n\feld, as shown in Fig. 1 (c)+(e). Within \u001c\u0018100 fs af-\nter time zero (t 0\u00185 ps) a sudden demagnetization step\nof the order of 0.1-0.2% of the total magnetization vec-\ntor is observed. The demagnetization step is followed by\na 'fast' relaxation process, \u001c\u00181 ps, and subsequently\nby a 'slow' recovery of the magnetization on a longer\ntimescale,\u001c\u0018100 ps. During and after magnetization\nrecovery, a relaxation precession (corresponding to the\nFMR) is superimposed, see Fig. 1 (d)+(f). The e\u000bect of\nreversing the externally applied magnetic \feld HExton\nthe MOKE measurements is illustrated in Fig. 1 (c)+(d).\nThis data shows that all the di\u000berent processes just iden-\nti\fed (demagnetization, coherent magnetization response\nin the range t 0<t.t0+ 2 ps, and FMR response)\nare indeed magnetic, as they all reverse their sign upon\nreversal of the sign of the bias magnetic \feld. Fig. 1\n(e)+(f) instead depict the e\u000bect of reversing the THz \feld\npolarity, while keeping the externally applied \feld con-\nstant. This data illustrate the symmetry of the magnetic\nresponse with respect to the THz magnetic \feld. The\ndemagnetization and FMR response, whose amplitude\nscales quadratically with the THz magnetic \feld HTHz,\nremain unchanged upon reversal of the terahertz mag-\nnetic \feld, while the coherent magnetization response,\nlinear in HTHz, reverses its sign.\nThe dependence of the THz-induced FMR response on\nthe magnitude of the magnetic bias \feld ( HExt) is shown\nin Fig. 2 (a). Oscillation amplitude and resonance fre-\nquency are summarized in Fig. 2 (b). The oscillation\namplitude is obtained from \ftting a damped sinusoidal\nfunction to the data shown in Fig. 2 (a), and the reso-\nnance frequency follows from subsequent Fourier trans-\nformation of that \ftting function. The relationship be-\ntween magnetic resonance \feld Hrand FMR frequency\nfFMR can be described by the phenomenological Kittel3\n0 200 400 600 800\nTime (ps)-1000100200300400500600700800MOKE signal (a.u.)2.0 2.2 2.4 2.6\nFrequency (GHz)0204060FMR amp. (a.u.)\n0102030405060\nMagnetic field (mT)02468Frequency (GHz)(a) (b)\n(c)\nFIG. 2. (Color online) Experimental data showing the mag-\nnetic \feld dependence of the FMR. (a) MOKE signal for\ndi\u000berent out-of-plane \felds \u00160HExt- 20, 50, 75, 100, 125,\n150, 175 and 185 mT in ascending order. The solid line is a\ndamped sine function \ftted to the experimental data. Both\n\ft and data curve are displaced on the y axis by a constant\no\u000bset. (b) Fourier transformation of the data shown in (a)\nvs. FMR amplitude. (c) FMR frequency as a function of the\nin-plane component of HExt. The solid line corresponds to\nKittel equation, Eq. (1).\nequation [28]\nfFMR =\r\u00160\n2\u0019p\nHr(Hr+Me\u000b) (1)\nwith gyromagnetic ratio \rand e\u000bective magnetization\nMe\u000b=MS\u0000H?\nK, where H?\nKis the perpendicular\nanisotropy \feld. From room-temperature magnetization\nmeasurements (see the Supplemental Material), a satu-\nration magnetization \u00160MS= 1.84 T and an anisotropy\n\feld\u00160H?\nK= 0.76 T are obtained. The value of H rrep-\nresents the small in-plane component of HExt(i.e. along\nthey-axis direction), which can be inferred from the mea-\nsurements in Fig. 2 (a) in order to get good agreement\nwith the experimental data, as shown in Fig. 2 (c).\nNumerical modelling. A detailed picture of the THz-\ninduced demagnetization and FMR response for an exter-\nnal magnetic \feld generated by a 0.45 T permanent mag-\nnet and a THz pulse of E THz= 18 MV/m ( \u00160HTHz\u001860\nmT) is presented in Fig. 3. Experimental data has been\nrecorded both on a short timescale after arrival of the\npump pulse showing coherent precession and demagneti-\nzation, Fig. 3 (a), and on long timescales showing mag-\nnetization recovery and FMR at a frequency of about 7.2\nGHz, Fig. 3 (b). By knowing the FMR frequency, the\nactual applied magnetic \feld can be deduced from the\ndata shown in Fig. 2 (a) and Eq. (1), as described in the\nprevious paragraph. Following this approach, we \fnd a\nvalue of 450 mT for the out-of-plane component and 56\nmT for the in-plane component of HExt, corresponding\nto the right-most point in Fig. 2 (c).\nThis complete set of good-quality data has been used\n(b)\n012345678\nTime (ps)53.853.954.054.154.254.354.4Mz/M 0(%)\n0 200 400 600 800\nTime (ps)(a) (b)FIG. 3. (Color online) Comparison of experimental data\n(open markers) and macrospin simulations (solid lines). (a)\nTHz-induced demagnetization for an external applied \feld of\n\u00160HExt\u0018450 mT and a THz pulse of E THz= 18 MV/m,\n(\u00160HTHz\u001860 mT). The blue markers/lines correspond to\n+HTHz, and the orange markers/lines to \u0000HTHz. (b) Exper-\nimental data and macrospin simulations under similar condi-\ntions as (a) showing FMR on a longer timescale.\nfor validating all the assumptions considered in our\nmacrospin simulations, whose results are shown as solid\nlines in Fig. 3. Indeed, magnetization dynamics of a uni-\nform ferromagnetic material can be modeled and under-\nstood from macrospin simulations where the description\nof the magnetic state is simpli\fed by a single-domain ap-\nproximation. The phenomenological LLG equation [22]\ncan be used to obtain a \frst approximate description of\nthe magnetization continuum precession and relaxation.\nSince the LLG equation preserves the length of the mag-\nnetization vector ( jMj= const.) [29], it does not account\nfor laser-induced demagnetization and it ignores relevant\nphysical phenomena such as scattering e\u000bects [11]. A\npossibility to account for optically- or thermally-induced\ndemagnetization is the use of the Landau-Lifshitz-Bloch\n(LLB) equation [30], which has been shown to describe\nultrafast demagnetization processes [31]. However, the\nparameters involved in the LLB equation depend on\ntemperature, and a description of their temporal evo-\nlution due to the interaction with ultrafast pulses is\nneeded. Such a description is often provided using a two-\ntemperature model [8, 31], which then has to be coupled\nto the LLB equation. However, such a two-temperature\nmodel relies on several phenomenological material pa-\nrameters, and little insight on the physics is gained by\nthis approach, in particular at longer time scales. Since\nhere we are interested in these time scales, we simplify\nthe problem by just assuming a phenomenological LLG\nequation with non-constant magnetization magnitude, a\nphysical quantity which can be readily measured.\nOur version of the LLG equation reads (see the Sup-4\nplemental Material)\ndM\ndt=\u0000\r0\u0002\nM\u0002He+\u000b\nMS(M\u0002(M\u0002He))\u0003\n\u0000c(M;He)M;\n(2)\nwith\r0=\r=(1 +\u000b2), where the gyromagnetic ratio is\nj\r=2\u0019j \u001928.025 GHz/T, and M= Mm, where M( t)\nis the magnetization amplitude and m(t) the unit vec-\ntor. The e\u000bective magnetic \feld Heincludes the exter-\nnal \feld HExt, THz \feld HTHzand demagnetization \feld\nHD. The Gilbert damping \u000b= 0:01 was independently\nmeasured with conventional FMR spectroscopy, as shown\nin the Supplemental Material. In Eq. (2), the \frst and\nsecond terms on the right-hand side describe coherent\nspin precession and the macroscopic spin relaxation, re-\nspectively, whereas the third term describes the fast de-\nmagnetization along the mdirection controlled by the\nfunctionc(M;He). A non-constant magnetization mag-\nnitude is also needed for reproducing the observed FMR\noscillation, as discussed in the Supplemental Material.\nHence, we propose a semi-empirical approach to iden-\ntifyc(M;He) and, consequently, the time evolution of\nthe magnetization vector length M( t) from experimental\ndemagnetization data. For this purpose, the change in\nM(t) as a function of time is calculated from the cumu-\nlative integral of the incident THz pulse [16]:\n\u0001M(t)/e\u0000t=\u001cRZt\n\u00001HTHz(\u0018)2d\u0018; (3)\nwhere the exponential term describes the recovery of the\nmagnetization on a timescale of \u0018100 ps for CoFeB. The\nshape of the incident THz pulse (in the time domain)\nis proportional to ETHz(t), which can be measured via\nelectro-optical sampling in GaP [32]. We notice that the\nTHz magnetic \feld HTHz(t) inside the material, to be\nused in Eq. (3), can be derived from the measured free-\nspace ETHz(t) after taking into account the whole sample\nstack, e.g. by means of the transfer matrix method [33].\nEventually, the function c(M;He) is obtained from Eq.\n(3) after re-scaling it with experimentally obtained de-\nmagnetization values (see Ref. [16] for more details).\nAfter determining c(M;He), all the terms in Eq. (2)\nare known, and the equation can be used to compute\nthe dynamics of the magnetization orientation in terms\nof the spherical angles \u0012(t); \u001e(t). Indeed, Eq. (2) is nu-\nmerically solved in spherical coordinates (M ;\u0012;\u001e) (see\nthe Supplemental Material for details) using a custom-\nmade Python solver based on the 4th-order Runge-Kutta\nmethod, which provides a simple implementation and\ngood accuracy of the numerical solution [29].\nIt is now interesting to explore the expected response of\nthe magnetization to THz \felds with strength larger than\nthe ones considered in this work, but nowadays accessi-\nble with table-top sources. For simplicity and generality,\nonly the free-space value of the THz peak electric \feld is\nreported in the below discussion. In case of THz \felds\n0 250 500 750 1000\nTime (ps)-0.4-0.200.2∆M y,z(%)\n0 5 10 15 20\nFrequency (GHz)0246Amplitude (a.u.)×10−9\nMz\nMy\n0 250 500 750 1000\nTime (ps)-40-20020∆M y,z(%)\n0 5 10 15 20\nFrequency (GHz)0246Amplitude (a.u.)×10−7\nMz\nMy(a)\n(c)(b)\n(d)FIG. 4. (Color online) Comparison of macrospin simulation\ndata for moderate (20 MV/m) and high (200 MV/m) THz-\npeak-\feld stimulus. The external magnetic \feld is set to\n\u00160HExt= 450 mT, i.e. replicating the experimental condi-\ntions from Fig. 3. Panel (a) shows the relative change of the\nMy(blue line) and M z(light blue line) magnetization com-\nponent for a E THzpeak \feld of 20 MV/m and (b) shows the\ncorresponding Fourier spectrum. (c) and (d) show the simu-\nlation data and corresponding Fourier spectrum for an E THz\npeak \feld of 200 MV/m, respectively.\nsmaller than E THz\u0018100 MV=m, the demagnetization\nfollows the square of the amplitude of the THz \feld, see\nRef. [16]. Lacking detailed experimental data, it is rea-\nsonable to assume that THz-induced demagnetization for\nhigher THz peak \felds (E THz>100 MV=m) can be de-\nscribed by the positive section of an error function, allow-\ning for a quadratic behavior for small demagnetization\nand a saturation for large demagnetization approaching\n100% [17]. From our experimental data, we derive a func-\ntional description of the demagnetization as a function of\nthe THz \feld such as Demag = f(E) = erf( A\u0001E2), with\nTHz peak \feld E and \ftting parameter A\u00196:0\u000110\u00006\nm2V\u00002. (See the Supplemental Material for further de-\ntails.)\nWith this assumption, the macrospin simulation re-\nsults for THz \felds E THz= 20 MV=m and E THz= 200\nMV=m are presented in Fig. 4 (a-b) and Fig. 4 (c-d),\nrespectively. For E THz = 200 MV =m, a clear nonlin-\near response of the magnetization to the THz \feld is\nfound, illustrated by the second harmonic oscillation in\nthe Mycomponent of the magnetization. The simulated\nTHz-induced demagnetization for E THz= 200 MV=m is\non the order of \u0001M z\u001820%. In Fig. 4 (b)+(d), the\nFourier spectrum of the FMR oscillation for the M yand\nMzcomponents of the magnetization at THz pump peak\n\felds of 20 MV/m and 200 MV/m are depicted. The\nFourier data of the 200 MV/m simulation shown in Fig. 4\n(c) clearly shows a second harmonic peak at \u001814 GHz,\npresent for M ybut not for M z. A similar behavior was5\nobserved recently by performing FMR spectroscopy of\nthin \flms irradiated with femtosecond optical pulses in-\nducing either ultrafast demagnetization [34], by exciting\nacoustic waves [35], and by two-dimensional THz mag-\nnetic resonance spectroscopy of antiferromagnets [36]. In\nour case, the high-harmonic generation process is solely\ndriven by the large amplitude of the terahertz magnetic\n\feld that is completely o\u000b-resonant with the uniform pre-\ncession mode. This would allow for exploring purely mag-\nnetic dynamics in regimes that are not accessible with\nconventional FMR spectroscopic techniques, where high-\namplitude dynamics are prevented by the occurrence of\nso-called Suhl's instabilities, i.e. non-uniform excitations\ndegenerate in energy with the uniform mode. Such non-\nresonant, high THz magnetic \felds are within the capa-\nbilities of recently developed table-top THz sources [37],\nand can also be generated in the near-\feld using meta-\nmaterial structures as described by Refs. [38{40].\nIn summary, we investigated magnetization dynam-\nics induced by moderate THz electromagnetic \felds in\namorphous CoFeB, in particular the ferromagnetic reso-\nnance response as a function of applied bias and THz\nmagnetic \felds. We demonstrate that semi-empirical\nmacrospin simulations, i.e. solving the Landau-Lifshitz-\nGilbert equation with a non-constant magnitude of the\nmagnetization vector to incorporate THz-induced de-\nmagnetization e\u000bect, are able to describe all the details\nof the experimental results to a good accuracy. Exist-\ning models of terahertz spin dynamics and spin pumping\nwould need to be extended to include the evidence pre-\nsented here [41, 42]. Starting from simulations describ-\ning experimental data for THz-induced demagnetization,\nwe extrapolate that THz \felds one order of magnitude\nlarger drive the magnetization into a nonlinear regime.\nIndeed, macrospin simulations with THz \felds on the or-\nder E THz\u0018200 MV=m (\u00160HTHz\u0018670 mT) predict a sig-\nni\fcant demagnetization of \u0001M z\u001820%, and a marked\nnonlinear behavior, apparent from second harmonic gen-\neration of the uniform precessional mode. We anticipate\nthat our results will stimulate further theoretical and ex-\nperimental investigations of nonlinear spin dynamics in\nthe ultrafast regime.\nM.H. gratefully acknowledges support from the\nSwedish Research Council grant E0635001, and the\nMarie Sk lodowska Curie Actions, Cofund, Project INCA\n600398s. The work of M.d'A. was carried out within the\nProgram for the Support of Individual Research 2017\nby University of Naples Parthenope. M.C.H. is sup-\nported by the U.S. Department of Energy, O\u000ece of Sci-\nence, O\u000ece of Basic Energy Sciences, under Award No.\n2015-SLAC-100238-Funding. M.P. and S.B. acknowledge\nsupport from the European Research Council, Starting\nGrant 715452 \\MAGNETIC-SPEED-LIMIT\".\u0003stefano.bonetti@fysik.su.se\n[1] M. Faraday, Philosophical Transactions of the Royal So-\nciety of London 136, 1 (1846).\n[2] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y.\nBigot, Phys. Rev. Lett. 76, 4250 (1996).\n[3] B. Koopmans, M. Van Kampen, J. T. 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Lett. 118, 257202 (2017)."    },    {        "title": "1001.4576v1.Effect_of_spin_conserving_scattering_on_Gilbert_damping_in_ferromagnetic_semiconductors.pdf",        "content": "arXiv:1001.4576v1  [cond-mat.mtrl-sci]  26 Jan 2010Effect of spin-conserving scattering on Gilbert damping in f erromagnetic\nsemiconductors\nK. Shen,1G. Tatara,2and M. W. Wu1,∗\n1Hefei National Laboratory for Physical Sciences at Microsc ale and Department of Physics,\nUniversity of Science and Technology of China, Hefei, Anhui , 230026, China\n2Department of Physics, Tokyo Metropolitan University, Hac hioji, Tokyo 192-0397, Japan\n(Dated: November 12, 2018)\nThe Gilbert damping in ferromagnetic semiconductors is the oretically investigated based on the\ns-dmodel. In contrast to the situation in metals, all the spin-c onserving scattering in ferromagnetic\nsemiconductors supplies an additional spin relaxation cha nnel due to the momentum dependent\neffective magnetic field of the spin-orbit coupling, thereby modifies the Gilbert damping. In the\npresence of a pure spin current, we predict a new contributio n due to the interplay of the anisotropic\nspin-orbit coupling and a pure spin current.\nPACS numbers: 72.25.Dc, 75.60.Ch, 72.25.Rb, 71.10.-w\nThe ferromagnetic systems have attracted much at-\ntention both for the abundant fundamental physics and\npromising applications in the past decade.1,2The study\non the collective magnetization dynamics in such sys-\ntems has been an active field with the aim to control\nthe magnetization. In the literature, the magnetization\ndynamics is usually described by the phenomenological\nLandau-Lifshitz-Gilbert (LLG) equation,3\n˙n=γHeff×n+αn×˙n, (1)\nwithndenoting the direction of the magnetization. The\nfirst and second terms on the right hand side of the equa-\ntion represent the precession and relaxation of the mag-\nnetization under the effective magnetic field Heff, respec-\ntively. The relaxation term is conventionally named as\nthe Gilbert damping term with the damping coefficient\nα. The time scale of the magnetization relaxation then\ncan be estimated by 1 /(αγHeff),4which is an important\nparameter for dynamic manipulations. The coefficient α\nis essential in determining the efficiency of the current-\ninduced magnetizationswiching, andexperimentaldeter-\nmination of αhas been carried out intensively in metals5\nand magnetic semiconductors.6\nTo date, many efforts have been made to clarify the\nmicroscopic origin of the Gilbert damping.7–12Kohno\net al.8employed the standard diagrammatic pertur-\nbation approach to calculate the spin torque in the\nsmall-amplitude magnetization dynamics and obtained a\nGilberttorquewiththedampingcoefficientinverselypro-\nportional to the electron spin lifetime. They showed that\nthe electron-non-magnetic impurity scattering, a spin-\nconserving process, does not affect the Gilbert damping.\nLater, they extended the theory into the finite-amplitude\ndynamics by introducing an SU(2) gauge field2and ob-\ntained a Gilbert torque identical to that in the case of\nsmall-amplitude dynamics.9In those calculations, the\nelectron-phonon and electron-electron scatterings were\ndiscarded. One may infer that both of them should be\nirrelevant to the Gilbert damping in ferromagnetic met-\nals, since they are independent of the electron spin re-laxation somewhat like the electron-non-magnetic impu-\nrity scattering. However, the situation is quite different\nin ferromagnetic semiconductors, where the spin-orbit\ncoupling (SOC) due to the bulk inversion asymmetry13\nand/or the structure inversion asymmetry14presents a\nmomentum-dependent effective magnetic field (inhomo-\ngeneous broadening15). As a result, any spin-conserving\nscattering, including the electron-electron Coulomb scat-\ntering,canresultinaspinrelaxationchanneltoaffectthe\nGilbert damping. In this case, many-body effects on the\nGilbert damping due to the electron-electron Coulomb\nscatteringshould be expected. Sinova et al.16studied the\nGilbert damping in GaMnAs ferromagnetic semiconduc-\ntors by including the SOC to the energy band structure.\nIn that work, the dynamics of the carrier spin coherence\nwas missed.17The issue of the present work is to study\nthe Gilbert damping in a coherent frame.\nIn this Report, we apply the gauge field approach to\ninvestigate the Gilbert damping in ferromagnetic semi-\nconductors. In our frame, all the relevant scattering pro-\ncesses, even the electron-electron scattering which gives\nrise to many-body effects, can be included. The goal\nof this work is to illustrate the role of the SOC and\nspin-conserving scattering on Gilbert damping. We show\nthat the spin-conserving scattering can affect the Gilbert\ndamping due to the contribution on spin relaxation pro-\ncess. We also discuss the case with a pure spin current,\nfrom which we predict a new Gilbert torque due to the\ninterplay of the SOC and the spin current.\nOur calculation is based on the s-dmodel with itiner-\nantsand localized delectrons. The collectivemagnetiza-\ntion arisingfrom the delectronsis denoted by M=Msn.\nThe exchange interaction between itinerant and local-\nized electrons can be written as Hsd=M/integraltext\ndr(n·σ),\nwhere the Pauli matrices σare spin operators of the\nitinerant electrons and Mis the coupling constant. In\norder to treat the magnetization dynamics with an ar-\nbitrary amplitude,9we define the temporal spinor oper-\nators of the itinerant electrons a(t) = (a↑(t),a↓(t))Tin\nthe rotation coordinate system with ↑(↓) labeling the2\nspin orientation parallel (antiparallel) to n. With a uni-\ntary transformation matrix U(t), one can connect the\noperators a↑(↓)with those defined in the lattice coor-\ndinate system c↑(↓)bya(t) =U(t)c. Then, an SU(2)\ngauge field Aµ(t) =−iU(t)†(∂µU(t)) =Aµ(t)·σshould\nbe introduced into the rotation framework to guarantee\nthe invariance of the total Lagrangian.9In the slow and\nsmooth precession limit, the gauge field can be treated\nperturbatively.9Besides, one needs a time-dependent\n3×3 orthogonal rotation matrix R(t), which obeys\nU†σU=Rσ, to transform any vector between the two\ncoordinate systems. More details can be found in Ref.\n2. In the following, we restrict our derivation to a spa-\ntially homogeneous system, to obtain the Gilbert damp-\ning torque.\nUp to the first order, the interaction Hamiltonian due\nto the gauge field is HA=/summationtext\nkA0·a†\nkσakand the spin-\norbit couping reads\nHso=1\n2/summationdisplay\nkhk·c†σc=1\n2/summationdisplay\nk˜hk·a†\nkσak,(2)\nwith˜h=Rh. Here, we take the Planck constant /planckover2pi1= 1.\nWe start from the fully microscopic kinetic spin Bloch\nequations of the itinerant electrons derived from the non-\nequilibrium Green’s function approach,15,18\n∂tρk=∂tρk/vextendsingle/vextendsingle\ncoh+∂tρk/vextendsingle/vextendsinglec\nscat+∂tρk/vextendsingle/vextendsinglef\nscat,(3)whereρkrepresenttheitinerantelectrondensitymatrices\ndefined in the rotation coordinate system. The coherent\nterm can be written as\n∂tρk/vextendsingle/vextendsingle\ncoh=−i[A·σ,ρk]−i[1\n2˜hk·σ+ˆΣHF,ρk].(4)\nHere [,] is the commutator and A(t) =A0(t)+Mˆzwith\nA0andMˆzrepresenting the gauge field and effective\nmagnetic filed due to s-dexchange interaction, respec-\ntively.ˆΣHFis the Coulomb Hartree-Fock term of the\nelectron-electron interaction. ∂tρk/vextendsingle/vextendsinglec\nscatand∂tρk/vextendsingle/vextendsinglef\nscatin\nEq.(3) include all the relevant spin-conserving and spin-\nflip scattering processes, respectively.\nThe spin-flip term ∂tρk/vextendsingle/vextendsinglef\nscatresults in the damping ef-\nfect was studied in Ref. 9. Let us confirm this by\nconsidering the case of the magnetic disorder Vm\nimp=\nus/summationtext\nj˜Sj·a†σaδ(r−Rj). The spin-flip part then reads\n∂tρk/vextendsingle/vextendsinglef\nscat=∂tρk/vextendsingle/vextendsinglef(0)\nscat+∂tρk/vextendsingle/vextendsinglef(1)\nscat, (5)\nwith∂tρk/vextendsingle/vextendsinglef(i)\nscatstanding for the i-th order term with re-\nspect to the gauge field, i.e.,\n∂tρk/vextendsingle/vextendsinglef(0)\nscat=−πnsu2\nsS2\nimp\n3/summationdisplay\nk1η1η2σαρ>\nk1(t)Tη1σαTη2ρ<\nk(t)δ(ǫk1η1−ǫkη2)−(>↔<)+H.c., (6)\n∂tρk/vextendsingle/vextendsinglef(1)\nscat=i2πnsu2\nsS2\nimp\n3εαβγAγ\n0(t)/summationdisplay\nk1η1η2σαρ>\nk1(t)Tη1σβTη2ρ<\nk(t)d\ndǫk1η1δ(ǫk1η1−ǫkη2)−(>↔<)+H.c.,(7)\nwhereTη(i,j) =δηiδηjfor the spin band η. Here\nρ>\nk= 1−ρk,ρ<\nk=ρk. (>↔<) is obtained by inter-\nchanging >and<from the first term in each equation.\nεijkis the Levi-Civita permutation symbol. The gauge\nfield term, ∂tρk/vextendsingle/vextendsinglef(1)\nscat, results from the spin correlation of\na single magnetic impurity at different times.9It induces\na spin polarization proportional to ˆz×A⊥\n0(t) which gives\na Gilbert torque. The damping coefficient is inversely\nproportional to the spin relaxation time τsdetemined by\nthe spin-flip scattering ∂tρk/vextendsingle/vextendsinglef(0)\nscat. The spin-flip scattering\nterm in Eq.(3) thus reproduces the result of Ref. 9.\nWe now demonstrate that the Gilbert damping torque\narises also from the spin-conserving scattering. For\nthe discussion of the spin-conserving term, it is suffi-\ncient to approximate the spin-flip term as ∂tρk/vextendsingle/vextendsinglef\nscat=\n−(ρk−ρe\nk)/τs, withρe\nkrepresenting the instantaneous\nequilibrium distribution (i.e., ρe\nkisρkwithout the gaugefield and Ps\nk). Equation(3) then reads\n∂tρk=−i[A·σ,ρk]−i[1\n2˜hk·σ,ρk]\n+∂tρk/vextendsingle/vextendsinglec\nscat−(ρk−ρe\nk)/τs+Ps\nk.(8)\nHere, we add an additional term, Ps\nk, to describe the\nsource of a pure spin current due to the magnetization\ndynamic pumping4or electrically injection19,20in order\nto discuss the system with a pure spin current. We ne-\nglect the Coulomb Hartree-Fock effective magnetic field\nsinceitisapproximatelyparalleltothe s-dexchangefield,\nbut with a smaller magnitude.\nBy averaging density matrices over the momentum\ndirection, one obtains the isotropic component ρi,k=/integraltextdΩk\n4πρk. The anisotropic component is then expressed\nasρa,k=ρk−ρi,k. It is obvious that this anisotropic\ncomponent does not give any spin torque in the absence\nof the SOC, since/summationtext\nkTr(σρa,k) = 0. Below, it is shown3\nthat this component leads to the damping when coupled\nto the spin-orbit interaction.\nBy denoting the isotropic component of the equi-\nlibrium part ( ρe\nk) asρe\ni,kand representing the non-\nequilibrium isotropic part by δρi,k=ρi,k−ρe\ni,k, we\nwrite the kinetic spin Bloch equations of the non-\nequilibrium isotropic density matrices δρi,kand those of\nthe anisotropic components ρa,kas\n∂tρi,k=−δρi,k\nτs−i[A·σ,δρi,k]−i[1\n2˜hk·σ,ρa,k]\n−i[A0·σ,ρe\ni,k], (9)\n∂tρa,k=∂tρa,k/vextendsingle/vextendsinglec\nscat−i[A·σ,ρa,k]−i[1\n2˜hk·σ,δρi,k]\n−i[1\n2˜hk·σ,ρa,k]+i[1\n2˜hk·σ,ρa,k]+Ps\nk,(10)\nrespectively. The overline in these equations presents a\nangular average over the momentum space.\nWe further define ρ(0)\na,kas the anisotropic density in the\nabsence of the gauge field, A0. As easily seen, it vanishes\nwhenPs\nk= 0. The anisotropic component involving the\ngauge field is denoted by ρ(1)\na,k=ρa,k−ρ(0)\na,k. Equation\n(10) is expressed in terms of these components as\n∂tρ(0)\na,k=−i[M·σ,ρ(0)\na,k]+∂tρ(0)\na,k/vextendsingle/vextendsinglec\nscat+Ps\nk\n−i[1\n2˜hk·σ,ρ(0)\na,k]+i[1\n2˜hk·σ,ρ(0)\na,k],(11)\n∂tρ(1)\na,k=∂tρ(1)\na,k/vextendsingle/vextendsinglec\nscat−i[A·σ,ρ(1)\na,k]−i[1\n2˜hk·σ,δρi,k]\n−i[1\n2˜hk·σ,ρ(1)\na,k]−i[A0·σ,ρ(0)\na,k].(12)\nWithin the elastic scattering approximation, the\nelectron-phonon scattering as well as the electron-non-\nmagnetic impurity scattering can be simply written as/summationtext\nl,mρ(1)\na,k,lmYlm/τl, where the density matrices are ex-\npanded by the spherical harmonics functions Ylm, i.e.,\nρ(1)\na,k,lm=/integraltextdΩk\n4πρ(1)\na,kYlm.τlis the effective momentum\nrelaxation time. The exact calculation of the Coulomb\nscattering is more complicated. Nevertheless, one can\nstill express this term in the form of ρ(1)\na,k/Fk(ρ), where\nFkis a function of the density matrices21and reflects\nmany-body effects. For simplification, we just introduce\na uniform momentum relaxation time τ∗\nlin the following\ncalculation. Expanding Eq.(12) by the spherical har-\nmonics functions, one obtains\n∂tρ(1)\na,k,lm=−i[A·σ,ρ(1)\na,k,lm]−i[1\n2˜hk,lm·σ,δρi,k]\n−i[A0·σ,ρ(0)\na,k,lm]−i[1\n2˜hk·σ,ρ(1)\na,k]lm−ρ(1)\na,k,lm\nτ∗\nl,(13)\nwhere the expanssion coefficient of any term fkis ex-\npressed as fk,lm=/integraltextdΩk\n4πfkYlm. In the strong scattering\nregime, i.e.,1\nτ∗\nl≫Mand1\nτ∗\nl≫ |hk|, the first and fourth\nterms are much smaller than the last term, hence can be\ndiscarded from the right side. By taking the fact that\nthe time derivative is a higher order term into account,\none also neglects ∂tρ(1)\na,k,lm. The solution of Eq.(13) canbe written as\nρ(1)\na,k,lm=−iτ∗\nl{[1\n2˜hk,lm·σ,δρi,k]+[A0·σ,ρ(0)\na,k,lm]}.(14)\nSubstituting it into Eq.(14) and rewriting the equation\nin the leading order, one obtains\n∂tρi,k=−i[A·σ,δρi,k]−i\n2[˜hk·σ,ρ(0)\na,k]−i[A0·σ,ρe\ni,k]\n−/summationdisplay\nlmτ∗\nl\n4/bracketleftBig\n˜hk,lm·σ,[˜hk,lm·σ,δρi,k]/bracketrightBig\n−δρi,k\nτs.(15)\nThe third term on the right hand side of the equation is\nproportional to the second order term of the SOC, which\ngives the spin dephasing channel due to the D’yakonov-\nPerel’ (DP) mechanism.22This term can be expressed\nbyτ−1\nDPδρi,kwithτ−1\nDPstanding for the spin dephas-\ning rate tensor, which can be written as ( τ−1\nDP)i,j=/summationtext\nl,m/an}b∇acketle{tτ∗\nl((hk,lm)2δij−hi\nk,lmhj\nk,lm)/an}b∇acket∇i}htby performingthe en-\nsemble averaging over the electron distribution. In the\nfollowing, we treat τDPas a scalar for simplification and\nthe total spin lifetime is hence given by\nτr= 1/(τ−1\nDP+τ−1\ns). (16)\nSimilar to the previous procedure, we discard ∂tρi,kin\nEq. (15) and obtain\ni[A·σ,δρi,k]+δρi,k/τr=−i[1\n2˜hk·σ,ρ(0)\na,k]−i[A0·σ,ρe\ni,k].\n(17)\nBy taking δ˜si=1\n2/summationtext\nkTr(σδρi,k),˜se\ni=1\n2/summationtext\nkTr(σρe\ni,k)\nand˜s(0)\na,k=1\n2Tr(σρ(0)\na,k), one can write the solution as\nδ˜si=˜v+2τrAטv+4τ2\nr(˜v·A)A\n1+4|A|2τ2r−˜se\ni,(18)\nwhere˜v=˜se\ni+τr/summationtext\nk˜hkטs(0)\na,k.˜se\niisjust the equilibrium\nspin density , which is parallel to the magnetization, i.e.,\n˜se\ni= ˜se\niˆz. Now, we pick up the transverse component in\ntheformof ˆz×A⊥\n0,δ˜s⊥, sinceonlythiscomponentresults\nin a Gilbert torque of the magnetization as mentioned\nabove. We come to\nδ˜s⊥= 2˜vz(A⊥\n0׈z)τ2\nexτr/(τ2\nr+τ2\nex),(19)\nwithτex= 1/(2M). By transforming it back to the lat-\ntice coordinate system with R(ˆz×A⊥\n0) =1\n2∂tn,9one\nobtains\nδs⊥=−˜vz(∂tn)τ2\nexτr/(τ2\nr+τ2\nex),(20)\nThis nonequilibrium spin polarization results in a spin\ntorqueperformed on the magnetizationaccordingto T=\n−2Mn×δs, i.e.,\nT= ˜vz(n×∂tn)τexτr/(τ2\nr+τ2\nex).(21)\nCompared with Eq.(1), the modification of the Gilbert\ndamping coefficient from this torque is\nα= ˜vzτexτr/(Msτ2\nr+Msτ2\nex), (22)4\nWe fisrt discuss the case without the source term of\nthe spin current. In this case, the anisotropic component\nρ(0)\na,kvanishesand ˜ vz= ˜se\ni. We seethat the Gilbert damp-\ning then arises from 1 /τr[Eq. (16)], i.e., from both the\nspin-flip scattering and the DP mechanism.22Our main\nmessage is that this DP contribution is affected by the\nspin-conservingscattering processes such as the electron-\nelectron interaction and phonons. The temperature de-\npendenceoftheGilbertdampingandthecurrent-induced\nmagnetization switching can thus be discussed quantita-\ntively by evaluating τr. We note that our result reduces\nto the results of previous works7,9when only the spin-flip\nscattering is considered.\nWe should point out that our formalism applies also\nto metals, by considering the case1\nτ∗\nl≪M. In this case,\nthe last term of Eq.(13) can be neglected and the effect\nof the spin-conserving scattering through τ∗\nlbecomes ir-\nrelevant.\nWhen the pure spin current is included, we found ad-\nditional contribution due to the interplay of the spin cur-\nrent and the SOC, since we have\n˜vz= ˜se\ni+/bracketleftBig\nR/parenleftBig\nτr/summationdisplay\nkhk×s(0)\na,k/parenrightBig/bracketrightBig\nz= ˜se\ni+ ˜ssc\nz,(23)\nwith the spin current associated term ˜ ssc\nzdefined ac-\ncordingly. The origin of ˜ ssc\nzcan be understood as fol-\nlows. The anisotropic spin polarization s(0)\na,karising from\nthe pure spin current rotates around the SOC effective\nmagnetic field hk, which is also anisotropic. This pre-cession finally results in an isotropic spin polarization\nssc=τr/summationtext\nkhk×s(0)\na,kin the presence of spin relaxation.\nThis term contributes to the spin polarization of the itin-\nerant electrons along the direction of the magnetization,\ni.e., ˜ssc\nz, thereby modifies the Gilbert damping term by\n˜ssc\nz/˜se\ni.\nThe additional Gilbert damping due to the spin cur-\nrent found here is different from the enhancement of the\ndamping in the spin pumping systems, where the exis-\ntence of the interface is essential.4In other words, what\ncontributes there is the divergence of the spin current, as\nis understood from the continuity equation for the spin,\nindicating that the spin damping is equal to ∇·js+ ˙s(s\nis the total spin density). In contrast, the damping found\nin the present paper arises even when the spin current is\nuniform if the spin-orbit interaction is there.\nIn summary, we have shown that the spin-conserving\nscatterings in ferromagnetic semiconductors, such as\nthe electron-electron, electron-phonon and electron-non-\nmagnetic impurity scatterings, contribute to the Gilbert\ndamping in the presence of the SOC because of the in-\nhomogeneous broadening effect. We also predict that a\nGilbert torque arises from a pure spin current when cou-\npled to the spin-orbit interaction.\nThis work wassupported by the Natural Science Foun-\ndation of China under Grant No. 10725417, the Na-\ntional Basic Research Program of China under Grant\nNo. 2006CB922005,the KnowledgeInnovation Projectof\nChinese Academy of Sciences, Kakenhi (1948027)MEXT\nJapan and the Hitachi Sci. Tech. Foundation.\n∗Author to whom correspondence should be addressed;\nElectronic address: mwwu@ustc.edu.cn.\n1Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n2G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468, 213\n(2008).\n3T. L. Gilbert, Phys. Rev. 100, 1243 (1955); L. D. Landau,\nE.M.Lifshitz, andL.P.Pitaevski, Statistical Physics, Part\n2, 3rd ed. (Pergamon, Oxford, 1980).\n4Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n5M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando,\nA. Sakuma, and T. Miyazaki, Jpn. J. Appl. Phys. 45, 3889\n(2006).\n6J.-P. Adam, N. Vernier, J. Ferr´ e, A. Thiaville, V. Jeudy,\nA. Lemaˆ ıtre, L. Thevenard, and G. Faini, Phys. Rev. B\n80, 193204 (2009).\n7P. Pi´ echon and A. Thiaville, Phys. Rev. B 75, 174414\n(2007).\n8H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn.\n75, 113706 (2006).\n9H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710\n(2007).\n10Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W.\nBauer, Phys. Rev. B 74, 144405 (2006).\n11J. Kuneˇ s and V. Kambersk´ y, Phys. Rev. B 65, 212411(2002).\n12D. Steiauf and M. F¨ ahnle, Phys. Rev. B 72, 064450 (2005).\n13G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n14Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039\n(1984); JETP Lett. 39, 78 (1984).\n15M. W. Wu and C. Z. Ning, Eur. Phys. J. B 18, 373 (2000);\nM. W. Wu, J. Phys. Soc. Jpn. 70, 2195 (2001).\n16J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J. K. Furdyna,\nW. A. Atkinson, and A. H. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n17Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004)\n18M. W. Wu and H. Metiu, Phys. Rev. B 61, 2945 (2000);\nFor review: M. W. Wu, J. H. Jiang, and M. Q. Weng,\narXiv:1001.0606; and references therein.\n19F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. A. Basel-\nmans, and B. J. van Wees, Nature 416, 713 (2002).\n20X. H. Lou, C. Adelmann, S. A. Crooker, E. S. Garlid, J. J.\nZhang, K. S. M. Reddy, S. D. Flexner, C. J. Palmstrom,\nand P. A. Crowell, Nature Phys. 3, 197 (2007).\n21M. M. Glazov and E. L. Ivchenko, JETP 99, 1279 (2004).\n22M. I. D’yakonov and V. I. Perel’, Zh. ´Eksp. Teor. Fiz. 60,\n1954(1971) [Sov.Phys.JETP 33, 1053(1971)]; Fiz.Tverd.\nTela (Leningrad) 13, 3581 (1971) [Sov. Phys. Solid State\n13, 3023 (1972)]."    },    {        "title": "1502.05687v3.Characterization_of_spin_relaxation_anisotropy_in_Co_using_spin_pumping.pdf",        "content": "arXiv:1502.05687v3  [cond-mat.mtrl-sci]  23 Nov 2016Characterization of spin relaxation anisotropy in Co using spin pumping\nY. Li, W. Cao and W. E. Bailey\nMaterials Science & Engineering, Department of Applied Phy sics and Applied Mathematics,\nColumbia University, New York NY 10027, USA\n(Dated: March 27, 2021)\nFerromagnets are believed to exhibit strongly anisotropic spin relaxation, with relaxation lengths\nfor spin longitudinal to magnetization significantly longe r than those for spin transverse to mag-\nnetization. Here we characterize the anisotropy of spin rel axation in Co using the spin pumping\ncontribution to Gilbert damping in noncollinearly magneti zed Py 1−xCux/Cu/Co trilayer structures.\nThe static magnetization angle between Py 1−xCuxand Co, adjusted under field bias perpendicular\nto film planes, controls the projections of longitudinal and transverse spin current pumped from\nPy1−xCuxinto Co. We find nearly isotropic absorption of pure spin curr ent in Co using this tech-\nnique; fits to a diffusive transport model yield the longitudi nal spin relaxation length <2 nm in Co.\nThe longitudinal spin relaxation lengths found are an order of magnitude smaller than those deter-\nmined by current-perpendicular-to-planes giant magnetor esistance measurements, but comparable\nwith transverse spin relaxation lengths in Co determined by spin pumping.\nA key question for spin electronics concerns the relax-\nation mechanisms for spin current injected into a vari-\nety of materials. Spin relaxation in ferromagnets (Fs),\ncentral for spin momentum transfer, is special because of\nthe anisotropyaxis presented by the spontaneousmagne-\ntizationM[1–10]. Longitudinal spin relaxation[1], with\nspin polarization σparallel (antiparallel) to M, causes\nspin accumulation to decrease exponentially with dis-\ntance over a scale greater than the electronic mean free\npath[2]. Transverse spin relaxation, with σorthogonal\ntoM, is governed by the dephasing process of spin-up\nand spin-down eigenmodes due to their different Fermi\nwavevectors, leading to oscillation and decay of spin ac-\ncumulation on a scale shorter than the electronic mean\nfree path[3, 10].\nThe characteristic length scales for the two different\nspin relaxation processes in ferromagnets, λL\nsrfor longi-\ntudinal and λT\nsrfor transverse spin relaxation, have been\nevaluated largely using two separate experimental tech-\nniques: magnetotransport[11] for λL\nsrand ferromagnetic\nresonance (FMR)[4, 5] for λT\nsr. These two measurements\ncharacterizecharge-accompaniedandchargelessspincur-\nrent, respectively[1, 4]. Estimates of λL\nsrcome from the F\nlayer thickness dependence of current-perpendicular-to-\nplanes giant magnetoresistance (CPP-GMR)[11–14]; ex-\ntracted values of λL\nsrrange from 5 nm for Ni 79Fe21up to\n40 nm for Co at room temperature. FMR measurements\nof spin pumping, for collinearly magnetized F 1/Cu/F 2\nstructures, show much shorter penetration depths ( λC)\nto fully absorb transverse spin current[15, 16]. Co has\nthe most anisotropic spin relaxation according to these\nseparate measurements, with λL\nsr/λT\nsr∼16 taking λT\nsr∼\n2λC= 2.4 nm[15, 17].\nIn this manuscript, we demonstrate that the longitu-\ndinal spin relaxation length, in addition to the trans-\nverse spin relaxation length[15], can also be character-\nized using a spin pumping measurement, enabling a mea-\nsurement of the anisotropy of spin relaxation in a givenferromagnetic layer. We present FMR measurements\nof the spin pumping contribution to Gilbert damping\nin noncollinearly magnetized Py 1−xCux/Cu/Co multi-\nlayers (Py=Ni 79Fe21). Using Py 1−xCuxalloys, which\nhave adjustably smaller saturation magnetization Ms\nthan Co, we can change the magnetization alignment of\nPy1−xCuxand Co from collinear for in-plane FMR to\nnear-orthogonalfor perpendicular FMR. As the angle θM\nbetween Py 1−xCuxand Co magnetization tends towards\nπ/2, one component of injected spin from Py 1−xCux\ntends towards the longitudinal direction (Fig. 1), allow-\ning us to probe anisotropy in spin relaxation through the\nlinewidth of the Py 1−xCuxlayer[18, 19]. We find, sur-\nprisingly, that spin relaxation, as measured through the\nspin pumping contribution to Gilbert damping, is mostly\nisotropic. In our Co films we estimate λL\nsr<2 nm for all\ndifferent Py 1−xCux/Cu/Co samples, which is compara-\nble to its transverse counterpart( ∼2.4 nm) but inconsis-\ntent with the much longer ( ∼40 nm) lengths reported\nfrom room-temperature CPP-GMR[11, 14].\nThree types of thin-film heterostructures were pre-\nHBnc \nIs\nωσσ\nF1=Py 1-x Cu x\nF2=Coσ\nσTωσL θMθM\nF1F2\nN\n// m1m1m1m2m2\nm1\nFIG. 1. Left:Noncollinear magnetization alignment of the\nF1/N/F2trilayer at the FMR condition for F 1.Right:m1\nis driven into precession, pumping spin current into m2, with\nspin components both longitudinal ( σL) and transverse ( σT)\nto them2magnetization.\npared by UHV sputtering and characterized by FMR.\nPseudo-spin-valve-type Py 1−xCux(t)/Cu(5 nm)/Co(52\nnm) trilayers were used to characterize the anisotropy\nof spin-current absorption in Co. Their response was\ncompared with two types of Py 1−xCux(t) alloy control\nsamples. Bilayers of Py 1−xCux(5 nm*)/Cu(5 nm) and\ntrilayers of Py 1−xCux(5 nm*)/Cu(5 nm)/Pt(3 nm) were\nused to characterizethe backgrounddamping of the alloy\nand the spin mixing conductance of the alloy/Cu inter-\nface, respectively. Co(5 nm)/Cu(5 nm)/Pt(3 nm) is also\ndeposited. For the alloy Cu contents x= 0 to 0.4 were\nprepared in each case, using confocal sputtering from Py\nand Cu targets[20]; thicker (10 nm*) alloy layers were\nused for x= 0.4. All layers were deposited on Si/SiO 2\nsubstrates, seeded by Ta(5 nm)/Cu(5 nm) and capped\nby Ta(2 nm). See Ref. [21] for details on preparation.\nRoom temperature, variable frequency (3-26 GHz),\nswept-field FMR measurements were used to character-\nize the samples, with instrumentation as described in\n[22]. In order to characterize FMR relaxation of the\nPy1−xCuxlayer under noncollinear magnetization align-\nment with Co, two types of measurements were car-\nried out. First, we compare the frequency-dependent\nlinewidths of Py 1−xCuxand Py 1−xCux/Cu/Co sam-\nples in both in-plane (parallel-condition, pc) and per-\npendicular (normal-condition, nc) FMR[23], for a series\nof four measurements at a given alloy content x; see\nFigs. 2c, 3, and 4. Here we expect the Co magneti-zation of trilayer samples to vary from fully perpendic-\nular to the film plane at high biasing field HB(high\nω/2π) to nearly parallel to the film plane at low HB\n(lowω/2π), while the Py 1−xCuxmagnetization is al-\nways perpendicular to the film plane. Second, we com-\npare the polar angle-dependent linewidths of Py 0.8Cu0.2\nand Py 0.8Cu0.2/Cu/Co samples at a fixed frequency of\nω/2π= 10 GHz; see Fig. 5. Here we expect the mis-\nalignment angle θMto change from zero to maximum as\nwe rotate the biasing field from in-plane (pc) to out-of-\nplane (nc).\nTheoretical models for the spin pumping contribution\nto damping under noncollinear magnetization alignment\nofsymmetricF 1/N/F1structuresweredevelopedinRefs.\n[18, 19]. We have extended these models to consider\nasymmetric F 1/N/F2structures where F 1=Py1−xCux\nand F 2=Co in our samples. In the spin valve structure\nthe spin-pumping damping enhancement ∆ αspof F1is\ncaused by the dissipation of spin current in F 2. If F1and\nF2are misaligned by an angle θM, whereθM=m1·m2\n(Fig. 1), during small-angle precession of F 1, the polar-\nization of spin current pumped into F 2will oscillate from\nfully transverse to maximally longitudinal. The instan-\ntaneous spin-pumping damping will then oscillate from\nαsp(0◦) = ∆α0טg↑↓\n2/(˜g↑↓\n1+˜g↑↓\n2), as given in the standard\ncollinear case[24], to a minimum value given by[25]:\n∆αsp(θM) = ∆α0g∗\n2(Asin2θM−BsinθMcosθM)+ ˜g↑↓\n2(Ccos2θM−BsinθMcosθM)\nAC−B2(1)\nHere ˜g↑↓\niandg∗\ni(i= 1, 2) are the effective trans-\nverse and longitudinal spin conductances, respectively;\n∆α0=γ¯h˜g↑↓\n1/(4πMstF) is the damping enhancement\nwith effective spin mixing conductance of ˜ g↑↓\n1[22];\nin Eq. 1 A(θM) =g∗\n1sin2θM+ ˜g↑↓\n1cos2θM+ ˜g↑↓\n2,\nB(θM) = (˜ g↑↓\n1−g∗\n1)sinθMcosθMandC(θM) =\ng∗\n1cos2θM+ ˜g↑↓\n1sin2θM+g∗\n2. We take the arithmetic\nmean of the two extreme cases as the effective damping\nenhancement, as found to be valid in Ref. [19]. See the\nSupplemental Materials for details.\nTo maximize the spin pumping anisotropy at finite\nθM, we use Co (5 nm) for F 2, where the dimension is\nchosen to be significantly thicker than the transverse\nspin penetration depth, λC= 1.2 nm[15], and thinner\nthan the reported longitudinal relaxation length λL\nsr,\n∼38 nm[11], resulting in a large expected asymmetry\nin spin relaxation. In the analysis of relaxation in non-\ncollinearly magnetized structures, we take spin mixing\nconductances ˜ gi↑↓as parameter inputs, determined from\nthe measurements on the Pt control structures, and\ntake the longitudinal spin relaxation length λL\nsras a fit\nparameter.Fig. 2 summarizes the results of fixed-angle nc-and\npc-FMR measurements for the three sample series. In\nFig. 2(a) we plot resonance fields µ0Hresas a function of\nfrequency for single layers and trilayers in nc-FMR. The\ngood agreement in the µ0Hresof Py1−xCuxmeasured in\nsingle layers and trilayers demonstrates that Py 1−xCux\nproperties are reproducible in deposition. In Fig. 2(b)\nthe effective magnetizations µ0Meff, extracted from fits\nto the linear Kittel equation ω/γ=µ0(Hres−Meff), are\nplotted as a function of x. The data show Slater-Pauling\ndilution of magnetic moment in the Py 1−xCuxlayer\nwith increasing Cu content x[20].\nInFig. 2(c)weplotfull-widthhalf-maximumlinewidth\nµ0∆H1/2as a function of ω/2πatx=0.2. Gilbert-type\ndamping, µ0∆H1/2=µ0∆H0+ 2αω/γ, with negligible\ninhomogeneous broadening µ0∆H0, is observed for both\npc- and nc-FMR in the single layer and for pc-FMR in\nthe trilayer. The linewidths agree closely for pc- and\nnc-FMR in the single layer, showing a negligible role\nfor two-magnon scattering in the linewidth[26]. In the\ntrilayer, nc and pc linewidths agree well for frequencies\nabove 10 GHz. These observations hold for samples3\n(nc)Single layer \n(nc)Trilayer \n(pc)Single layer \n(pc)Trilayer \n= 0.20\n0.1\n0.2\n0.3\n0.4Co satura/g415on: 1.4 T \nSingle layer \nTrilayer \nTrilayer(Co) Single layer \nTrilayer 1/2 (a) (b)\n(c) (d)\nFIG. 2. (a) Perpendicular (nc-FMR) resonance field µ0Hres\nfor Py 1−xCuxsingle layers and Py 1−xCux/Cu/Co trilayers,\nx= 0 - 0.4, as a function of frequency ω/2π. (b) Effective\nmagnetization µ0Meffextracted from (a) as a function of x.\n(c) Resonance linewidths µ0∆H1/2of the Py 0.8Cu0.2single\nlayer and trilayer as a function of frequency ω/2π. The spin\npumping enhancement is clearly visible in the increased slo pe\n(α) of the trilayer data; the low-frequency deviation is dis-\ncussed in Fig. 3. (d) Effective spin mixing conductances g↑↓\neff\nof Py1−xCux/Cu/Co, Py 1−xCux/Cu/Pt and Co/Cu/Pt.\nwith all Cu content 0 ≥x≥0.4; the deviations at\nlow frequency are discussed in Fig. 3. The effective\nspin mixing conductances g↑↓\neffof trilayer samples are\nextracted from ∆ αsp=γ¯hg↑↓\neff/(4πMstM), shown above\nwhere ∆ αspis the difference in αbetween trilayers and\nsingle layers. In Fig. 2(d) we show the extracted g↑↓\neff\nfor Py 1−xCux/Cu/Co and Py 1−xCux/Cu/Pt structures\nas a function of x. We also plot g↑↓\neffof Co/Cu/Pt\nfor reference. The lower level of g↑↓\neff∼7 nm−2for\nPy1−xCux/Cu/Co, compared with ∼15 nm−2measured\nin Ref. [15], is likely to be from a more resistive Cu layer,\nwhich adds an additional resistance of (2 e2/h)tCu/σCu\nto the inverse of total spin mixing conductance where\nσCuis the Cu conductivity. Using these g↑↓\neffvalues,\nwe extract the effective spin mixing conductance of\nPy1−xCux/Cu and Co/Cu interfaces, shown in the\nSupplemental Materials[25]. These parameters will\nbe used to determine the longitudinal spin relaxation\nlengths from the spin pumping data in Figs. 4 and 5.\nIn the measurements presented in Fig. 2(c), the nc-\nFMR linewidths are measured at applied fields below the\nsaturation field for Co, µ0Meff= 1.4 T. The saturation\nfield corresponds to a nc-FMR resonance frequency for\nPy0.8Cu0.2of 25 GHz, as shown in Fig. 2(a). With the\nresultant noncollinear magnetization alignment in the\ntrilayer, we expect to see spin-pumping damping ∆ αsp\nfor Py 0.8Cu0.2reduced in nc-FMR compared with the\nvalues in pc-FMR. Instead, we find that the linewidths\nof the trilayer measured in pc- and nc-FMR agree closely\nwhenω/2π >10 GHz. Furthermore, in nc-FMR there isan additional broadening from 2-10 GHz in the trilayers\nwhich is not predicted by the model.\nIn order to determine whether the low-frequency 1/2 \u001e\u001e\n\u000e\u000e\u001e \u000e = 0\nPy/Cu/Co Py/Cu/MgO/Co Py/Cu/MgO pc nc \nFIG. 3. pc- and nc-FMR linewidths for single (Py) and\ntrilayer (Py/Cu/Co) structures, introducing MgO interlay ers\nto suppress spin pumping. Dashed lines are linear fits to pc-\nFMR linewidths. Solid curves assume (magnetostatic) inter -\nlayer coupling of 10 mT acting on Py and reproduce the low-\nfrequencyupturnin linewidth, seen to be present equally wi th\nand without MgO. Inset:enhancements of nc-FMR linewidth\nover pc-FMR linewidth for the three samples.\nbroadening is related to spin pumping, we have also\nmeasured pc- and nc-FMR linewidths of Py(5 nm)/Cu(5\nnm)/MgO(2 nm) and Py(5 nm)/Cu(3 nm)/MgO(2\nnm)/Co(5 nm) structures, deposited with the same\nseed and capping layers. MgO interlayers are known to\nsuppress spin pumping[27]. Introducing MgO between\nPy and Co, we show in Fig. 3 that the pc linewidths of\nPy in trilayer Py/Cu/Co (blue crosses) are restored to\nthose of single-layerPy/Cu/MgO(overlappinggreen and\nred crosses), demonstrating suppression of spin pumping\nbetween Py and Co. However, we see a very similar\nupturn in low-frequency ( <10 GHz) Py linewidth in\nnc-FMR (red circles), similar to that shown in Fig.\n2(c). We attribute this low-frequency behavior to an\ninterlayer coupling which cants the magnetization of\nPy a few degrees off the film normal when Co is not\nfully saturated along the film normal (i.e. HB< Meff).\nThe solid curves in Fig. 3 assume a coupling field of\n10 mT on Py, parallel to the local Co magnetization,\nwhich reproduce the linewidth broadening of nc-FMR.\nThe peak-like features around 3 GHz show the maximal\nGilbert damping enhancement when the Py magnetiza-\ntion is canted, as demonstrated in Fig. 5 inset.\nFig. 4 shows the central result of the paper. We\ncompare the spin-pumping linewidth enhancements,\nµ0(∆Htri\n1/2−∆Hsingle\n1/2), between pc- and nc-FMR (crosses\nand circles) in Fig. 4(a-d). Here ∆ Hsingle\n1/2and ∆Htri\n1/2\nare the linewidths of Py 1−xCuxin Py1−xCux/Cu single4\nlayers and Py 1−xCux/Cu/Co trilayers, respectively. The\nspin pumping linewidths are quite linear as a function of\nfrequency for the pc-FMR data, as expected. However,\nabove 10 GHz (shaded regions), they are also quite\nlinear in nc-FMR, which is not expected. Collinear\nand noncollinear spin pumping linewidths agree closely.\nThis behavior is in contrast to the predicted behavior\nusingλL\nsr= 38 nm for Co, measured by CPP-GMR[11],\nand calculated in dashed curves according to the theory\nin the Supplemental Materials. From the evident\nagreement between pc- and nc-linewidths above 10 GHz,\nfor all Cu content x, we find no evidence for anisotropy\nin spin relaxation in our Co films. Best fits to the data\nyield longitudinal spin relaxation lengths λL\nsr<2 nm\nin each of the four cases, approximately equal to the\npreviously measured transverse length λT\nsr= 2.4 nm[15].\nOur model has assumed single-domain (macrospin)\n1/2 1/2 1/2 1/2 \n1/2 1/2 1/2 1/2 pc-FMR\npc-FMR(linear fit)nc-FMR\nnc-FMR(theo, λsr L=38 nm)\n(a) x=0.1 (b) x=0.2\n(c) x=0.3 (d) x=0.4(a) x=0.1 (b) x=0.2\n(c) x=0.3 (d) x=0.4\nFIG. 4. Spin pumping contribution to linewidth in pc- and\nnc-FMR. (a-d) Linewidth enhancement of Py 1−xCuxbetween\nsingle layers and trilayers in pc- and nc-FMR, x= 0.1-0.4.\nSolid lines are linear fits to the pc data (crosses); dashed\ncurves are predicted from Eq. (1) using λL\nsr=38 nm. The\nshadows at ω/2π≤10 GHz denote where the low-frequency\nlinewidth broadenings are significant.\nbehavior in both Co and Py 1−xCuxlayers. For\nPy1−xCuxunder field bias well in excess of Ms, the\nmagnetization is well saturated, but for the Co layer,\nwith higher Ms, nonuniform magnetization is possible.\nFor greater control over the Co domain state, we have\nalso carried out angle-dependent, fixed-frequency FMR\nmeasurements on Py 0.8Cu0.2and Py 0.8Cu0.2/Cu/Co.\nHere the Co layer can be saturated more easily be-\ncause the biasing field is canted away from the normal\ncondition. The frequency is set to 10 GHz, where the\nlow-frequency linewidth broadening of Py 0.8Cu0.2is\ninsignificant (Figs. 3 and 4). As the field angle θH\ngoes from 90◦to 0◦(pc to nc), the angle between the\nmagnetizations of Py 0.8Cu0.2and Co changes from zeroto maximum noncollinearity ( ∼50◦) and ∆ αspwould\nbe expected to decrease significantly where the spin\nrelaxation length in Co is markedly anisotropic.\nFig. 5 Insetshows the angular dependence of1/2 1/2 λsr L=38 nmΔH tri/ΔH single\n=1.50 ± 0.02\nθH\n1/2 Single layer \nTrilayer x = 0.2 \nFIG. 5. Angle dependent linewidth ratio ∆ Htri\n1/2/∆Hsingle\n1/2.\nThe shadowed region shows the average with errorbar\n(1.50±0.02).Inset:Angular dependence of µ0∆H1/2for\nPy0.8Cu0.2and Py 0.8Cu0.2/Cu/Co at ω/2π= 10 GHz. Solid\nlines are macrospin calculations.\n∆Hsingle\n1/2(red) and ∆ Htri\n1/2(blue) for Py 0.8Cu0.2.\nThe data can be reproduced through a macrospin\nmodel[28, 29] as shown in the solid curves, using similar\nmagnetizations and isotropic dampings extracted from\nFig. 2(a) and (c) ( µ0Meff= 0.53 T,α1= 0.0114\nfor the single layer, µ0Meff= 0.55 T,α3= 0.0168\nfor the trilayer). The inhomogeneous broadenings are\nnegligible, shown in Fig. 2(c). For small enough θH,\nthe resonance field of the Co starts to fall below the\nexpected macrospin value, as shown in the Supplemental\nMaterials, Section C[25]. We take the angle at which\nthis behavior appears (at θH∼18◦) to be the limit above\nwhich we have the greatest confidence in single-domain\nordering of Co.\nIn the main panel of Fig. 5 we replot the trilayer\nand single-layer linewidths for Py 0.8Cu0.2, shown in\nthe inset, as the ratio ∆ Htri\n1/2/∆Hsingle\n1/2. Because the\ninhomogeneous linewidths are negligible for the struc-\ntures (<0.5 mT), the linewidth ratio for isotropic\nspin pumping would be approximated well through the\nratio of the Gilbert damping for the two configurations,\n∆Htri\n1/2/∆Hsingle\n1/2= 1 + ∆ αsp/α1. We find that the\nlinewidth ratio is in fact constant within experimental\nerror, shown by the shaded region in Fig. 5. The\nblue dashed curve shows the expected behavior for\nanisotropic spin relaxation, assuming λL\nsr= 38 nm, with\na marked decrease in the linewidth ratio for low angles\nθH. A best fit to these data returns λL\nsr<1.1 nm. If\nwe restrict our attention to field angles θH≥18◦, above\nwhich we have confidence in macrospin behavior of the5\nCo layer, the best fit is not changed greatly, with λL\nsr≤4\nnm, within experimental error of the transverse length\nλT\nsr.\nExtrinsic effects, i.e. issues of sample quality, may\nplay some role in the results. First, longitudinal spin\nrelaxation lengths λL\nsr, if equated with the spin diffusion\nlengthλsd, are inversely proportional to (defect-related)\nresistivity[30]. However, four-point probe measure-\nments of the resistivity of our Co (5 nm) films show\n25µΩ·cm, comparable with the 18 µΩ·cm reported\nin the room-temperature CPP-GMR experiment[11],\nand therefore comparably long spin diffusion lengths\nshould be expected. Second, we see that the spin mixing\nconductances g↑↓\neffof Py1−xCux/Cu/Co measured here\nare lower than those measured in Ref. [15], on structures\ndeposited elsewhere. The most plausible source of the\nreduction is a more resistive Cu layer, which adds an\nadditional resistive term[31, 32] (2 e2/h)tCuρCutog↑↓\neff.\nHere however the bulk Cu properties should have little\ninfluence over either spin relaxation length and should\nnot affect the anisotropy of spin relaxation strongly.\nOur estimate of λL\nsrin Co is consistent with a general\nobservation that spin relaxation as measured in spin\npumping/FMR is shorter-ranged than it is as measured\nin magnetotransport. In Pd and Pt, the characteristic\nrelaxation lengths for dynamically pumped spin current\nare measured as 1-5 nm[15, 33–35], whereas in GMR\nthey are closer to 10-20 nm[36, 37]. We suggest therefore\nthat the quantities revealed by the two types of measure-\nments may differ in some respect. For example, robust\nspin-pumping effects have been found in ferrimagnetic\ninsulators such as yttrium iron garnet (YIG). These\neffects clearly have little to do with electronic transport\nin YIG, and their characteristic lengths would refer\nto scattering mechanisms distinct from those involved\nin CPP-GMR. A second possibility, alluded to in the\nreview in Ref. [14], is that the room-temperature spin\ndiffusion length of 38 nm in [11] is an overestimate due\nto technical issues of the CPP-GMR measurement in\nCo multilayers; the majority of such measurements in\nvarious ferromagnets show <10 nm[14]. Our results, in\nthis scenario, may alternately imply that the short spin\ndiffusion length observed in Py is not far away from that\nof Co.\nIn summary, we have experimentally demonstrated\nthat the spin relaxation in Co, as measured by non-\ncollinear spin pumping, is largely isotropic. The\nestimated longitudinal spin relaxation length, <2 nm,\nis an order of magnitude smaller than measured by\nmagnetotransport but comparable to the transverse spin\nrelaxation length. 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Ebels and W. E.\nBailey,Appl. Phys. Lett. 98, 052508 (2011).\n[33] W. Zhang, V. Vlaminck, J. E. Pearson, R. Divan, S. D.\nBader and A. Hoffmann, Appl. Phys. Lett. 103, 242414\n(2013).\n[34] V. Vlaminck, J. E. Pearson, S. D. Bader and A. Hoff-\nmann,Phys. Rev. B 88, 064414 (2013).\n[35] M. Caminale, A. Ghosh, S. Auffret, U. Ebels, K. Ollefs,F. Wilhelm, A. Rogalev and W. E. Bailey, Phys. Rev. B\n94, 014414 (2016).\n[36] H. Kurt, R. Loloee, K. Eid, W. P. Pratt Jr. and J. Bass,\nAppl. Phys. Lett. 81, 4787 (2002).\n[37] M. Morota, Y. Niimi, K. Ohnishi, D. H. Wei, T. Tanaka,\nH. Kontani, T. Kimura and Y. Otani, Phys. Rev. B 83,\n174405 (2011).arXiv:1502.05687v3  [cond-mat.mtrl-sci]  23 Nov 2016Supplemental Material to ”Characterization of spin relaxa tion anisotropy in Co using\nspin pumping”\nY. Li, W. Cao and W. E. Bailey1\nDept. of Applied Physics & Applied Mathematics, Columbia Un iversity,\nNew York NY 10027, USA\n(Dated: 27 March 2021)\n1A. Calculation of spin-pumping damping for noncollinearly magnetized,\nasymmetric trilayers\nN\nm1m2\nF1F2θ\nIs1 pump \nθ\nxy\nIs1 pump θ\nxy\nm2 (0,1)\nm1 (-sinθ,cosθ)\nμsN (μ x\nsN ,μ y\nsN )Is1 pump \n(cosθ,sinθ)\nμsN (0, 0, μz\nsN )m1 (-sinθ,cosθ) m2 (0,1)Case 1 Case 2(a)\n(b) (c)Hrf (ωt)\nFIG. 1. (a) Magnetization configuration of the asymmetric F 1/N/F2trilayer. (b) An instant in\nwhich the spin polarization of Ipump\ns1is orthogonal to both m1andm2.µsNis also orthogonal to\nm1andm2. (c) An instant in which the spin polarization of Ipump\ns1is in the same plane of m1and\nm2.µsNis also in the same plane of m1andm2\nConsider an asymmetric ferromagnet / noble metal / ferromagnet (F1/N/F2) spin-valve\ntrilayer structure, shown in Fig. 1(a). The time-averaged magnet ization of F 1is pictured\nalong the film-normal, although it can take any angle with respect to t he film-normal. We\nassume that F 1undergoes small-angle precession. F 2is noncollinearly magnetized with\nrespect to F 1, whereθis the angle of noncollinearity or misalignment; θ= 0 for parallel\nmagnetizations m1=m2, wheremi,i= 1 or 2, is the unit vector of the magnetization Mi\nof Fi. The magnetizaton of F 2is taken to be stationary. The spin current flows from the N\nspacer to each of the F layers F 1, F2are1–4:\nIN→F1\ns=g∗\n1\n4πm1(µsN·m1) +˜g↑↓\n1\n4πm1×µsN×m1 (1)\n2IN→F2\ns=g∗\n2\n4πm2(µsN·m2) +˜g↑↓\n2\n4πm2×µsN×m2 (2)\nwhereµsNis the spin accumulation vector in the N layer, g∗\niand ˜g↑↓\niare the effective\nlongitudinal spin conductance and transverse spin mixing conducta nce for F i/N interface,\nrespectively. Here the spin current vector denotes the direction of spin polarization, the\ndirection of current flow always being normal to interfaces. The co nservation of spin angular\nmomentum, assuming spin-current conservation (negligible dissipat ion) N, gives:\nIN→F1\ns+IN→F2\ns=Ipump\ns1 (3)\nwhereIpump\ns1is the pumped spin current from F 1into N2,5:\nIpump\ns1=¯h\n4π˜g↑↓\n1m1×˙ m1 (4)\nSubstituting Eq. (1), (2) and (4) into the continuity expression (3 ), we obtain a vector\nequation in terms of the vector spin accumulation µsN. To calculate the spin pumping\ndamping enhancement, we seek solutions for µsNin order to find the spin current flow into\nm2, which is absorbed by m2.\nThe vector Ipump\ns1, proportional to m1×˙ m1, rotates in the plane with normal given by m1.\nAssuming a finite misalignment angle θbetween m1andm2,Ipump\ns1will oscillate between\nfully orthogonal to m2(Fig. 1b) and canted away from orthogonality by θ(Fig. 1c). We\nconsider these two extreme cases during the precession of m1. In case 1 (Fig. 1b), Ipump\ns1is\nperpendicular to both m1andm2. In case 2 (Fig. 1c), Ipump\ns1is in the same plane as m1\nandm2and has the largest longitudinal component along m2.\nIncase 1,IN→F1s,IN→F2sandµsNare all parallel to Ipump\ns1. In Eqs. (1) and (2) the first\nterms become zero and only the second terms remain. The solution o f Eqs. (1)-(3) has a\nscalar form along the direction ˆ zorthogonal to both F 1and F 2:\nIN→F1{F2}\ns,z =/parenleftBigg\n˜g↑↓\n1{2}\n˜g↑↓\n1+ ˜g↑↓\n2/parenrightBigg\nIpump\ns1, µz\nsN=/parenleftbigg4π\n˜g↑↓\n1+ ˜g↑↓\n2/parenrightbigg\nIpump\ns1 (5)\nIt has been shown previously that2,5the dissipation of spin angular momentum due to a\ntransverse spin mixing conductance g↑↓leads to an additional Gilbert damping term ∆ α=\nγ¯hg↑↓/4πMsd. With only IN→F2sdissipated, the spin-pumping damping enhancement can be\nexpressed as:\n∆αsp=Iback\ns2·∆α0\nIpump\ns1= ∆α0·˜g↑↓\n2\n˜g↑↓\n1+ ˜g↑↓\n2(6)\n3with ∆α0=γ¯h˜g↑↓\n1/4πMs1d. Eq. (6) is identical to the collinear spin pumping case with an\neffective spin mixing conductance (˜ g↑↓\neff)−1= (˜g↑↓\n1)−1+(˜g↑↓\n2)−1.\nIncase 2,µsNhas only a component coplanar with m1andm2(µx\nsNandµy\nsN). In Eq.\n(1) and (2) both terms need to be considered. The ˆ xand ˆycomponents of Eq. (3) can be\nwritten as:\n4πIpump\ns1/parenleftbiggcosθ\nsinθ/parenrightbigg\n=g∗\n1(−µx\nsNsinθ+µy\nsNcosθ)/parenleftbigg−sinθ\ncosθ/parenrightbigg\n+ ˜g↑↓\n1(µx\nsNcosθ+µy\nsNsinθ)/parenleftbiggcosθ\nsinθ/parenrightbigg\n+g∗\n2µy\nsN/parenleftbigg0\n1/parenrightbigg\n+ ˜g↑↓\n2µx\nsN/parenleftbigg1\n0/parenrightbigg\n(7)\nThe solution of Eq. (7) can be expressed as:\nµx\nsN=4πIpump\ns1(Ccosθ−Bsinθ)\nAC−B2(8a)\nµy\nsN=4πIpump\ns1(Asinθ−Bcosθ)\nAC−B2(8b)\nwhere\nA(θ) =g∗\n1sin2θ+ ˜g↑↓\n1cos2θ+ ˜g↑↓\n2 (9a)\nB(θ) = (˜g↑↓\n1−g∗\n1)sinθcosθ (9b)\nC(θ) =g∗\n1cos2θ+ ˜g↑↓\n1sin2θ+g∗\n2 (9c)\nThe spin torque is equal to the conponent of IN→F2stransverse to m1, or the component\nwhich is parallel to Ipump\ns1. Thus the spin-pumping damping enhancement can be written in\nterms of the defined misalignment-dependent quantities A(θ),B(θ),C(θ) as:\n∆αsp(θ) =IN→F2s·Ipump\ns1\nIpump\ns1·∆α0\nIpump\ns1\n=∆α0·g∗\n2(Asin2θ−Bsinθcosθ)+ ˜g↑↓\n2(Ccos2θ−Bsinθcosθ)\nAC−B2(10)\nIt is easy to verify that at θ= 0◦Eq. (10) recovers Eq. (6), same as the collinear spin\npumping.\nHaving treated the two special spin current orientations, we need to take the average of\nall the orientation possibilities. We refer to the calculation by Taniguc hi, et al.3, that in a\nsymmetric spin valve (F 1=F2) the small-precession limit of averaged spin-pumping damping\nenhancement is equal to the arithmetic mean of damping enhanceme nt with out-of-plane\n4Ipump\ns1(case 1) and in-plane Ipump\ns1(case 2). The Eq. 13 in Ref.3can be simplified, at small\nprecession angle, as:\n∆αsp= ∆α0/bracketleftbigg\n1−(ν/2)sin2θ\n1−ν2cos2θ/bracketrightbigg\n(11)\nwhich is the average of ∆ α0and ∆α0[1−νsin2θ/(1−ν2cos2θ)] (Eq. 5 in Ref.3). We apply\nit to the asymmetric spin valve condition: all the theoretical curves in the main text are\ncalculated from the mean of Eq. (6) and Eq. (10).\nThe theoretical curves in Fig. 4 and 5 of the main text are calculated using the routine,\nassuming λL\nsr= 38 nm for Co. The new estimation of λL\nsr(<2 nm) in the manuscript takes\nthe best value that fits the damping calculation to the experimental data.\nB. Values of g∗and˜g↑↓\nIn this section we calculate the value of the two effective spin conduc tances. The trans-\nverse spin mixing conductance ˜ g↑↓(Sharvin correction includedtserkovnyakRMP2005) of\neachinterfacecanbecalculatedfromtheeffectivespinmixingcondu ctanceofPy 1−xCux/Cu/Co\nstructures and the comparison measurements of Py 1−xCux/Cu/Pt and Co/Cu/Pt (Table I).\nFor Py 1−xCux/Cu/Co, the total spin mixing conductance can be expressed as:\n1\ng↑↓\nPy1−xCux/Cu/Co=1\n˜g↑↓\nPy1−xCux/Cu+1\n˜g↑↓\nCo/Cu(12)\nFor F/Cu/Pt (F=Py 1−xCuxor Co), the effective spin mixing conductance can be formulated\nas:\n1\ng↑↓\nF/Cu/Pt=1\n˜g↑↓\nF/Cu+1\n˜g↑↓\nCu/Pt(13)\nIn the experiment the thicknesses of Pt are kept the same and we c an treat ˜ g↑↓\nCu/Ptas a\nconstant. Solving Eq. (12) and (13) we obtain:\n1\n˜g↑↓\nPy1−xCux/Cu=1\ng↑↓\nPy1−xCux/Cu/Co+1\ng↑↓\nPy1−xCux/Cu/Pt−1\ng↑↓\nCo/Cu/Pt(14a)\n1\n˜g↑↓\nCo/Cu=1\ng↑↓\nPy1−xCux/Cu/Co−1\ng↑↓\nPy1−xCux/Cu/Pt+1\ng↑↓\nCo/Cu/Pt(14b)\nIn Table II we list the calculated values of ˜ g↑↓\nPy1−xCux/Cuand ˜g↑↓\nCo/Cu. Forx= 0.1 and 0.3\nwe take the linear interpolated values to evaluate g↑↓\nPy1−xCux/Cu/Pt. In addition, we also show\n5the values compensating the Sharvin correction, with 1 /g↑↓\ni= 1/˜g↑↓\ni+ 1/2gSh\nCu,gSh\nCu= 15\nnm−2.\nCompared with previous measurements10,11, we find smaller values of g↑↓\nPy1−xCux/Cu/Coand\ng↑↓\nPy1−xCux/Cuforx= 0. However we argue that the spin mixing conductances of Co/Cu/ Pt\nin Table I and Co/Cu interfaces in Table II are reasonable, which ensu res a good Co/Cu\ninterface crucial for the study of spin relaxation anisotropy. It is also possible that a resistive\nCu spacer contributes an additional resistance, (2 e2/h)tCuρCu2, to the right side of Eq. (13).\nTo reduce the spin mixing conductance of Py/Cu/Co from 15.0 nm−2in Ref.10to 7.6 nm−2\nin Table I, one needs to take ρCu= 16.8µΩ·cm. However we point out that this resistive\nscattering will contribute to both transverse and longitudinal spin conductance by the same\namount, and the anisotropy of spin relaxation should not be affecte d. In practice, we take\nthe effective interfacial spin mixing conductance into the model for the estimation of λL\nsrand\nand use the values of λT\nsrfrom Ref.10.\n(Unit: nm−2)x= 0x= 0.1x= 0.2x= 0.3x= 0.4\ng↑↓\nPy1−xCux/Cu/Co7.6 5.6 7.3 6.8 6.8\ng↑↓\nPy1−xCux/Cu/Pt6.0 - 5.0 - 5.0\ng↑↓\nCo/Cu/Pt7.9\nTABLE I. Experimental values of (effective) spin mixing condu ctance of Py1−xCux/Cu/Co,\nPy1−xCux/Cu/Pt and Co /Cu/Pt samples, extracted from spin-pumping linewidth enhance ments.\nThe effective longitudinal spin conductance g∗can be expressed as1:\n1\ng∗=g↑↑+g↓↓\n2g↑↑g↓↓+1\ngsdtanh(tF/λLsr)(15)\nIn the first term, g↑↑{↓↓}\niis the interfacial spin-up {spin-down }conductance. g↑↑{↓↓}can be\ncalculated by 1 /g↑↑{↓↓}= (e2/h)AR↑{↓}\nF/NwhereAR↑{↓}\nF/Nis the electron interface resistance.\nWe use the experimental value from GMR measurements: 2 AR∗= (AR↑+AR↓)/2 = 1.04\nfΩ·m2for Co/Cu6and 1.0 fΩ ·m2for Py/Cu7. We can calculate that 2 g↑↑g↓↓/(g↑↑+g↓↓) = 26\n6(Unit: nm−2)x= 0x= 0.1x= 0.2x= 0.3x= 0.4\n˜g↑↓\nPy1−xCux/Cu11.7 8.6 9.5 9.1 9.1\n˜g↑↓\nCo/Cu21.9 16.2 31.5 27.2 27.2\ng↑↓\nPy1−xCux/Cu8.4 6.7 7.2 7.0 7.0\ng↑↓\nCo/Cu12.7 10.5 15.4 14.3 14.3\nTABLE II. “˜ g↑↓\ni”: Sharvin-corrected spin mixing conductance of Py1−xCux/Cu and Co /Cu in-\nterfaces, calculated from Eq. (14). “ g↑↓\ni”: interfacial spin mixing conductance compensating the\nSharvin conductance of Cu layer. i= Py1−xCux/Cu or Co/Cu.\nnm−2for both interfaces.\nIn the second term, gsdhas been expressed in Ref.1as:\ngsd=h\ne2λL\nsr2σ↑σ↓\nσ↑+σ↓(16)\nwhereσ↑,↓are the spin-up/down electron conductivity in F, his the Planck constant and\neis the electronic charge. Here we simply take σ↑=σ↓=σ/2 (σis the total electrical\nconductivity), which has also been done in Eq. (74) of Ref.2. Following this treatment, the\nterm 2σ↑σ↓/(σ↑+σ↓) is replaced by σ/2. Taking ρCo= 25µΩ·cm andρPy= 30µΩ·cm from\nour four-point probe measurements and λL\nsr= 38 nm for Co8and 4.3 nm for Py9from the\nliteratures, we calculate gsdtanh(tF/λL\nsr) to be 0.18 nm−2for Co and 8.3 nm−2for Py when\nthe F thickness is 5 nm; the large disagreement comes from the expe cted difference in λL\nsr.\nAs a result, g∗= 0.18 nm−2for Co and 6.2 nm−2for Py are obtained from Eq. (15) and\nused to produce the theoretical curves in the manuscript.\nIn the experiment, we do not find the anisotropic response of spin p umping predicted\nabove. According to our model, the lack of anisotropic response ca n be explained best\nthrough a difference in the longitudinal spin conductance g∗for Co/Cu, as this is the most\ndominant terminEq. (10)andsensitive to λL\nsr. This isbecauseintheexperiments wechoose\nthe thickness of Co to be much less than 38 nm in order to examine the spin relaxation\nanisotropy.\nFrom Py to Py 1−xCux, we should expect that both g↑↑{↓↓}andσ↑{↓}will increase due to a\nbetter conducting ability of Cu than Py. λL\nsrmay also vary. However we emphasize that in\nEq. (10), the anisotropy is dominated by g∗\n2and ˜g↑↓\n2and not sensitive to g∗\n1. For example in\n7the angular dependence of linewidth ration for x= 0.2 (Fig. 5 of the main text), increasing\ng∗\nPy0.8Cu0.2/Cuby a factor of two will change the single-domain estimation of λL\nsrfrom 1.8±2.7\nto 2.1±2.8, still much smaller than the GMR measurements. Thus for simplicity we keep\nusing the value of g∗of Py for Py 1−xCuxlayers.\nC. Single-domain limit determination\n= 18 deg = 9 deg\nPy 0.8Cu 0.2 , 10 GHz Co, 14.8 GHz Co, 15.0 GHz \nPy 0.8Cu 0.2 , 10 GHz Py 0.8Cu 0.2 Py 0.8Cu 0.2 Co Theory (f=15 GHz) Theory (f=14.8 GHz)\nFIG. 2. Resonance peak of Co and Py 0.8Cu0.2independently measured in Py 0.8Cu0.2/Cu/Co with\nθH= 18◦(a) and 9◦(b). The resonance frequency of Py 0.8Cu0.2are both 10 GHz. The resonance\nfrequency of Co is adjusted so that the µ0Hresof Co is equal to Py 0.8Cu0.2. Dashed curves show\nthe theoretical prediction of Co resonance signals.\nTo determine whether the Co layer is in a single-domain state in Py 0.8Cu0.2/Cu/Co when\nthe Py 0.8Cu0.2layer is at resonance, we have measured the FMR signal of Co at diffe rentθH.\nFirst we measure the FMR signal of Py 0.8Cu0.2at one angle and determine the resonance\nfieldµ0Hres. Next we adjust the frequency so that the Co FMR signal can be me asured at\nthe same field. Then we compare the lineshape with the macrospin mod el prediction12. In\nFig. 2(a), when θH= 18◦the resonance field µ0Hresfor Py 0.8Cu0.2is 0.53 T at 10 GHz.\nFor Co, the resonance field is located at 0.53 T for 14.8 GHz. The macr ospin model for\nangle-dependent FMR shows ω/2π= 14.8 GHz, identical to the experiment, showing the\nCo can indeed be treated as a macrospin. However for θH= 9◦(Fig. 2b), we find that the\nCo resonance is located at 15.0 GHz, quite different from the macros pin prediction of 12.3\nGHz. To see the difference more clearly, we plot (dashed lines) the ma crospin prediction for\nboth Py 0.8Cu0.2and Co resonances at 15.0 GHz, based on the magnetizations and line widths\n8measured from perpendicular FMR. The Py 0.8Cu0.2peak matches with experiment. The\ncalculated Co peak deviates from experiment in both resonance field and linewidth. Thus\nwe determine the single-domain limit of θHto be somewhere between 9◦and 18◦in the\nsample. The upper bound 18◦is used in the manuscript for the single-domain limit.\nREFERENCES\n1Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys. Rev. B 67, 140404(R) (2003).\n2Y. Tserkovnyak, A. Brataas, G. E. W. Bauer and B. I. Halperin, Rev. Mod. Phys 77, 1375\n(2005).\n3T. Taniguchi and H. Imamura, Phys. Rev. B 76, 092402 (2007).\n4T. Taniguchi, S. Yakata, H. Imamura and Y. Ando, Appl. Phys. Express 1, 031302 (2008).\n5Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).\n6A. C. Reilly, W.-C. Chiang, W. Park, S. Y. 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Phys. 10409, 580 (2001).\n9"    },    {        "title": "1806.00151v1.Dirac_Surface_State_Modulated_Spin_Dynamics_in_a_Ferrimagnetic_Insulator_at_Room_Temperature.pdf",        "content": "  \nDirac -Surface -State Modulated Spin Dynamics in a  Ferrimagnetic Insulator  at Room \nTemperature  \n \nChi Tang1†, Qi Song2, 8†, Cui -Zu Chang3, 4, Yadong Xu1, Yuichi Ohnuma5, Mamoru \nMatsuo5, 6, Yawen Liu1, Wei Yuan2, 8, Yunyan Yao2, 8, Jagadeesh S. Moodera3, 7, \nSadamichi Maekawa5, Wei Han2, 8 and Jing Shi1* \n1Department of Physics & Astronomy, University of California, Riverside, Riverside, CA 92521, \nUSA  \n2International Center for Quantum Materials, School  of Physics , Peking University, Beijing \n100871, China  \n3Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, MA \n02139, USA  \n4Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA  \n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319 -1195, Ibaraki, \nJapan  \n6Advanced Institute for Materials Research, Tohoku University, Sendai 980 -8577, Miyagi, Japan \n7Department of Physics, Massachusetts Institute of Technology, C ambridge, MA 02139, USA  \n8Collaborative Innovation Center of Quantum Matter, Beijing 100871, China  \n†These are co -first authors.  \n*Correspondence to:  jing.shi@ucr.edu .   \n \n \n \nAbstract: This work demonstrates d ramatic ally modified spin dynamics of  magnetic \ninsulator (MI) by the spin -momentu m locked Dirac surface states of  the adjacent \ntopological insulator (TI) which can be harnessed for spintronic applica tions. As the \nBi-concentration x is systematically tuned in  5 nm thick (Bi xSb1-x)2Te3 TI film, the \nweight of the surface relative to bulk states peaks at x = 0.32 when the chemical \npotential approaches the Dirac point. At this concentration, the Gilbert damping \nconstant of the  precessing magnetization in  10 nm thick Y 3Fe5O12 MI film in the \nMI/TI heterostructures  is enhanced by an order of magnitude, the largest amo ng all \nconcentrations. In addition, the MI acquires additional strong magnetic anisotropy \nthat favors the in -plane orientation with similar Bi -concentration dependence. These \nextraordinary effects  of the Dirac surface states distinguish TI from other mater ials \nsuch as heavy metals in modulat ing spin dynamics of the neighboring magnetic layer.   INTRODUCTION  \n \nTopological insulators (TI) are a new state  of quantum matter with unique spin and charge properties \nowing to the non -trivial band topology an d strong spin -orbit  coupling  (1). These properties can lead to a \nvariety of exotic phenomena including topological magneto -electric effects  (2), quantum anomalous Hall \neffect  (3), image  magnetic monopoles  (4), etc. A remarkable feature that profoundly affect s spin and \ncharge transport in TI is that electrons in Dirac surface states have their  spin locked orthogonally  to the ir \nmomenta in two dimensions  (5, 6). Various spin and charge transport effects of such spin -momentum \nlocking have been observed in spin valves with TI  (7), spin Seebeck effect  (8), inverse Edelstein effect  (9-\n13) and spin -orbit torque switching  (14, 15). Clearly, strong coupling between ele ctron spin and \ntranslational degrees of freedom can be exploited as an efficient way of manipulating spins and vice versa, \nwhich are essential for spintronics.  \nDevices revealing the aforementioned effects are often constructed in heterostructures containin g TI and a \nferromagnetic layer which serves as either a pure spin current source or a spin detector. Any spin -\ndependent effect on transport properties is conveniently measured through the metallic surface states. In \nthe same heterostructures, the reverse e ffect, i.e. the effect of the spin -momentum locked Dirac surface \nstates on the spin dynamics of the magnetic layer, has not yet been systematically studied. It is known that \na thin heavy metal such as Pt or W in contact with a magnetic material can cause b roadening of the \nferromagnetic resonance (FMR) linewidth, which is mainly attributed to the excess flow of spin current \ndue to the spin pumping effect  (16, 17); however, the linewidth broadening is generally insignificant (~10 \nOe) (18, 19). In this regime, the effect is quantitatively described by the momentum sum of the imaginary \npart of the dynamical transverse spin susceptibility  (16) or alternatively the spin -mixing conductance at the \ninterface  (17).  \nIn this work, we investigate the room temperature spin dynamics of yttrium iron garnet ( Y3Fe5O12 or YIG), \na ferrimagnetic insulator, in heterostructures containing (Bi xSb1-x)2Te3, a TI material . By systematically \ntuning the chemical potential of the TI  with a fixed thickness, i.e. five quintuple layers (QL) or ~ 5  nm, via \nvarying Bi concentration x, we control the weight of the surface states  relative to the bulk states  (20). An \nalternative way of controlling the surface state weight is to use electrostatic gating which requires \nextensive materials work. In TI layers dominated by surface states,  the direction of spins pumped into the \nDirac  surface states by FMR has to be locked in the plane. On the other hand, the spins in TI and those of \nYIG are exchange coupled to each other at the interface . As a result,  the direction of the  precessing spins \nin YIG is forced to be aligned in the plane (Fig. 1(a)). Consequently, spin pumping into Dirac surface \nstates  is expected to give rise to  strong damping of the precessing magnetization . Indeed, we observe \nunprecedentedly large FMR linewidth broadening of YIG (up to 111 Oe at 9.6 GHz) when the chemical \npotential of the TI is tuned close to the Dirac point which corresponds to a 15 times larger Gilbert damping \nconstant. This dramatic enhancement in damping is accompanied by an anomalously large increase (~ \n100%) in the easy-plane anisotropy of the YIG layer.    \nRESULTS  \n \nWe choose YIG for the magnetic constituent in our heterostructures for several reasons  to be described in \nthe Materials and Methods section . In this study, we have fabricated seven heterostructure samples with \nvarious Bi concentration s, x, ranging from 0 to 1 .  x = 0.32 sam ple is most insulating as indicated by the \nlargest resistance at 300 K (Fig. 1(b)) and the largest negative slope in its temperature dependence (Fig. \n1(c)), suggesting that the chemical potential is located close to the Dirac point. With x deviating from 0 .32, \nthe chemical potential is tuned away from the Dirac point  (20). In Bi 2Te3 (x = 1) and Sb 2Te3 (x = 0), the \nchemical potential is located in the bulk conduction and valence bands, respectively .  \nWe measure FMR spectra with both cavity and broad -brand co -planar waveguide FMR setups. Prior to TI \nlayer deposition,  cavity  FMR  measurements are performed on all 10 nm thick YIG films at room \ntemperature using a Bruker 9. 6 GHz X -band EMX EPR spectrometer. Fig. 2( b) (red curve) shows a FMR \nderivative absorption spectrum of a  representative YIG film  with an in -plane magnetic field, which can be \nwell fitted with a single Lorentzian derivative. T he peak -to-peak  linewidth ∆𝐻𝑝𝑝 of 7.0 Oe and  resonance \nfield 𝐻𝑟𝑒𝑠 of 2434 Oe  are obtained from the fitting . Such a narrow linewidth indicates  high YIG f ilm \nquality . The mean values of both ∆𝐻𝑝𝑝 and 𝐻𝑟𝑒𝑠 for all seven  samples are 10.2 ± 3.1 Oe and 2394.5 ± 40.2 \nOe, respectively. When FMR is performed as a function of the polar angles 𝜃𝐻 (defined in Fig. 2(a )), the \n𝜃𝐻-dependence of both quantitie s shows very small variatio ns for all seven YIG samples (fig. S2 ), \nindicating a tight control over the YIG film quality.  \nTo study the effect of the  TI Dirac surface state s on YIG spin dynamics, we compare the YIG FMR spectra \ntaken before and after the TI growth. Fig. 2( b) shows a direct comparison between the two representative \nFMR spectra taken with in -plane fields before and after the growth of a 5 QL Sb 2Te3 (x = 0) film. Two \ndistinct differences stand out. First, 𝐻𝑟𝑒𝑠 is shifted to a lower field  by 207 Oe , i.e. from 2434  Oe to 2227 \nOe, indicating a large effect on magnetic anisotropy upon adding the 5 QL Sb 2Te3. Second, the F MR \nlinewidth ∆𝐻𝑝𝑝 is broadened  by seven times. To more accurately determine the effective anisotropy field \nchange, we me asure  FMR of  both YIG/Sb 2Te3 and a YIG reference sample as a function of  polar angle  𝜃𝐻 \n(21, 22). Fig. 2(c) shows the spectra at selected polar angles between the  in-plane ( 𝜃𝐻 = 90˚) and out -of-\nplane ( 𝜃𝐻 = 0˚) magnetic field orientations. The YIG reference sample has  relatively narrow FMR  \nlinewidth  for all polar angles.  In the meantime, 𝐻𝑟𝑒𝑠 decreases monotonically from ~5560 Oe for 𝐻 out-of-\nplane to ~ 2434  Oe for 𝐻 in-plane,  consistent with the beh avior of nanometer thick YIG films with easy-\nplane magnetic anisotropy  (23). With 5 QL Sb 2Te3 on top, however, dramatic differences can be readily \nidentified  as shown in the top panel of Fig. 2(c): significant broadening of ∆𝐻𝑝𝑝 and large shift in 𝐻𝑟𝑒𝑠. \n𝐻𝑟𝑒𝑠 shift occurs at all angles; hence, the overall 𝐻𝑟𝑒𝑠 range is greatly expanded, i.e. between 2227 Oe and \n6000 Oe. Such marked  effects are not  seen in heterostructures containing thin heavy metal layers such as \nPt (18).   The differences  in both ∆𝐻𝑝𝑝 and 𝐻𝑟𝑒𝑠 caused by the 5 QL Sb 2Te3 suggest  their origin in  TI’s band \nstructure. In order to study the effects of the TI Dirac surface states, we compare the FMR results in all \nseven samples  in which the surface -to-bulk ratio systematically varies . We first study the effect of varying \nx on the easy-plane magnetic anisotropy. We plot 𝐻𝑟𝑒𝑠 as a function of θH in Fig. 3(a)  for all seven \nYIG/(Bi xSb1-x)2Te3 samples plus a YIG reference film. T he angular dependence data can be fitted \nreasonably well with the shape anisotropy plus  a uniaxial anisotropy term, both in the form of cos2𝜃. As is \nroutinely done in literature (21), we solve three transcendental equations  numerically and seek  the least -\nsquare fitting results . Note that if the uniaxial anisotropy  term is negative, it simply represents  easy-plane \nanisotropy. We find that  it is indeed negative, i.e.  there is additional  easy-plane anisotropy  in all seven \nsamples .  By fitting the p olar angle dependence, we obtain the two best -fit parameters, 4𝜋𝑀 𝑒𝑓𝑓 and the g-\nfactor for each sample. The effective anisotropy field is defined as  4𝜋𝑀 𝑒𝑓𝑓= 4𝜋𝑀 𝑠+ 𝐻𝑎𝑛, where 4𝜋𝑀 𝑠 \nand 𝐻𝑎𝑛 denote the demagnetizing field and an effective easy-plane (positive) magnetic anisotropy field, \nrespectively  (24). The best -fit parameters are s ummarized  in table. S1 . The extracted 4𝜋𝑀 𝑒𝑓𝑓 is plotted in \nFig. 3(b). Since the demagnetizing field  4𝜋𝑀 𝑠 is about 1780 Oe, 4𝜋𝑀 𝑒𝑓𝑓 is clearly larger than 4𝜋𝑀 𝑠 in all \nsamples. Furthermore, 4𝜋𝑀 𝑒𝑓𝑓 depends on x. For the two most metallic TI films, i.e. x = 0 and x = 1, \n4𝜋𝑀 𝑒𝑓𝑓 is increased by 420 and 460 Oe, which accounts for 23% and 26% of its demagnetizing field \n4𝜋𝑀 𝑠, respectively. In comparison, the corresponding increase is only 5%  in YIG/Pt (fig. S3(a)). More \ninterestingly, as the chemical potential is tuned into the band gap, i.e. surface states becoming dominant, \n4𝜋𝑀 𝑒𝑓𝑓 increases further and peaks in the most insulating sample ( x = 0.32), reaching 3800 Oe. This \nincrease repr esents nearly a 100% enhancement over the YIG demagnetizing field 4𝜋𝑀 𝑠.  \nA common origin of enhanced magnetic anisotropy in thin films is related to the interface strain. In \n(Bi xSb1-x)2Te3, the interaction between the neighboring Te -Bi/Sb -Te-Bi/Sb -Te qu intuple layers is of the \nvan der Waals type. Between TIG and TI , there is no  epitaxial relation due to widely different lattice \nstructures; therefore, the strain and strain -induced anisotropy are expected to be small at the YIG -TI \ninterface for all samples . Another possibility of the enhanced 4𝜋𝑀 𝑒𝑓𝑓 is an increased demagnetizing field. \nIf part of TI becomes ferromagnetic, it can in principle cause an increase in 4𝜋𝑀 𝑠. We exclude this \npossibility with the following arguments. First, s uch a proximity induced moment, if exists, can only come  \nfrom a few atomic layers at the interface and is clearly too small to account for the  observed  100% \nincrease. Our magnetization measurements do not support this possibility either (f ig. S5) . The same  \nmagnetometry results do not show any clear  Bi-concentration dependence  within experimental uncertainty.  \nMore importantly, the proximity effect in YIG/TI heterostructures occurs at much lower temperatures (< \n150 K) as reported in previous studies  (8, 25). Additionally , the Tc of the induced ferromagnetism in TI \nwas found to be uncorrected with TI’s chemical potential position  (25). In our data, the 4𝜋𝑀 𝑒𝑓𝑓 \nenhancement follows the same trend as the re sistivity as shown in Fig. 1(c) . From these analyses, w e \nconclude that the enhanced effective  anisotropy originates from the Dirac surface states.   The cavity -FMR meas urements have already indicated anomalously broadened FMR linewidth at a \nparticular microwave frequency. In order to extract the Gilbert damping constant α, we perform broad -\nband FMR measurements using a coplanar waveguide setup for all YIG/TI samples up t o 12 GHz. \nRepresentative transmission data 𝑆21 in Fig. 4(a) show the FMR absorption of  YIG/Sb 2Te3 with several \nfrequencies. Both FMR resonance field shift and linewidth broadening display the same trend in Fig. 4(b). \nThe half width at half maximum, ∆𝐻= √3∆𝐻𝑝𝑝/2 , is extracted by fitting a Lorentzian function to each \n𝑆21 spectrum up to 12 GHz, and then plotted in Fig. 4(c) for all  samples . A linear relation between ∆𝐻 and \nfrequency is observed, and α can be calculated by (24) \n \n∆𝐻=2𝜋\n𝛾𝛼𝑓+ ∆𝐻0 ,      (1) \n \nwhere 𝛾 and ∆𝐻0 are the gyromagnetic ratio and  the inhomogeneity linewidth broadening, respectively. \nFig. 4(d) shows α vs. x for all samples. I nterestingly, 𝛼 peaks at x = 0.32 as well, i.e. in the most insulating \nsample, similar to the resistivity and the effective  anisotropy field. Compared to the two most metallic \nsamples with 𝛼 ×for x = 0 (or Sb 2Te3) and 𝛼 ×for x = 1 (or Bi 2Te3), 𝛼 reaches the \nmaximum value of 2.2 ×  for x = 0.32, which is an order of magnitude larger than that of the bare YIG \nfilms (average value of ×). In comparison, 𝛼 in YIG/Pt is only twice as large as that in the YIG \nreference, as shown in f ig. S4(b). Additionally, 𝛼 shows the same trend as that of the resistivity and \neffective  anisotropy. These fact s suggest a common origin, i.e. the special band structures of the TI surface \nstates rather than the spin -orbit coupling of the constituent elements. The latter effect would imply a \nmonotonically increasing trend as more Bi atoms are incorporated.  \n \nDISCUSS ION  \n \nWe now show that these three effects are actually connected and given by the spin -momentum locking \nproperties of  the TI Dirac surface state s. The spins ( 𝜎⃗) of TI and the spins ( 𝑆⃗) of YIG are coupled by the \nexchange interaction expressed as  (26, 27), 𝐻=𝐽𝑠𝑑𝜎⃗ ∙ 𝑆⃗, with 𝐽𝑠𝑑 being the exchange constant at the \ninterface.  We note that the spins in TI lie  in the plane due to the spin -momentum locking.  Therefore, the \nspins in YIG are pulled towards  the plane as well.  This pulling effect induces th e easy -plane anisotropy \ngiven by  \n \n                            𝐸𝑎𝑛= −1\n2 (𝜒∕∕−𝜒⊥)𝑀2                                                (2) \n     \nwhere 𝜒∕∕ and 𝜒⊥ are the in -plane  and out -of-plane components of the spin susceptibility in TI, \nrespectively, and M is the magnetization of  YIG.  In particular, when the chemical potential  is close to  the  Dirac point , 𝜒⊥ is strongly suppressed due to the gap (28) and, thus, 𝜒∕∕≫ 𝜒⊥. TI’s spin susceptibility \ngives rise to the easy-plane magnetic anisotropy field by  𝐻𝑎𝑛= −𝜕𝐸𝑎𝑛\n𝜕𝑀. \nThe same interfacial exchange interaction also affects  spin pumping  (16); the spin precession of YIG i n \nFMR results in the motion of spins in TI.  The induced spin current ( 𝐼𝑠) in TI is expressed as  \n  \n     𝐼𝑠 ∝ 𝐽𝑠𝑑2𝐼𝑚𝜒+−(𝜔)     (3) \n \nwhere 𝜒+−(𝜔) is the transverse component in TI to the mag netization direction of YIG, and  ω is the FMR \nfrequenc y.  We note that since the spin Seebeck effect is due to the spin pumping by heat  (26, 27), it is also \ngiven by 𝜒+−(𝜔) but integrated over ω in the range of the thermal distribution of spin fluctuations.   The \nsusceptibility is calculated by taking into account the di rect transition near the Fermi level  because of the \nspin-momentum locking in TI (see SI  for details).  Since the resonance frequency  (a few GHz) is much \nsmaller than the energy gap of TI (~ 0.3  eV) (29), the direct transition is more effective when the Fermi \nenergy is near the Dirac point.  As a result, the spin pumping (the Gilbert damping) is significantly \nenhanced near the Dirac point. This model also explain s enhanced spin Seebeck effect reported previously  \n(8). \nIn summary , we have observe d dramatic modifications of YIG spin dynamics by spin -momentum locked \nsurfaces of a thin TI layer in high -quality YIG/TI heterostructures  with different Bi/Sb ratios . The spin -\nmomentum locking in TI provides not only a sensitive detection of the magnetic s tate in magnetic \nmaterials serving as a spin current source, but also an active way of manipulating ultrafast magnetization \ndynamics and magnetic anisotropy with the unique properties of the topological Dirac surface states, \nwhich offers exciting opportuni ties for potential spintronic applications.    \n \nMATERIALS  AND METHODS  \n \nChoice of YIG : We choose 10 nm thick YIG films as the MI layers in all MI/TI heterostructures. First, \nYIG in general has a very small Gilbert damping constant 𝛼~ 3 × 10-5 in crystals  and ~ 10-3 in 10 nm \nthick YIG films); and the FMR linewidth is relatively narrow (~ 10 Oe  at ~ 10 GHz for thin films). \nTherefore, small linewidth changes can be easily detected. Second, YIG films are prepared  first with high \ntemperatures (~ 800 ºC) with pulsed laser deposition and rapid thermal annealing  and the TI layers are \ngrown at much lower temperatures (~230 ºC) after  with molecular beam epitaxy . This growth sequence \nand the large temperature difference prevent serious intermixing across the interface. In our \nheterostructures, YIG is atomically flat, which ensures the flat YIG -TI interface. Third, similar to our \nprevious spin Seebeck effect study  (8), here we conduct FMR measurements at room temperature which is \nwell above that of the induced ferromagnetism in the TI surface layer. Consequently, the dynamic behavior \nof YIG is not affected by the induced ferromagnetism in TI  (18).   Heterostructure growth:  Thin YIG films are grown on epi -ready lattice -matched single crystal GGG \n(111) substrates via pulsed laser deposition. The detailed growth recipe of YIG films is described in a \nprevious p aper (30). To fabricate high -quality YIG/(Bi xSb1-x)2Te3 heterostructures with various Bi  \nconcentrations (x = 0, 0.22, 0.27, 0.32, 0.46, 0.67 and 1 in this study) , YIG (111) films are then transferred \nto an ultra-high vacuum molecular beam epitaxy (MBE) system with the base pressure better than 5×10-10 \nTorr for TI growth. High -purity Bi (99.999%), Sb (99.9999%) and Te (99.9999%) are evaporated from \nKnudsen effusion cells. During the growth, the YIG substrate  is kept at 230 °C and the growth rate  of TI  is \n~0.2 QL/min. The heterostructure film is covered with a 5 nm Te protection layer before taken out of the \nMBE chamber for the FMR measurements.    \nFMR measurements:  The polar angle dependent FMR  measurements fo r all samples are performed using \na Bruker 9.6 GHz X -band EMX EPR spectrometer. Samples can be rotated with respect to the static field \ndirection from the in -plane to out -of-plane geometry with a protractor  reading the angle precisely.  \nThe Gilbert damping  constant measurements are conducted by a broad -band FMR using coplanar \nwaveguide setup. The forward amplitude of complex transmission coefficients ( S21) is recorded by the \nvector network analyzer (VNA, Agilent E5071C) connected to a straight -line coplanar  waveguide  (31). \nSample is attached to the waveguide and the measurement is performed with the frequency sweeping from \n1 GHz to 12 GHz at a fixed magnetic field which can be varied up to 4000 Oe.  \n \n \nSUPPLEMENTARY  MATERIALS  \n \nfig. S1. Crystal structure and surface morphology.  \nfig. S2. FMR resonance field and linewidth for all YIG films.  \nfig. S3. Effect of Pt on YIG resonance characteristics.  \nfig. S4. Comparison of FMR between YIG/TI and YIG/Pt.  \nfig. S5. Comparison of total effective anisotropy field and demagnetizing field.  \nfig. S6. High -resolution transmission electron microscope image of a representative YIG/TI sample.  \nfig. S7 . TI surface state dispersion.  \nfig. S8 . Direct transition of TI conduction electrons driven by spi n pumping.  \ntable S1. Two parameters (i.e. 4𝜋𝑀 𝑒𝑓𝑓 and γ/2π) obtained from fitting.  \n  REFERENCES AND NOTES  \n \n1. X. L. Qi, S. C. Zhang, Topological Insulators and Superconductors. Rev. Mod. Phys.  83, \n1057 (2011).  \n2. X. L. Qi, T. L. Hughes, S. C. Zhang, Topological Field Theory of Time -Reversal Invariant \nInsulators. Phys. Rev. B  78, 195424 (2008).  \n3. C. Z. Chang  et al. , Experimental Observation of the Quantum Anomalous Hall Effect in a \nMagnetic Topological Insulator. Science  340, 167 (2013).  \n4. X. L. Qi, R. D. Li, J. D. Zang, S. C. Zhang, Inducing a Magnetic Monopole with \nTopological Surface States. Science  323, 1184 (2009).  \n5. D. 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Maekawa, Enhanced Dc Spin Pumping into a \nFluctuating Ferromagnet near T c. Phys. Rev. B  89, 174417 (2014).  \n17. Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, Spin Pumping and Magnetization Dynamics \nin Metallic Multilayers. Phys. Rev. B  66, 224403 (2002).  \n18. Y. Y. Sun  et al. , Damping in Yttrium Iron Garnet Nanoscale Films C apped by Platinum. \nPhys. Rev. Lett.  111, 106601 (2013).  \n19. C. H. Du, H. L. Wang, F. Y. Yang, P. C. Hammel, Enhancement of Pure Spin Currents in \nSpin Pumping Y 3fe5o12/Cu/Metal Trilayers through Spin Conductance Matching. Phys. \nRev. Applied  1, 044004 (2014).  \n20. J. S. Zhang  et al. , Band Structure Engineering in (Bi 1-Xsbx)2te3 Ternary Topological \nInsulators. Nat. Commun.  2, 574 (2011).  \n21. S. Mizukami, Y. Ando, T. Miyazaki, The Study on Ferromagnetic Resonance Linewidth \nfor Nm/80nife/Nm (Nm = Cu, Ta, Pd  and Pt) Films. Jpn J Appl Phys 1  40, 580 (2001).   22. X. Liu  et al. , Perpendicular Magnetization Reversal, Magnetic Anisotropy, Multistep Spin \nSwitching, and Domain Nucleation and Expansion in Ga 1-Xmn xas Films. J. Appl. Phys.  98, \n063904 (2005).  \n23. C. Tang  et al. , Anomalous Hall Hysteresis in Tm 3fe5o12/Pt with Strain -Induced \nPerpendicular Magnetic Anisotropy. Phys. Rev. B  94, 140403(R) (2016).  \n24. S. Vonsovskii, Ferromagnetic Resonance Pergamon, Oxford ,  (1966).  \n25. Z. L. Jiang  et al. , Independent Tuning of  Electronic Properties and Induced \nFerromagnetism in Topological Insulators with Heterostructure Approach. Nano Lett.  15, \n5835 (2015).  \n26. H. Adachi, J. Ohe, S. Takahashi, S. Maekawa, Linear -Response Theory of Spin Seebeck \nEffect in Ferromagnetic Insulator s. Phys. Rev. B  83, 094410 (2011).  \n27. H. Adachi, K. Uchida, E. Saitoh, S. Maekawa, Theory of the Spin Seebeck Effect. Rep. \nProg. Phys.  76, 036501 (2013).  \n28. M. M. Vazifeh, M. Franz, Spin Response of Electrons on the Surface of a Topological \nInsulator. Phys. Rev. B  86, 045401 (2012).  \n29. Y. L. Chen  et al. , Experimental Realization of a Three -Dimensional Topological Insulator \nBi2te3. Science  325, 178 (2009).  \n30. C. Tang  et al. , Exquisite Growth Control and Magnetic Properties of Yttrium Iron Garnet \nThin F ilms. Appl. Phys. Lett.  108, 102403 (2016).  \n31. Y. Zhao  et al. , Experimental Investigation of Temperature -Dependent Gilbert Damping in \nPermalloy Thin Films. Sci Rep  6, 22890 (2016).  \n32. Y. Ohnuma, H. Adachi, E. Saitoh, S. Maekawa, Magnon Instability Driven  by Heat \nCurrent in Magnetic Bilayers. Phys. Rev. B  92, 224404 (2015).  \n33. S. Maekawa, H. Adachi, K. Uchida, J. Ieda, E. Saitoh, Spin Current: Experimental and \nTheoretical Aspects. J. Phys. Soc. Jpn.  82, 102002 (2013).  \n \n(Refs. 32 and 3 3 are for supplementary materials)  \n \n \nAcknowledgments  \n \nWe acknowledge  the assistance from N. Samarth and J. Kally for the sample preparation and useful \ndiscussions with Z. Shi, J. Li, V. Ortiz and M. Aldosary .  \nFunding: This work was  supported as part of the  SHINES, an Energy Frontier Research Center funded by \nthe U.S. Department of Energy, Office of Science, Basic Energy Sci ences under Award No. SC0012670  \n(CT, YDX, YWL  and JS) . QS, WY, YYY and WH  acknowledge the support of the National Basic \nResearch Program s of China (973 Grants 2014CB920902 and 2015CB921104) and the National Natural \nScience Foundation of China (NSFC Grant 11574006).  CZC and JSM acknowledge the support from NSF \nGrants No. DMR -1207469, ONR Grant No. N00014 -16-1-2657, and the STC Center for In tegrated \nQuantum Materials under NSF Grant No. DMR -1231319.  \nAuthor contributions:  JS conceived and supervised the experiments. CT grew the YIG magnet ic insulator \nthin films  and performed cavity FMR measurements and data analysis with the help of YWL . QS \nperformed the broadband FMR measurements and data analysis with the help of WY and YYY and the \nsupervision of WH. YDX performed TEM sample preparation. CZC grew the topological insulator thin  films on YIG to form heterostructures  with JSM’s supervision . YO, MM and SM  did theoretical \ncalculations . All authors  participated in the preparation of the final manuscript.  \nCompeting interests:  The authors declare that they have no competing interests. Data and materials \navailability: All data needed to evaluate t he conclusions in the paper are present in the paper and/or the \nSupplementary Materials. Additional data related to this paper may be requested from the authors.  \n  FIGURES AND TABLES  \n \n \n \n \n \nFig. 1.  FMR measurement principle  and TI properties .  (a) Schematic drawing of \nmagnetization dynamics in YIG interfaced with TI in which the spin of the surface state electron is \nlocked to momentum. (b) Room temperature sheet resistance of (Bi xSb1-x)2Te3 with different Bi \nconcentrations. (c) Temperature depe ndence of the sheet resistance of seven YIG (10 nm)/(Bi xSb1-\nx)2Te3 (5 QL) heterostructures.  \n  \n \n \nFig. 2.  YIG FMR spectra with and without TI .  (a) Definition of polar angles, 𝜃𝐻  and 𝜃𝑀 in \nFMR measurements.  (b) FMR derivative absorption spectra of YIG/Sb 2Te3 and YIG reference \nsample at a frequency of 9.6 GHz with magnetic field applied in -plane ( 𝜃𝐻 = 90˚). The solid lines \nare the best fits to extract the resonance field 𝐻𝑟𝑒𝑠 and peak -to-peak linewidth ∆𝐻𝑝𝑝. (c) FMR \nderivative absorption spe ctra of YIG/Sb 2Te3 and YIG reference sample with the polar angle θH \nranging from 0˚(out -of-plane) to 90˚ (in -plane) at 300 K. The extra peak -like feature on the high \nfield side of the resonance at 0˚ and 10˚ is also observed some other samples, which could  be \ncaused by minor inhomogeneity change in YIG due to the presence of the TI layer.  \n  \n \n \n \nFig. 3. Extracted 4𝜋𝑀 𝑒𝑓𝑓 from FMR polar angle dependence fitting.  (a) Polar angle 𝜃𝐻 \ndependence of FMR resonance field 𝐻𝑟𝑒𝑠 for all seven YIG (10 nm)/(Bi xSb1-x)2Te3 (5 QL) samples \nand YIG reference sample.  Solid curves are the best fits.  (b) Bi -concentration dependence of \nextracted effective anisotropy field 4𝜋𝑀 𝑒𝑓𝑓 obtained from fitting in (a) for all seven YIG (10 \nnm)/(Bi xSb1-x)2Te3 (5 QL) samples. Th e black dash ed line is the 4𝜋𝑀 𝑒𝑓𝑓 value for the YIG \nreference sample.  \n \n  \n Fig. 4.  Extracted Gilbert damping from FMR linewidth fitting . (a) FMR transmission spectra \n𝑆21 for YIG/Sb 2Te3 for different chosen frequencies: 4, 6, 8, 10 and 12 GHz at 300 K after \nbackground subtraction. (b) Normalized FMR spectra 𝑆21at a fixed frequency of 10 GHz with an \napplied in -plane static field for YIG/(Bi xSb1-x)2Te3 samples with different Bi concentrat ions and \nYIG reference sample. (c) Frequency dependence of FMR linewidth for all seven YIG/TI samples \nand YIG reference sample. The resonance peak height is reduced in samples with increased \ndamping constant , which causes poor Lorent zian fitting and conseq uently large apparent noise in \nextracted linewidth. (d) Bi concentration ( x)-dependence of the Gilbert damping constant α \nextracted from the slope of the straight lines in (c).  \n \n "    },    {        "title": "2002.08723v2.Stoner_Wohlfarth_switching_of_the_condensate_magnetization_in_a_dipolar_spinor_gas_and_the_metrology_of_excitation_damping.pdf",        "content": "Stoner-Wohlfarth switching of the condensate magnetization in a dipolar spinor gas\nand the metrology of excitation damping\nSeong-Ho Shinn,1Daniel Braun,2and Uwe R. Fischer1\n1Department of Physics and Astronomy, Seoul National University, 08826 Seoul, Korea\n2Eberhard-Karls-Universit at T ubingen, Institut f ur Theoretische Physik, 72076 T ubingen, Germany\n(Dated: May 13, 2020)\nWe consider quasi-one-dimensional dipolar spinor Bose-Einstein condensates in the homogeneous-\nlocal-spin-orientation approximation, that is with unidirectional local magnetization. By analyti-\ncally calculating the exact e\u000bective dipole-dipole interaction, we derive a Landau-Lifshitz-Gilbert\nequation for the dissipative condensate magnetization dynamics, and show how it leads to the Stoner-\nWohlfarth model of a uni-axial ferro-magnetic particle, where the latter model determines the stable\nmagnetization patterns and hysteresis curves for switching between them. For an external magnetic\n\feld pointing along the axial, long direction, we analytically solve the Landau-Lifshitz-Gilbert equa-\ntion. The solution explicitly demonstrates that the magnetic dipole-dipole interaction accelerates\nthe dissipative dynamics of the magnetic moment distribution and the associated dephasing of the\nmagnetic moment direction. Under suitable conditions, dephasing of the magnetization direction\ndue to dipole-dipole interactions occurs within time scales up to two orders of magnitude smaller\nthan the lifetime of currently experimentally realized dipolar spinor condensates, e.g., produced with\nthe large magnetic-dipole-moment atoms166Er. This enables experimental access to the dissipation\nparameter \u0000 in the Gross-Pitaevski\u0014 \u0010 mean-\feld equation, for a system currently lacking a complete\nquantum kinetic treatment of dissipative processes and, in particular, an experimental check of the\ncommonly used assumption that \u0000 is a single scalar independent of spin indices.\nI. INTRODUCTION\nEver since a phenomenological theory to describe the\nbehavior of super\ruid helium II near the \u0015point has\nbeen developed by Pitaevski\u0014 \u0010 [1], the dynamics of Bose-\nEinstein condensates (BEC) under dissipation has been\nintensely studied, see, e.g., [2{8]. Experimentally, the im-\npact of Bose-Einstein condensation on excitation damp-\ning and its temperature dependence has for example been\ndemonstrated in [9{12].\nDissipation in the form of condensate loss is de\fned\nby a dimensionless damping rate \u0000 entering the left-\nhand side of the Gross-Pitaevski\u0014 \u0010 equation, replacing the\ntime derivative as i@t!(i\u0000\u0000)@t. While a micro-\nscopic theory of condensate damping is comparatively\nwell established in the contact-interaction case, using var-\nious approaches, cf., e.g., [5, 13{15], we emphasize the\nabsence of a microscopic theory of damping in dipolar\nspinor gases. While for scalar dipolar condensates, par-\ntial answers as to the degree and origin of condensate-\nexcitation damping have been found see, e.g., Refs. [16{\n19], in spinor or multicomponent gases the interplay of\nanisotropic long-range interactions and internal spinor or\nmulticomponent degrees of freedom leads to a highly in-\ntricate and di\u000ecult-to-disentangle many-body behavior\nof condensate-excitation damping.\nIn this paper, we propose a method to experimen-\ntally access \u0000 in a dipolar spinor condensate by using\nthe dynamics of the unidirectional local magnetization in\na quasi-one-dimensional (quasi-1D) dipolar spinor BEC\nin the presence of an external magnetic \feld. To this\nend, we \frst derive an equation of motion for the mag-\nnetization of the BEC that has the form of a Landau-\nLifshitz-Gilbert (LLG) equation [20{22], with an addi-tional term due to the dipole-dipole interaction between\nthe atoms. The LLG equation is ubiquitous in nano-\nmagnetism, where it describes the creation and dynam-\nics of magnetization. The static limit of this equation\nis, in the limit of homogeneous local spin-orientation, de-\nscribed by the well-known Stoner-Wolfarth (SW) model\n[23{25] of a small magnetic particle with an easy axis of\nmagnetization. We then investigate the magnetization\nswitching after \ripping the sign of the external magnetic\n\feld, and demonstrate the detailed dependence of the\nswitching dynamics on the dissipative parameter \u0000.\nFor a quasi-2D spinor BEC with inhomogeneous local\nmagnetization, Ref. [26] has studied the magnetic domain\nwall formation process by deriving a LLG type equa-\ntion. Here, we derive the LLG equation in a quasi-1D\nspinor BEC with unidirectional local magnetization, in\norder to establish a most direct connection to the orig-\ninal SW model. In distinction to [27], which studied\nthe e\u000bective quasi-1D dipole-dipole interaction resulting\nfrom integrating out the two transverse directions within\na simple approximation, we employ below an exact ana-\nlytic form of the dipole-dipole interaction. In Section II,\nwe establish the quasi-1D spinor Gross-Pitaevski\u0014 \u0010 (GP)\nequation with dissipation, and equations of motion for\nthe magnetization direction (unit vector) M. Section V\nshows how the LLG equation and the SW model result,\nand Section VI derives analytical solutions to the equa-\ntions of motion for Mwhen the external magnetic \feld\npoints along the long, zaxis. We summarize our results\nin section VII.\nWe defer two longer derivations to Appendices. The\nanalytical form of the e\u000bective dipole-dipole interaction\nenergy is deduced in Appendix A, and the quasi-1D GP\nmean-\feld equation with dissipation is described in detailarXiv:2002.08723v2  [cond-mat.quant-gas]  12 May 20202\nin Appendix B. Finally, in Appendix C, we brie\ry discuss\nto which extent relaxing the usual simplifying assumption\nthat dissipation even in the spinor case is described by\na single scalar changes the LLG equation, and whether\nthis a\u000bects the SW model and its predictions.\nII. GENERAL DESCRIPTION OF DAMPING IN\nBECS\nThe standard derivation of the quantum kinetics of\nBose-Einstein condensate damping [5] starts from the\nmicroscopic Heisenberg equation of motion for the quan-\ntum \feld operator ^ (r;t), for a scalar (single compo-\nnent) BEC in the s-wave scattering limit. Using their\nresults, [28] obtained a mean-\feld equation to describe\nthe dissipation of scalar BEC, whose form is\n(i\u0000\u0000)~@ \n@t=H (1)\nwhere is the (in the large Nlimit) dominant mean-\feld\npart upon expanding the full bosonic \feld operator ^ .\nIn Ref. [1], Pitaevski\u0014 \u0010 obtained a similar but slightly\ndi\u000berent form of the dissipative mean-\feld equation\nbased on phenomenological considerations, i~@ \n@t=\n(1\u0000i\u0000)H , by parametrizing the deviation from exact\ncontinuity for the condensate fraction while minimizing\nthe energy [1]. The latter deviation is assumed to be\nsmall, which is equivalent to assuming that \u0000 remains\nsmall. This provides a clear physical interpretation of the\ndamping mechanism, namely one based on particle loss\nfrom the condensate fraction. The version of Pitaevski\u0014 \u0010\ncan be written as\n(i\u0000\u0000)~@ \n@t=\u0000\n1 + \u00002\u0001\nH : (2)\nIt can thus be simply obtained by rescaling time with a\nfactor 1 + \u00002compared to (1). Hence, as long as one\ndoes not predict precisely \u0000, the two dissipative equa-\ntions (1) and (2) cannot be distinguished experimentally\nfrom the dynamics they induce. From the data of [11],\n[4] estimated typical values of \u0000 '0:03 for a scalar BEC\nof23Na atoms (see also [12]), which shows that to distin-\nguish between (1) and (2) experimentally the theoretical\npredictions of \u0000 would need to be precise to the order of\n10\u00004.\nHow eqs.(1) and (2) can be generalized to the dipolar\nspinor gases is comparatively little investigated. Using a\nsymmetry-breaking mean-\feld approach by writing the\nquantum \feld operator as ^ (r;t) as ^ (r;t) = (r;t) +\n\u000e^ (r;t), with (r;t) =h^ (r;t)iandh\u000e^ (r;t)i= 0, [5]\nand [28] showed that \u0000 is derived from the three-\feld\ncorrelation function h\u000e^ y(r;t)\u000e^ (r;t)\u000e^ (r;t)iin a ba-\nsis whereh\u000e^ (r;t)\u000e^ (r;t)i= 0. From this microscopicorigin, based on correlation functions, it is clear that in\nprinciple \u0000 might depend on the spin indices in a spinor\nBECs and hence become a tensor (see Appendix C for\na corresponding phenomenological generalization). Nev-\nertheless, it is commonly assumed cf.,e.g., [26, 29], that\n\u0000 does not depend on spin indices, and the scalar value\nfound speci\fcally in [4] for a scalar BEC of23Na atoms\nis commonly used, while a clear justi\fcation of this as-\nsumption is missing.\nExtending the microscopic derivations in [5] and [28] to\nthe spinor case would be theoretically interesting, but is\nbeyond the scope of the present paper. Here, we instead\nfocus on the question whether the standard assumption\nthat the damping of each spinor component can be de-\nscribed by the mean-\feld equation [28] leads to exper-\nimentally falsi\fable dynamical signatures. It will turn\nout that this assumption introduces an additional strong\ndephasing in the spin-degrees of freedom, ampli\fed by\nthe dipolar interaction. Hence, even on time scales on\nwhich the decay of the condensate fraction according to\n(1) can be neglected, the relaxation of the magnetization\nof the BEC potentially o\u000bers valuable insights whether\nthe scalar-\u0000 assumption is justi\fed. Indeed, in [30] it\nwas shown experimentally that on the time scale of the\nswitching dynamics of the magnetization the number of\nparticles in the condensates remains approximately con-\nstant. One might wonder, then, which dissipative mech-\nanism is left. However, as we will show, by assuming the\nsame GP equation for each component of the spinor as\nfor scalar bosons, additional dephasing occurs that is in\nfact much more rapid than the decay of condensate den-\nsity due to dephasing accelerated by the dipole-dipole\ninteraction.\nIII. MEAN-FIELD DYNAMICS OF DAMPING\nIN DIPOLAR SPINOR BECS\nFor a spinor BEC, linear and quadratic Zeeman inter-\nactions are commonly included in the Hamiltonian. The\nquadratic Zeeman interaction is related to a second-order\nperturbation term in the total energy that can be induced\nby the interaction with an external magnetic \feld ( qB)\nas well as with the interaction with a microwave \feld\n(qMW) [31]. Speci\fcally, by applying a linearly polarized\nmicrowave \feld, one can change qMWwithout changing\nqB[32, 33]. Hence, we will assume that the quadratic\nZeeman term can be rendered zero by suitably changing\nqMW.\nFollowing [26], we thus assert that for a dipolar spinor\nBEC without quadratic Zeeman term, the mean-\feld\nequation can be written as3\n(i\u0000\u0000)~@ (r;t)\n@t=\u0014\n\u0000~2\n2mr2+Vtr(r) +c0j (r;t)j2\u0000~fb\u0000bdd(r;t)g\u0001^f\u0015\n (r;t)\n+SX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zF\u00171;\u00172;\u0001\u0001\u0001;\u0017k(r;t)^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k (r;t): (3)\nwhere (r;t) is a vector quantity whose \u000b-th component\nin the spinor basis is  \u000b(r;t) (spin-space indices from\nthe beginning of the Greek alphabet such as \u000b;\f;\r;:::\nare integers running from \u0000StoS). In this expres-\nsion, ~^fis the spin- Soperator where the spin ladder\nis de\fned by ^fzj\u000bi=\u000bj\u000biandh\u000bj\fi=\u000e\u000b;\f, while\nF\u00171;\u00172;\u0001\u0001\u0001;\u0017k(r;t):= y(r;t)^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k (r;t) are the\ncomponents of the expectation value of ^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k.\nThe Larmor frequency vector reads b=gF\u0016BB=~\n(with Land\u0013 e g-factor gF, Bohr magneton \u0016B, and\nthe external magnetic induction B),~bdd(r;t)\u0001e\u0017=\ncddR\nd3r0P\n\u00170=x;y;zQ\u0017;\u00170(r\u0000r0)F\u00170(r0;t):Here,cdd=\n\u00160(gF\u0016B)2=(4\u0019) ande\u0017is a unit vector along the \u0017\naxis [31] (by convention, indices from the middle of the\nGreek alphabet such as \u0014;\u0015;\u0016;\u0017;::: =x;y;z denote spa-\ntial indices), and Q\u0017;\u00170is the spin-space tensor de\fned in\nEq. (A2) of Appendix A. Finally, mis the boson mass,\nc0the density-density interaction coe\u000ecient, and c2kthe\ninteraction coe\u000ecient parametrizing the spin-spin inter-\nactions, where kis an positive integer running from 1\ntoS[26]. For example, c2is the spin-spin interaction\ncoe\u000ecient of a spin-1 gas ( S= 1).\nTo develop a simple and intuitive physical approach,\nwe consider a quasi-1D gas for which one can perform\nanalytical calculations. We set the trap potential as\nVtr(x;y;z ) =1\n2m!2\n?\u0000\nx2+y2\u0001\n+V(z); (4)\nso that the long axis of our gas is directed along the z\naxis and the gas is strongly con\fned perpendicularly.\nFor a harmonic trap along all directions, i.e. when\nV(z) =m!2\nzz2=2, we set!?\u001d!z. For a box trap along\nz, i.e. when V(z) = 0 forjzj \u0014LzandV(z) =1\nFIG. 1. Schematic of the considered geometry in a quasi-1D\ngas (shaded ellipsoid). The length of the red magnetization\narrows, all pointing in the same direction (homogeneous local-\nspin-orientation limit), represents jd(z;t)j.forjzj> Lz, our gas will be strongly con\fned along z\nas long as the quasi-1D condition is satis\fed, where we\nwill discuss below whether the condition is satis\fed, in\nsection VI A.\nSingle-domain spinor BECs have been already real-\nized, for example, using spin-187Rb [34]. This single-\ndomain approximation is common in nanomagnetism, see\nfor example [24], by assuming magnetic particles much\nsmaller than the typical width of a domain wall. The\nlocal magnetization is related to the expectation value\n~F(r;t)\u0011~ y(r;t)^f (r;t) of the spatial spin density\noperator byd(r;t) =gF\u0016BF(r;t). An unidirectional\nlocal magnetization d(z;t) is then given by\ndx(z;t) =d(z;t) sin\u0012(t) cos\u001e(t);\ndy(z;t) =d(z;t) sin\u0012(t) sin\u001e(t); (5)\ndz(z;t) =d(z;t) cos\u0012(t);\nwhered\u0017(z;t) =d(z;t)\u0001e\u0017is the\u0017-th component of\nd(z;t),d(z;t) =jd(z;t)j,\u0012(t) is polar angle of d(z;t),\nand\u001e(t) is azimuthal angle of d(z;t). For an illustra-\ntion of the geometry considered, see Fig. 1. For a single\ncomponent dipolar BEC, F(r;t) has a \fxed direction.\nTo study the relation of the Stoner-Wohlfarth model, in\nwhichF(r;t) changes its direction, with a dipolar BEC,\na multi-component dipolar BEC should therefore be em-\nployed.\nIn the quasi-1D approximation, the order parameter\n \u000b(r;t) is commonly assumed to be of the form\n \u000b(r;t) =e\u0000\u001a2=(2l2\n?)\nl?p\u0019\t\u000b(z;t): (6)\nwherel?is the harmonic oscillator length in the x\u0000\u0000y\nplane and\u001a=p\nx2+y2. Assuming our gas is in the\nhomogeneous local spin-orientation limit, we may also\napply a single mode approximation in space so that\n\t\u000b(z;t) = \t uni(z;t)\u0010\u000b(t). The time-dependent spinor\npart is\n\u0010\u000b(t) =h\u000bje\u0000i^fz\u001e(t)e\u0000i^fy\u0012(t)jSi; (7)\nfor spin-Sparticles [26, 31] and the normalization reads\nj\u0010(t)j2:=\u0010y(t)\u0010(t) = 1. Finally, due to the ( i\u0000\u0000)\nfactor on the left-hand side of Eq. (3), for the ease of\ncalculation, we may make the following ansatz for the\n \u000b(r;t), cf. Ref. [35],\n \u000b(r;t) =e\u0000\u001a2=(2l2\n?)\nl?p\u0019\t (z;t)\u0010\u000b(t)e\u0000(i+\u0000)!?t=(1+\u00002):\n(8)4\nFrom our ans atze in Eq. (7) and (8), one concludes that\nthe expectation value of the (spatial) spin-density oper-\nator is\n~Fx(r;t) =~Se\u0000\u001a2=l2\n?\n\u0019l2\n?j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002)\n\u0002sin\u0012(t) cos\u001e(t);\n~Fy(r;t) =~Se\u0000\u001a2=l2\n?\n\u0019l2\n?j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002)\n\u0002sin\u0012(t) sin\u001e(t);\n~Fz(r;t) =~Se\u0000\u001a2=l2\n?\n\u0019l2\n?j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002)\n\u0002cos\u0012(t): (9)\nThe above equations lead to unidirectional local magneti-\nzation, which has been assumed in Eqs. (5), in the quasi-\n1D limit (after integrating out the strongly con\fning xandyaxes). Note however that our ansatz in Eq. (8) is\nsu\u000ecient, but not necessary for the homogeneous-local-\nspin-orientation limit, and the homogeneous-local-spin-\norientation ansatz is thus designed to render our ap-\nproach as simple as possible.\nBecause we are not assuming any speci\fc form of\n\t (z;t) in our ansatz in Eq. (8),we cover every possible\ntime behavior of j (r;t)j2:= y(t) (t):\nj (r;t)j2=e\u0000\u001a2=l2\n?\n\u0019l2\n?j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002):(10)\nEq. (10) explicitly shows that Eq. (8) does not imply\nan exponentially decaying wavefunction with time since\nj\t (z;t)j2can be any physical function of time t. How-\never, the ansatz (8) simpli\fes the resulting equation for\n\t(z;t), Eq.(11) below.\nBy integrating out the xandydirections, the GP equa-\ntion for a quasi-1D spin- SBEC can be written as (see\nfor a detailed derivation Appendix B)\n(i\u0000\u0000)~@f\t (z;t)\u0010\u000b(t)g\n@t=\u001a\n\u0000~2\n2m@2\n@z2+V(z) +c0\n2\u0019l2\n?n(z;t)\u001b\n\t (z;t)\u0010\u000b(t)\n+~[\u0000b+SfM(t)\u00003Mz(t)ezgPdd(z;t)]\u00018\n<\n:SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f\t (z;t)\u0010\f(t)9\n=\n;\n+SX\nk=1c2k\n2\u0019l2\n?n(z;t)X\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSM\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t)8\n<\n:SX\n\f=\u0000S\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\t (z;t)\u0010\f(t)9\n=\n;;\n(11)\nwhere we de\fned the two functions\nM\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t):=1\nSSX\n\u000b;\f=\u0000S\u0010y\n\u000b(t)\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\u0010\f(t); (12)\nPdd(z;t):=cdd\n2~l3\n?Z1\n\u00001dz0n(z0;t)\u001a\nG\u0012jz\u0000z0j\nl?\u0013\n\u00004\n3\u000e\u0012z\u0000z0\nl?\u0013\u001b\n; (13)\nwith the axial density n(z;t):=R\nd2\u001aj (r;t)j2=\nj\t (z;t)j2e\u00002\u0000!?t=(1+\u00002), whereR\nd2\u001a :=R1\n\u00001dxR1\n\u00001dy. Finally, the function Gappearing\ninPddis de\fned as\nG(\u0015):=r\u0019\n2\u0000\n\u00152+ 1\u0001\ne\u00152=2Erfc\u0012\u0015p\n2\u0013\n\u0000\u0015:(14)\nWe plotG(\u0015) as a function of \u0015in Fig. 2. Eq.(11) rep-\nresents our starting point for analyzing the dynamics of\nmagnetization. We will now proceed to show how it leads\nto the LLG equation and the Stoner-Wolfarth model.IV. EFFECTIVE LANGRANGIAN\nDESCRIPTION\nTo provide a concise phase space picture of the conden-\nsate magnetization dynamics, we discuss in this section a\ncollective coordinate Lagrangian appropriate to our sys-\ntem.\nLetM(t):=d(z;t)=d(z;t) where the magne-\ntizationd(z;t) is de\fned in Eq. (5). Explicitly,\nthe local magnetization direction reads M(t) =\n(sin\u0012(t) cos\u001e(t);sin\u0012(t) sin\u001e(t);cos\u0012(t)). Then, from\nEqs. (9) and (10), F(r;t) =SM(t)j (r;t)j2and one5\n1 2 3 4 50.20.40.60.81.01.2\nFIG. 2. The function G(\u0015) de\fned in Eq. (14). Note that\nG(\u0015)'2=\u00153+O\u0000\n\u0015\u00005\u0001\nfor\u0015\u001d1, soG(\u0015) is always positive\nfor\u0015\u00150.\nobtains (see for a detailed derivation Appendix B)\n@M\n@t=M\u0002fb+S\u00030\ndd(t)Mzezg\u0000\u0000M\u0002@M\n@t;(15)\nwhere the renormalized interaction function \u00030\ndd(t) reads\n\u00030\ndd(t) =3\nN(t)Z1\n\u00001dzn(z;t)Pdd(z;t);(16)\nandN(t):=R\nd3rj (r;t)j2=R1\n\u00001dz n (z;t). From\nEqs. (A9), (A12), and (13), \u00030\ndd(t) is connected to the\ndipole-dipole interaction contribution Vdd(t) by\nVdd(t) =3\n2~S2\u001a\nsin2\u0012(t)\u00002\n3\u001bZ1\n\u00001dzn(z;t)Pdd(z;t)\n=~\n2S2N(t) \u00030\ndd(t)\u001a1\n3\u0000cos2\u0012(t)\u001b\n: (17)\nWe note that in order to obtain the e\u000bective quasi-1D\ndipolar interaction (17), we did not use, in distinction\nto Ref. [27], any simplifying approximation. A detailed\nderivation is provided in Appendix A.\nEq.(15) is the LLG equation with the external mag-\nnetic \feld in z-direction modi\fed by the magnetiza-\ntion inz-direction due to the dipole-dipole interac-\ntion. The corresponding term in units of magnetic \feld,\n~S\u00030\ndd(t)Mzez=(gF\u0016B), can be seen as an additional\nmagnetic \feld that is itself proportional to the magne-\ntization inz-direction, and which leads to an additional\nnonlinearity in the LLG equation.\nFrom Eqs. (13) and (16), to get how \u00030\ndd(t) depends\non timet, one has to calculate the double integral\nZ\ndzZ\ndz0n(z;t)n(z0;t)\u001a\nG\u0012jz\u0000z0j\nl?\u0013\n\u00004\n3\u000e\u0012z\u0000z0\nl?\u0013\u001b\n:\n(18)\nTo achieve a simple physical picture, we assume that\nn(z;t) does not depend on time twithin the time range\nwe are interested in. Then we may write \u00030\ndd(t) = \u00030\ndd.\nThe lifetime of a typical dipolar BEC with large atomicmagnetic dipole moments such as164Dy [36],162Dy and\n160Dy [37], or166Er [30] is of the order of seconds. Since\ntaking into account the time dependence of n(z;t) gen-\nerally requires a numerical solution of Eq. (11), we here\nconsider the case where n(z;t) is constant in time tas\nin [26], to predominantly extract the e\u000bect of magnetic\ndipole-dipole interaction per se .\nWe also neglect the possible e\u000bect of magnetostriction.\nThe latter e\u000bect, amounting to a distortion of the aspect\nratio of the condensate in a harmonic trap as a func-\ntion of the angle of the external magnetic \feld with the\nsymmetry axis of the trap, was measured in a conden-\nsate of Chromium atoms [38] (with a magnetic moment\nof 6\u0016B). The magnetostriction e\u000bect in that experiment\nwas of the order of 10%. For alkali atoms with spin-1\nthe e\u000bect should be a factor 62smaller. In addition, the-\noretical analyses in the Thomas-Fermi limit show that\nmagnetostriction in harmonic traps becomes particularly\nsmall for very small or very large asymmetries of the trap\n[39, 40].\nMore speci\fcally, Ref. [41] has shown that magne-\ntostriction is due to the force induced by the dipole-dipole\nmean-\feld potential \b dd(r;t). In Appendix D, we ap-\nply the approach of [41] to a dipolar spinor BEC. From\nEqs. (16), (17), (A1), and (D5), \u00030\ndd(t) contains \b dd(z;t)\n[the quasi-1D form of \b dd(r;t) de\fned in Eq. (D5)] by\nS2\b\n1\u00003M2\nz(t)\t\nN(t)~\u00030\ndd(t)\n= 3Z1\n\u00001dzn(z;t) \bdd(z;t):(19)\nHence, our LLG-type equation in Eq. (15) e\u000bectively con-\ntains the dipole-dipole mean-\feld potential which causes\nmagnetostriction and the form of Eq. (15) itself will not\nbe changed whether the e\u000bect of magnetostriction is large\nor not. Only the value of \u00030\ndd(t) will be changed because\nmagnetostriction changes the integration domain. Fur-\nthermore, we show in Appendix D that for our quasi-1D\nsystem, the e\u000bect of magnetostriction is smaller in a box\ntrap than in harmonic trap. In fact, for the box trap,\nthis e\u000bect can be neglected if Lz=l?is su\u000eciently large.\nThus, we may neglect the e\u000bect of magnetostriction un-\nder suitable limits for both box and harmonic traps.\nTo get a simple physical idea of the dynamical behavior\nof our system, let us, for now, assume that there is no\ndamping, \u0000 = 0. When the external magnetic \feld is\nchosen to lie in the x\u0000zplane,B= (Bx;0;Bz), Eq. (15)\nbecomes\nd\u0012\ndt=bxsin\u001e;\nd\u001e\ndt=bxcot\u0012cos\u001e\u0000bz\u0000S\u00030\nddcos\u0012: (20)\nwhere we already de\fned the Larmor frequency vector\nb=gF\u0016BB=~below Eq. (3).\nBy using the Lagrangian formalism introduced in [42],6\nthe Lagrangian Lof this system then ful\flls\nL\n~=_\u001ecos\u0012+bxsin\u0012cos\u001e+bzcos\u0012+S\n4\u00030\nddcos (2\u0012);\n(21)\nwhere _\u001e=d\u001e=dt . The equations of motion are\n1\n~@L\n@\u0012=\u0000_\u001esin\u0012+bxcos\u0012cos\u001e\u0000bzsin\u0012\u0000S\n2\u00030\nddsin (2\u0012);\n@L\n@_\u0012= 0;1\n~@L\n@\u001e=\u0000bxsin\u0012sin\u001e;1\n~@L\n@_\u001e= cos\u0012:(22)\nOne easily veri\fes that Eq. (21) is indeed the Lagrangian\nwhich gives Eqs. (20). Let p\u0018be the conjugate momen-\ntum of the coordinate \u0018. Sincep\u0012= 0 andp\u001e=~cos\u0012\n(~times thezcomponent of M), the Hamiltonian His\ngiven by\nH=\u0000bxq\n~2\u0000p2\n\u001ecos\u001e\u0000bzp\u001e+~2\u00002p2\n\u001e\n4~S\u00030\ndd:(23)\nNote that the energy ~E:=H\u0000~S\u00030\ndd=4 is conserved.\nHence, if we put p\u001e= (p\u001e)inand\u001e=\u0019=2 at some time\nt=t0,~E=\u0000bz(p\u001e)in\u0000S\u00030\ndd(p\u001e)2\nin=2~. We can then\nexpress\u001eas a function of p\u001eas\ncos\u001e=\u0000~E+bzp\u001e+1\n2~S\u00030\nddp2\n\u001e\nbxq\n~2\u0000p2\n\u001e\n=\b\n(p\u001e)in\u0000p\u001e\tbz+S\u00030\ndd(p\u001e)in+p\u001e\n2~\nbxq\n~2\u0000p2\n\u001e: (24)\nThe canonical momentum p\u001eremains the initial ( p\u001e)in\nwhenbx= 0, implying that \u0012does not change when\nbx= 0, consistent with Eqs. (20). If jbxjis larger than\njbz\u0006S\u00030\nddj, we can have p\u001e6= (p\u001e)inwithjcos\u001ej\u00141,\nwhich allows for the switching process of the magneti-\nzation. Below a threshold value of jbxjthat depends on\nbzandS\u00030\ndd,p\u001ehas to remain constant for Eq. (24) to\nbe satis\fed, which corresponds to simple magnetization\nprecession about the zaxis.\nWhenp\u001eis a function of time, there are two important\ncases:\n(a)jbzj\u001dS\u00030\ndd: cos\u001e=bz\nbx(p\u001e)in\u0000p\u001eq\n~2\u0000p2\n\u001e;\n(b)jbzj\u001cS\u00030\ndd: cos\u001e=S\u00030\ndd\n2bx(p\u001e)2\nin\u0000p2\n\u001e\n~q\n~2\u0000p2\n\u001e:(25)\nWe plot the corresponding phase diagrams ( \u0012vs\u001e) in\nFig. 3.\nLet (p\u001e)in=~cos\u0012in,bx=bsin\u00120, andbz=bcos\u00120.\nWhen case (a) holds jbzj\u001dS\u00030\ndd, one concludes that\ncos\u00120cos\u0012+ sin\u00120sin\u0012cos\u001e= cos\u00120cos\u0012in, which is\nconstant. Since d\u0001b=db(cos\u00120cos\u0012+ sin\u00120sin\u0012cos\u001e),\nin case (a) the magnetization dprecesses around the ex-\nternal magnetic \feld B, as expected. When (b) holds,\nSW switching can occur, to the description of which we\nproceed in the following.\n0.5 1.0 1.5 2.0\n-1.0-0.50.51.0(p\u001e)in=~=2 and\u001ein=\u0019=2\n0.5 1.0 1.5 2.0\n-1.0-0.50.51.0\n(p\u001e)in=\u0000~=2and\u001ein=\u0019=2\nFIG. 3.p\u001e=~vs\u001e=\u0019when \u0000 = 0 (no dissipation), with initial\nvalues (p\u001e)inand\u001ein(initial value of \u001e) as shown. (1) Dashed\nblue:bz=bx= 0:2 andjbzj\u001dS\u00030\ndd. (2) Black line: bz=bx=\n0:2 andS\u00030\ndd=bx= 0:6. (3) Dash-Dotted red: S\u00030\ndd=bx= 0:6\nandjbzj\u001cS\u00030\ndd. (4) Dotted orange horizontal line: bx= 0.\nV. CONNECTION TO STONER-WOHLFARTH\nMODEL\nThe phenomenological SW model can be directly read\no\u000b from the equations in the preceding section. From\nEq. (23), ~H:=H+~S\u00030\ndd=4 is given by\n~H\n~=\u0000bxsin\u0012cos\u001e\u0000bzcos\u0012+S\u00030\ndd\n2sin2\u0012:\n(26)\nLet (b\u0017)crbe the value of b\u0017at the stability limit where\n@~H=@\u0012 = 0 and@2~H=@\u00122= 0. Then one obtains the\ncritical magnetic \felds\n(bx)crcos\u001e=S\u00030\nddsin3\u0012;(bz)cr=\u0000S\u00030\nddcos3\u0012:\n(27)\nwhich satisfy the equation\nf(bx)crcos\u001eg2=3+ (bz)2=3\ncr=fS\u00030\nddg2=3: (28)\nWe coin the curve in the ( bx;bz)-plane described by\nEq. (28) the switching curve, in accordance with the ter-\nminology established in [43]. Because \u001echanges in time\n[see Eqs. (20) and Fig. 3], the switching curve depends\nin general on the timing of the applied external magnetic\n\felds. We note that, for \u001e= 0, Eqs. (26) and (28) are\nidentical to the SW energy functional\nHSW\n~=\u0000bxsin\u0012\u0000bzcos\u0012+Ksin2\u0012 (29)7\nand the SW astroid [43], respectively, if we identify K=\nS\u00030\ndd=2.\nThe LLG equation in Eq. (15) has stationary solu-\ntions withMparallel to the e\u000bective magnetic \feld\n~fb+S\u00030\ndd(t)Mzezg=(gF\u0016B). Since we set bto lie\nin thexzplane,\u001ewill go to zero for su\u000eciently large\ntimes. Thus Eq. (26) leads to the SW model (29) due\nto the damping term in (15) if \u0000 >0. In Appendix C,\nwe demonstrate that a more general tensorial damping\ncoe\u000ecient \u0000 introduces additional terms on the right-\nhand side of the LLG equation (15), which involve time\nderivatives . While these will thus not a\u000bect the SW phe-\nnomenology, which results from the steady states as func-\ntion of the applied magnetic \felds, and which is thus gov-\nerned by the vanishing (in the stationary limit) of the \frst\nterm on the right-hand side of the LLG equation, they\na\u000bect the detailed relaxation dynamics of the magneti-\nzation and its time scales. These deviations can hence\ncan be used to probe deviations from assuming a single\nscalar \u0000.\nBefore we move on to the next section, we show the\ncharacteristic behavior of \u00030\nddde\fned in Eq. (16), for\na box-trap scenario de\fned by n(z;t) =N=(2Lz) for\n\u0000Lz\u0014z\u0014Lzandn(z;t) = 0 otherwise ( Nis number\nof particles).\nWe stress that due to the \fnite size of the trap along\nthe \\long\" zdirection, in variance with the Hohenberg-\nMermin-Wagner theorem holding for in\fnitely extended\nsystems in the thermodynamic limit, a quasi-1D BEC can\nexist also at \fnite temperatures [44]. This remains true\nup to a ratio of its proper length to the de-Broglie wave-\nlength [45], beyond which strong phase \ructuations set\nin [46]. In fact, these strongly elongated quasi-1D BECs\nat \fnite temperature have been \frst realized already long\nago, cf., e.g. [47].\nFor the box trap, \u00030\ndd= \u0003dd(Lz=l?) where\n\u0003dd(\u0015) =3Ncdd\n2~l3\n?1\n\u0015(Z2\u0015\n0dv\u0010\n1\u0000v\n2\u0015\u0011\nG(v)\u00002\n3)\n:\n(30)\nFrom Eq. (14), G(v)'2=v3+O\u0000\nv\u00005\u0001\nforv\u001d1, so that\n\u0003dd(\u0015)'Ncdd\n2~l3\n?1\n\u0015for\u0015=Lz\nl?\u001d1: (31)\nHence \u0003dd(\u0015) is a slowly decreasing function of the\ncigar's aspect ratio \u0015(keeping everything else \fxed). We\nwill see below that for the parameters of experiments\nsuch as [30], the e\u000bective magnetic \feld due to dipolar\ninteractions greatly exceeds the externally applied mag-\nnetic \felds (in the range relevant for SW switching to be\nobserved) [48].\nVI. ANALYTICAL RESULTS FOR AXIALLY\nDIRECTED EXTERNAL MAGNETIC FIELD\nWithout dissipation, when bx= 0,p\u001e=~cos\u0012=~Mz\nis rendered constant; see Eq. (20). However, in the pres-ence of dissipation, Mzchanges in time even if bx= 0.\nBy employing this change, we propose an experimental\nmethod to measure \u0000.\nFor simplicity, we will assume that the number density\nis constant in time (also see section IV) and the external\nmagnetic \feld points along the zdirection,B=Bzez.\nLet a critical (see for a detailed discussion below) value\nof the magnetization be\n(Mz)cr:=\u0000bz\nS\u00030\ndd: (32)\nThen Eq. (15) can be written as\n@M\n@t=S\u00030\nddM\u0002ezfMz\u0000(Mz)crg\u0000\u0000M\u0002@M\n@t\n=M\u0002ez(bz+S\u00030\nddMz)\u0000\u0000M\u0002@M\n@t:(33)\nSinceM\u0001@M\n@t= 0, by taking the cross product with M\non both sides of Eq. (15), one can derive an expression\nforM\u0002@M\n@t:\n@Mz\n@t=\u0000\u0000S\u00030\ndd\n1 + \u00002fMz\u0000(Mz)crg\u0000\nM2\nz\u00001\u0001\n=\u0000\u0000\n1 + \u00002(bz+S\u00030\nddMz)\u0000\nM2\nz\u00001\u0001\n:(34)\nSinceMis the scaled magnetization, jMj= 1 with a\ncondensate. Hence, \u00001\u0014Mz\u00141. Also, according to\nthe discussion below Eq. (26), the generally positive SW\ncoe\u000ecient (with units of frequency) KisS\u00030\ndd=2.\nFrom Eq. (34), for time-independent \u00030\ndd, one con-\ncludes that there are three time-independent solutions,\nMz= (Mz)crandMz=\u00061. For a box-trapped BEC\nand constant number density, \u00030\ndd= \u0003ddwhich is always\npositive in the quasi-1D limit (cf. Eq. (30) and the discus-\nsion following it). For some arbitrary physical quasi-1D\ntrap potential, in which the number density is not con-\nstant in space, from Eqs. (13), (16), and Fig. 2, one can\ninfer that \u00030\ndd>0, due to the fact that the quasi-1D num-\nber density n(z;t)>0,n(z;t) has its maximum value\nnearz= 0 for a symmetric trap centered there, and then\nG(\u0015) also has its maximum value near \u0015= 0. Then, if\nj(Mz)crj<1,Mz= (Mz)cris an unstable solution and\nMz=\u00061 are stable solutions. When j(Mz)crj<1 and\n\u00001< Mz<(Mz)cr,Mzgoes to\u00001. Likewise, Mzgoes\nto 1 when ( Mz)cr< Mz<1. This bifurcation does not\noccur ifj(Mz)crj>1. For simplicity, we assume that\nj(Mz)crj<1. This is the more interesting case due to\nthe possibility of a bifurcation of stable solutions leading\nto SW switching.\nLet (Mz)inbe the value of Mzatt= 0. The analytic8\nsolution of Eq. (34) satis\fes\nt=1 + \u00002\n\u0000S\u00030\ndd\"\n1\nf(Mz)crg2\u00001ln\u001a(Mz)in\u0000(Mz)cr\nMz\u0000(Mz)cr\u001b\n\u00001\n2f1\u0000(Mz)crgln\u001a1\u0000Mz\n1\u0000(Mz)in\u001b\n+1\n2f1 + (Mz)crgln\u001a1 + (Mz)in\n1 +Mz\u001b\u0015\n=1 + \u00002\n\u0000\"\nS\u00030\ndd\nb2z\u0000(S\u00030\ndd)2ln\u001abz+S\u00030\ndd(Mz)in\nbz+S\u00030\nddMz\u001b\n\u00001\n2 (bz+S\u00030\ndd)ln\u001a1\u0000Mz\n1\u0000(Mz)in\u001b\n\u00001\n2 (bz\u0000S\u00030\ndd)ln\u001a1 + (Mz)in\n1 +Mz\u001b\u0015\n: (35)\nThe above equation tells us that, if ( Mz)in6= (Mz)crand\n(Mz)in6=\u00061,Mzgoes to its stable time-independent\nsolution (jMzj= 1) at time t=1. Thus, we de\fne\nacritical switching time tcrto be the time when jMzj=\n0:99. Also, note that the form of LLG equation (Eq. (33))\ndoes not change whether BEC is con\fned in a quasi-\n1D, quasi-2D, or a three-dimensional geometry. This is\nbecause one can \fnd a connection between \u00030\nddand the\ne\u000bective dipole-dipole-interaction potential Ve\u000b, so one\ncan measure \u0000 even if the BEC is e\u000bectively con\fned in\na space with dimension higher than one, using Eq. (35).\nWe point out, in particular, that tcris inversely pro-\nportional to \u00030\ndd. Hence, for a constant density quasi-\n1D BEC con\fned between \u0000Lz\u0014z\u0014Lz, \u00030\ndd=\n\u0003dd(Lz=l?), and thus tcris also inversely proportional\nto the linear number density along z. This follows from\nthe relation between \u0003 dd(Lz=l?) and the linear numberdensity along zdisplayed in Eq. (30).\nFor large dipolar interaction, the asymptotic expres-\nsion fortcris, assuming \u0000\u001c1\ntcr'1\n\u0000S\u00030\nddln\"\n5p\n2(1\u0000(Mz)2\nin)\nj(Mz)in\u0000(Mz)crj#\n(36)\nprovidedS\u00030\ndd\u001djbzj () j (Mz)crj\u001c1:\nThe above tcrdiverges at ( Mz)in= (Mz)cror\u00061, as\nexpected, since Mz= (Mz)crandMz=\u00061 are time-\nindependent solutions of the LLG equation. We stress\nthat Eq. (36) clearly shows that the magnetic dipole-\ndipole interaction accelerates the decay of Mz. Hence, by\nusing a dipolar spinor BEC with large magnetic dipole\nmoment such as produced from164Dy or166Er one may\nobserve the relaxation of Mzto the stable state within\nthe BEC lifetime, enabling the measurement of \u0000.\nBefore we show how the critical switching time tcr\ndepends on ( Mz)inand \u0000, we will qualitatively discuss\nwhen our quasi-1D assumption and homogeneous-local-\nspin-orientation assumption are valid. Typically, spin-\nspin-interaction couplings are much smaller than their\ndensity-density-interaction counterparts, by two orders\nof magnitude. For spin 123Na BEC or spin 187Rb BEC,\nc0'100jc2j[31, 34]. Thus we may neglect to a \frst ap-\nproximation the S2timesc2kterms in Eq. (11) (see the\ndiscussion at the end of Appendix D). We also require\nj(Mz)crj<1. Thus, we may additionally neglect the b\nterm compared to the Pdd(z;t) term since, for b=bzez,\nS\u00030\ndd>jbjshould be satis\fed to make j(Mz)crj<1 (see\nEq. (32)) and \u00030\nddis related to Pdd(z;t) by Eq. (16).\nWhen \u0000 = 0, using our ansatz in Eq. (8) and integrating\nout thexandydirections, Eq. (D4) can be approximated\nby the expression\n\u0016(t) \t (z;t) =\u001a\n\u0000~2\n2m@2\n@z2+V(z) +c0\n2\u0019l2\n?j\t (z;t)j2+ \bdd(z;t)\u001b\n\t (z;t); (37)\nwhere, from Eqs. (D5), (A1), and (17), the dipole-dipole interaction mean-\feld potential reads\n\bdd(z;t) =~S2\b\n1\u00003M2\nz(t)\t\nPdd(z;t)\n=cdd\n2l3\n?S2\b\n1\u00003M2\nz(t)\tZ1\n\u00001dz0j\t (z0;t)j2\u001a\nG\u0012jz0\u0000zj\nl?\u0013\n\u00004\n3\u000e\u0012z0\u0000z\nl?\u0013\u001b\n=cdd\n2\u0019l2\n?\u0019S2\b\n1\u00003M2\nz(t)\t\u001aZ1\n\u00001d\u0016zj\t (z+ \u0016zl?;t)j2G(j\u0016zj)\u00004\n3j\t (z;t)j2\u001b\n: (38)\nFrom Fig. 2, the function G(\u0015) is positive and decreases\nexponentially as \u0015increases. Thus, if l?is small enough\nsuch thatj\t (z+ \u0016zl?;t)j2does not change within the\nrangej\u0016zj\u00145, one may conclude that\n\bdd(z;t)'2\u0019\n3S2\b\n1\u00003M2\nz(t)\tcdd\n2\u0019l2\n?j\t (z;t)j2;(39)due to the propertyR1\n0d\u0015G (\u0015) = 1.\nA spinor (S= 6) dipolar BEC has been realized using\n166Er [30]. For this BEC, c0= 4\u0019~2a=m wherea'67aB\n(aBis Bohr radius) and 2 \u0019S2cdd=3 = 0:4911c0. Due to\njMz(t)j\u00141 from the de\fnition of M(t), the maximum\nvalue of the chemical potential \u0016(t) is achieved when9\nMz(t) = 0, where\n\u0016(t)'V(z) +\u0012\nc0+2\u0019\n3S2cdd\u0013n(z;t)\n2\u0019l2\n?: (40)\nFrom above Eq. (40), we may regard the 3D number\ndensity as n(z;t)=\u0000\n2\u0019l2\n?\u0001\n. In [30], N= 1:2\u0002105,\n!?=(2\u0019) =p156\u0002198 Hz = 175 :75 Hz,!z=(2\u0019) =\n17:2 Hz,l?= 0:589\u0016m, and the measured peak number\ndensity \u0016npeakis 6:2\u00021020m\u00003. Using Eq. (37) and (39),\nby denoting Lzas the Thomas-Fermi radius along z,\n(\u0000Lz\u0014z\u0014Lz) withV(z) =m!2\nzz2=2, one derives\nLz=(\n3\u0000\nc0+ 2\u0019S2cdd=3\u0001\nN\n4\u0019m!2zl2\n?)1=3\n; (41)\nand the mean number density \u0016 n= (N=2Lz)=\u0000\n2\u0019l2\n?\u0001\n=\n6:721\u00021020m\u00003as well as chemical potential \u0016=(~!?) =\nm!2\nzL2\nz=(2~!?) = 23:22. Note that \u0016 n'1:1 \u0016npeak. Be-\ncause\u0016is not less than ~!?, the experiment [30] is not\nconducted within the quasi-1D limit.\nThe homogeneous-local-spin-orientation approxima-\ntion is valid when the system size is on the order of the\nspin healing length \u0018sor less, which has been experimen-\ntally veri\fed in in [34]. Using c0'100jc2j,\u0018s'10\u0018d\nwhere\u0018d=p\n~2=(2mc0\u0016n) is the density healing length\nand\u0018s=p\n~2=(2mjc2j\u0016n) is the spin healing length.\nThus, ifLzis on the order of 10 \u0018d, the homogeneous-\nlocal-spin-orientation approximation is justi\fed.\nUsing the S= 6 element166Er, we can provide\nnumerical values which satisfy both the quasi-1D and\nhomogeneous-local-spin-orientation limits, as well as\nthey enable us to explicitly show how tcrdepends on\n(Mz)inin a concretely realizable setup. We consider be-\nlow two cases: (A) box trap along z[49] and (B) harmonic\ntrap alongz.\nA. Box traps\nWe setV(z) = 0 forjzj< Lzand1other-\nwise. Then n(z;t) =N=(2Lz) and we estimate \u0016'\u0000\nc0+ 2\u0019S2cdd=3\u0001\nN=\u0000\n4\u0019l2\n?Lz\u0001\nfrom Eq. (40). In this\ncase, \u00030\ndd= \u0003dd(Lz=l?) as is calculated in Eq. (30).\nFixingBz=\u00000:03 mG and N= 100, we con-\nsider the following two cases: (1) !?=(2\u0019) = 2:4\u0002\n104Hz andLz= 3:125\u0016m. Then Lz=l?= 62:03,\n\u0016=(~!?) = 0:1692, and Lz=\u0018d= 29:55. Thus,\nthe system is in both the quasi-1D and homogeneous-\nlocal-spin-orientation limit. S\u0003dd(Lz=l?) = 4:074\u0002\n103Hz,~S\u0003dd(Lz=l?)=(gF\u0016B) = 0:3969 mG, and \u0012cr:=\ncos\u00001(Mz)cris 85:67\u000e.\n(2)!?=(2\u0019) = 1:2\u0002104Hz andLz= 6:250\u0016m.\nThenLz=l?= 87:72,\u0016=(~!?) = 0:0846, and\nLz=\u0018d= 29:55. Thus, again the system is\nin both the quasi-1D and homogeneous-local-spin-\norientation limits. S\u0003dd(Lz=l?) = 1:028\u0002103Hz,\n~S\u0003dd(Lz=l?)=(gF\u0016B) = 0:1002 mG, and \u0012cr= 72:57\u000e.\nFig. 4 shows the relation between tcrand (Mz)in.\n-1.0 -0.5 0.5 1.00.050.100.15!?=(2\u0019) = 2:4\u0002104Hz,Lz= 3:125\u0016m, andl?= 0:0504\u0016m\nwhereN=\u0000\n4\u0019Lzl2\n?\u0001\n= 10:03\u00021020m\u00003((Mz)cr= 0:0756).\n-1.0 -0.5 0.5 1.00.20.40.6\n!?=(2\u0019) = 1:2\u0002104Hz,Lz= 6:250\u0016m, andl?= 0:0712\u0016m\nwhereN=\u0000\n4\u0019Lzl2\n?\u0001\n= 2:508\u00021020m\u00003((Mz)cr= 0:2995).\nFIG. 4.tcras a function of ( Mz)inwhenB=Bzezwhere\nBz=\u00000:03 mG and particle number N= 100. From top to\nbottom: Red for \u0000 = 0 :01, black for \u0000 = 0 :03, and blue for\n\u0000 = 0:09. Lines are from exact analytic formula in Eq. (35),\nand dot-dashed are from asymptotic expression in Eq. (36).\nGenerally,tcrdecreases as \u0000 increases. Also, note that tcr\ndiverges as ( Mz)in!(Mz)cr. For larger mean number den-\nsityN=\u0000\n4\u0019Lzl2\n?\u0001\n(top), the asymptotic expression of tcris\nessentially indistinguishable from the exact analytic formula\noftcr.\nB. Harmonic traps\nWe setV(z) =m!2\nzz2=2. Using the Thomas-Fermi\napproximation, from Eq. (40), \u0016=m!2\nzL2\nz=2 whereLz\nis given by Eq. (41).\u0000\nc0+ 2\u0019S2cdd=3\u0001\nn(z;t)=\u0000\n\u0019l2\n?\u0001\n=\nm!2\nz\u0000\nL2\nz\u0000z2\u0001\nforjzj\u0014Lzandn(z;t) = 0 forjzj>Lz.\nFrom thisn(z;t), we performed a numerical integration\nto calculate \u00030\nddin Eq. (16). Fixing Bz=\u00000:03 mG, we\nconsider the following two cases:\n(1)N= 240,!?=(2\u0019) = 2000 Hz, and !z=(2\u0019) =\n50 Hz, for which Lz= 5:703\u0016m andLz=l?= 32:68.\nWe obtain again the quasi-1D and homogeneous-local-\nspin-orientation limits since \u0016=(~!?) = 0:3337 and\nLz=\u0018d= 17:85. Furthermore, S\u00030\ndd= 1:644\u0002103Hz,\n~S\u00030\ndd=(gF\u0016B) = 1:602\u000210\u00001mG, and\u0012cr= 79:21\u000e.\n(2)N= 340,!?=(2\u0019) = 1000 Hz, and !z=(2\u0019) =\n25 Hz, where Lz= 8:070\u0016m andLz=l?= 32:70. Again,\nwe have the quasi-1D and with homogeneous-local-spin-\norientation limits ful\flled due to \u0016=(~!?) = 0:3341 and10\n-1.0 -0.5 0.5 1.00.10.20.30.4\nN= 240,!?=(2\u0019) = 2000 Hz, and !z=(2\u0019) = 50 Hz.\nLz= 5:703\u0016m andl?= 0:1745\u0016m where\nN=\u0000\n4\u0019Lzl2\n?\u0001\n= 1:010\u00021020m\u00003((Mz)cr= 0:1873).\n-1.0 -0.5 0.5 1.00.20.40.60.8\nN= 340,!?=(2\u0019) = 1000 Hz, and !z=(2\u0019) = 25 Hz.\nLz= 8:070\u0016m andl?= 0:2468\u0016m where\nN=\u0000\n4\u0019Lzl2\n?\u0001\n= 0:550\u00021020m\u00003((Mz)cr= 0:3741).\nFIG. 5.tcras a function of ( Mz)inwhenB=Bzezwhere\nBz=\u00000:03 mG, for two particle numbers Nas shown. From\ntop to bottom: Red for \u0000 = 0 :01, black for \u0000 = 0 :03, and\nblue for \u0000 = 0 :09. Lines are from exact analytic formula in\nEq. (35), and dot-dashed are from asymptotic expression in\nEq. (36). Generally, tcrdecreases as \u0000 increases. Also, note\nthattcrdiverges as ( Mz)in!(Mz)cr. For larger mean number\ndensityN=\u0000\n4\u0019Lzl2\n?\u0001\n(top), the asymptotic expression of tcris\nessentially indistinguishable from the exact analytic formula\noftcr.\nLz=\u0018d= 17:87. In addition, S\u00030\ndd= 8:230\u0002102Hz,\n~S\u00030\ndd=(gF\u0016B) = 8:019\u000210\u00002mG, and\u0012cr= 68:03\u000e.\nFig. 5 shows for the harmonic traps the relation be-\ntweentcrand (Mz)in.\nC. Measurability of critical switching time\nFigs. 4 and 5 demonstrate that the critical switching\ntimetcris much smaller than the lifetime of BEC (sev-\neral seconds [30]) and thus, by measuring tcrby varying\n(Mz)in, one will be able to obtain the value of \u0000, pro-\nvided \u0000 indeed does not depend on spin indices as for\nexample Refs. [26, 29] have assumed. Conversely, if one\nobtains from the measurements a di\u000berent functional re-\nlation which does not follow Eq. (35), this implies that \u0000\nmay depend on spin indices.\nNote that both \fgures, Figs. 4 and 5, show that tcrisinversely proportional to the mean number density\nN=\u0000\n4\u0019Lzl2\n?\u0001\n. Eq. (36) states that tcris inversely pro-\nportional to \u00030\ndd, but except for the box trap case, in\nwhich one can analytically calculate \u00030\ndd= \u0003dd(Lz=l?)\nin Eq. (30), the dependence of \u00030\nddand the mean number\ndensityN=\u0000\n4\u0019Lzl2\n?\u0001\nis not immediately apparent. Thus,\nat least for harmonic traps, and in the Thomas-Fermi ap-\nproximation, one may use the box trap results of Eq. (30)\nfor provide an approximate estimate of the behavior of\ntcr.\nVII. CONCLUSION\nFor a quasi-1D dipolar spinor condensate with\nunidirectional local magnetization (that is in the\nhomogeneous-local-spin-orientation limit), we provided\nan analytical derivation of the Landau-Lifshitz-Gilbert\nequation and the Stoner-Wohlfarth model. For an exter-\nnal magnetic \feld along the long axis, we obtained an\nexact solution of the quasi-1D Landau-Lifshitz-Gilbert\nequation. Our analytical solution demonstrates that the\nmagnetic dipole-dipole interaction accelerates the relax-\nation of the magnetization to stable states and hence\nstrongly facilitates observation of this process within the\nlifetime of typical dipolar spinor BECs. Employing this\nsolution, we hence propose a method to experimentally\naccess the dissipative parameter(s) \u0000.\nWe expect, in particular, that our proposal provides a\nviable tool to verify in experiment whether \u0000 is indeed\nindependent of spin indices, as commonly assumed, and\ndoes not have to be replaced by a tensorial quantity for\nspinor gases. We hope that this will stimulate further\nmore detailed investigations of the dissipative mechanism\nin dipolar BECs with internal degrees of freedom.\nWe considered that the magnetization along z,Mz, has\ncontributions solely from the atoms residing in the con-\ndensate, an approximation valid at su\u000eciently low tem-\nperatures. When the magnetization from noncondensed\natoms is not negligible, as considered by Ref. [5] for a\ncontact interacting scalar BEC, correlation terms mix-\ning the noncondensed part and the mean \feld, such asPS\n\f=\u0000S \u0003\n\f(r;t)h\u000e^ \u000b(r;t)\u000e^ \f(r;t)iwill appear on the\nright-hand side of Eq. (3). Here, \u000e^ \u000b(r;t) is the\u000b-th\ncomponent of quantum \feld excitations above the mean-\n\feld ground state in the spinor basis. Considering the\ne\u000bect of these terms is a subject of future studies.\nACKNOWLEDGMENTS\nThe work of SHS was supported by the National\nResearch Foundation of Korea (NRF), Grant No.\nNRF-2015-033908 (Global PhD Fellowship Program).\nSHS also acknowledges the hospitality of the Uni-\nversity of T ubingen during his stay in the summer\nof 2019. URF has been supported by the NRF11\nunder Grant No. 2017R1A2A2A05001422 and Grant No. 2020R1A2C2008103.\nAppendix A: Derivation of the e\u000bective potential Ve\u000b\nThe dipole-dipole interaction term Vdd(t) in the total energy is given by [31]\nVdd(t) =cdd\n2Z\nd3rZ\nd3r0X\n\u0017;\u00170=x;y;zF\u0017(r;t)Q\u0017;\u00170(r\u0000r0)F\u00170(r0;t); (A1)\nwherecddis dipole-dipole interaction coe\u000ecient, F\u0017(r;t) = y(r;t)^f\u0017 (r;t), andQ\u0017;\u00170(r) is de\fned as the tensor\nQ\u0017;\u00170(r):=r2\u000e\u0017;\u00170\u00003r\u0017r\u00170\nr5(A2)\nin spin space, where r=jrjandr\u0017=r\u0001e\u0017, withe\u0017being the unit vector along the \u0017axis. From now on, we de\fne\n\u001a= (x;y) such that dxdy =d2\u001a=d'd\u001a\u001a where tan'=y=x.\nUsing the convolution theorem, the dipole-dipole interaction term Vdd(t) can be expressed by\nVdd(t) =cdd\n2(2\u0019)D=2Z\nd3k~n(k;t) ~n(\u0000k;t)~Udd(k;t) (A3)\nwith the Fourier transform\nUdd(\u0011;t) =1\nn(r;t)n(r0;t)X\n\u0017;\u00170=x;y;zF\u0017(r;t)Q\u0017;\u00170(\u0011)F\u00170(r0;t); (A4)\nwhere ~g(k;t) = (2\u0019)\u0000D=2R\ndrg(r;t)eik\u0001ris the Fourier transform of the function g(r;t) inD-dimensional space r\n(in our case, D= 3),\u0011=r\u0000r0, andn(r;t) =j (r;t)j2.\nBy denoting k= (k\u001a;kz), wherek\u001a= (kx;ky) withk\u001a=q\nk2x+k2yand tan'k\u001a=ky=kx, with our mean-\feld\nwavefunction in Eq. (8), one derives\n~n(k;t) =1\n\u0019l2\n?1\n(2\u0019)3=2Z\nd2\u001aZ1\n\u00001dze\u0000(\u001a=l?)2n(z;t)ei\u001a\u0001k\u001aeikzz=1\n2\u0019~n(kz;t)e\u0000k2\n\u001al2\n?=4; (A5)\nwheren(z;t):=j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002). Note the factor of (2 \u0019)\u00001appearing, when compared to Eq. (12) in\nRef. [27], which is stemming from our de\fnition of Fourier transform.\nDenoting\u0011=j\u0011j, by writinge\u0011for the unit vector along \u0011, we obtain\nUdd(\u0011;t) =\u00001\n\u00113r\n6\u0019\n5hn\nY2\n2(e\u0011)e\u00002i\u001e(t)+Y\u00002\n2(e\u0011)e2i\u001e(t)o\nS2sin2\u0012(t)\n\u0000n\nY1\n2(e\u0011)e\u0000i\u001e(t)\u0000Y\u00001\n2(e\u0011)ei\u001e(t)o\nS2sinf2\u0012(t)gi\n+1\n\u00113r\n6\u0019\n5Y0\n2(e\u0011)r\n2\n3S2\b\n3 sin2\u0012(t)\u00002\t\n; (A6)\nwhereYm\nl(e\u0011) are the usual spherical harmonics. Its Fourier transform ~Udd(k;t) is\n~Udd(k;t) =1\n(2\u0019)3=24\u0019\n3S2\u001a\n1\u00003\n2sin2\u0012(t)\u001b\u0012\n3k2\nz\nk2\u001a+k2z\u00001\u0013\n+1p\n2\u0019k2\n\u001a\nk2\u001a+k2zS2sin2\u0012(t) cos\b\n2'k\u001a\u00002\u001e(t)\t\n+r\n2\n\u0019k\u001akz\nk2\u001a+k2zS2sinf2\u0012(t)gcos\b\n'k\u001a\u0000\u001e(t)\t\n:(A7)\nBy plugging Eq. (A5) and Eq. (A7) into Eq. (A3), we \fnally obtain Vdd(t) as\nVdd(t) =cdd\n2p\n2\u0019Z1\n\u00001dkz~n(kz;t) ~n(\u0000kz;t)2S2\nl2\n?p\n2\u0019\u001a\n1\u00003\n2sin2\u0012(t)\u001b\u001a\u0000\nk2\nzl2\n?=2\u0001\nek2\nzl2\n?=2E1\u0000\nk2\nzl2\n?=2\u0001\n\u00001\n3\u001b\n;(A8)12\nwhereE1(x) =R1\nxdue\u0000u=uis exponential integral.\nNote that Eq. (A8) can be also written as\nVdd(t) =cdd\n2p\n2\u0019Z1\n\u00001dkz~n(kz;t) ~n(\u0000kz;t)~Ve\u000b(kz;t) =cdd\n2Z1\n\u00001dzZ1\n\u00001dz0n(z;t)n(z0;t)Ve\u000b(z\u0000z0;t):(A9)\nDue to the fact that ~Ve\u000b(kz;t) can be obtained by Eq. (A8), we can get Ve\u000b(z;t) by inverse Fourier transform. As a\npreliminary step, we \frst write down some integrals of E1(x) as follows:\nZ1\n\u00001dxex2E1\u0000\nx2\u0001\ne\u0000ikx=Z1\n\u00001dxe\u0000ikxZ1\nx2dte\u0000(t\u0000x2)\nt= (\u0019)3=2ek2=4Erfc (jkj=2): (A10)\nDi\u000berentiating Eq. (A10) with respect to ktwo times results in\nZ1\n\u00001dxx2ex2E1\u0000\nx2\u0001\ne\u0000ikx=\u0000(\u0019)3=2\u001a1\n2\u0012k2\n2+ 1\u0013\nek2=4Erfc (jkj=2)\u0000jkj\n2p\u0019\u00002p\u0019\u000e(k)\u001b\n: (A11)\nTherefore,Ve\u000b(z;t) can be calculated as\nVe\u000b(z;t) =1p\n2\u0019Z1\n\u00001dkz2S2\nl2\n?p\n2\u0019\u001a\n1\u00003\n2sin2\u0012(t)\u001b\u001a\u0000\nk2\nzl2\n?=2\u0001\nek2\nzl2\n?=2E1\u0000\nk2\nzl2\n?=2\u0001\n\u00001\n3\u001b\ne\u0000ikzz\n=S2\nl3\n?\u001a3\n2sin2\u0012(t)\u00001\u001b\u001a\nG(jzj=l?)\u00004\n3\u000e(z=l?)\u001b\n; (A12)\nwhereG(x) is de\fned in Eq. (14), and \u000e(x) is the Dirac delta function.\nThe Fourier transform of Eq. (A12) acquires the form\n~Ve\u000b(kz;t) =1p\n2\u0019Z1\n\u00001dzV e\u000b(z;t)eikzz=r\n2\n\u0019S2\nl2\n?\u001a3\n2sin2\u0012(t)\u00001\u001b\u001aZ1\n0dvG (v) cos (kzl?v)\u00002\n3\u001b\n=r\n2\n\u0019S2\nl2\n?\u001a3\n2sin2\u0012(t)\u00001\u001b\u0014Z1\n0dunp\u0019\u0000\n2u2+ 1\u0001\neu2Erfc (u)\u00002uo\ncos\u0010p\n2kzl?u\u0011\n\u00002\n3\u0015\n:(A13)\nFrom [50], the following integral involving the complementary error function is\nZ1\n0dueu2Erfc (u) cos (bu) =1\n2p\u0019eb2=4E1\u0000\nb2=4\u0001\n: (A14)\nBy di\u000berentiating Eq. (A14) two times with respect to b, we get\nZ1\n0duu2eu2Erfc (u) cos (bu) =\u00001\n2p\u0019\u001a1\n2\u0012b2\n2+ 1\u0013\neb2=4E1\u0000\nb2=4\u0001\n\u00001 +2\nb2\u001b\n: (A15)\nHence, Eq. (A13) becomes\n~Ve\u000b(kz;t) =r\n2\n\u0019S2\nl2\n?\u001a3\n2sin2\u0012(t)\u00001\u001b\u0014\n\u0000\u001a1\n2\u0000\nk2\nzl2\n?+ 1\u0001\nek2\nzl2\n?=2E1\u0000\nk2\nzl2\n?=2\u0001\n\u00001 +1\nk2zl2\n?\u001b\n+1\n2ek2\nzl2\n?=2E1\u0000\nk2\nzl2\n?=2\u0001\n+1\nk2zl2\n?\u00002\n3\u0015\n=2S2\nl2\n?p\n2\u0019\u001a\n1\u00003\n2sin2\u0012(t)\u001b\u001a\u0000\nk2\nzl2\n?=2\u0001\nek2\nzl2\n?=2E1\u0000\nk2\nzl2\n?=2\u0001\n\u00001\n3\u001b\n: (A16)\nComparing Eq. (A8) with Eq. (A16), one veri\fes that Eq. (A12) is the correct result for the e\u000bective interaction of\nthe quasi-1D dipolar spinor gas.13\nAppendix B: Quasi-1D Gross-Pitaevski\u0014 \u0010 equation with dissipation\nBy introducing an identical damping coe\u000ecient for each component of the spinor, cf., e.g. Refs. [26, 29] (i.e. as if\neach component e\u000bectively behaves as a scalar BEC [28]), and neglecting a possible quadratic Zeeman term, the GP\nequation for a spin- SBEC can be written as [26]\n(i\u0000\u0000)~@ \u000b(r;t)\n@t=\u001a\n\u0000~2\n2mr2+Vtr(r) +c0j (r;t)j2\u001b\n \u000b(r;t)\u0000~SX\n\f=\u0000Sfb\u0000bdd(r;t)g\u0001\u0010\n^f\u0011\n\u000b;\f \f(r;t)\n+SX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zF\u00171;\u00172;\u0001\u0001\u0001;\u0017k(r;t)SX\n\f=\u0000S\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f \f(r;t);\n(B1)\nwhere \u000b(r;t) is the\u000b-th component of the mean-\feld wavefunction  (r;t) (the spin-space index \u000bis an integer\ntaking 2S+ 1 values running from \u0000SandS),F\u00171;\u00172;\u0001\u0001\u0001;\u0017k(r;t):= y(r;t)^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k (r;t),~^fis the spin- S\noperator,b=gF\u0016BB=~(gFis the Land\u0013 e g-factor, \u0016Bis the Bohr magneton, and Bthe external magnetic \feld).\nFinally, ~bdd(r;t)\u0001e\u0017=cddR\nd3r0P\n\u00170=x;y;zQ\u0017;\u00170(r\u0000r0)F\u00170(r0;t), wheree\u0017is the unit vector along the \u0017axis\n(\u0017=x;y;z ) [31]. Applying the formalism of Ref. [1] to a spinor BEC assuming that \u0000 does not depend on spin\nindices, one just needs to transform t!\u0000\n1 + \u00002\u0001\ntin Eq. (B1) and (8). We then integrate out the xandydirections\nin Eq. (B1) to obtain the quasi-1D GP equation.\nFrom Eq. (8) in the main text, we have\nZ\nd2\u001aSX\n\f=\u0000Se\u0000\u001a2=(2l2\n?)\nl?p\u0019\u001a\n~bdd(r)\u0001\u0010\n^f\u0011\n\u000b;\f\u001b\n \f(r;t)\n=cdd\n2l3\n?Z1\n\u00001dz0n(z0;t)\u001a\nG\u0012jz\u0000z0j\nl?\u0013\n\u00004\n3\u000e\u0012z\u0000z0\nl?\u0013\u001b\n\t (z;t)e\u0000i+\u0000\n1+\u00002!?tSfM(t)\u00003Mz(t)ezg\u0001SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f\u0010\f(t);\n(B2)\nwhereR\nd2\u001a:=R1\n\u00001dxR1\n\u00001dyandn(z;t):=R\nd2\u001aj (r;t)j2=j\t (z;t)j2e\u00002\u0000!?t=(1+\u00002).\nFor a spin-SBEC, from Eq. (B1), for the trap potential given in Eq. (4) and if we use Eq. (8), by integrating out\nthexandydirections, one acquires the expression\n(i\u0000\u0000)~@f\t (z;t)\u0010\u000b(t)g\n@t=\u001a\n\u0000~2\n2m@2\n@z2+V(z) +c0\n2\u0019l2\n?n(z;t)\u001b\n\t (z;t)\u0010\u000b(t)\n+ [\u0000~b+~SfM(t)\u00003Mz(t)ezgPdd(z;t)]\u00018\n<\n:SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f\t (z;t)\u0010\f(t)9\n=\n;\n+SX\nk=1c2k\n2\u0019l2\n?n(z;t)X\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSM\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t)8\n<\n:SX\n\f=\u0000S\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\t (z;t)\u0010\f(t)9\n=\n;;\n(B3)\nwhereM\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t) is de\fned in Eq. (12) and\nPdd(z;t) =cdd\n2~l3\n?Z1\n\u00001dz0n(z0;t)\u001a\nG\u0012jz\u0000z0j\nl?\u0013\n\u00004\n3\u000e\u0012z\u0000z0\nl?\u0013\u001b\n=cdd\n~S2\b\n3 sin2\u0012(t)\u00002\tZ1\n\u00001dz0n(z0;t)Ve\u000b(z\u0000z0;t);\n(B4)\nwithVe\u000bde\fned in (A12). It is already clear from Eq. (B3) that, besides particle loss from the condensate encoded\nin a decayingj\t(z;t)j, dissipation also leads to a dephasing , i.e. the decay of \u0010(t) due to the term \u0000\u0000@\u0010(t)=@t.\nFrom now on, if there is no ambiguity, and for brevity, we drop the arguments such as x;y;z;t from the functions.14\nFrom Eq. (B3), we then get\n~@\u0010\u000b\n@t=\u0000~\n\t@\t\n@t\u0010\u000b\u0000\u0000 +i\n1 + \u00002\u0012\n\u0000~2\n2m1\n\t@2\t\n@z2+V+c0\n2\u0019l2\n?n\u0013\n\u0010\u000b+\u0000 +i\n1 + \u00002f~b\u0000S(M\u00003Mzez)~Pddg\u00018\n<\n:SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f\u0010\f9\n=\n;\n\u0000\u0000 +i\n1 + \u00002SX\nk=1c2k\n2\u0019l2\n?nX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSM\u00171;\u00172;\u0001\u0001\u0001;\u0017k8\n<\n:SX\n\f=\u0000S\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\u0010\f9\n=\n;; (B5)\nSince@j\u0010j2\n@t= 0 due to the normalization j\u0010j2= 1, we then have\n0 = 2Re\u001a\n\u0000~\n\t@\t\n@t\u0000\u0000\n1 + \u00002\u0012\n\u0000~2\n2m1\n\t@2\t\n@z2+V+c0\n2\u0019l2\n?n\u0013\u001b\n+i\n1 + \u00002~2\n2m\u00121\n\t@2\t\n@z2\u00001\n\t\u0003@2\t\u0003\n@z2\u0013\n+2\u0000\n1 + \u00002f~b\u0000S(M\u00003Mzez)~Pddg\u0001SM\u00002\u0000\n1 + \u00002SX\nk=1c2k\n2\u0019l2\n?nX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zS2M2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k: (B6)\nHence the dynamics of the magnetization direction follows the equation\n~S@M\u0017\n@t= 2Re8\n<\n:SX\n\u000b;\f=\u0000S\u0010y\n\u000b\u0010\n^f\u0017\u0011\n\u000b;\f\u0012\n~@\u0010\f\n@t\u00139\n=\n;\n=\u00002\u0000\n1 + \u00002S2M\u0017f~b\u0000S(M\u00003Mzez)~Pddg\u0001M+2\u0000\n1 + \u00002M\u0017SX\nk=1c2k\n2\u0019l2\n?nX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zS3M2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k\n+\u0000\n1 + \u00002X\n\u0016=x;y;zf~b\u0016\u0000S(M\u0016\u00003Mz\u000e\u0016;z)~PddgSf\u000e\u0016;\u0017+ (2S\u00001)M\u0016M\u0017g\n\u00001\n1 + \u00002X\n\u0016;\u0014=x;y;zf~b\u0016\u0000S(M\u0016\u00003Mz\u000e\u0016;z)~Pddg\u000f\u0017;\u0016;\u0014SM\u0014\n\u00002Re8\n<\n:\u0000 +i\n1 + \u00002SX\nk=1c2k\n2\u0019l2\n?nX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSM\u00171;\u00172;\u0001\u0001\u0001;\u0017kSX\n\u000b;\f=\u0000S\u0010y\n\u000b\u0010\n^f\u0017^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\u0010\f9\n=\n;; (B7)\nsince the scalar product \u0010y\u0010\n^f\u000b^f\f+^f\f^f\u000b\u0011\n\u0010=Sf\u000e\u000b;\f+ (2S\u00001)M\u000bM\fg[26].\nBy direct comparison, we can identify Eq. (B8) below as being identical to Eq. (B21) in [26], the only di\u000berence\nconsisting in the de\fnition of M\u00171;\u00172;\u0001\u0001\u0001;\u0017k: We employ a scaled version of M\u00171;\u00172;\u0001\u0001\u0001;\u0017k, which is normalized to Sin\n[26]. From Eq. (7) in the main text,\nX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM\u00171;\u00172;\u0001\u0001\u0001;\u0017kSX\n\u000b;\f=\u0000S\u0010y\n\u000b\u0010\n^f\u0017^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\u0010\f=X\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017kS2M\u0017; (B8)\nwhich is real. Therefore, Eq. (B7) can be written in the following form\n@M\n@t=\u0000\u0000\n1 + \u00002M\u0002[M\u0002fb\u0000S(M\u00003Mzez)Pddg] +1\n1 + \u00002M\u0002fb\u0000S(M\u00003Mzez)Pddg\n=1\n1 + \u00002M\u0002(b+ 3SPddMzez)\u0000\u0000\n1 + \u00002M\u0002[M\u0002(b+ 3SPddMzez)]\n=M\u0002(b+ 3SPddMzez)\u0000\u0000M\u0002@M\n@t; (B9)\nsinceM\u0001@M\n@t= 0 holds.\nAsPis a function of zandt, butMis independent of z[Mis the scaled local magnetization and our aim is to\nstudy a dipolar spinor BEC with unidirectional local magnetization (the homogeneous-local-spin-orientation limit)],\nby multiplying with n(z;t) both sides of Eq. (B9) and integrating along z, we \fnally get the LLG equation\n@M\n@t=M\u0002(b+S\u00030\nddMzez)\u0000\u0000M\u0002@M\n@t; (B10)\nwhere \u00030\nddis de\fned in Eq. (16). Note here that \u00030\nddbecomes \u0003 dd(Lz=l?) de\fned in Eq. (30) when n(z;t) =N=(2Lz)\nfor\u0000Lz\u0014z\u0014Lzandn(z;t) = 0 otherwise.15\nAppendix C: Modi\fcation of the LLG equation for \u0000a spin-space tensor\nWhen \u0000 depends on spin indices, i.e. is a tensor, Eq. (B3) can be generalized to read\nSX\n\f=\u0000S(i\u000e\u000b;\f\u0000\u0000\u000b;\f)~@f\t (z;t)\u0010\f(t)g\n@t=SX\n\f=\u0000SH\u000b;\f\t (z;t)\u0010\f(t): (C1)\nThe spinor part of the wavefunction is normalized to unity, j\u0010j2= 1. Hence, we know that@j\u0010j2\n@t= 0. Therefore, from\nEq. (C1), we derive the expression\nSX\n\u000b;\f=\u0000SRe\u0014\n\u0000i\u0010\u0003\n\u000b\u0000\u000b;\f@\u0010\f\n@t\u0000i\u0010\u0003\n\u000b\u0000\u000b;\f\u0010\f1\n\t@\t\n@t\u0000i1\n~\t\u0010\u0003\n\u000bH\u000b;\f\t\u0010\f\u0015\n\u0000Re\u00141\n\t@\t\n@t\u0015\n= 0: (C2)\nThis then leads us to\n@M\u0017\n@t=2\nSSX\n\u000b;\f;\r =\u0000SRe\u0014\n\u0000i\u0010\u0003\n\u000b\u0010\n^f\u0017\u0011\n\u000b;\f\u0000\f;\r@\u0010\r\n@t\u0000i\u0010\u0003\n\u000b\u0010\n^f\u0017\u0011\n\u000b;\f\u0000\f;\r\u0010\r1\n\t@\t\n@t\u0000i1\n~\t\u0010\u0003\n\u000b\u0010\n^f\u0017\u0011\n\u000b;\fH\f;\r\t\u0010\r\u0015\n\u00002Re\u0014\nM\u00171\n\t@\t\n@t\u0015\n: (C3)\nFor scalar \u0000, \u0000 \u000b;\f!\u0000\u000e\u000b;\f, the equation above becomes Eq. (B7).\nFrom Eqs. (C2) and (C3), one concludes that the stationary solution M\u0017of Eq. (C3) is independent of \u0000. In other\nwords, whether \u0000 depends on spin indices or not, the SW model (29) is left una\u000bected, also see the discussion in\nSection V of the main text.\nAppendix D: Description of magnetostriction\nFor a dipolar spinor BEC without quadratic Zeeman term, when there is no dissipation (\u0000 = 0), the mean-\feld\nequation in Eq. (3) can be written as\n\u0016\u000b(t) \u000b(r;t) =8\n<\n:\u0000~2\n2mr2+Vtr(r) +c0SX\n\f=\u0000Sj \f(r;t)j29\n=\n; \u000b(r;t)\u0000~fb\u0000bdd(r;t)g\u0001SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f \f(r;t)\n+SX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSX\n\u000b1;\f1;\f=\u0000S\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b1;\f1\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f \u0003\n\u000b1(r;t) \f1(r;t) \f(r;t):\n(D1)\nwhere we have substituted i~@ \u000b(r;t)\n@t=\u0016\u000b(t) \u000b(r;t).\nSince we consider the homogeneous-local-spin-orientation limit, we may write  \u000b(r;t) = \t uni(r;t)\u0010\u000b(t). In this\nlimit, we have\nj (r;t)j2:= y(r;t) (r;t) =SX\n\u000b=\u0000S y\n\u000b(r;t) \u000b(r;t) =j\tuni(r;t)j2; (D2)\nsinceSX\n\u000b=\u0000Sj\u0010\u000b(t)j2= 1 from the de\fnition of \u0010\u000b(t) in Eq. (7). Thus j\tuni(r;t)j2is equal to the number density.\nThen Eq. (D1) can be written as\n\u0016\u000b(t)\u0010\u000b(t) \tuni(r;t) =\u001a\n\u0000~2\n2mr2+Vtr(r) +c0j\tuni(r;t)j2\u001b\n\u0010\u000b(t) \tuni(r;t)\n\u0000~fb\u0000bdd(r;t)g\u0001SX\n\f=\u0000S\u0010\n^f\u0011\n\u000b;\f\u0010\f(t) \tuni(r;t)\n+SSX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zSX\n\f=\u0000SM\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t)\u0010\n^f\u00171^f\u00172\u0001\u0001\u0001^f\u0017k\u0011\n\u000b;\f\u0010\f(t)j\tuni(r;t)j2\tuni(r;t):\n(D3)16\nNow, we decompose the chemical potential \u0016(t) as\u0016(t):=SX\n\u000b=\u0000S\u0016\u000b(t)j\u0010\u000b(t)j2. Then one obtains\n\u0016(t) \tuni(r;t) =\"\n\u0000~2\n2mr2+Vtr(r) +(\nc0+S2SX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t))\nj\tuni(r;t)j2#\n\tuni(r;t)\n+ [\bdd(r;t)\u0000S~fb\u0001M(t)g] \tuni(r;t);\n(D4)\nwhere\n\bdd(r;t):=S2cdd2\n4Z\nd3r08\n<\n:X\n\u0017;\u00170=x;y;zM\u0017(t)Q\u0017;\u00170(r\u0000r0)M\u00170(t)9\n=\n;j\tuni(r0;t)j23\n5; (D5)\nis the dipole-dipole mean-\feld potential [41] following from the de\fnition of bddbelow Eq. (3) in the main text.\nDue toMx(t) = sin\u0012(t) cos\u001e(t),My(t) = sin\u0012(t) sin\u001e(t), andMz(t) = cos\u0012(t), from Eqs. (A4) and (A6), we\nhave\nX\n\u0017;\u00170=x;y;zM\u0017(t)Q\u0017;\u00170(\u0011)M\u00170(t) =\u00001\n\u00113r\n6\u0019\n5hn\nY2\n2(e\u0011)e\u00002i\u001e(t)+Y\u00002\n2(e\u0011)e2i\u001e(t)o\nsin2\u0012(t)\n\u0000n\nY1\n2(e\u0011)e\u0000i\u001e(t)\u0000Y\u00001\n2(e\u0011)ei\u001e(t)o\nsinf2\u0012(t)gi\n+1\n\u00113r\n6\u0019\n5Y0\n2(e\u0011)r\n2\n3\b\n3 sin2\u0012(t)\u00002\t\n; (D6)\nwhereYm\nl(e\u0011) are the usual spherical harmonics.\nBy using Eq. (A2), an alternative form of Eq. (D6) can be obtained:\nX\n\u0017;\u00170=x;y;zM\u0017(t)Q\u0017;\u00170(\u0011)M\u00170(t) =X\n\u0017;\u00170=x;y;z\u00112\u000e\u0017;\u00170\u00003\u0011\u0017\u0011\u00170\n\u00115M\u0017(t)M\u00170(t) =\u00112jM(t)j2\u00003f\u0011\u0001M(t)g2\n\u00115\n=\u00112\u00003f\u0011\u0001M(t)g2\n\u00115: (D7)\nThus, \bdd(r;t) can be written as\n\bdd(r;t) =S2cdd\"Z\nd3r0jr\u0000r0j2\u00003f(r\u0000r0)\u0001M(t)g2\njr\u0000r0j5j\tuni(r0;t)j2#\n=S2cdd\"Z\nd3\u0016r0j\u0016r\u0000\u0016r0j2\u00003f(\u0016r\u0000\u0016r0)\u0001M(t)g2\nj\u0016r\u0000\u0016r0j5j\tuni(\u0016r0;t)j2#\n=\u00003\n2S2cddsin2\u0012(t)Z\nd3\u0016\u0011j\tuni(\u0016\u0011+\u0016r;t)j21\n\u0016\u00115h\n\u0016\u00112\u0000\u0016\u00112\nz\u00002f\u0016\u0011xsin\u001e(t)\u0000\u0016\u0011ycos\u001e(t)g2i\n\u00003S2cddsinf2\u0012(t)gZ\nd3\u0016\u0011j\tuni(\u0016\u0011+\u0016r;t)j2\u0016\u0011z\n\u0016\u00115f\u0016\u0011xcos\u001e(t) + \u0016\u0011ysin\u001e(t)g\n+1\n2S2cdd\b\n1\u00003 cos2\u0012(t)\tZ\nd3\u0016\u0011j\tuni(\u0016\u0011+\u0016r;t)j21\n\u0016\u00115\u0000\n3\u0016\u00112\nz\u0000\u0016\u00112\u0001\n: (D8)\nwhere \u0016r:=r=LwithLbeing some length which scales r(so that \u0016ris a dimensionless vector). For example, in\nquasi-1D with trap potential being Eq. (4), L=l?. Note that, in the special case where M(t) =Mz(t)ez, the form\nof Eq. (D8) becomes identical to Eq.(6) in Ref. [40].\nSince we concentrate on quasi-1D gases, with trap potential given by Eq. (4) in the main text, we will explicitly\ncompute the form of \b dd(r;t) for the quasi-1D setup. By writing\nj\tuni(r;t)j2=e\u0000\u001a2=l2\n?\n\u0019l2\n?j\t (z;t)j2; (D9)17\n-20 -10 10 20\n-0.4-0.20.20.40.60.81.0\n-60 -40 -20 20 40 60\n-0.4-0.20.20.40.60.81.0\nFIG. 6. Scaled dipole-dipole mean-\feld potential \u0016\bdd(z) as a function of zfor a quasi-1D box trap. (Left) Lz=l?= 10.\n(Right)Lz=l?= 30.\nand integrating out xandydirections, one can get the quasi-1D dipole-dipole-interaction mean-\feld potential \b dd(z;t)\nas follows (which is in Eq. (38)):\n\bdd(z;t) =cdd\n2l2\n?S2\b\n1\u00003M2\nz(t)\t\u001aZ1\n\u00001d\u0016zj\t (z+ \u0016zl?;t)j2G(j\u0016zj)\u00004\n3j\t (z;t)j2\u001b\n: (D10)\nNow, let us consider box trap in quasi-1D case, i.e. V(z) = 0 forjzj\u0014LzandV(z) =1forjzj>LzwhereV(z)\nis in Eq. (4). Then we may write\nj\t (z;t)j2=2\n64N\n2Lzforjzj\u0014Lz,\n0 forjzj>Lz,(D11)\nsinceV(z) = 0 for\u0000Lz\u0014z\u0014Lz. Thus, \b dd(z;t) can be written as\n\bdd(z;t) =2\n666664\u0016\bdd(t)(Z(Lz\u0000z)=l?\n\u0000(Lz+z)=l?d\u0016zG(j\u0016zj)\u00004\n3)\nforjzj\u0014Lz,\n\u0016\bdd(t)Z(Lz\u0000z)=l?\n\u0000(Lz+z)=l?d\u0016zG(j\u0016zj) forjzj>Lz,(D12)\nwhere \u0016\bdd(t):=NcddS2\b\n1\u00003M2\nz(t)\t\n=\u0000\n2Lzl2\n?\u0001\n. \bdd(z;t) is discontinuous at z=\u0006Lzbecause of the sudden change\nof the density at the boundary ( z=\u0006Lz) due to box trap potential.\nDe\fning the scaled density-density mean-\feld potential \u0016\bdd(z):= \bdd(z;t)=\u0016\bdd(t), we obtain Fig. 6, for two\ndi\u000berent axial extensions, Lz=l?= 10 and 30. As Fig. 6 clearly illustrates, in a box-trapped quasi-1D gas, \b dd(z;t)\nbecomes approximately constant for jzj<LcandLc!LzforLz=l?\u001d1. Depending on the value of M(t), \bdd(r;t)\nwill introduce either a repulsive or an attractive force. This force will however exist only near the boundary for a box\ntrap, where it can lead to a slight modi\fcation of the density of atomes. Its relative in\ruence decreases with increasing\nextension of the trapped gas along the zaxis, and can therefore be consistently neglected in the approximation of\nconstant particle-density.\nHowever, to assess whether signi\fcant magnetostriction occurs, one has to consider, in addition to \b dd, the trap\npotentialVtrand the `quasi' density-density interaction mean \feld potential \b 0de\fned as\n\b0(r;t):=(\nc0+S2SX\nk=1c2kX\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t))\nj\tuni(r;t)j2: (D13)\nWe can coin \b 0(r;t) a `quasi' density-density interaction mean \feld potential because only c0is a density-density\ninteraction coe\u000ecient ( c2kare interaction coe\u000ecients parametrizing the spin-spin interactions for spin- Sgas wherek\nis an integer with 1 \u0014k\u0014S. For example, c2is the spin-spin interaction coe\u000ecient of a spin-1 gas). In our quasi-1D\ncase, this \b 0(r;t) potential is \b 0(z;t) where\n\b0(z;t):=(\nc0\n2\u0019l2\n?+S2SX\nk=1c2k\n2\u0019l2\n?X\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t))\nj\t (z;t)j2: (D14)18\nIn the main text, we assume that c0\u001dS2PS\nk=1c2kP\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t). For spin-123Na or87Rb,S= 1\nandc0'100jc2j[31, 34], so this is an appropriate assumption (note thatPS\nk=1P\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k=x;y;zM2\n\u00171;\u00172;\u0001\u0001\u0001;\u0017k(t) = 1).\nThe values of the c2kare not yet established for166Er. We therefore tacitly assume in the main text, when calculating\nconcrete numerical examples for166Er, that the above condition also still holds, despite the prefactor S2enhancing\nthe importance of spin-spin interactions in \b 0(z;t). 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Ketterle,\n\\Realization of Bose-Einstein Condensates in Lower Di-\nmensions,\" Phys. Rev. Lett. 87, 130402 (2001).\n[48] This fact is equivalent to the critical dimensionless mag-\nnetization in Eq. (32) being always much less than unity.\n[49] We note, while box traps so far have been created for\nscalar BECs with contact interaction only, there is no\nobstacle in principle to create them as well for dipolar\nspinor gases (J. Dalibard, private communication).\n[50] M. Geller, \\A table of integrals of the error functions,\"\nJ. Res. Nat. Bureau Stand., B 73, 1{20 (1969)."    },    {        "title": "1503.07043v5.Spin_dynamics_and_frequency_dependence_of_magnetic_damping_study_in_soft_ferromagnetic_FeTaC_film_with_a_stripe_domain_structure.pdf",        "content": "Spin dynamics and frequency dependence of magnetic damping study in soft\nferromagnetic FeTaC \flm with a stripe domain structure\nB. Samantaray1a), Akhilesh K. Singh2, A.Perumal2, R. Ranganathan1and P. Mandal1, 2\n1)Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, Indiaa)\n2)Department of Physics, Indian Institute of Technology Guwahati, Guwahati - 781039,\nIndia\n(Dated: 12 November 2021)\nPerpendicular magnetic anisotropy (PMA) and low magnetic damping are the key factors for the free layer\nmagnetization switching by spin transfer torque technique in magnetic tunnel junction devices. The mag-\nnetization precessional dynamics in soft ferromagnetic FeTaC thin \flm with a stripe domain structure was\nexplored in broad band frequency range by employing micro-strip ferromagnetic resonance technique. The\npolar angular variation of resonance \feld and linewidth at di\u000berent frequencies have been analyzed numeri-\ncally using Landau-Lifshitz-Gilbert equation by taking into account the total free energy density of the \flm.\nThe numerically estimated parameters Land\u0013 e g-factor, PMA constant, and e\u000bective magnetization are found\nto be 2.1, 2\u0002105erg/cm3and 7145 Oe, respectively. The frequency dependence of Gilbert damping parameter\n(\u000b) is evaluated by considering both intrinsic and extrinsic e\u000bects into the total linewidth analysis. The value\nof\u000bis found to be 0.006 at 10 GHz and it increases with decreasing precessional frequency.\nSpin transfer torque (STT) has grater credibility com-\npared to other techniques towards ultrafast spin dynam-\nics in ferromagnet by electric current induced magneti-\nzation reversal of spin valves and magnetic tunnel junc-\ntions (MTJ).1The current researchers are more keen to\nfocus on STT technology for its high density magnetic\nrandom access memories (MRAM),2,3STT-driven do-\nmain wall devices4and perpendicular magnetic record-\ning media5applications. In order to make this technol-\nogy more e\u000ecient, lowering the critical current density is\nessential which requires the material speci\fcations with\nlow saturation magnetization ( MS), high spin polariza-\ntion, large uniaxial perpendicular magnetic anisotropy\n(PMA) constant and low magnetic damping.6{8The mag-\nnetic damping parameter ( \u000b) can be described well by\nthe phenomenological Landau-Lifshitz-Gilbert equation\nand is known as the Gilbert damping.9,10Several at-\ntempts have been made for understanding the origin of\nGilbert damping in spin dynamics relaxation in single\nlayer as well as multilayered magnetic alloys, which arises\nfrom both intrinsic and extrinsic parts of the material.\nThe intrinsic contribution to the Gilbert damping pa-\nrameter has been studied by tuning the strength of the\nspin-orbit coupling.8,11,12Recently, Ikeda et al.13have re-\nported that CoFeB-MgO based MTJ with PMA would be\nreliable for high-density non-volatile memory application\ndue to its high thermal stability and e\u000eciency towards\nSTT technology. The investigation on magnetic dynam-\nics, PMA and the apparent magnetic damping have been\nstudied extensively in CoFeB based soft ferromagnetic\nthin \flm by ferromagnetic resonance (FMR) and time-\nresolved magneto-optical Kerr e\u000bect.14,15Malinowski et\nal.16have reported a large increase in Gilbert damping\nwith applied magnetic \feld in perpendicularly magne-\ntized CoFeB thin \flm.\na)Electronic mail: iitg.biswanath@gmail.comIn this letter, we focus on amorphous FeTaC layer due\nto its interesting soft ferromagnetic (FM) properties.17,18\nThe amorphous soft FM layer reduces the number of pin-\nning centers which may lead to the STT-driven domain\nwall motion along with high tunneling magnetoresistance\nratio (TMR). The transcritical loop along with the stripe\ndomain structure, which are the manifestation of PMA\ncomponent were reported on FeTaC thin \flm with thick-\nness of 200 nm.18,19To shed some more light onto its dy-\nnamic magnetic properties, we have further studied this\n\flm by using ferromagnetic resonance technique. Though\nthe magnetic anisotropy and Gilbert damping have been\nstudied by FMR technique in several magnetic thin\n\flms like Heusler alloys, permalloy, soft magnetic ma-\nterials and multilayered (FM/antiferromagnetic or non-\nmagnetic/FM) magnetic \flms for magnetic recording,\nMTJ and TMR reader applications, most of the reports\nare limited to single frequency due to the measurements\nin X-band electron-spin-resonance spectrometer where\nthe cavity resonates at particular frequency.6,8,15,20{23In\nthis report, spin dynamics and magnetic relaxation are\nstudied at di\u000berent magnetization precessional frequen-\ncies.\nSoft ferromagnetic Fe 80Ta8C12single layer \flm with\nthickness 200 nm was deposited by dc magnetron sput-\ntering technique and the details of growing environment\nwere reported earlier.18The static and dynamic magnetic\nproperties were explored by using a vector network ana-\nlyzer (VNA) based homemade micro-strip ferromagnetic\nresonance (MS-FMR) spectrometer. The micro-strip line\nwhich was coupled to VNA and Schottky diode detector\n(Agilent 8473D) through high frequency coaxial cables\nwas mounted in between the pole pieces of the electro-\nmagnet. The magnetic thin \flm with \flm side down-\nward was mounted on the strip line. The frequency of\nthe microwave signal was \fxed by using an Agilent Tech-\nnologies made VNA (Model PNA-X, N-5242A) with a\nconstant microwave power of 5 dBm. The \frst deriva-arXiv:1503.07043v5  [cond-mat.mtrl-sci]  14 May 20152\nFIG. 1. Schematic diagram of MandHvectors in spherical\npolar coordinate system. 'Mand'Hare the in-plane angle\nof magnetization ( M) and external magnetic \feld ( H) with\nrespect toxaxis, while \u0012Mand\u0012Hare out-of-plane angles\nwith respect to zaxis.\ntive of the absorption spectrum with respect to magnetic\n\feld (H) was collected by \feld modulation and lock-in\ndetection technique. The FM thin \flm was treated in-\nplane and out-of-plane orientations. The magnetic \feld\nsweeping FMR spectra were recorded by varying two pa-\nrameters: precessional frequency and the angle between\nHand normal of the \flm. The frequency ( f) and po-\nlar angle (\u0012H) dependence of resonance \feld ( Hr) and\nlinewidth (\u0001 HPP) were extracted from each FMR spec-\ntrum and the numerical calculations were carried out by\nmathematica program for di\u000berent relaxation processes.\nThe precession of magnetization ( M) in the sample\nplane under the in\ruence of microwave and external mag-\nnetic \feld is illustrated in Fig. 1 in a polar coordinate\nsystem.'H('M) is the in-plane angle between H(M)\nandxaxis and\u0012H(\u0012M) is the polar angle between zaxis\nandH(M). The uniform precession of magnetization\ncan be described by the Landau-Lifshitz-Gilbert (LLG)\nequation of motion,9,10\n@\u0000 !M\n@t=\u0000\r\u0010\u0000 !M\u0002\u0000 !Heff\u0011\n+G\n\rM2\nS\"\n\u0000 !M\u0002@\u0000 !M\n@t#\n(1)\nThe \frst term corresponds to the precessional torque in\nthe e\u000bective magnetic \feld and the second term is the\nGilbert damping torque. \r=g\u0016B=~is denoted as gyro-\nmagnetic ratio and written in terms of Land\u0013 e gfactor,\nBohr magneton \u0016B, and Planck constant ~.G=\r\u000bMS\nis related to the intrinsic relaxation rate of the material.\n\u000bis the dimensionless Gilbert damping parameter. The\nfree energy density of a single magnetic thin \flm can be\nwritten as,\nE=\u0000MSH[sin\u0012Hsin\u0012Mcos('H\u0000'M) + cos\u0012Hcos\u0012M]\n\u00002\u0019M2\nSsin2\u0012M+K?sin2\u0012M\n(2)\nwhere the \frst term is analogous to the Zeeman energy,\nthe second term is dipolar demagnetization energy, the\nthird term signi\fes the anisotropy energy, MSis the sat-\nuration magnetization, K?is the PMA constant with\ncorresponding anisotropic \feld H?= 2K?=MS. The\nresonance frequency frof the uniform precession mode is\nFIG. 2. (a) shows typical FMR spectra at di\u000berent frequencies\nin planar orientation, (b) shows fdependence of Hrfor in-\nplane applied magnetic \feld. Experimentally and numerically\ncalculatedHrvalues are shown as open circles and solid line\nrespectively and (c) shows the plot of room temperature M\u0000\nHloop and domain images reproduced from Ref. [19].\ndeduced from the energy density by using the following\nexpression,24\nf2\nr=\u0010\r\n2\u0019\u00112 1\nM2\nSsin2\u0012M\"\n@2E\n@\u00122\nM@2E\n@'2\nM\u0000\u0012@2E\n@\u0012M@'M\u00132#\n(3)\nwhere the derivatives are evaluated at equilibrium posi-\ntions ofMandH.\nFor the in-plane orientation, the typical FMR spectra\nat di\u000berent frequencies are shown in Fig. 2(a). The mea-\nsurements were carried out by varying the frequency from\n1 to 18 GHz with an interval of 0.5 GHz. In the lower\nfrequency range 1-6 GHz, the FMR spectra show two res-\nonance peaks. The low-\feld resonance peaks named as\nsecondary mode and are marked by 2 and 2\u0003for 2.5 and 6\nGHz, respectively in Fig. 2(a). This mode arises from the\nlinear unsaturated zone of the transcritical M(H) loop19\nand is usually observed in stripe-domain structure.25,26\nThe transcritical loop along with the domain structure\nare reproduced from earlier report19and is shown in\nFig. 2(c). The dense stripe-domain structure observed in\nthis \flm con\frms the presence of perpendicular magnetic\nanisotropy. The primary modes usually called uniform\nmode are marked as 1 and 1\u0003for 2.5 and 6 GHz, respec-\ntively. The value of Hrfor secondary resonance peak\nincreases with the increase in fup to 4.5 GHz and then\nfollows the reverse trend as depicted in Fig. 2(b). This3\nFIG. 3. Equilibrium angle of the magnetization, \u0012M, as a\nfunction of the applied \feld direction, \u0012H, in out-of-plane con-\n\fguration at di\u000berent frequencies.\ncould be explained on the basis that the value of Hrof\nthe uniform mode above 4.5 GHz overcomes the parallel\nsaturation \feld, i.e., 280 Oe as observed in M(H) curve.\nAbove 6 GHz, Hrexceeds the parallel saturation \feld in\nlarge extent and this could be the reason for the strong\nattenuation of secondary phase. In planar con\fguration\n(\u0012M=\u0012H=\u0019=2), the solution for the in-plane resonance\nfrequency can be calculated by incorporating the total\nenergy in Eq. 3 and is given by,\nfr=\r\n2\u0019[(4\u0019M+Hcos ('H\u0000'M)) (Hcos ('H\u0000'M))]1\n2\n(4)\nThe value of 'Mcan be calculated by using the solution\nofHat equilibrium condition, i.e.,@E\n@'M=0. However,\nfor the present thin \flm, we could not \fnd any planar\nanisotropy from the 'Hdependence of Hrand hence\nconclude,'H='M. Thefdependence of Hris nu-\nmerically calculated by using Eq. 4 and is shown as a\nsolid line in Fig. 2(b). The numerically calculated values\nyielded a good \ft and the parameters are found to be re-\nliable with 4 \u0019MS= 7791\u000610 Oe and\r= 2.95 MHz/Oe\nwith ag-factor of 2.1. The deduced value of saturation\nmagnetization is very close to earlier reported value from\nM(H) loop measurement.18\nIn out-of-plane con\fguration, the solution for the res-\nonance frequency is deduced from Eq. 3 by employing\nthe conditions, 'H='M=0\u000eand is represented in Eq. 5.\nf2\nr=\u0000\r\n2\u0019\u00012h\nHcos (\u0012M\u0000\u0012H)\u0000\u0010\n4\u0019MS\u00002K?\nMS\u0011\ncos 2\u0012Mi\nh\nHcos (\u0012M\u0000\u0012H)\u0000\u0010\n4\u0019MS\u00002K?\nMS\u0011\ncos2\u0012Mi\n(5)\nThe equilibrium angle \u0012Mis numerically calculated for\nFIG. 4. Angular dependence of resonance \feld Hrin out-\nof-plane con\fguration at di\u000berent frequencies. ( \u000e) shows the\nexperimental points and the line (-) shows the modeled data.\neach value of \u0012Hby minimizing the energy, i.e.,@E\n@\u0012M= 0\nand is depicted in Fig. 3 for di\u000berent frequencies. Fig.\n3 demonstrates that magnetization suddenly attempts to\nalign in planar direction as the magnetic \feld goes away\nfrom the\u0012H=0\u000eand 180\u000e. Fig. 4 shows one complete\nround of\u0012Hdependence of Hrat di\u000berent frequencies.\nUniaxial PMA is found to be observed along with singu-\nlarity at\u0012H= 0\u000eand\u0012H= 180\u000e, which signi\fes that\nin\fnite magnetic \feld is required to turn the Mvector\nparallel toHin perpendicular con\fguration. The depen-\ndence ofHron\u0012His modeled at di\u000berent frequencies\nstarting from 4 to 10 GHz with 2 GHz intervals by using\nEq. 5 and the interpolated values of \u0012Mfrom Fig. 3. The\nmodeled values of Hrare plotted as a solid line in Fig.\n4 and a very good agreement with experimental data is\nobserved. The parameters deduced from this calculation\nare found to be K?=2\u0002105erg/cm3and 4\u0019Meff=7145\nOe.\nFinally, the damping of magnetization precession has\nbeen analyzed from linewidth of FMR spectra. The \u0012H\ndependence of \u0001 HPPas shown in Fig. 5 was extracted\nfrom the polar angle variation of FMR spectrum at di\u000ber-\nent frequencies in the range of 4-10 GHz with an interval\nof 2 GHz. In order to get better clarity of Fig. 5, the\ndata for the 4 GHz frequency are not shown. The total\nlinewidth broadening due to the intrinsic and extrinsic\nparts of the material has been expressed in the following\nequation,8,27\n\u0001HPP= \u0001H(\u000b) + \u0001H(\u00014\u0019Meff) + \u0001H(\u0001\u0012H)\n=2p\n31\nj@!\n@Hrj\u000b\r\nMS\u0010\n@2E\n@\u00122\nM+1\nsin2\u0012M@2E\n@'2\nM\u0011\n+\n1p\n3\u0010\f\f\f@H\n@4\u0019Meff\f\f\f\u00014\u0019Meff\u0011\n+1p\n3\u0010\f\f\f@H\n@\u0012H\f\f\f\u0001\u0012H\u0011\n(6)\nwhere 4\u0019Meff= 4\u0019MS\u00002K?=MS, is the e\u000bective mag-4\nFIG. 5. Out-of-plane angular dependence of total linewidth,\n\u0001Hppat di\u000berent frequencies. ( \u000e) shows the experimental\npoints and the line ( \u0000) shows the modeled data.\nFIG. 6.\u0012Hdependence of \u0001 Hpp, \u0001H(\u000b), \u0001H(\u00014\u0019Meff)\nand \u0001H(\u0001\u0012H) modeled data at 6 GHz frequency.\nnetization. \u0001 H(\u000b) arises from the intrinsic Gilbert type\ndamping and has large contribution towards linewidth\nbroadening. The parameter \u000bsigni\fes how fast the pre-\ncessional energy is dissipated into the lattice. The terms\n\u0001H(4\u0019\u0001Meff) and \u0001H(\u0001\u0012H) represent the linewidth\nbroadening due to the spatial dispersion of the magni-\ntude and direction of Meff, respectively. The \u0012Hdepen-\ndence of \u0001 HPPwas modeled by using Eq. 6 and the\ninterpolated values of \u0012Mfrom Fig. 3 at di\u000berent fre-\nquencies. The numerically calculated values of \u0001 HPP\nare shown as solid lines in Fig. 5. The individual contri-\nbutions towards the total linewidth (\u0001 Hpp) is also shown\nin Fig. 6. The curves are shown for a single frequency\nfor better clarity. The linewidth broadening is observed\nmainly due to the intrinsic Gilbert damping. The ex-trinsic contribution is found to be negligible when \u0012H\nis away from 0\u000eand 180\u000ebut it is large near the per-\npendicular con\fguration. The Gilbert damping param-\neter at di\u000berent frequencies for the FeTaC thin \flm is\nplotted in the inset of Fig. 6. It shows that the \u000bde-\ncrease monotonically with the increase in frequency. The\nlow value of damping parameter observed in the present\nthin \flm can be more relevant towards the STT tech-\nnology or MTJ applications. Such an increase of \u000bby\ndecreasing precessional frequency has been attributed to\nthe inhomogeneous linewidth broadening due to the dis-\npersion of anisotropic \feld.28,29The dispersion in mag-\nnitude and direction of e\u000bective magnetization are found\nto be \u00014\u0019Meff=0.1 KOe, \u0001 \u0012H\u00191\u000210\u00004degree. The\nvalue of Gilbert damping constant for the present Fe-\nTaC thin \flm is found to be comparable to those re-\nported in Fe-based magnetic thin \flms, such as FePd\nternary alloy11, permalloy30, NiFe/CoFeB/CoFe multi-\nlayered sturucture15and (FeCo) 1\u0000xGdx8. However, the\nMn- and Co- based thin \flms6,7have larger damping pa-\nrameter as compared to the present \flm which could be\nunderstood on the basis of spin-orbit coupling.\nIn conclusion, PMA and Gilbert damping which\nare very important and crucial parameters for STT,\nSTT-MRAM and TMR applications, have been analyzed\nin FeTaC soft ferromagnetic thin \flm with a striped\ndomain structure by using MS-FMR technique in broad\nband frequency range. The precise estimation of Land\u0013 e\ng-factor, PMA constant and 4 \u0019Meffwere carried out\nby using total energy density function for magnetic\nthin \flm. 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Otani, \\Spin wave contributions to the high-frequency mag-\nnetic response of thin \flms obtained with inductive methods,\"\nJournal of Applied Physics, 95, 5646{5652 (2004)."    },    {        "title": "0705.0508v1.Effective_attraction_induced_by_repulsive_interaction_in_a_spin_transfer_system.pdf",        "content": "arXiv:0705.0508v1  [cond-mat.mtrl-sci]  3 May 2007Effective attraction induced by repulsive interaction in a s pin-transfer system\nYa. B. Bazaliy\nInstituut Lorentz, Leiden University, The Netherlands,\nDepartment of Physics and Astronomy, University of South Ca rolina, Columbia, SC, and\nInstitute of Magnetism, National Academy of Science, Ukrai ne.\n(Dated: April, 2006)\nIn magnetic systems with dominating easy-plane anisotropy the magnetization can be described\nby an effective one dimensional equation for the in-plane ang le. Re-deriving this equation in the\npresence of spin-transfer torques, we obtain a description that allows for a more intuitive under-\nstanding of spintronic devices’ operation and can serve as a tool for finding new dynamic regimes.\nA surprising prediction is obtained for a planar “spin-flip t ransistor”: an unstable equilibrium point\ncan be stabilized by a current induced torque that further re pels the system from that point. Stabi-\nlization by repulsion happens due to the presence of dissipa tive environment and requires a Gilbert\ndamping constant that is large enough to ensure overdamped d ynamics at zero current.\nPACS numbers: 72.25.Pn, 72.25.Mk, 85.75.-d\nIn physics, there are cases where due to the presence\nof complex environment a repulsive force can lead to ac-\ntual attraction of the entities. A well known example\nis a superconductor, where the Cooper pairs are formed\nfrom electrons repelled by the Coulomb forces due to the\ndynamical elastic environment. Here we report a phe-\nnomena of effective attraction induced by the repulsive\nspin-transfer torque in the presence of highly dissipative\nenvironment. The spin-transfer effect producing the re-\npulsivetorqueis a non-equilibriuminteractionthat arises\nwhen a current of electrons flows through a non-collinear\nmagnetic texture [1, 2, 3]. This interaction can become\nsignificant in nanoscopic magnets and is nowadays stud-\nied experimentally in a variety of systems. Its manifesta-\ntions - either current induced magnetic switching [4] or\nmagnetic domain wall motion [5] - serve as an underlying\nmechanism for a number of suggested memory and logic\napplications.\nHere we consider a conventional spin-transfer device\nconsisting of a a magnetic polarizer (fixed layer) and a\nsmall magnet (free layer) with electric current flowing\nfrom one to another (Fig 1). Both layers can be de-\nscribed by a macro-spin model due to large exchange\nstiffness. The free layer is influenced by the spin transfer\ntorque,whilethepolarizeristoolargetofeelit. Magnetic\ndynamics of the free layer is described by the Landau-\nLifshitz-Gilbert (LLG) equation with the spin transfer\ntorque term [2, 6].\nThe solutions of LLG are easy to find for the simplest\neasy axis magnetic anisotropy of the free layer. There\nexists a critical current at which the free layer either\nswitches between the two minima of magnetic energy,\nor goes into a state of permanent precession, powered by\nthe current source [2, 6, 7]. The same basic processes\nhappen in the case of realistic anisotropies, however the\ncomplexity of the calculations increases substantially. In\na nanopillar device [8] one additionally finds that stabi-\nlization of magnetic energy maxima is possible (“cantedstates”[6]) andthat multiple precessionmodesexist with\ntransitions between them happening as the current is in-\ncreased [7, 9, 10]. The anisotropy of a nanopillar device\nis a combination of a magnetic easy plane and magnetic\neasy axis directed in that plane. Experimentally, the\neasy plane anisotropy energy is usually much larger than\nthe easy axis energy, i.e. the system is in the regime\nof a planar spintronic device [11] (Fig. 1). This limit of\ndominatingeasyplane energyis characterizedby another\nsimplification of the dynamic equations [12, 13], which\ncomes not from the high symmetry of the problem, but\nfrom the existence of a small parameter: the ratio of the\nenergy modulation within the plane to the easy plane en-\nergy. The deviation of the magnetization from the plane\nbecomes small, making the motion effectively one dimen-\nsional.\nIn this paper we present a general form of effective\nplanar equation describing a macrospin free layer in the\npresence of spin transfer torques. Its relationship to the\nfirst order expansion in the current magnitude used in\nRef. 13 is discussed at the end. We then use this equa-\ntion tostudy the “spin-fliptransistor”: aplanardevice in\nwhich the spin polarizer is perpendicular to the direction\nfavored by the magnetic anisotropy energy. It was pre-\ndicted [14] that the competition between the anisotropy\nand spin transfer torques leads to a 90 degrees jump of\ns\n A B\njnjn\ns\nFIG. 1: Planar spin-transfer devices. Hashed parts of the\ndevices are ferromagnetic, white parts are made from a non-\nmagnetic metal.2\nthe magnetization at the critical current. Whether the\njump happens into the parallel or antiparallel state with\nrespect to the polarizer is determined by the direction of\nthe current.\nHere it is shown that the behavior of the spin-flip tran-\nsistor is more complicated than expected from the simple\npicture above. Namely, the current inducing a jump into\nthe parallel direction can also stabilize the antiparallel\ndirection. This conclusion is certainly counter-intuitive\nbecause the spin torque repels the magnetization from\nthis already unstable saddle point of the energy. How-\never, a combination of two destabilizing torques manages\nto result in a stable equilibrium. We will see that this\nhappens due to the dissipation terms and a sufficiently\nlarge(but still smallcomparedto unity) Gilbert damping\nconstant is required to observe the phenomena.\nThe magnetization of the free layer M=Mnhas a\nconstant absolute value Mand a direction given by a\nunit vector n(t). The LLG equation [2, 6] reads:\n˙n=γ\nM/bracketleftbigg\n−δE\nδn×n/bracketrightbigg\n+u(n)[n×[s×n]]+α[n×˙n].(1)\nHereγis the gyromagnetic ratio, E(n) is the magnetic\nenergy of the free layer, and αis the Gilbert damping\nconstant. The second term on the right is the spin trans-\nfer torque, where sis a unit vector along the direction\nof the polarizer, and the spin transfer strength u(n) is\nproportional to the electric current I[6, 13]. In general,\nspin transfer strength is a function of the angle between\nthe polarizer and the free layer u(n) =f[(n·s)]I, with\nthe function f[(n·s)] being material and device specific.\nEquation (1) can be written in polar angles ( θ(t),φ(t)):\n˙θ+α˙φsinθ=−γ\nMsinθ∂E\n∂φ+u(s·eθ)≡Fθ,\n˙φsinθ−α˙θ=γ\nM∂E\n∂θ+u(s·eφ)≡Fφ, (2)\nwith tangent vectors eφ= [ˆz×n]/sinθ,eθ= [eφ×n].\nThe easy plane is chosen at θ=π/2, and the mag-\nnetic energy has the form E= (K⊥/2)cos2θ+Er(θ,φ),\nwhereEris the “residual” energy. In the planar limit,\nK⊥→ ∞, the energy minima are very close to the easy\nplane and the low energysolutionsofLLG havethe prop-\nertyθ(t) =π/2 +δθwithδθ→0. Equations (2) can\nthen be expanded in small parameters |Er|/K⊥≪1,\n|u(n)|/K⊥≪1. Assuming time-independent uandswe\nobtain an effective equation of the in-plane motion\n1\nω⊥¨φ+αeff˙φ=−γ\nM∂Eeff\n∂φ, (3)\nwhich has has the form of the Newton’s equation of mo-\ntion for a particle in external potential Eeff(φ) with a\nvariable viscous friction coefficient αeff(φ). The expres-\nsions for the effective friction and energy are\nαeff(φ) =α−(Γφ+Γθ)/ω⊥, (4)\nΓφ= (∂Fφ/∂φ)θ=π/2,Γθ= (∂Fθ/∂θ)θ=π/2,and\nEeff(φ) =Er(π/2,φ)+∆E(φ), (5)\n∆E=−M\nγ/integraldisplayφ/bracketleftbigg\nu(n)(s·eθ)−Γθ\nω⊥Fφ/bracketrightbigg\nθ=π\n2dφ′.\nEquation (3) with definitions (4,5) gives a general de-\nscription of a planar device in the presence of spin trans-\nfer torque. At non-zero current the effective friction can\nbecome negative (see below), and the effective energy is\nnot necessarily periodic in φ(e.g. in the case of “mag-\nnetic fan”[13, 15]). Physicallythis reflectsthe possibility\nof extracting energy from the current source, and thus\ndeveloping a “negative dissipation” in the system.\nIn many planar devices the polarizer direction slies\nin the easy plane, θs=π/2, with a direction defined\nby the azimuthal angle φs. At the same time the resid-\nual energy has a property ( ∂Er/∂θ)θ=π/2= 0, i.e. does\nnot shift the energy minima away from the plane. We\nwill also use the simplest form f[(n·s)] = const for the\nspin transfer strength. A more realistic function will not\nchange the result qualitatively and can be easily used if\nneeded. With these restrictions the effective friction and\nthe energy correction get the form:\nαeff=α+2ucos(φs−φ)\nω⊥(6)\n∆E=−Mu2\n2γω⊥cos2(φs−φ).\nIn a spin-flip transistor the polarizer direction is given\nbyφs=π/2. Following Ref. 14, we consider in-plane\nanisotropy energy Er(π/2,φ) =−(K||/2)cos2φcorre-\nsponding to an easy axis. Then the effective friction\nisαeff=α+ (2usinφ)/ω⊥and effective energy equals\n(γ/M)Eeff=−[(ω||−u2/ω⊥)/2]cos2φ+ const with\nω||=γK||/M. Equilibrium points φ= 0,±π/2,πare\nthe minima and maxima of the effective energy, and do\nnot depend on u. Stability of any equilibrium in one di-\nmension depends on whether it is a minimum or a maxi-\nmum ofEeffand on the sign of αeffat the equilibrium\npoint. It is easy to check, that out of four possibilities\nonly an energy minimum with αeff>0 is stable. In the\ncaseofaspin-fliptransistortheenergylandscapechanges\nabove a threshold |u|>√ω||ω⊥: the energy minima at\nφ= 0,πbecome maxima, and, vice versa, the energy\nmaxima at φ=±π/2 switch to minima. Effective fric-\ntion atφ= 0,πis positive independent of u, while at\nφ=±π/2 it changes sign at u=∓αω⊥/2.\nThe behavior of the spin-flip transistor is summarized\nin a switching diagram Fig. 2 plotted on the plane of the\nmaterial characteristic αand the experimental parame-\nteru∼I. For definiteness we will discuss a current with\nu >0. The effect of the opposite current is completely\nsymmetric. For small values of Gilbert damping one ob-\nserves stabilization of the φ=π/2 (parallel) equilibrium3\n+π/2\n−π/20 πα*u\nααEeff\n0+π/2 π −π −π/2effαEeff\n0+π/2 π −π −π/2eff αEeff\n0+π/2 π −π −π/2eff\nabcd\nFIG. 2: Switching diagram of the spin-flip transistor. In eac h\nzone one or two arrows show the possible stable directions of\nthe free layer magnetization. Directions of the easy axis an d\nspin polarizer are defined in the right bottom corner. Angula r\ndependencies of αeffandEeffare givenin insets. Stable sub-\nregions ��b” and “c” differ in overdamped vs. underdamped\napproach to the equilibrium.\nto which the spin torque attracts the magnetization of\nthe free layer, while the opposite (antiparallel) direction\nremains unstable. This is in accord with the results of\nRef. 14. However, when the damping constant is larger\nthan the critical value α∗= 2/radicalbig\nω||/ω⊥, a window of sta-\nbility of the antiparallel equilibrium opens on the dia-\ngram. Since α≪1, a sufficiently large easy plane energy\nis required to achieve α∗< α≪1.\nIf one thinks about the stability of the ( θ,φ) =\n(π/2,−π/2)equilibriumfor u >0intermsofEq.(1), this\nprediction seems completely unexpected. The anisotropy\ntorques do not stabilize this equilibrium because it is a\nsaddle point of the total magnetic energy E, and the\nadded spin transfer torque repels nfrom this point as\nwell. The whole phenomena may be called “stabilization\nby repulsion”. To check the accuracy of the planar ap-\nproximation (3), the result was verified using the LLG\nequations (2) with no approximations for the axis-and-\nplane energy E= (K⊥/2)cos2θ−(K||/2)sin2θcos2φ.\nCalculatingtheeigenvaluesofthelinearizeddynamicma-\ntrices [6] at the equilibrium points ( π/2,±π/2) we ob-\ntained the same switching diagram and confirmed the\nstabilization of the antiparallel direction. Typical trajec-\ntoriesn(t) numerically calculated from the LLG equa-\ntion with no approximations are shown in Fig. 3 to illus-−0.5π −0.5π −0.7π −0.3π −0.7π −0.3π0.54π0.46π\n0.5π\n0.54π0.46π\n0.5π\nFIG. 3: Typical trajectories of n(t) forω||/ω⊥= 0.01,α=\n1.5α∗. The plot labels correspond to the regions in Fig. 2, the\ncurrent magnitude is given in the units of u/p\nω||/ω⊥and we\nlook at the stability of the φ=−π/2 equilibrium: (a) 0.93,\nunstable (b): 1.08, stabilized with overdamped approach (c ):\n1.38, stable, butwith oscillatory approach (d): 1.53, unst able;\na stable cycle is formed around the equilibrium.\ntrate the predictions. At u >√ω||ω⊥theφ=−π/2\nequilibrium is stabilized. In accord with the predic-\ntions of Eqs. (3),(6), the wedge of its stability consists\nof two regions (b) and (c) characterized by overdamped\nand underdamped dynamics during the approach to the\nequilibrium. The dividing dashed line is given by u=\nω||/α+αω⊥/4. It was checked that small deviations\nof the polarizer sfrom the ( π/2,π/2) direction do not\nchange the behavior qualitatively. Larger deviations\neventually destroy the effect, especially the out-of-plane\ndeviation which produces the “magnetic fan” effect [15]\nleading to the full-circle rotation of φin the plane.\nAs the current is further increased to u > αω ⊥/2, the\nantiparallel state looses stability and the trajectory ap-\nproaches a stable precession cycle (Fig. 3(d)). The exis-\ntence of the precession state is easy to understand from\n(3) viewedas anequation fora particlein externalpoten-\ntial. Just above the stability boundary the effective fric-\ntionαeff(φ) is negative in a small vicinity of φ=−π/2,\nand positive elsewhere. Within the αeff<0 region the\ndissipation is negative and any small deviation from the\nequilibrium initiates growing oscillations. As their am-\nplitude exceeds the size of that region, part of the cycle\nstarts to happen with positive dissipation. Eventually\nthe amplitude reaches a value at which the energy gain\nduring the motion in the αeff<0 region is exactly com-\npensated by the energy loss in the αeff>0 region: thus\na cycle solutionemerges. The effective planardescription\nallows for the analysis of the further evolution of the cy-\nclewithtransitionsintodifferentprecessionmodes,which\nwill be a subject of another publication.4\nThe fact that α > α ∗condition is required for the\nstabilization means that dissipation terms play a crucial\nrole entangling two types of repulsion to produce a net\nattraction to the reversed direction. Note that an in-\nterplay of a strong easy plane anisotropy and dissipa-\ntion terms produces unexpected effects already in con-\nventional ( u= 0) magnetic systems. The effective planar\nequation (3) at u= 0 was discussed in Ref. 12. It was\nfound that the same threshold α∗represents a bound-\nary between the oscillatory and overdamped approaches\nthe equilibrium. Above α∗the familiar precession of a\nmagnetic moment in the anisotropy field is replaced by\nthe dissipative motion directed towards the energy mini-\nmum. When the easy plane anisotropy is strong enough\nto ensure α≫α∗, one can drop the second order time\nderivative term in Eq. (3) and use the resulting first or-\nder dissipative equation. In the presence of spin transfer,\nαeff(φ,u) depends on the current and can assume small\nvalues even for α≫α∗, thus no general statement about\nthe¨φterm can be made.\nThe simplest easy axis energy expression Er(π/2,φ) =\n−(K||/2)cos2φhappens to have the same angular de-\npendence as ∆ E(φ) given by Eq. (6). Due to this spe-\ncial property the energy profile flips upside down at\nu=√ω||ω⊥. For a generic Er(π/2,φ) with minima at\nφ= 0,πand maxima at φ=±π/2 the nature of equilib-\nria will change at different current thresholds. This will\nmake the switching diagram more complicated, but will\nnotaffectthestabilizationbyrepulsionphenomena. Sim-\nilarcomplicationswillbe introducedbyageneric f[(n·s)]\nangular dependence of the spin transfer strength.\nIn Ref. 13 the known switching diagram for the\ncollinear ( φs= 0) devices [6, 9, 10] were reproduced\nby equation (3) with Eeff=Er(π/2,φ). The ∆ E\nterm (6) was dropped as being second order in small\nu. This approximation gives a correct result for the\nfollowing reason. In a collinear device ( γ/M)Eeff=\n−[(ω||+u2/ω⊥)/2]cos2φ+const and the current never\nchanges the nature of the equilibrium from a maximum\nto a minimum. Consequently, dropping ∆ Edoes not af-\nfect the results. As was already noted in Ref. 13, the first\norder expansion in uis insufficient for the description of\na spin-flip transistor, where the full form (6) is required.\nIn summary, we derived a general form of the effec-\ntive planar equation (3) for a macrospin free layer in\nthe presence of spin transfer torque produced by a fixed\nspin-polarizerandtime-independent current. Qualitative\nunderstanding of the solutions of planar equation is ob-\ntained by employing the analogy with a one-dimensional\nmechanical motion of a particle with variable friction co-\nefficient in an external potential. The resulting predic-\ntive power is illustrated by the discovery of the stabi-\nlization by repulsion phenomena in the spin-flip device.\nSuch stabilization relies on the form of the dissipative\ntorquesin the LLG equationand happens onlyfora large\nenough Gilbert damping constant. The new stable stateand the corresponding precession cycle can be used to\nengineer novel memory or logic devices, and microwave\nnano-generators with tunable frequency.\nTo observe the phenomena experimentally, one has to\nfabricate a device with α > α∗, and initially set it into a\nparallel or antiparallel state by external magnetic field.\nThenthecurrentisturnedonandthefieldisswitchedoff.\nBoth states should be stabilized by a moderate current√ω||ω⊥< u < αω ⊥/2, but cannot yet be distinguished\nby their magnetoresistive signals. The difference can be\nobserved as the current is increased above the αω⊥/2\nthreshold: the parallel state will remain a stable equilib-\nrium, while the antiparallel state will transform into a\nprecession cycle and an oscillating component of magne-\ntoresistance will appear.\nThe author wishes to thank C. W. J. Beenakker, G.\nE. W. Bauer, and Yu. V. Nazarov for illuminating dis-\ncussions. Research at Leiden University was supported\nby the Dutch Science Foundation NWO/FOM. Part of\nthis work was performed at KITP Santa Barbara sup-\nported by the NSF grant No. PHY99-07949, and at As-\npen PhysicsInstitute duringthe Winterprogramof2007.\n[1] L. Berger, J. Appl. Phys., 49, 2160 (1978); Phys. Rev. B\n33, 1572 (1986); J. Appl. Phys. 63, 1663 (1988).\n[2] J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[3] Ya. B. Bazaliy et al., Phys. Rev. B, 57, R3213 (1998).\n[4] S. Kaka et al., Nature 437, 389 (2005); M. R. Pufall et\nal., Phys.Rev. Lett. 97, 087206 (2006); M. L. Schneider\net al., Appl. Phys. Lett., 90, 092504 (2007); X. Jiang\net al., Phys. Rev. Lett. 97, 217202 (2006); W. Chen et\nal., Phys. Rev. B, 74, 144408(2006); B. Ozyilmaz et al.,\nPhys. Rev. Lett., 93, 176604 (2004); I. N. Krivorotov et\nal.Science, 307, 228 (2005); N. C. Emley et al.Phys.\nRev. Lett., 96, 247204 (2006); J. C. Sankey, et al., Phys.\nRev. Lett., 96, 227601 (2006).\n[5] G. Beach et al., Phys. Rev. Lett., 97, 057203 (2006); Na-\nture Materials, 4, 741 (2005); M. Klaui et al., Phys. Rev.\nLett.,95, 026601 (2005); M. Laufenberg et al., Phys.\nRev. Lett., 97, 046602 (2006); L. Thomas et al., Science,\n315, 1553 (2007); M. Hayashi et al., Phys. Rev.Lett., 98,\n037204 (2007); Nature Physics, 3, 21 (2007); Phys. Rev.\nLett.,97, 207205 (2006); M. Yamanouchi et al.Nature,\n428, 539 (2004); Phys. Rev. Lett., 96, 096601 (2006).\n[6] Ya. B. Bazaliy et al., Phys. Rev. B, 69, 094421 (2004).\n[7] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[8] J. A. Katine et al., Phys. Rev. Lett., 84, 3149 (2000).\n[9] S. I. Kiselev et al., Nature, 425, 380 (2003).\n[10] J. Xiao et al., Phys. Rev. B, 72, 014446 (2005)\n[11] A. Brataas et al., Phys. Rep., 427, 157 (2006).\n[12] C. Garcia-Cervera et al., J. Appl. Phys., 90, 370 (2001).\n[13] Ya. B. Bazaliy et al., arXiv:0705.0406v1 (2007), to be\npublished in J. Nanoscience and Nanotechnology.\n[14] A. Brataas et al., Phys. Rev. Lett. 84, 2481 (2000);\nX. Wang et al., Japan. J. Appl. Phys., 45, 3863 (2006).\n[15] X. Wang et al., Phys. Rev. B, 73, 054436 (2006)."    },    {        "title": "0904.1455v1.Evaluating_the_locality_of_intrinsic_precession_damping_in_transition_metals.pdf",        "content": "arXiv:0904.1455v1  [cond-mat.mtrl-sci]  9 Apr 2009Evaluating the locality of intrinsic precession damping in transition metals\nKeith Gilmore1,2and Mark D. Stiles1\n1Center for Nanoscale Science and Technology\nNational Institute of Standards and Technology,\nGaithersburg, MD 20899-6202\n2Maryland Nanocenter, University of Maryland,\nCollege Park, MD 20742-3511\n(Dated: December 4, 2018.)\nThe Landau-Lifshitz-Gilbert damping parameter is typical ly assumed to be a local quantity, in-\ndependent of magnetic configuration. To test the validity of this assumption we calculate the\nprecession damping rate of small amplitude non-uniform mod e magnons in iron, cobalt, and nickel.\nAt scattering rates expected near and above room temperatur e, little change in the damping rate is\nfound as the magnon wavelength is decreased from infinity to a length shorter than features probed\nin recent experiments. This result indicates that non-loca l effects due to the presence of weakly\nnon-uniform modes, expected in real devices, should not app reciably affect the dynamic response of\nthe element at typical operating temperatures. Conversely , at scattering rates expected in very pure\nsamples around cryogenic temperatures, non-local effects r esult in an order of magnitude decrease\nin damping rates for magnons with wavelengths commensurate with domain wall widths. While this\nlow temperature result is likely of little practical import ance, it provides an experimentally testable\nprediction of the non-local contribution of the spin-orbit torque-correlation model of precession\ndamping. None of these results exhibit strong dependence on the magnon propagation direction.\nMagnetization dynamics continues to be a techno-\nlogically important, but incompletely understood topic.\nHistorically, field induced magnetization dynamics have\nbeen described adequately by the phenomenological\nLandau-Lifshitz (LL) equation [1]\n˙m=−|γM|m×H+λˆm×(m×H),(1)\nor the mathematically equivalent Gilbert form [2, 3].\nEquation 1 accounts for the near equilibrium dynamics\nof systems in the absence of an electrical current. γM\nis the gyromagnetic ratio and λis the phenomenological\ndamping parameter, which quantifies the decay of the\nexcited system back to equilibrium. The LL equation\nis a rather simple approximation to very intricate dy-\nnamic processes. The limitations of the approximations\nentering into the LL equation are likely to be tested by\nthe next generation of magnetodynamic devices. While\nmanygeneralizationsforthe LLequationarepossible, we\nfocus on investigatingthe importance of non-local contri-\nbutions to damping. It is generally assumed in both ana-\nlyzing experimental results and in performing micromag-\nnetic simulations that damping is a local phenomenon.\nWhile no clearexperimental evidence exists to contradict\nthis assumption, the possibility that the damping is non-\nlocal – that it depends, for example, on the local gradient\nofthemagnetization–wouldhaveparticularimplications\nfor experiments that quantify spin-current polarization\n[4], for storage [5] and logic [6] devices based on using\nthis spin-current to move domain-walls, quantifying vor-\ntex [7] and mode [8] dynamics in patterned samples, and\nthe behavior of nano-contact oscillators [9, 10].\nWhile several viable mechanisms have been proposed\nto explain the damping process in different systems\n[11, 12, 13, 14, 15, 16, 17], werestrictthescopeofthispa-\nper to investigating the degree to which the assumptionof local damping is violated for small amplitude dynam-\nics within pure bulk transition metal systems where the\ndominant source of damping is the intrinsic spin-orbit in-\nteraction. For such systems, Kambersk´ y’s [14] spin-orbit\ntorque-correlationmodel, which predicts a decay rate for\nthe uniform precession mode of\nλ0=π¯hγ2\nM\nµ0/summationdisplay\nnm/integraldisplay\ndk/vextendsingle/vextendsingleΓ−\nnm(k)/vextendsingle/vextendsingle2Wnm(k),(2)\nhasrecentlybeen demonstratedtoaccountforthe major-\nity of damping [18, 19]. The matrix elements |Γ−\nnm(k)|2\nrepresent a scattering event in which a quantum of the\nuniformmodedecaysintoasinglequasi-particleelectron-\nhole excitation. This annihilation of a magnon raises the\nangularmomentum of the system, orienting the magneti-\nzation closer to equilibrium. The excited electron, which\nhas wavevector kand band index m, and the hole, with\nwavevector kand band index n, carry off the energy and\nangular momentum of the magnon. This electron-hole\npair is rapidly quenched through lattice scattering. The\nweighting function Wnm(k) measures the rate at which\nthe scattering event occurs. The very short lifetime of\nthe electron-hole pair quasiparticle (on the order of fs at\nroom temperature) introduces significant energy broad-\nening (several hundred meV). The weighting function,\nwhich is a generalization of the delta function appearing\nin a simple Fermi’s golden rule expression, quantifies the\nenergy overlap of the broadened electron and hole states\nwith each other and with the Fermi level.\nEquation 2, which has been discussed extensively\n[14, 18, 19, 20], considers only local contributions to the\ndamping rate. Non-local contributions to damping may\nbe studied through the decay of non-uniform spin-waves.\nAlthough recent efforts have approached the problem of2\nnon-localcontributionsto the dissipationofnon-collinear\nexcited states [21, 22] the simple step of generalizing\nKambersk´ y’s theory to non-uniform mode magnons has\nnot yet been taken. We fill this obvious gap, obtaining a\ndamping rate of\nλq=π¯hγ2\nM\nµ0/summationdisplay\nnm/integraldisplay\ndk/vextendsingle/vextendsingleΓ−\nnm(k,k+q)/vextendsingle/vextendsingle2Wnm(k,k+q)\n(3)\nfor a magnon with wavevector q. We test the impor-\ntance of non-local effects by quantifying this expression\nfor varying degrees of magnetic non-collinearity (magnon\nwavevector magnitude). The numerical evaluation of\nEq. 3 for the damping rate of finite wavelength magnons\nin transition metal systems, presented in Fig. 1, and the\nensuing physical discussion form the primary contribu-\ntion of this paper. We find that the damping rate ex-\npected inverypuresamplesatlowtemperatureisrapidly\nreduced as the magnon wavevector |q|grows, but the\ndamping rate anticipated outside of this ideal limit is\nbarely affected. We provide a simple band structure ar-\ngument to explain these observations. The results are\nrelevant to systems for which the non-collinear excita-\ntion may be expanded in long wavelength spin-waves,\nprovided the amplitude of these waves is small enough to\nneglect magnon-magnon scattering.\nCalculations for the single-mode damping constant\n(Eq. 3) as a function of electron scattering rate are pre-\nsented in Fig. 1 for iron, cobalt, and nickel. The Gilbert\ndamping parameter α=λ/γMis also given. Damp-\ning rates are given for magnons with wavevectors along\nthe bulk equilibrium directions, which are /angbracketleft100/angbracketrightfor Fe,\n/angbracketleft0001/angbracketrightfor Co, and /angbracketleft111/angbracketrightfor Ni. Qualitatively and quan-\ntitatively similar results were obtained for other magnon\nwavevector directions for each metal. The magnons re-\nportedoninFig.1constitutesmalldeviationsofthemag-\nnetization transverse to the equilibrium direction with\nwavevectormagnitudes between zero and 1 % of the Bril-\nlouin zone edge. This wavevector range corresponds to\nmagnon half-wavelengths between infinity and 100 lat-\ntice spacings, which is 28.7 nm for Fe, 40.7 nm for Co,\nand 35.2 nm for Ni. This range includes the wavelengths\nreported by Vlaminck and Bailleul in their recent mea-\nsurement of spin polarization [4].\nResults for the three metals are qualitatively similar.\nThemoststrikingtrendisadramatic,orderofmagnitude\ndecreaseofthe damping rate at the lowestscatteringrate\ntested as the wavevector magnitude increases from zero\nto 1 % of the Brillouin zone edge. This observation holds\nin each metal for every magnon propagation direction\ninvestigated. For the higher scattering rates expected in\ndevices at room temperature there is almost no change\nin the damping rate as the magnon wavevector increases\nfrom zero to 1 % of the Brillouin zone edge in any of the\ndirections investigated for any of the metals.\nTo understand the different dependences of the damp-\ning rate on the magnon wavevector at low versus high\nscattering rates we first note that the damping rate10121013101410151091010\n0.010.1λ (s-1)\nγ (s-1)\n α 1081090.01λ (s-1) \n α1090.01 1.0 0.1\n0.01\n λ (s-1)\n αhγ (eV)\n10-310-3\n10-30.0\n0.001\n0.003\n0.005\n0.01\n0.0\n0.001\n0.003\n0.005\n0.01\n0.0\n0.001\n0.003\n0.005\n0.01Fe\nCo\nNiq\nq\nq\nFIG. 1: Damping rates versus scattering rate. The preces-\nsion damping rates for magnons in iron, cobalt, and nickel\nare plotted versus electron scattering rate for several mag non\nwavevectors. A dramatic reduction in damping rate is ob-\nserved at the lowest scattering rates. The Landau-Lifshitz λ\n(Gilbert α) damping parameter is given on the left (right)\naxes. Electron scattering rate is given in eV on the top axis.\nMagnon wavevector magnitudes are given in units of the Bril-\nlouin zone edge and directions are as indicated in the text.\n(Eqs. 2 & 3) is a convolution of two factors: the torque\nmatrix elements and the weighting function. The ma-\ntrix elements do not change significantly as the magnon\nwavevector increases, however, the weighting function\ncan change substantially. The weighting function\nWnm(k,k+q)≈An,k(ǫF)Am,k+q(ǫF) (4)3\n789101112\n789101112\n Energy  (eV)\nH F P\nFIG. 2: Partial band structure of bcc iron. The horizontal\nblack line indicates the Fermi level and the shaded region\nrepresents the degree of spectral broadening. The solid dot is\na hypothetical initial electron state while the open circle is a\npotential final scattering state. (Initial and final state wa ve-\nvector separations are exaggerated for clarity of illustra tion.)\nThe intraband magnon decay rate diminishes as the energy\nseparation of the states exceeds the spectral broadening.\ncontains a product of the initial and final state electron\nspectral functions\nAn,k(ǫ) =1\nπ¯hγ\n(ǫ−ǫn,k)2+(¯hγ)2, (5)\nwhichareLorentziansinenergyspace. Thespectralfunc-\ntion for state |n,k/angbracketright, which has nominal band energy ǫn,k,\nis evaluated within a verynarrowrangeofthe Fermi level\nǫF. The width of the spectral function ¯ hγis given by\nthe electron scattering rate γ= 1/2τwhereτis the\norbital lifetime. (The lifetimes of all orbital states are\ntaken to be equal for these calculations and no specific\nscattering mechanism is implied.) The weighting func-\ntion restricts the electron-hole pair generated during the\nmagnon decay to states close in energy to each other and\nnear the Fermi level. For high scattering rates, the elec-\ntron spectral functions are significantly broadened and\nthe weighting function incorporates states within an ap-\npreciablerange(severalhundredmeV) ofthe Fermi level.\nFor low scattering rates, the spectral functions are quite\nnarrow (only a few meV) and both the electron and hole\nstate must be very close to the Fermi level.\nThe second consideration useful for understanding the\nresults of Fig. 1 is that the sum in Eqs. 2 & 3 can be\ndivided into intraband ( n=m) and interband ( n/negationslash=m)\nterms. For the uniform mode, these two contributions\ncorrespond to different physical processes with the intra-\nband contributiondominatingatlowscatteringratesand\nthe interband terms dominating at high scattering rates\n[14, 18, 19, 20].101210131014101510-3\n10-3\n10-4\n1071081090.01 0. 11\n0.01\n 0.0\n 0.001\n 0.003\n 0.005\n 0.01λintra,inte r (s-1)\nγ (s-1)q in ( π/a)interband\nintraband\n αintra,inter hγ (eV)Increasing q\nFIG. 3: Intraband and interband damping contributions in\niron. Theintrabandandinterbandcontributionstothedamp -\ning rate of magnons in the /angbracketleft100/angbracketrightdirection in iron are plot-\nted versus scattering rate for several magnitudes of magnon\nwavevector. Magnitudes are given in units of the Brillouin\nzone edge.\nFor intraband scattering, the electron and hole occupy\nthe sameband and must haveessentiallythe sameenergy\n(within ¯hγ). The energy difference between the electron\nand hole states may be approximated as ǫn,k+q−ǫn,k≈\nq·∂ǫn,k/∂k. The generation of intraband electron-hole\npairs responsible for intraband damping gets suppressed\nasq·∂ǫn,k/∂kbecomes largecomparedto ¯ hγ. Unless the\nbands are very flat at the Fermi level there will be few lo-\ncations on the Fermi surface that maintain the condition\nq·∂ǫn,k/∂k<¯hγfor low scattering rates as the magnon\nwavevectorgrows. (See Fig. 2). Indeed, at low scattering\nrates when ¯ hγis only a few meV, Fig. 3 shows that the\nintraband contribution to damping decreases markedly\nwith only modest increase of the magnon wavevector.\nSince the intraband contribution dominates the inter-\nband term in this limit the total damping rate also de-\ncreases sharply as the magnon wavevector is increased\nfor low scattering rates. For higher scattering rates, the\nelectronspectralfunctionsaresufficientlybroadenedthat\nthe overlap of intraband states does not decrease appre-\nciably as the states are separated by finite wavevector\n(q·∂ǫn,k/∂k<¯hγgenerally holds over the Fermi sur-\nface). Therefore, the intraband contribution is largelyin-\ndependentofmagnonwavevectorathighscatteringrates.\nThe interband contribution to damping involves scat-\ntering between states in different bands, separated by the\nmagnon wavevector q. Isolating the interband damping\ncontribution reveals that these contributions are insensi-\ntive to the magnon wavevector at higher scattering rates\nwhere they form the dominant contribution to damp-\ning (see Fig. 3). To understand these observations we\nagain compare the spectral broadening ¯ hγto the quasi-\nparticle energy difference ∆m,k+q\nn,k=ǫm,k+q−ǫn,k. The\nquasiparticle energy difference may be approximated as4\n∆m,k\nn,k+q·∂∆m,k\nn,k/∂k. The interband energy spacings\nare effectively modulated by the product of the magnon\nwavevector and the slopes of the bands. At high scatter-\ning rates, when the spectral broadening exceeds the ver-\ntical band spacings, this energy modulation is unimpor-\ntant and the damping rate is independent of the magnon\nwavevector. At low scattering rates, when the spec-\ntral broadening is less than many of the band spacings,\nthis modulation can alter the interband energy spacings\nenough to allow or forbid generation of these electron-\nhole pairs. For Fe, Co, and Ni, this produces a modest\nincrease in the interband damping rate at low scattering\nrates as the magnon wavevector increases. However, this\neffect is unimportant to the total damping rate, which\nremains dominated by the intraband terms at low scat-\ntering rates.\nLastly, we describe the numerical methods employed\nin this study. Converged ground state electron densities\nwere first obtained via the linear-augmented-plane-wave\nmethod. The Perdew-Wang functional for exchange-\ncorrelation within the local spin density approximation\nwas implemented. Many details of the ground state den-\nsity convergence process are given in [23]. Densities were\nthen expanded into Kohn-Sham orbitals using a scalar-\nrelativistic spin-orbit interaction with the magnetiza-\ntion aligned along the experimentally determined mag-\nnetocrystalline anisotropy easy axis. The Kohn-Sham\nenergies were artificially broadened through the ad hoc\nintroduction of an electron lifetime. Matrix elements of\nthe torque operator Γ−= [σ−,Hso] were evaluated sim-\nilarly to the spin-orbit matrix elements [24]. ( σ−is the\nspin lowering operator and Hsois the spin-orbit Hamil-\ntonian.) The product of the matrix elements and the\nweightingfunction wereintegratedover k-spaceusingthe\nspecial points method with a fictitious smearing of the\nFermi surface for numerical stability. Convergence wasobtained by sampling the full Brillouin zone with 1603\nk-points for Fe and Ni, and 1602x 91 points for Co.\nIn summary, we have investigated the importance of\nnon-local damping effects by calculating the intrinsic\nspin-orbit contribution to precession damping in bulk\ntransition metal ferromagnets for small amplitude spin-\nwaveswith finite wavelengths. Results ofthe calculations\ndo not contradict the common-practice assumption that\ndamping is a local phenomenon. For transition metals,\nat scattering rates corresponding to room temperature,\nwe find that the single-mode damping rate is essentially\nindependent of magnon wavevector for wavevectors be-\ntween zero and 1 % of the Brillouin zone edge. It is not\nuntil low temperatures in the most pure samples that\nnon-local effects become significant. At these scatter-\ning rates, damping rates decrease by as much as an or-\nder of magnitude as the magnon wavevector is increased.\nThe insensitivity of damping rate to magnon wavevector\nat high scattering rates versus the strong sensitivity at\nlow scattering rates can be explained in terms of band\nstructure effects. Due to electron spectral broadening at\nhigh scattering rates the energy conservation constraint\nduring magnon decay is effectively relaxed, making the\ndamping rate independent of magnon wavevector. The\nminimal spectral broadening at low scattering rates –\nseenonlyinverypureandcoldsamples–greatlyrestricts\nthe possible intraband scattering processes, lowering the\ndamping rate. The prediction of reduced damping at low\nscattering rates and non-zero magnon wavevectors is of\nlittle practical importance, but could provide an accessi-\nble test of the torque-correlationmodel. Specifically, this\nmight be testable in ferromagnetic semiconductors such\nas (Ga,Mn)As forwhich manyspin-waveresonanceshave\nbeen experimentally observed at low temperatures [25].\nThis work has been supported in part through NIST-\nCNST / UMD-Nanocenter cooperative agreement.\n[1] L.LandauandE. Lifshitz, Phys.Z.Sowjet. 8, 153 (1935).\n[2] T. L. Gilbert, Armour research foundation project No.\nA059, supplementary report, unpublished (1956).\n[3] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[4] V. Vlaminck and M. Bailleul, Science 322, 410 (2008).\n[5] S.S.P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n[6] M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and\nS.S.P. Parkin, Science 320, 209 (2008).\n[7] B.E. Argyle, E. Terrenzio, and J.C. Slonczewski,\nPhys. Rev. Lett. 53, 190 (1984).\n[8] R.D. McMichael, C.A. Ross, and V.P. Chuang,\nJ. Appl. Phys. 103, 07C505 (2008).\n[9] S. Kaka, M.R. Pufall, W.H. Rippard, T.J. Silva,\nS.E. Russek, and J.A. Katine, Nature 437, 389 (2005).\n[10] F.B. Mancoff, N.D. Rizzo, B.N. Engel, and S. Tehrani,\nNature437, 393 (2005).\n[11] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y,\nPhys. Stat. Sol. 23, 501 (1967).\n[12] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).[13] V. Korenman and R.E. Prange, Phys. Rev. B 6, 2769\n(1972).\n[14] V. Kambersk´ y, Czech. J. Phys. B 26, 1366 (1976).\n[15] J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J.K. Furdyna ,\nW.A. Atkinson, and A.H. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n[16] Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[17] M. Zwierzycki, Y. Tserkovnyak, P.J. Kelly, A. Brataas,\nand G.E.W. Bauer, Phys. Rev. B 71, 064420 (2005).\n[18] K. Gilmore, Y.U. Idzerda, and M.D. Stiles,\nPhys. Rev. Lett. 99, 027204 (2007).\n[19] V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007).\n[20] K.Gilmore, Y.U.Idzerda, andM.D.Stiles, J.Appl.Phys .\n103, 07D303 (2008).\n[21] M. F¨ ahnle and D. Steiauf, Phys. Rev. B 73, 184427\n(2006).\n[22] J. Foros, A.Brataas, Y.Tserkovnyak,andG.E.W. Bauer,\nPhys. Rev. B 78, 140402(R) (2008).\n[23] L.F. Mattheiss and D.R. Hamann, Phys. Rev. B 33, 8235\n(1986).\n[24] M.D. Stiles, S.V. Halilov, R.A. Hyman, and A. Zangwill,\nPhys. Rev. B 64, 104430 (2001).\n[25] S.T.B. Goennenwein, T. Graf, T. Wassner, M.S. Brandt,M. Stutzmann, A. Koeder, S. Frank, W. Schoch, and\nA. Waag, Journal of Superconductivity 16, 75 (2003)."    },    {        "title": "2205.06399v1.Precession_dynamics_of_a_small_magnet_with_non_Markovian_damping__Theoretical_proposal_for_an_experiment_to_determine_the_correlation_time.pdf",        "content": "arXiv:2205.06399v1  [cond-mat.mes-hall]  13 May 2022Precession dynamics of a small magnet with non-Markovian da mping: Theoretical\nproposal for an experiment to determine the correlation tim e,✩✩\nHiroshi Imamura, Hiroko Arai, Rie Matsumoto, Toshiki Yamaj i, Hiroshi Tsukahara✩\nNational Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan\nAbstract\nRecent advances in experimental techniques have made it pos sible to manipulate and measure the magnetization dynamics on\nthe femtosecond time scale which is the same order as the corr elation time of the bath degrees of freedom. In the equations of\nmotion of magnetization, the correlation of the bath is repr esented by the non-Markovian damping. For development of th e science\nand technologies based on the ultrafast magnetization dyna mics it is important to understand how the magnetization dyn amics\ndepend on the correlation time. It is also important to deter mine the correlation time experimentally. Here we study the precession\ndynamics of a small magnet with the non-Markovian damping. E xtending the theoretical analysis of Miyazaki and Seki [J. C hem.\nPhys. 108, 7052 (1998)] we obtain analytical expressions of the prece ssion angular velocity and the e ffective damping constant for\nany values of the correlation time under assumption of small Gilbert damping constant. We also propose a possible experi ment for\ndetermination of the correlation time.\nKeywords: non-Markovian damping, generalized Langevin equation, LL G equation, ultrafast spin dynamics, correlation time\n1. Introduction\nDynamics of magnetization is the combination of precession\nand damping. The precession is caused by the torque due to\nthe internal and external magnetic fields. Typical time scal e\nof the precession around the external field and the anisotrop y\nfield is nanosecond. The damping is caused by the coupling\nwith the bath degrees of freedom such as conduction electron s\nand phonons. The typical time scale of the relaxation of con-\nduction electrons and phonons is picosecond or sub-picosec ond\nwhich is much faster than precession. In typical experiment al\nsituations such as ferromagnetic resonance and magnetizat ion\nprocess, the time correlation of the bath degrees of freedom\ncan be neglected and the magnetization dynamics is well repr e-\nsented by the Landau-Lifshitz-Gilbert (LLG) equation with the\nMarkovian damping term[1–3].\nRecent advances in experimental techniques such as fem-\ntosecond laser pulse and time-resolved magneto-optical Ke rr\neffect measurement have made it possible to manipulate and\nmeasure magnetization dynamics on the femtosecond time\nscale[4–11]. In 1996, Beaurepaire et al. observed the femto sec-\nond laser pulse induced sub-picosecond demagnetization of a\nNi thin film[4], which opens the field of ultrafast magnetiza-\ntion dynamics. The all-optical switching of magnetization in a\n✩Permanent address: High Energy Accelerator Research Organ ization\n(KEK), Institute of Materials Structure Science (IMSS), Ts ukuba, Ibaraki 305-\n0801, Japan\n✩✩This work is partly supported by JSPS KAKENHI Grant Numbers\nJP19H01108 and JP18H03787.\nEmail addresses: h-imamura@aist.go.jp (Hiroshi Imamura),\narai-h@aist.go.jp (Hiroko Arai)ferrimagnetic GdFeCo alloy was demonstrated by Stanciu et a l.\nusing a 40 femtosecond circularly polarized laser pulse[5] . The\nhelicity-dependent laser-induced domain wall motion in Co /Pt\nmultilayer thin films was reported by Quessab et al.[11].\nTo understand the physics behind the ultrafast magnetizati on\ndynamics it is necessary to take into account the time correl a-\ntion of bath in the equations of motion of magnetization. The\nfirst attempt was done by Kawabata in 1972[12]. He derived the\nBloch equation and the Fokker-Planck equation for a classic al\nspin interacting with the surroundings based on the Nakajim a-\nZwanzig-Mori formalism[13–15]. In 1998, Miyazaki and Seki\nconstructed a theory for the Brownian motion of a classical\nspin and derived the integro-di fferential form of the generalized\nLangevin equation with non-Markovian damping[16]. They\nalso showed that the generalized Langevin equation reduces to\nthe LLG equation with modified parameters in a certain limit.\nAtxitia et al. applied the theory of Miyazaki and Seki to the\natomistic model simulations and showed that materials with\nsmaller correlation time demagnetized faster[17].\nDespite the experimental and theoretical progresses to dat e\nlittle attention has been paid to how to determine the correl a-\ntion time experimentally. For development of the science an d\ntechnologies based on the ultrafast magnetization dynamic s it\nis important to determine the correlation time experimenta lly\nas well as to understand how the magnetization dynamics de-\npend on the correlation time.\nIn this paper the precession dynamics of a small magnet with\nnon-Markovian damping is theoretically studied based on th e\nmacrospin model. The magnet is assumed to have a uniaxial\nanisotropy and to be subjected to an external magnetic field\nparallel to the magnetization easy axis. The non-Markovian ity\nPreprint submitted to Journal of Magnetism and Magnetic Mat erials May 16, 2022enhances the precession angular velocity and reduces the da mp-\ning. Assuming that the Gilbert damping constant is much\nsmaller than unity, the analytical expressions of the prece ssion\nangular velocity and the e ffective damping constant are derived\nfor any values of the correlation time by extending the analy sis\nof Miyazaki and Seki[16]. We also propose a possible exper-\niment for determination of the correlation time. The correl a-\ntion time can be determined by analyzing the external field at\nwhich the enhancement of the precession angular velocity is\nmaximized.\nThe paper is organized as follows. Section 2 explains the\ntheoretical model and the equations of motion. Section 3 giv es\nthe numerical and theoretical analysis of the precession dy nam-\nics in the absence of an anisotropy field. The e ffect of the\nanisotropy field is discussed in Sec. 4. A possible experimen t\nfor determination of the correlation time is proposed in Sec . 5.\nThe results are summarized in Sec. 6.\n2. Theoretical model\nWe calculate the magnetization dynamics in a small mag-\nnet with a uniaxial anisotropy under an external magnetic fie ld\nbased on the macrospin model. The magnetization easy axis\nis taken to be z-axis and the magnetic field is applied in the\npositive z-direction. In terms of the magnetization unit vector,\nm=(mx,my,mz), the energy density is given by\nE=K(1−m2\nz)−µ0MsH m z, (1)\nwhere Kis the effective anisotropy constant including the crys-\ntalline, interfacial, and shape anisotropies. µ0is the vacuum\npermeability, Msis the saturation magnetization, His the exter-\nnal magnetic field. The e ffective field is obtained as\nHeff=(Hkmz+H)ez, (2)\nwhere ezis the unit vector in the positive zdirection and Hk=\n2K/(µ0Ms) is the effective anisotropy field.\nThe magnetization precesses around the e ffective field with\ndamping. The energy and angular momentum are absorbed by\nthe bath degrees of freedom such as conduction electrons and\nphonons until the magnetization becomes parallel to the e ffec-\ntive field. The equations of motion of mcoupled with the bath\nis given by the Langevin equation with the stochastic field re p-\nresenting the bath degrees of freedom. If the time scale of th e\nbath is much smaller than the precession frequency the stoch as-\ntic field can be treated as the Wiener process[18] as shown by\nBrown[3].\nSince we are interested in the ultrafast magnetization dyna m-\nics of which time scale is the same order as the correlation ti me\nof the bath degrees of freedom, the stochastic field should be\ntreated as the Ornstein-Uhlenbeck process[18, 19]. As show n\nby Miyazaki and Seki [16] the equations of motion of mtakes\nthe following integro-di fferential form:\n˙m=−γm×(Heff+r)+αm×/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′,(3)whereγis the gyromagnetic ratio, αis the Gilbert damping con-\nstant, and ris the stochastic field. The first term represents the\nprecession around the sum of the e ffective field and the stochas-\ntic field, and the second term represents the non-Markovian\ndamping. The memory function in the non-Markovian damping\nterm is defined as\nν(t−t′)=1\nτcexp/parenleftigg\n−|t−t′|\nτc/parenrightigg\n, (4)\nwhereτcis the correlation time of the bath degrees of freedom.\nThe stochastic field, r, satisfies/angbracketleftri(t)/angbracketright=0 and\n/angbracketleftrj(t)rk(t′)/angbracketright=µ\n2δj,kν(t−t′), (5)\nwhere/angbracketleft/angbracketrightrepresents the statistical mean, and\nµ=2αkBT\nγMsV. (6)\nThe subscripts jandkstand for x,y, orz,kBis the Boltzmann\nconstant, Tis the temperature, Vis the volume of the mag-\nnet, andδj,kis Kronecker’s delta. The LLG equation with the\nMarkovian damping derived by Brown [3] is reproduced in the\nlimit ofτc→0 because lim τc→0ν(t−t′)=2δ(t−t′), where\nδ(t−t′) is Dirac’s delta function. Equation (3) is equivalent to\nthe following set of the first order di fferential equations,\n˙m=−γm×[Heff+δH] (7)\n˙δH=−1\nτcδH−α\nτ2cm−γ\nτcR, (8)\nwhere Rrepresents the stochastic field due to thermal agita-\ntion. Equations (7), (8) are used for numerical simulations . The\nstochastic field, R, satisfies/angbracketleftRj(t)/angbracketright=0 and\n/angbracketleftRj(t)Rk(t′)/angbracketright=µδj,kδ(t−t′). (9)\n3. Precession dynamics in the absence of an anisotropy field\nIn this section the precession dynamics in the absence of an\nanisotropy field, i.e. Hk=0, is considered. The initial di-\nrection of magnetization is assumed to be m=(1,0,0). The\nnumerical simulation shows that the non-Markovian damping\nenhances the precession angular velocity and reduces the da mp-\ning. The numerical results are theoretically analyzed assu ming\nthatα≪1. The analytical expressions of the precession an-\ngular velocity and the e ffective damping constant are obtained.\nThe case with Hk/nequal0 will be discussed in Sec. 4.\n3.1. numerical simulation results\nWe numerically solve Eqs. (7), (8) for H=10 T,α=0.05,\nandτc=1 ps. The temperature is assumed to be low enough\nto set R=0 in Eq. (8). Figure 1(a) shows the trajectory of m\non a unit sphere. The initial direction is indicated by the fil led\ncircle. The plot of the temporal evolutions of mx,my, and mzare\nshown in Fig. 1(b). The magnetization relaxes to the positiv ez\ndirection with precessing around the external field. The res ults\n2t [ps] 0.00 0.01 0.03 \n0.02 0.04 \n100 200 0\nt [ps] \na) b) \nc) d) z\nx yHφ [rad / ps] \n1.76 1.77 1.79 \n1.78 1.80 100 200 0\n100 200 0-1 1\n0mx, m y, m z\nt [ps] \nφ00.05 \nαmzmymx\nyαeff \nFigure 1: (a) Trajectory of mon a unit sphere. The external field of H=\n10 T is applied in the positive zdirection. The initial direction is assumed to\nbem=(1,0,0) as indicated by the filled circle. The other parameters are\nτc=1 ps, andα=0.05. (b) Temporal evolution of mx,my,mz. (c) Temporal\nevolution of the precession angular velocity, ˙φ. The solid red curve shows the\nsimulation result. The dotted black line indicates the resu lt of the Markovian\nLLG equation, i.e. ˙φ0=γH/(1+α2). (d) Temporal evolution of the e ffective\ndamping constant, αeff. The solid red curve shows the simulation result. The\ndotted black line indicates α=0.05.\nare quite similar to that of the Markovian LLG equation, whic h\nimplies that the non-Markovianity in damping causes renorm al-\nization of the gyromagnetic ratio and the Gilbert damping co n-\nstant in the Markovian LLG equation.\nThe renormalized value of the gyromagnetic ratio can be\nobserved as a variation of the precession angular velocity, ˙φ,\nwhere the polar and azimuthal angles are defined as m=\n(sinθcosφ,sinθsinφ,cosθ). Figure 1(c) shows that temporal\nevolution of ˙φ(solid red) together with the precession angular\nvelocity without non-Makovianity, ˙φ0=γH/(1+α2), (dotted\nblack). The precession angular velocity increases with inc rease\nof time and saturates to a certain value around 1.798. The sha pe\nof the time dependence of ˙φis quite similar to that of mzshown\nin Fig. 1(b), which suggests that the non-Markovian damping\nacts as an effective anisotropy field in the precession dynamics.\nThe renormalization of the Gilbert damping constant can be\nobserved as a variation of the temporal evolution of the pola r\nangle, ˙θ. Rearranging the LLG equation for ˙θ, the effective\ndamping constant can be defined as\nαeff=−˙θ/(γHsinθ). (10)\nIn Fig. 1(d)αeffis shown by the red solid curve as a function of\ntime. The effective damping constant is reduced to about one-\nfifth of the original value of α=0.05 (dotted black). Contrary\nto˙φ,αeffdoes not show clear correlation with the dynamics of\nm. During the precession, αeffis kept almost constant.\nThe enhancement of the precession angular velocity and the\nreduction of the Gilbert damping constant due to the non-\nMarkovian damping will be explained by deriving the e ffectiveLLG equation that is valid up to the first order of αin the next\nsubsection.\n3.2. Theoretical analysis\nSince the Gilbert damping constant, α, of a conventional\nmagnet is of the order of 0 .01∼0.1, it is natural to take the\nfirst order ofαto derive the effective equations of motion for\nm. The other parameters related to the motion of mareγ,H,\nandτc. Multiplying these parameters we can obtain the dimen-\nsionless parameter, ξ=γHτc, which represents the increment\nof the precession angle during the correlation time.\nIn the case ofξ<1 Miyazaki and Seki dereived the e ffective\nLLG equation using time derivative series expansion[16]. W e\nfirst briefly review their analysis. Then we derive the e ffective\nLLG equation forξ>1 using the time-integral series expansion\nand show that the e ffective LLG equation has the same form for\nbothξ< 1 andξ> 1. Therefore, it is natural to assume that\nthe derived effective LLG equation is valid for any values of ξ\nincludingξ=1.\n3.2.1. Brief review of Miyazaki and Seki’s derivation of the ef-\nfective LLG equation for ξ<1\nIn Ref. 16, Miyazaki and Seki derived the e ffective LLG\nequation with renormalized parameters using the time deriv a-\ntive series expansion. Similar analysis of the LLG equation\nwas also done by Shul in the study of the damping due to\nstrain[20, 21]. The following is the brief summary of the der iva-\ntion.\nSuccessive application of the integration by parts using ν(t−\nt′)=τc[dν(t−t′)/dt′] gives the following time derivative se-\nries expansion:\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay\nn=1(−τc)n−1dnm\ndtn. (11)\nThen the non-Markovian damping term in Eq. (3) is expressed\nas\nα∞/summationdisplay\nn=1(−τc)n−1/parenleftigg\nm×dnm\ndtn/parenrightigg\n. (12)\nThe first derivative, n=1, is given by\n˙m=−γHm×ez+O(α), (13)\nwhere Ois the Bachmann–Landau symbol. For n=2, substi-\ntution of Eq. (13) into the time derivative of Eq. (13) gives\n¨m=(−γH)2(m×ez)×ez+O(α). (14)\nThe n-th order time derivative is obtained by using the same\nalgebra as\ndn\ndtnm=(−γH)n/bracketleftbig(m×ez)×ez.../bracketrightbig+O(α), (15)\nwhere ezappears ntimes. Expanding the vector products we\nobtain for even order time derivatives\nd2nm\ndt2n=(−1)n(γH)2n/bracketleftbigm−mzez/bracketrightbig+O(α), (16)\n3and for odd order time derivatives\nd2n+1m\ndt2n+1=(−1)n(γH)2n˙m+O(α). (17)\nSubstituting Eqs. (16) and (17) into Eq. (12) the non-\nMarkovian damping term is expressed as\n−∞/summationdisplay\nn=1γ2nm×ez+∞/summationdisplay\nn=0α2n+1m×˙m, (18)\nwhere\nγ2n=αγH m z(−1)n−1ξ2n−1(19)\nα2n+1=α(−1)nξ2n. (20)\nThe sums in Eq. (18) converge for ξ<1. Introducing\n˜γ=γ/parenleftigg\n1+αmzξ\n1+ξ2/parenrightigg\n(21)\n˜α=α\n1+ξ2, (22)\nEq. (3) can be expressed as the following e ffective LLG equa-\ntion with renormalized gyromagnetic ratio, ˜ γ, and damping\nconstant, ˜α:\n˙m=−˜γm×(H+r)+˜αm×˙m+O(α2). (23)\n3.2.2. Derivation of the e ffective LLG equation for ξ>1\nForξ>1 we expand Eq. (3) in power series of 1 /ξusing the\ntime integral series expansion approach. Using the integra tion\nby parts with dν(t−t′)/dt′=ν(t−t′)/τcthe integral part of the\nnon-Markovian damping can be written as\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=1\nτc/integraldisplayt\n−∞˙m(t′)dt′\n−1\nτc/integraldisplayt\n−∞ν(t−t′)/bracketleftigg/integraldisplayt′\n−∞˙m(t′′)dt′′/bracketrightigg\ndt′. (24)\nSuccessive application of the integration by parts gives\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=−∞/summationdisplay\nn=1/parenleftigg\n−1\nτc/parenrightiggn\nJn, (25)\nwhere Jnis the nth order multiple integral defined as\nJn=/integraldisplayt\n−∞/integraldisplayt1\n−∞···/integraldisplaytn−1\n−∞˙m(tn)dtn···dt2dt1. (26)\nFrom Eq. (17), on the other hand, ˙ mis expressed as\n˙m=1\n(−1)n(γH)2nd2n\ndt2n˙m+O(α). (27)\nSubstituting Eq. (27) into Eq. (26) the multiple integrals a re\ncalculated as\nJ2n=1\n(−1)n(γH)2n˙m (28)\nJ2n−1=1\n(−1)n(γH)2n¨m. (29)Then Eq. (25) becomes\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay\nn=11\n(−1)n−1ξ2n˙m\n+∞/summationdisplay\nn=1τc\n(−1)nξ2n¨m. (30)\nSubstituting Eq. (30) into the second term of Eq. (3) the non-\nMarkovian damping term is expressed as\nα∞/summationdisplay\nn=11\n(−1)n−1ξ2nm×[ ˙m+τc¨m]+O(α2). (31)\nFrom Eq. (16) ¨ mis expressed as\n¨m=(−1)(γH)2/bracketleftbigm−mzez/bracketrightbig. (32)\nSubstituting Eqs. (31) and (32) into Eq. (3) we obtain\n˙m=−γH∞/summationdisplay\nn=1/bracketleftigg\n1+αmz\n(−1)n−1ξ2n−1/bracketrightigg\nm×ez−γm×r\n+α∞/summationdisplay\nn=11\n(−1)n−1ξ2nm×˙m+O(α2). (33)\nThe sums converge for ξ>1, and the effective LLG equation\nforξ>1 has the same form as ξ<1, i.e. Eq. (23). Since the\neffective LLG equation has the same form for both ξ< 1 and\nξ>1, it is natural to Eq. (23) is valid for any values of ξ.\nAs pointed out by Miyazaki and Seki, and independently by\nSuhl the effect of the non-Markovian damping on the precession\ncan be regarded as the renormalization of the e ffective field [16,\n20, 21]. Equation (23) can be expressed as\n˙m=−γm×/parenleftigg\nH+αHξ\n1+ξ2mz/parenrightigg\nez−γm×r\n+˜αm×˙m+O(α2). (34)\nThe second term in the bracket represents the fictitious unia xial\nanisotropy field originated from the non-Markovian damping .\nThe fictitious anisotropy field increases with increase of ξfor\nξ<1 and takes the maximum value of αH m z/2 atξ=1, i.e.\nγHτc=1. Forξ>1 the fictitious anisotropy field decreases\nwith increase ofξand vanishes in the limit of ξ→∞ because\nthe non-Markovian damping term vanishes in the limit of τc→\n∞. The precession angular velocity, ˙φ, is expected to have the\nsameξdependence as the fictitious anisotropy field and to have\nthe same temporal evolution as mzas shown in Figs. 1(b) and\n1(c).\n3.2.3. The Correlation time dependence of the precession an -\ngular velocity, and e ffective damping constant\nEquation (21) tells us that up to the first order of αthe pre-\ncession angular velocity can be approximated as\n˙φ≃˜γH=γH/bracketleftigg\n1+αmzγHτc\n1+(γHτc)2/bracketrightigg\n, (35)\n4τc’ =1/( γH) \n0.1 1 10 0.01 \nτc [ps] τc [ps] a) b) \nφ [rad / ps] \n1.76 1.77 1.79 \n1.78 1.80 \n0.1 1 10 0.01 \nα, αeff ~\nαeffα0.04 \n0.00 0.02 0.05 \n0.01 0.03 \nααα\neffeffeffαeffαeff~sim. approx. \nFigure 2: (a) The correlation time, τc, dependence of the precession angular\nvelocity,δ˙φ, atθ=5◦forH=10 T. The solid yellow curve shows the ap-\nproximation result, ˜ γH. The dotted black curve shows the simulation results\nobtained by numerically solving Eqs. (7) and (8). The thin ve rtical dotted line\nindicates the critical value of the correlation time, τ′\nc=1/(γH). (b)τcdepen-\ndence of ˜α(solid yellow) andαeff(dotted black). The parameters and the other\nsymbols are the same as panel (a).\nwhere the second term in the square bracket represents the en -\nhancement due to the fictitious anisotropy field.\nIn Fig. 2(a) the approximation result of Eq. (35) at θ=5◦\nwhere ˙φis almost saturated is plotted as a function of τcby the\nsolid yellow curve. The external field and the Gilbert damp-\ning constant are assumed to be H=10 T andα=0.05, re-\nspectively. The corresponding simulation results obtaine d by\nnumerically solving Eqs. (7) and (8) are shown by the dotted\nblack curve. Both curves agree well with each other because\nαis as small as 0.05. The precession angular velocity is maxi-\nmized at the critical value of the correlation time τ′\nc=1/(γH).\nFigure 2(b) shows the τcdependence of ˜α(solid yellow) and\nαeff(dotted black) for the same parameters as panel (a). Both\ncurves agree well with each other and are monotonic decreasi ng\nfunctions ofτc. They vanish in the limit of τc→∞ similar to\nthe non-Markovian damping term.\n4. Effect of an anisotropy field on precession dynamics\nThe theoretical analysis given in the previous section can\nbe applied to the case with Hk/nequal0 by replacingξwithξk=\nγ(H+Hkmz)τc. Following the same procedure as for Hk=0\nEq. (3) can be expressed as\n˙m=−γm×/parenleftig\nH+αHξk\n1+ξ2\nkmz+αHkξk\n1+ξ2\nkm2\nz/parenrightig\nez\n−γm×r+α\n1+ξ2\nkm×˙m+O(α2). (36)\nThe second and the third terms in the bracket can be regarded\nas the fictitious uniaxial and unidirectional anisotropy fie lds\ncaused by the non-Markovian damping. Similar to the re-\nsults for Hk=0 the precession angular velocity is maximized\natξk=1. The renormalized damping constant is given by\nα/(1+ξ2\nk) which is a monotonic decreasing function of ξkand\nvanishes in the limit of ξk→∞ .c) d) a) b) \n24 6 14 12 10 8 0\nH [T] δφ /φ0 [%] 2\n013\nδφ /φ0 [%] 2\n01\n24 6 14 12 10 8 0\nH [T] Hk = 0 H’=1/( γτ c) \nτc = 1 ps \nθ = 5 oθ = 5 oHk = 1 T \nτc = 1 ps H’=1/( γτ c) − HkmzH = 6, 7, 8, 9 T \nH = 2, 3, 4, 5 T \n = 1 ps \n = 5  = 0 \n = 1 ps τ\nθH\nτ = 1 ps  = 1 ps  = 1 ps τc = 1 ps τ = 1 ps τ = 1 ps \nθτH\nτ\n = 5  = 1 ps  = 1 T \n = 1 ps  = 1 ps  = 1 T \n = 1 ps  = 1 ps τc = 1 ps ττ = 1 ps  = 1 ps 3δφ /φ0 [%] 2\n013\nδφ /φ0 [%] 2\n013\nt [ps] 100 200 0\nt [ps] 100 200 0Hk = 0 \nτc = 1 ps Hk = 0 \nτc = 1 ps \nFigure 3: (a)τcdependence ofδ˙φ/˙φ0atθ=5◦. From top to bottom the\nexternal field is H=2,3,4,5 T. The parameters are Hk=0, andτc=1 ps. (b)\nThe same plot as panel (a) for H≥5 T. From top to bottom the external field is\nH=6,7,8,9 T. (c) Hdependence ofδ˙φ/˙φ0atθ=5◦obtained by solving Eqs.\n(7) and (8). The parameters are Hk=0, andτc=1 ps. The critical value of\nthe external field, H′=1/(γτc), is indicated by the thin vertical dotted line. (d)\nThe same plot as panel (c) for Hk=1 T. The thin vertical dotted line indicates\nthe critical value of the external field, H′=1/(γτc)−Hkmz.\n5. A possible experiment to determine the correlation time\nBased on the results shown in Secs. 3 and 4 we propose a\npossible experiment to determine the correlation time, τc. Sim-\nilar to the previous sections we first discuss the case withou t\nanisotropy field, i.e. Hk=0, and then extend the discussion to\nthe case with Hk/nequal0.\nIn Figs. 3(a) and 3(b) we show the temporal evolution of the\nenhancement of angular velocity, δ˙φ/˙φ0, obtained by the solv-\ning Eqs. (7) and (8) for various values of H. The increment\nof the precession angular velocity is defined as δ˙φ=˙φ−˙φ0.\nThe initial state and the correlation time are assumed to be\nm=(1,0,0) andτc=1 ps, respectively. As shown in Fig.\n3(a),δ˙φ/˙φ0increases with increase of HforH≤5T. Once\nthe external field exceeds the critical value of 1 /(γτc)=5.7 T,\nδ˙φ/˙φ0decreases with increase of Has shown in Fig. 3 (b). The\nresults suggest that correlation time can be determined by a na-\nlyzing the external field that maximizes the enhancement of t he\nprecession angular velocity.\nFigure 3(c) shows the Hdependence ofδ˙φ/˙φ0atθ=5◦\nwhereδ˙φ/˙φ0is almost saturated. The enhancement is maxi-\nmized at the critical value of the external field, H′=5.7 T. The\ncorrelation time is calculated as τc=1/(γH′)=1 ps.\nIf the system has a uniaxial anisotropy field, Hk, the en-\nhancement of the precession angular velocity is maximized a t\nH′=1/(γτc)−Hkmzas shown in Fig. 3(d). The correlation\ntime is obtained as τc=1/γ(H′+Hkmz).\nThe above analysis is expected to be performed experimen-\n5tally using the time resolved magneto optical Kerr e ffect mea-\nsurement technique. In the practical experiments the analy sis\ncan be simplified as follows. The polar angle of the initial st ate\nis not necessarily large. It can be small as far as the preces-\nsion angular velocity can be measured. Instead of analyzing\nδ˙φ/˙φ0, one can analyze ˙φ/Hor˙φ/(H+Hkmz) because they are\nmaximized at the same value of Hasδ˙φ/˙φ0. Since the required\nmagnetic field is as high as 10 T, a superconducting magnet [22 ]\nis required.\n6. Summary\nIn summary we theoretically analyze the ultrafast precessi on\ndynamics of a small magnet with non-Markovian damping. As-\nsumingα≪1, we derive the effective LLG equation valid for\nany values ofτc, which is a direct extension of Miyazaki and\nSeki’s work[16]. The derived e ffective LLG equation reveals\nthe condition for maximizing ˙φin terms of Handτc. Based on\nthe results we propose a possible experiment for determinat ion\nofτc, whereτccan be determined from the external field that\nmaximizesδ˙φ/˙φ0.\nReferences\n[1] L. Landau, E. 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Larbalestier, High field magnets with HTS conductors, I EEE\nTransactions on Applied Superconductivity 20 (3) (2010) 57 6–582.\ndoi:10.1109/TASC.2010.2043080 .\n6"    },    {        "title": "1608.00984v2.Ferromagnetic_Damping_Anti_damping_in_a_Periodic_2D_Helical_surface__A_Non_Equilibrium_Keldysh_Green_Function_Approach.pdf",        "content": "arXiv:1608.00984v2  [cond-mat.mes-hall]  13 Aug 2016Ferromagnetic Damping/Anti-damping in a Periodic 2D Helic al surface; A\nNon-Equilibrium Keldysh Green Function Approach\nFarzad Mahfouzi1,∗and Nicholas Kioussis1\n1Department of Physics, California State University, North ridge, California 91330-8268, USA\nIn this paper, we investigate theoretically the spin-orbit torque as well as the Gilbert damping for\na two band model of a 2D helical surface state with a Ferromagn etic (FM) exchange coupling. We\ndecompose the density matrix into the Fermi sea and Fermi sur face components and obtain their\ncontributions to the electronic transport as well as the spi n-orbit torque (SOT). Furthermore, we\nobtain the expression for the Gilbert damping due to the surf ace state of a 3D Topological Insulator\n(TI) and predicted its dependence on the direction of the mag netization precession axis.\nPACS numbers: 72.25.Dc, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nThe spin-transfer torque (STT) is a phenomenon in\nwhich spin current of large enough density injected into\na ferromagnetic layer switches its magnetization from\none static configuration to another [1]. The origin of\nSTT is absorption of itinerant flow of angular momen-\ntum components normal to the magnetization direc-\ntion. It represents one of the central phenomena of the\nsecond-generation spintronics, focused on manipulation\nof coherent spin states, since reduction of current den-\nsities (currently of the order 106-108A/cm2) required\nfor STT-based magnetization switching is expected to\nbring commercially viable magnetic random access mem-\nory (MRAM) [2]. The rich nonequilibrium physics [3]\narising in the interplay of spin currents carried by fast\nconduction electrons and collective magnetization dy-\nnamics, viewed as the slow classical degree of freedom,\nis of great fundamental interest.\nVery recent experiments [4, 5] and theoretical stud-\nies[6] havesoughtSTT innontraditionalsetupswhich do\nnot involvetheusual two(spin-polarizingandfree) F lay-\ners with noncollinear magnetizations [3], but rely instead\non the spin-orbit coupling (SOC) effects in structures\nlacking inversion symmetry. Such “SO torques” [7] have\nbeen detected [4] in Pt/Co/AlO xlateral devices where\ncurrent flows in the plane of Co layer. Concurrently, the\nrecent discovery [8] of three-dimensional (3D) topologi-\ncal insulators (TIs), which possess a usual band gap in\nthe bulk while hosting metallic surfaces whose massless\nDirac electrons have spins locked with their momenta\ndue to the strong Rashba-type SOC, has led to theoreti-\ncal proposals to employ these exotic states of matter for\nspintronics [9] and STT in particular [10]. For example,\nmagnetizationofa ferromagneticfilm with perpendicular\nanisotropy deposited on the TI surface could be switched\nby interfacial quantum Hall current [10].\nIn this paper, we investigate the dynamical properties\nof a FM/3DTI heterostructure, where the F overlayer\n∗farzad.mahfouzi@gmail.comcovers a TI surface and the device is periodic along in-\nplanex−ydirections. The effect of the F overlayer is a\nproximityinduced exchangefield −∆surf/vector m·/vectorσ/2superim-\nposed on the Dirac cone dispersion. For a partially cov-\nered FM/TI heterostructure, the spin-momentum-locked\nDirac electrons flip their spin upon entering into the in-\nterface region, thereby inducing a large antidamping-like\nSOT on the FM [15–17]. The antidamping-like SOT\ndriven by this mechanism which is unique to the sur-\nface of TIs has been predicted in Ref. [17], where a time-\ndependent nonequilibrium Green function [18] (NEGF)-\nbased framework was developed. The formalism made it\npossible to separate different torque components in the\npresenceofarbitraryspin-flipprocesseswithinthedevice.\nSimilaranti-dampingtorqueshasalsobeen predicted[19]\nto exist due to the Berry phase in periodic structures\nwhere the device is considered infinite in in-plane direc-\ntions and a Kubo formula was used to describe the SOT\nas a linear response to homoginiuos electric field at the\ninterface. However, the connection between the two ap-\nproaches is not clear and one of the goals of the current\npaper is to address the similarities and the differences\nbetween the two. In the following we present the theo-\nretical formalism of the SOT and damping in the regime\nofslowlyvaryingparametersofaperiodicsysteminspace\nand time.\nGenerally, in a quantum system with slowly varying\nparameters in space and/or time, the system stays close\ntoits equilibrium state ( i.e.adiabaticregime)and the ef-\nfects of the nonadiabaticity is taken into account pertur-\nbatively using adiabatic expansion. Conventionally, this\nexpansion is performed using Wigner representation [20]\nafter the separation of the fast and slow variations in\nspace and/or time. [21] The slow variation implies that\nthe NEGFs vary slowly with the central space ( time ),\n/vector xc= (/vector x+/vector x′)/2(tc= (t+t′)/2 ), while they changefast\nwith the relative space (time), /vector xr=/vector x−/vector x′(tr=t−t′).\nHere we use an alternative approach, where we consider\n(x,t) and (/vector xr,tr) as the natural variables to describe the\nclose to adiabatic apace-time evolution of NEGFs and\nthen perform the following Fourier transform\nˇG(/vector xt;/vector x′t′) =/integraldisplaydE\n2πd/vectork\nΩkeiE(t−t′)+i/vectork·(/vector x−/vector x′)ˇG/vectorkE(/vector xt).(1)2\nwhere, Ω kis the volume of the phase space that the\n/vectork-integration is being performed. The standard Dyson\nequation of motion for ˇG(/vector xt;/vector x′t′) is cumbersome to ma-\nnipulate[22,23]orsolvenumerically,[24]sotheyareusu-\nally transformed to some other representation.[11] Gen-\neralizing the equation to take into account slowly varying\ntime and spatial dependence of the Hamiltonian we ob-\ntain,\nˇG=/parenleftbigg\nGrG<\n0Ga/parenrightbigg\n, (2)\n=/parenleftbigg\nGr,−1\nad−iDxtΣ<\n0 Ga,−1\nad−iDxt/parenrightbigg−1\n,\nwhere,\nGr,−1\nad= (E−iη)1−H(/vectork,t)−µ(/vector x),(3a)\nΣ<=−2iηf(E−i∂\n∂t−µ(/vector x)), (3b)\nDxt=∂\n∂t+∂H\n∂/vectork·/vector∇, (3c)\nand,η=/planckover2pi1/2τis the phenomenological broadening pa-\nrameter, where τis the relaxation time. It is worth\nmentioning that for a finite ηthe number of particles\nis not conserved, and a more accurate interpretation of\nthe introduced broadening might be to consider it as an\nenergy-independent scape rate of electrons to fictitious\nreservoirs attached to the positions /vector x. Consequently, a\nfinite broadening could be interpreted as the existence\nof an interface in the model between each atom in the\nsystem and the reservoir that is spread homogeneously\nalong the infinite periodic system.\nEq. (2) shows that the effect of the space/time varia-\ntion is to replace E→E−i∂/∂tand/vectork→/vectork−i/vector∇in the\nequation of motion for the GFs in stationary state. To\nthe lowest order with respect to the derivatives we can\nwrite,\nˇG=ˇGad−i∂ˇGad\n∂E∂ˇG−1\nad\n∂tˇGad−i∂ˇGad\n∂/vectork·/vector∇ˇG−1\nadˇGad,\n(4)\nwhere,\nˇG−1\nad=/parenleftbigg\nGr,−1\nad−2iηf(E−µ(/vector x))\n0 Ga,−1\nad/parenrightbigg\n.(5)\nFor the density matrix of the system, ρ(t) =1\niG<(t,t),\nwe obtain,\nρneq\n/vectork,t≈ −/integraldisplaydE\n2πℜ/parenleftbigg\n[D(Gr\nad),Gr\nad]f+2iηD(Gr\nad)Ga\nad∂f\n∂E/parenrightbigg\n(6)\nwhereD=∂\n∂t−/vector∇µ·∂\n∂/vectorkis the differential operator act-\ning on the slowly varying parameters in space and time.The details of the derivation is presented in Appendix.A.\nThe density matrix in Eq. (6) is the central formula of\nthe paper and consists of two terms; the first term con-\ntains the equilibrium Fermi distribution function from\nthe electrons bellow the Fermi surface occupying a slowly\n(linearly) varying single particle states that has only in-\nterband contributions and can as well be formulated in\nterms of the Berry phase as we will show the following\nsections, and; the second term corresponds to the elec-\ntrons with Fermi energy (at zero temperature we have,\n∂f/∂E=δ(E−EF)) which are the only electrons al-\nlowed to get excited in the presence of the slowly varying\nperturbations. The fact that the first term originates\nfrom the assumption that the electric field is constant\ninside the metallic FM suggests that this term might dis-\nappearoncethescreeningeffect isincluded. Onthe other\nhand, duetothe factthat thesecondtermcorrespondsto\nthe nonequilibrium electrons injected from the fictitious\nreservoirs attached to the device through the scape rate\nη, it might capture the possible physical processes that\noccur at the contact region and makes it more suitable\nfor the calculation of the relevant physical observables in\nsuch systems.\nUsing the expression for the nonequilibrium density\nmatrix the local spin density can be obtained from,\n/vectorSneq(t) =/angb∇acketleft/vector σ/angb∇acket∇ightneq=1\n4π2/integraldisplay\nd2/vectorkTr[ρneq\n/vectork,t/vector σ],(7)\nwhere/angb∇acketleft.../angb∇acket∇ightneqrefers to the ensemble average over many-\nbody states out of equilibrium demonstrated by the\nnonequilibrium density matrix of the electrons and, Tr\nrefers to the trace. In this case the time derivative in the\ndifferential operator Dleads to the damping of the dy-\nnamics of the ferromagnet while the momentum deriva-\ntive leads to either damping or anti-damping of the FM\ndynamics depending on the direction of the applied elec-\ntric field. In the followingsection we apply the formalism\nto a two band helical surface state model attached to a\nFM.\nII. SOT AND DAMPING OF A HELICAL 2D\nSURFACE\nA two band Hamiltonian model for the system can be\ngenerally written as,\nH(/vectork,t) =ε0(/vectork)1+/vectorh(/vectork,t)·/vectorσ (8)\nwhere,/vectorh=/vectorhso(/vectork)+∆xc(/vectork)\n2/vector m(t), with/vectorhso(/vectork) =−/vectorhso(−/vectork)\nand ∆ xc(/vectork) = ∆ xc(−/vectork) being spin-orbit and magnetic\nexchange coupling terms respectively. In particular in\nthe case of Rashba type helical states we have /vectorhso=\nαsoˆez×/vectork. In this case for the adiabatic single particle\nGF we have,\nGr\nad(E,t) =(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2(9)3\nFrom Eq. (7) for the local spin density, we obtain (See\nAppendix B for details),\n/vectorSneq(t) =/integraldisplayd2/vectork\n4π2/parenleftBigg/vectorh×D/vectorh\n2|/vectorh|3(f1−f2)−(/vector∇µ·/vector v0)/vectorh\n2η|/vectorh|(f′\n1−f′\n2)\n+(/vectorh×D/vectorh\n2|/vectorh|2+ηD/vectorh−1\nη(/vectorh·D/vectorh)/vectorh\n2|/vectorh|2)(f′\n1+f′\n2)/parenrightBigg\n(10)\nwhere,f1,2=f(ε0± |/vectorh|) and/vector v0=∂ε0/∂/vectorkis the group\nvelocity of electrons in the absence of the SOI. Here, we\nassumeη≪ |/vectorh|which corresponds to a system close to\nthe ballistic regime. In this expression we kept the ηD/vectorh\nbecause of its unique vector orientation characteristics.\nAs it becomes clear in the following, the first term in\nEq. (10) is a topological quantity which in the presence\nof an electric field becomes dissipative and leads to an\nanti-damping torque. The second term in this expression\nleads to the Rashba-Edelstein field-like torque which is a\nnondissipative observable. The third term has the exact\nformasthe firstterm with the difference that it is strictly\na Fermi surface quantity. The fourth term, also leads to\na field like torque that as we will see in the following has\nsimilar features as the Rashba-Edelstein effect. It is im-\nportant to pay attention that unlike the first term, the\nrest of the terms in Eq. (10) are solely due to the flow\nof the non-equilibrium electrons on the Fermi surface.\nFurthermore, we notice that the terms that lead to dissi-\npation in the presence of an electric field ( D ≡/vector∇µ·∂\n∂/vectork)\nbecome nondissipative when we consider D ≡∂/∂tand\nvice versa.\nA. Surface State of a 3D-TI\nIn the case of the surface state of a 3D-TI, as an ap-\nproximation we can ignore ε0(/vectork) and consider the helical\nterm as the only kinetic term of the Hamiltonian. In this\ncasethelocalchargecurrentandthenonequilibriumlocal\nspin density share a similar expression, /vectorI=/angb∇acketleft∂(/vectorh·/vector σ)/∂/vectork/angb∇acket∇ight.\nFor the conductivity, analogous to Eq. (10), we obtain,\nσij=e/integraldisplayd2/vectork\n4π2\n/vectorh·∂/vectorh\n∂ki×∂/vectorh\n∂kj\n2|/vectorh|2/parenleftBigg\nf1−f2\n|/vectorh|+f′\n1+f′\n2/parenrightBigg\nδi/negationslash=j\n+−η|∂/vectorh\n∂ki|2+1\nη(∂|/vectorh|2\n∂ki)2\n2|/vectorh|2(f′\n1+f′\n2)δij\n(11)\nThis shows that the Fermi sea component of the density\nmatrixcontributesonlytotheanomalousHallconductiv-\nity which is in terms of a winding number. On the other\nhand, the second term is finite only for the longitudinal\ncomponents of the conductivity and can be rewritten in\nterms of the group velocity of the electrons in the system\nwhich leads to the Drude-like formula.Should the linear dispersion approximation for the ki-\nnetic term in the Hamiltonian be valid in the range of\nthe energy scale corresponding to the magnetic exchange\ncoupling ∆ xc(i.e. when vF≫∆xc), the effect of the in-\nplane component of the magnetic exchange coupling is to\nshift the Dirac point (i.e. center of the k-space integra-\ntion) which does not affect the result ofthe k-integration.\nIn this case after performing the partial time-momentum\nderivatives, ( D(/vectorh) =∆xc\n2∂/vector m\n∂t−vFˆez×/vector∇µ), we use\n/vectorh(/vectork,t) =vFˆez×/vectork+∆xc\n2mz(t)ˆez, to obtain,\n/vectorSneq(t) =/integraldisplaykdk\n4π|/vectorh|2/parenleftBigg\n/vectorS1f1−f2\n|/vectorh|+(/vectorS1+/vectorS2)(f′\n1+f′\n2)/parenrightBigg\n,\n(12)\nwhere,\n/vectorS1(/vectork,t) =∆2\nxc\n4mz(t)ˆez×∂/vector m\n∂t+∆xcvF\n2mz(t)/vector∇µ(13)\n/vectorS2(/vectork,t) =∆xc\n4η(2η2−v2\nF|k|2)(∂mx\n∂tˆex+∂my\n∂tˆey)\n+∆xc\n4η(2η2−∆2\nxcm2\nz\n2)∂mz\n∂tˆez\n−vF\nη(η2−v2\nF|k|2\n2)ˆez×/vector∇µ (14)\nThe dynamics of the FM obeys the LLG equation where\nthe conductions electrons insert torque on the FM mo-\nments through the magnetic exchange coupling,\n∂/vector m\n∂t=/vector m×\nγ/vectorBext+∆xc\n2/vectorSneq(t)−/summationdisplay\nijαij\n0∂mi\n∂tˆej\n\n(15)\nwhere,αij\n0=αji\n0, withi,j=x,y,z, is the intrinsic\nGilbert damping tensor of the FM in the absence of\nthe TI surface state and /vectorBextis the total magnetic field\napplied on the FM aside from the contribution of the\nnonequilibrium electrons.\nWhile the terms that consist of /vector∇µare called SOT,\nthe ones that contain∂/vector m\n∂tare generally responsible for\nthe damping of the FM dynamics. However, we no-\ntice that ˆ ez×∂/vector m\n∂tterm in Eq. (13) which arises from\nthe Berry curvature, becomes mz∂/vector m\n∂tin the LLG equa-\ntion that does not contribute to the damping and only\nrenormalizes the coefficient of the left hand side of the\nEq. (15). The second term in the Eq. (13), is the\nanti-damping SOT pointing along ( ez×/vector∇µ)-axis. The\ncone angle dependence of the anti-damping term can\nbe checked by assuming an electric field along the x-\naxis when the FM precesses around the y-axis, (i.e.\n/vector m(t) = cos(θ)ˆey+sin(θ)cos(ωt)ˆex+sin(θ)sin(ωt)ˆez). In\nthis case the average of the SOT along the y-axis in one\nperiod of the precession leads to the average of the an-\ntidamping SOT that shows a sin2(θ) dependence, which\nis typical for the damping-like torques. Keeping in mind4\nthat in this section we consider vF≫∆xc, the first and\nsecond terms in Eq. (14) show that the Gilbert damp-\ning increases as the precession axis goes from in-plane ( x\nory) to out of plane ( z) direction. Furthermore, when\nthe precession axis is in-plane (e.g. along y-axis), the\ndamping rate due to the oscillation of the out of plane\ncomponent of the magnetization ( ∂mz/∂t) has a sin4(θ)\ndependence that can be ignored for low power measure-\nment of the Gilbert damping θ≪1. This leaves us with\nthe contribution from the in-plane magnetization oscilla-\ntion (∂mx∂t) only. Therefore, the Gilbert damping for\nin-plane magnetization becomes half of the case when\nmagnetization is out-of-plane. The anisotropic depen-\ndence of the Gilbert damping can be used to verify the\nexistence of the surface state of the 3DTI as well as the\nproximity induce magnetization at the interface between\na FM and a 3DTI. Finally, the third term in Eq. (14)\ndemonstrates a field like SOT with the same vector field\ncharacteristics as the Rashba-Edelstein effect.\nIII. CONCLUSION\nIn conclusion, we have developed a linear response\nNEGF framework which provides unified treatment of\nboth spin torque and damping due to SOC at interfaces.\nWe obtained the expressions for both damping and anti-\ndamping torques in the presence of a linear gradiance of\nthe electric field and adiabatic time dependence of the\nmagnetization dynamics for a helical state correspond-\ning to the surface state of a 3D topological insulator.\nWe present the exact expressions for the damping/anti-\ndamping SOT as well as the field like torques and showed\nthat, (i); Both Fermi surface and Fermi sea contribute\nsimilarly to the anti-damping SOT as well as the Hall\nconductivity and, ( ii); The Gilbert damping due to the\nsurface state of a 3D TI when the magnetization is in-\nplane is less than the Gilbert damping when it is in the\nout-of-plane direction. This dependence can be used as\na unique signature of the helicity of the surface states\nof the 3DTIs and the presence of the proximity induced\nmagnetic exchange from the FM overlayer.\nACKNOWLEDGMENTS\nWe thank Branislav K. Nikoli´ c for the fruitful discus-\nsions. F. M. and N. K. were supported by NSF PREM\nGrant No. 1205734.Appendix A: Derivation of the Density Matrix\nUsing Eqs. (2) and . (4) it is straightforwardto obtain,\nG<=(Gr\nad−Ga\nad)f−2ηf′∇µ·∂Gr\nad\n∂kGa\nad\n+i∂G<\nad\n∂E∂H\n∂tGa\nad+i∂Gr\nad\n∂E∂H\n∂tG<\nad\n+i∂G<\nad\n∂k·∇HGa\nad+i∂Gr\nad\n∂k·∇HG<\nad(A1)\nWe plug in the expression for the adiabatic lesser GF in\nequilibrium, G<\nad= 2iηfGr\nadGa\nad= (Gr\nad−Ga\nad)f, and\nobtain,\nG<= (Gr\nad−Ga\nad)f−2ηf′∇µ·∂Gr\nad\n∂kGa\nad\n+if∂(Gr\nad−Ga\nad)\n∂E∂H\n∂tGa\nad+if∂Gr\nad\n∂E∂H\n∂t(Gr\nad−Ga\nad)\n+if′(Gr\nad−Ga\nad)∂H\n∂tGa\nad+if∂(Gr\nad−Ga\nad)\n∂k·∇HGa\nad\n+if∂Gr\nad\n∂k·∇H(Gr\nad−Ga\nad). (A2)\nExpanding the terms, leads to,\nG<=/parenleftbigg\nGr\nad−Ga\nad+iGa\nad∂H(t)\n∂t∂Ga\nad\n∂E\n−iGr\nad∂H\n∂k·∇µ(x)∂Gr\nad\n∂E−i∂Ga\nad\n∂E∂H(t)\n∂tGa\nad\n+i∂Gr\nad\n∂E∂H\n∂k·∇µ(x)Gr\nad/parenrightbigg\nf\n+if′Gr\nad∇µ·∂H\n∂k(Gr\nad−Ga\nad)\n+if′(Gr\nad−Ga\nad)∂H\n∂tGa\nad, (A3)\nwhere,forthefirstandthirdlineswehaveusedtheiranti-\nHermitian forms instead. Since to calculate the density\nmatrix we integrate G<over energy, we can use integra-\ntion by parts and obtain,\nG<=/parenleftbigg\nGr\nad−Ga\nad+2iGa\nad∂H(t)\n∂t∂Ga\nad\n∂E\n−2iGr\nad∂H\n∂k·∇µ(x)∂Gr\nad\n∂E/parenrightbigg\nf\n−if′/parenleftbigg\nGr\nad∇µ·∂H\n∂kGa\nad−Gr\nad∂H\n∂tGa\nad/parenrightbigg\n= (Gr\nad−Ga\nad)f\n+i/parenleftbigg\n2Gr\nadD(H)∂Gr\nad\n∂Ef+Gr\nadD(H)Ga\nadf′/parenrightbigg\n.(A4)\nwhere we define, D=∂\n∂t− ∇µ·∂\n∂k. Finally, using the\nidentity,\n2Gr\nadD(H)∂Gr\nad\n∂E=Gr\nadD(H)∂Gr\nad\n∂E−∂Gr\nad\n∂ED(H)Gr\nad\n+∂(Gr\nadD(H)Gr\nad)\n∂E, (A5)5\nand performing the differential over the energy, we arrive\nat Eq. (6).\nAppendix B: Derivation of the Local Spin Density\nFrom Eqs. (6) and (7), the local spin density can be written as, /vectorSneq=/vectorS(1)\nneq+/vectorS(2)\nneq, where /vectorS(1)\nneqis due to the\nnonequilibrium electrons at the Fermi surface and for a two band mo del can be calculated as the following,\n/vectorS(1)\nneq≈−2ηℑ/integraldisplaydE\n2πTr\n/parenleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig/parenleftBig\n(E−ε0+iη)1+/vectorh·/vectorσ/parenrightBig\n((E−ε0−iη)2−|/vectorh|2)((E−ε0+iη)2−|/vectorh|2)/vectorσ\n+D(1\n(E−ε0−iη)2−|/vectorh|2)((E−ε0+iη)1+/vectorh·/vectorσ)/vectorσ((E−ε0−iη)1+/vectorh·/vectorσ)\n(E−ε0+iη)2−|/vectorh|2/bracketrightBigg\nf′(E) (B1)\nPreforming the trace over the Pauli matrix, we obtain,\n/vectorS(1)\nneq≈ −4η/integraldisplaydE\n2π/parenleftBigg\nηD/vectorh+D(/vectorh)×/vectorh\n((E−ε0−iη)2−|/vectorh|2)(E−ε0+iη)2−|/vectorh|2)\n+4ℑ(/parenleftBig\n/vectorh·D(/vectorh)+(E−ε0−iη)(/vector∇µ·/vector v0)/parenrightBig\n(E−ε0)/vectorh\n((E−ε0−iη)2−|/vectorh|2)2((E−ε0+iη)2−|/vectorh|2))\nf′(E) (B2)\nIn the limit of small broadening, η≪ |/vectorh|, we obtain,\n/vectorS(1)\nneq≈−ηD/vectorh−/vectorh×D/vectorh+1\nη/vectorh·D(/vectorh)/vectorh\n2|/vectorh|2/parenleftBig\nf′(ε0+|/vectorh|)+f′(ε0−|/vectorh|)/parenrightBig\n−1\nη(/vector∇µ·/vector v0)/vectorh\n2|/vectorh|/parenleftBig\nf′(ε0+|/vectorh|)−f′(ε0−|/vectorh|)/parenrightBig\n(B3)\nFor the Fermi sea contribution to the nonequilibrium local spin densit y we have,\n/vectorS(2)\nneq≈ℜ/integraldisplaydE\n2πTr/bracketleftBigg\n∂\n∂E/parenleftBigg\n(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/parenrightBigg/parenleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/vectorσ\n−(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/parenleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01/parenrightBig∂\n∂E/parenleftBigg\n(E−ε0−iη)1+/vectorh·/vectorσ\n(E−ε0−iη)2−|/vectorh|2/parenrightBigg\n/vectorσ/bracketrightBigg\nf(E). (B4)\nSimilarly, we obtain,\n/vectorS(2)\nneq≈ℜ/integraldisplaydE\n2πTr\n/bracketleftBig\nD(/vectorh·/vectorσ)+/vector∇µ·/vector v01,(E−ε0−iη)1+/vectorh·/vectorσ/bracketrightBig\n((E−ε0−iη)2−|/vectorh|2)2/vectorσ\nf(E)\n≈2ℑ/integraldisplaydE\nπD(/vectorh)×/vectorh\n((E−ε0−iη)2−|/vectorh|2)2f(E)\n≈−1\n2D(/vectorh)×/vectorh\n|/vectorh|3/parenleftBig\nf(ε0+|/vectorh|)−f(ε0−|/vectorh|)/parenrightBig\n(B5)\n[1] D. Ralph and M. Stiles, Journal of Magnetism and Mag-\nnetic Materials 320, 1190 (2008).[2] J. Katine and E. E. Fullerton, Journal of Magnetism and\nMagnetic Materials 320, 1217 (2008).6\n[3] C. Wang et al., Nature Physics 7, 496 (2011).\n[4] I. M. Miron et al., Nature Materials 9, 230 (2010).\n[5] L. Liu et al., Phys. Rev. Lett. 109, 096602 (2012).\n[6] A.Manchon andS. Zhang, Physical ReviewB 78, 212405\n(2008).\n[7] P. Gambardella and I. M. Miron, Philos. Transact. A\nMath. Phys. Eng. Sci. 369, 3175 (2011).\n[8] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[9] D. Pesin and A. H. MacDonald, Nature Mater. 11, 409\n(2012).\n[10] I. Garate and M. Franz, Phys. Rev. Lett. 104, 146802\n(2010).\n[11] F. Mahfouzi, J. Fabian, N. Nagaosa, and B. K. Nikoli´ c,\nPhys. Rev. B 85, 054406 (2012).\n[12] A. Manchon, Phys. Rev. B 83, 172403 (2011).[13] E. Zhao, C. Zhang, and M. Lababidi, Phys. Rev. B 82,\n205331 (2010).\n[14] N. P. Butch et al., Phys. Rev. B 81, 241301 (2010).\n[15] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Nat Phys 4,\n273 (2008).\n[16] F. Mahfouzi, B. K.Nikoli´ c, S.-H.Chen, andC.-R.Chang ,\nPhys. Rev. B 82, 195440 (2010).\n[17] F. Mahfouzi, B. K. Nikoli´ c, and N. Kioussis, Phys. Rev.\nB93, 115419 (2016).\n[18] L. Keldysh, Sov. Phys. JETP 20, 1018 (1965).\n[19] H. K. et al, Nature Nanotechnology 9, 211 (2014).\n[20] H. Haug and A.-P. Jauho, Quantum kinetics in transport\nand optics of semiconductors (Springer-Verlag, Berlin,\nADDRESS, 2008).\n[21] N. Bode et al., Phys. Rev. B 85, 115440 (2012).\n[22] L. Arrachea, Phys. Rev. B 75, 035319 (2007).\n[23] A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B\n50, 5528 (1994).\n[24] B. Gaury et al., Physics Reports 534, 1 (2014)."    },    {        "title": "2112.12967v2.Skyrmion_nucleation_on_the_surface_of_a_topological_insulator.pdf",        "content": "Skyrmion nucleation on the surface of a topological insulator\nDaichi Kurebayashi1, 2,\u0003and Oleg A. Tretiakov1,y\n1School of Physics, The University of New South Wales, Sydney 2052, Australia\n2Center for Emergent Matter Science, RIKEN, Wako 351-0198, Japan\nSkyrmion nucleation induced by spin-transfer torques at an interface of a topological insulator and a ferro-\nmagnetic insulator is investigated. Due to strong spin-orbit coupling on a surface of topological insulators, which\nenhances the e \u000bect of spin torques, e \u000ecient manipulation of skyrmions is expected, and therefore, topological\ninsulators could provide the ideal platform to achieve high-performance skyrmionic devices. Using micromag-\nnetic simulations and energetics, we evaluate properties of the skyrmion nucleation on a surface of topological\ninsulators, such as nucleation time, critical electric field, and skyrmion numbers. We show that the nucleation\ntime is inversely proportional to the applied electric field. We also identify the Gilbert damping and temperature\ndependencies of the critical field. Furthermore, we analytically evaluate the e \u000bect of the Dzyaloshinskii-Moriya\ninteraction and demonstrate that the temperature dependence can be explained by the reduction of a magnon\nexcitation gap due to the self-energy corrections.\nA magnetic skyrmion is a real-space topological object de-\nfined by its non-coplanar spin texture [1–7]. Because of their\ntopological spin structure, skyrmions exhibit unique dynam-\nics [8, 9] and transport properties associated with a nontriv-\nial quantum Berry phase resulting in emergent electromag-\nnetic fields [10–12]. On the one hand, a topological insulator\n(TI) is a momentum-space topological object characterized\nby its nontrivial band structure [13, 14]. Due to this topo-\nlogically nontrivial structure, transport phenomena related to\nthe momentum-space Berry curvature, such as the quantum\nspin Hall e \u000bect [15–17] and the quantum anomalous Hall ef-\nfect [18–20], are realized and experimentally observed. An-\nother consequence of the nontrivial bands is the appearance\nof metallic gapless surface states. The surface Dirac elec-\ntrons mediate strong correlations between spin and current as\ntheir spin and momentum have a one-to-one correspondence,\nknown as the spin-momentum locking. Although both TIs\nand skyrmions separately have been recent emergent topics in\ncondensed matter physics, a combination of them could be an\nideal platform to study the interplay of real- and momentum-\nspace topology.\nRecently, in a heterostructure consisting of a TI and a mag-\nnetic insulator, the ferromagnetic (FM) skyrmion formation\nhas been observed by transport measurements [21–23]. In\naddition to the conventional anomalous Hall e \u000bect, an extra\nHall signal has been observed, which was attributed to the\ntopological Hall e \u000bect arising from the emergent electromag-\nnetic field of skyrmions. Beyond that, a real-space observation\nby a scanning transmission X-ray microscopy has been made\nand confirmed the formation of antiferromagnetically coupled\nN´eel-type skyrmions at a TI interface with a ferrimagnet [24].\nSkyrmions on a TI surface are exhilarating not only from\nthe viewpoint of emerging physics arising from the interplay\nof real- and momentum-space topologies but also due to their\nprospects for spintronic nanodevices [6, 7]. It has been theo-\nretically proposed that a skyrmion on a TI surface is accom-\npanied by a nonzero charge density due to the chiral edge\n\u0003d.kurebayashi@unsw.edu.au\nyo.tretiakov@unsw.edu.austates [25–28]. Because of this additional charge attached to\nit, a skyrmion can be manipulated by external electric fields\nwithout Ohmic losses from currents. Another mechanism\nhas been proposed to manipulate skyrmions by utilizing spin-\ntransfer torques, which are greatly enhanced due to the spin-\nmomentum locking on the TI surface [29, 30]. Consequently,\nthe dynamics of skyrmions is expected to be faster, which is\nalso highly favorable for memory applications.\nAlthough the skyrmion dynamics on a TI surface has been\nintensively investigated, the missing ingredient for success-\nful applications of skyrmions in TIs is their nucleation stud-\nies. In conventional FMs, skyrmion nucleation has been ex-\nplored [31], for example, by employing geometric structures\nand local magnetic fluctuations, such as notches, edges, or\nimpurity sites [32–38] as well as by utilizing local injection\nof charge and spin currents [39–42]. However, the e \u000bect of\ndissipation on the nucleation process, including the influence\nof the Gilbert damping and thermal fluctuations, has not been\nunderstood well and therefore requires further investigation.\nMoreover, skyrmion nucleation on a TI surface might be sig-\nnificantly di \u000berent from the one in conventional FMs because\nof the spin-momentum locking. With increasing interest in\nskyrmionics with TIs, detailed studies on the skyrmion nu-\ncleation process are highly demanded and therefore are the\nsubject of this paper. We investigate the general properties of\nskyrmion nucleation on a TI surface, such as nucleation time\nand critical field. We perform micromagnetic simulations at\na finite temperature and demonstrate the Gilbert damping and\ntemperature dependences of the nucleation process. To give\nphysical understanding, we describe analytically the e \u000bect of\ntemperature based on a self-energy renormalization. By treat-\ning the Dzyaloshinskii-Moriya interaction (DMI) as a pertur-\nbation within the random-phase approximation, we succeed in\nreproducing the temperature dependence obtained in our mi-\ncromagnetic simulations.\nStochastic magnetization dynamics model.– Magnetization\ndynamics at finite temperature is analyzed with the stochastic\nLandau-Lifshitz-Gilbert (sLLG) equation [43, 44],\n@n\n@t=\u0000\r\n1+\u000b2\u0002n\u0002Be+\u000bn\u0002(n\u0002Be)\u0003+T; (1)\nwhere\ris the gyromagnetic ratio, \u000bis the Gilbert damp-arXiv:2112.12967v2  [cond-mat.mes-hall]  15 Nov 20222\ning constant, n(r)=M(r)=Msis the normalized mag-\nnetization, Msis the saturation magnetization, Be(r)=\n\u0000(1=Ms)\u000eFM=\u000en(r)+Bthis the e \u000bective magnetic field, Bth\nis the thermal field, FMis the magnetic free energy, and T\nis the spin-transfer torque. On a surface of TIs, the mag-\nnetic free energy FM=Fex+Fani+FZ+FDMI, where\nFex=(2J0S2=la)R\ndV(rn)2,Fani=\u0000KR\ndV(nz)2,FZ=\n\u0000MsR\ndV B znz, and FDMIare the exchange, anisotropy, Zee-\nman, and DMI energies, respectively. Here J0is the ex-\nchange constant between local magnetic moments, lais the\nmagnetic lattice constant, Sis the amplitude of the local\nmoments, Kis the easy-axis anisotropy constant along z-\naxis, and Bzis the magnetic field perpendicular to the film\n[45]. On a TI surface, electrons mediate anisotropic ex-\nchange interaction, namely the DMI, reflecting an inversion\nsymmetry breaking. Then the DMI takes the form FDMI=\n(D=\u0018l)R\ndV[nz(@xnx+@yny)\u0000nx@xnz\u0000ny@ynz], where\u0018lis the\npenetration length of a TI surface state into the magnetic in-\nsulator, D=\u0000J2[\u0002(jJj+EF)\u0000\u0002(jJj\u0000EF)]=(8\u0019vF) is the\nDMI constant mediated by the surface Dirac electrons [46],\nwhere Jis the s-dcoupling constant between electron’s spins\nand local moments, vFis the Fermi velocity, EFis the Fermi\nenergy, and \u0002(x) is the Heaviside function. The e \u000bective mag-\nnetic field is then given by\nBe\nx;y=4J0S2\nMsla\u0001nx;y+2D\nMs\u0018l@x;ynz+Bth\nx;y; (2)\nBe\nz=4J0S2\nMsla\u0001nz+Bz\u00002D\nMs\u0018lr\u0001n+2K\nMsnz+Bth\nz:(3)\nIt is known that the spin-transfer torques on a TI surface that\ncouples to local magnetic moments are substantially modified\ndue to the strong spin-orbit coupling:\nT=\u0000e\r\nMs\u0018l(\u000beˆx+\feˆy) (r\u0001n)Ex; (4)\nwhere\u000be=\u001cJ3sgn(EF)\u0002(E2\nF\u0000J2)=(8\u0019E2\nF) and\fe=\n\u001c2J2(E2\nF\u0000J2)sgn( EF)\u0002(E2\nF\u0000J2)=(8\u0019E2\nF) are dimensionless\ncoe\u000ecients [29, 30]. These specific spin-transfer torques oc-\ncur at the TI /magnetic insulator interface or in magnetic TI\nthin films. A detailed derivation of the spin-transfer torques\non the TI surface is given in the Appendix. Note that we ne-\nglected the e \u000bect of the spin-orbit torques (SOT) on the mag-\nnetization dynamics in Eq. (1). On the TI surface, the SOT\nplays an important role in the magnetic dynamics because of\nthe spin-momentum locking, however, for the skyrmion nu-\ncleation, which requires spatial inhomogeneity, the SOT does\nnot give rise to any qualitative di \u000berences as the SOT are uni-\nform contribution. Instead, the SOT under the DC current is\nequivalent to a static uniform in-plane magnetic field, there-\nfore reducing the energy barrier between the uniform FM and\nskyrmion states and thus contributing to the reduction of the\ncritical field.\nFor micromagnetic simulations, we discretize a space into\na square lattice by using relations @in(r)jrj\u0019(nj+ei\u0000\nnj\u0000ei)=(2la),@2\nin(r)\f\f\frj\u0019(nj+ei\u00002nj+nj\u0000ei)=l2\na, and impos-\ning periodic boundary conditions in x- and y-directions. To\nnumerically simulate finite temperature, the thermal field is\n050100150200y/la(a)t= 0 ns\n (b)t= 0.2 ns\n0 50 100 150 200\nx/la050100150200y/la(c)t= 1 ns\n0 50 100 150 200\nx/la(d)t= 2 ns\n−1.0−0.50.00.51.0\nnz\n−1.0−0.50.00.51.0\nnzFIG. 1. Magnetic profile at (a) t=0 s when no current is applied,\n(b)t=0:2 ns after current pulse is applied, (c) t=1 ns after current\npulse is applied, and (d) after the current pulse is switched o \u000b. The\ncolor code and arrows show the z- and in-plane components of mag-\nnetization, respectively. The parameters are \u000b=0:04,T=0:2K, and\nEx=6:5\u0002104V/m.\ndefined by Bth=\u0011p\n2\u000bkBT=(Ms\rV\u0001t), where \u0011is the ran-\ndom vector drawn from a standard normal distribution, Vis\nthe average magnetic-ion volume, and \u0001tis the simulation\ntime-step. As an integration scheme, we have employed the\nHeun’s method with the time-step \u0001t\u00183 fs. Since these\nsimulations are all stochastic, all numerical data are obtained\nas the statistical average over 50 independent simulations.\nFor typical parameters of a magnetic TI, we have used val-\nues estimated from the first-principle calculations and exper-\niments: vF=2:55 eV Å, J=0:15 eV , J0=1:38\u000210\u000023\nJ,K=7:25\u000210\u000027J/Å3,Ms=1:16\u0002104J/(Tm3), and\nla=\u0018l=8:1 Å [47, 48].\nSkyrmion nucleation.– We first examine the skyrmion nu-\ncleation with uniform currents. Figure 1 shows the magnetic\nprofiles under a uniform current pulse. Here, we chose the\nparameters as \u000b=0:04,T=0:2 K, and Ex=6:5\u0002104\nV/m. At t=0 ns, when no current is applied, the magne-\ntization is along the z-axis. After application of the current\npulse, it first develops particle-like small fluctuations as shown\nin Fig. 1(b). Eventually, these particle-like fluctuations grow\ninto N ´eel skyrmions once their radius reaches the critical one\ndetermined by J,D, and Bz[49, 50], otherwise they collapse\nback into a uniform state. Note that, during an early stage of\nthe nucleation, a skyrmion - antiskyrmion pair is created due\nto the topological number conservation [11, 51]. However, be-\ncause our DMI stabilizes only skyrmions, the antiskyrmions\nquickly decay into the uniform state [52]. After the current\npulse is switched o \u000b, only the skyrmions survive, see Fig. 1\n(d). These results clearly show that skyrmions can be nucle-3\nFIG. 2. (a) The nucleation time and (b) the skyrmion number as a\nfunction of electric field Exare plotted for various Gilbert damping\n\u000bandT=1 K. The dots are numerical data and solid lines are\nfitting functions. The inset of panel (a) shows the Gilbert damping\ndependence of the critical field Ecin the units of 104V/m.\nated by uniform current pulses at TI /FM interfaces [53].\nE\u000bect of Gilbert damping.– There is always a delay be-\nfore the first skyrmion is nucleated. To investigate this nu-\ncleation time, we first study its Gilbert damping \u000bdepen-\ndence. Since \u000bdepends on various factors such as disor-\nder, it is important to understand how it a \u000bects the nucle-\nation process. The skyrmion nucleation time tnfor various\n\u000bis shown in Fig. 2 (a) as a function of applied field Ex.\nWe define the tnas a time before the total skyrmion num-\nber,Nsk=\f\f\f!\ndxdy n\u0001(@xn\u0002@yn)\f\f\f=(4\u00192);exceeds one. The\nskyrmion nucleation is absent for small Ex, i.e., there is a\ncritical field Ecfor the nucleation process. In this regime,\nthe energy dissipation caused by the Gilbert damping exceeds\nthe energy influx due to the spin-transfer torque [54], such\nthat the total accumulated energy is insu \u000ecient to nucleate a\nskyrmion.\nIn terms of the nucleation time, Fig. 2 (a) shows diverg-\ning behavior at Ecand monotonically decreases with Ex. This\ncan be explained based on the energy considerations. Since\nthe energy influx per unit time to the system is linearly pro-\nportional to Exas the coupling to electrons is treated within\nthe linear response theory, the total accumulated energy is\nFtot/Ex. The nucleation rate 1 =tnis, then, proportional to\nthe energy di \u000berence between the total accumulated energy\nFtotand the nucleation energy of a single skyrmion Fsk, i.e.,\n1=tn/Ftot\u0000Fsk\u0018Ex\u0000Ec. Thus, the nucleation time scales\nastn/(Ex\u0000Ec)\u00001. Indeed, the numerical data is in excellent\nagreement with it, see Fig. 2 (a), when fitted by,\ntn(\u000b)=A[jExj\u0000Ec(\u000b)]\u00001; (5)\nwhere Ais a coe \u000ecient and Ec(\u000b) is the critical field at the\ngiven\u000b. As shown in the inset of Fig. 2 (a), the critical field Ec\nis linear in\u000b. This is because the energy dissipation is linear in\n\u000band the energy influx is /Ex. Thus, as the total accumulated\nenergy is determined as the di \u000berence of the energy influx\ndue to spin-transfer torques and the Gilbert dissipation, the\nrequired energy influx to nucleate a skyrmion should linearly\nincrease with \u000b.\nWe also examined the total skyrmion number nucleated af-\nter 1 ns pulse as a function of applied field Exfor several\n\u000b. As shown in Fig. 2 (b), the nucleated skyrmion number\nNsklinearly increases with Exin the vicinity of the critical\n0 5 10 15 20 25\nEx[104V/m]0.00.20.40.60.81.0t[ns]\n(a)\nSteady\nnucleationTurbulence\nregime\n05101520Nsk\n(b)\n(c)FIG. 3. (a) The skyrmion number Nskas a function of time and elec-\ntric field Ex, for\u000b=0:04 and T=0:2 K. (b), (c) Magnetic profile at\nt=0:5 ns and Ex=7\u0002104V/m, and (c) t=0:5 ns and Ex=22\u0002104\nV/m.\nfield. However, as Exincreases further, Nskdeviates from\na linear slope and saturates. The saturation occurs because\nskyrmions start overlapping and merging into large domains\nas the skyrmion density increases. By further increasing Ex,\nskyrmion states are almost destroyed by strong magnon exci-\ntations. Time evolution of Nskat each applied field and corre-\nsponding magnetic profiles are shown in Fig. 3. As seen from\nFig. 3 (a), Nskstays constant after reaching a steady state for\nthe fields below Ex\u00181:5\u0002105m/V . In this regime, all nucle-\nated skyrmions are well separated, as shown in Fig. 3 (b). On\nthe other hand, for larger Excorresponding to the turbulence\nregime in Fig. 3 (a), Nskdecreases as Exincreases and oscil-\nlates in time. Figure 3 (c) shows a typical magnetic profile in\nthe strong field regime, swirling magnetic structures no longer\nsurvive [53], and larger magnetic domains are formed because\nof strong magnon excitations. Note that it is a crossover, not a\nphase transition, between the steady nucleation and turbulence\nregimes.\nE\u000bect of temperature.– Next, we examine the temperature\ne\u000bects on the nucleation phenomenon. The nucleation time\nfor various temperatures is presented in Fig. 4. One can notice\nthat the temperature Tonly a \u000bects the critical field, see Ec(\u000b)\nin Eq. (5), whereas the functional form of Exis hardly mod-\nified. The critical field linearly decreases with T, as shown\nin the inset of Fig. 4. Phenomenologically, the linear depen-\ndence on Tcan be understood as follows. The thermal fluc-\ntuations supply the energy \u0018kBTto the system, where kBis\nthe Boltzmann constant. Due to this additional contribution,\nthe energy required to create a skyrmion reduces linearly with\nT. We note that the critical field vanishes around T\u00188 K;\nabove this temperature, skyrmions are nucleated even without\nthe spin-transfer torques due to the thermal fluctuations.\nAlthough phenomenological energy considerations explain\nthe temperature e \u000bect on the nucleation process, we move one\nstep further and try to explain this phenomenon in terms of\nferromagnetic magnon excitations. Introducing the Holstein-\nPrimako \u000brepresentation, the free energy of the system can be4\nFIG. 4. The nucleation time as a function of electric field Exfor\nvarious temperatures and \u000b=0:04. The dots are numerical data,\nwhile the solid curves are fitting functions given by Eq. (5). The\ninset shows the temperature dependence of the critical field Ec.\ntransformed to the magnonic Hamiltonian as\nˆHm=X\nkay\nk\u0010\nJk+˜K+˜B\u0000tk\u0011\nak\n+X\nk;q\u0010\nDqay\nqay\nk\u0000q=2ak+q=2+D\u0003\nqay\nk+q=2ak\u0000q=2aq\u0011\n;(6)\nwhere akis the magnon annihilation operator with the wave\nnumber k,Jk=8J0S2P\ni=x;y(1\u0000coskj) is the exchange en-\nergy, ˜K=2Kl3\nais the easy-axis anisotropy, ˜B=MsBzl3\nais\nthe Zeeman energy, Dk=2iDla(sinkx+isinky) is the DMI,\nandtk=4eExla(\fesinky+\u000besinkx) describes the e \u000bect of\nthe spin-transfer torque. Note that we have retained the lin-\near in Exterm and neglected the higher-order terms. The first\nterm gives a single-particle magnon dispersion as it already\nhas the bilinear form in magnon operators. On the other hand,\nthe DMI in the second line of Eq. (6) contains three-magnon\noperators and has to be treated perturbatively.\nThe full Green’s function is given by Gk(D)=h\u001ek¯\u001eki=R\nD(¯\u001e;\u001e)\u001ek¯\u001eke\u0000S[\u001e;¯\u001e]\nR\nD(¯\u001e;\u001e)e\u0000S[\u001e;¯\u001e], where S[\u001e;¯\u001e]=P\nk¯\u001ek(\u0000i!n)\u001ek+Hm[\u001e;¯\u001e]\nis the imaginary-time action, \u001ek=(!n;k)is the eigenvalue of\nthe magnon operator ak, andHmis the Hamiltonian in the\nmagnon coherent states basis. Within the random phase ap-\nproximation, the Green’s function Gk\u0019(\u0000i!n+Jk+˜K+˜B\u0000\ntk\u0000\u0006k)\u00001, where \u0006kis the self-energy induced by the DMI.\nThe real part of the self-energy modifies the magnon disper-\nsion, while the imaginary part gives a finite lifetime. There-\nfore, the e \u000bective magnon dispersion including the e \u000bect of\nDMI takes the form\n!e\u000b(k)=Jk+˜K+˜B\u0000tk\u0000Re[\u0006k]: (7)\nAs depicted in Fig. 5, the self-energy has two contributions\nFIG. 5. The Feynman diagrams contributing to the magnon self-\nenergy. (a) and (b) show the density-density and the pair correlations,\nrespectively.\n\u0006k= \u0006 k;d+ \u0006 k;pwith\n\u0006k;d=\u00004jDkj2X\nqfB(!q)\u0000fB(!k+q)\n!q\u0000!k+q+i0+; (8)\n\u0006k;p=4X\nqjDqj21+fB(!q)+fB(!k\u0000q)\n!q+!k\u0000q\u0000i0+; (9)\nwhere fBis the Bose-Einstein distribution and !k=Jk+˜K+\n˜B\u0000tkis the bare magnon dispersion. Equations (8) and (9) cor-\nrespond to the magnon density-density response function, \u0006d,\nand pair-correlation function, \u0006p. From Eqs. (8) and (9), one\nnotices that the real part of the self-energy is always positive,\nnamely, the magnon correlations always reduce the magnon\nexcitation gap min( !e\u000b). As the Bose-Einstein distribution\nfunction can be expanded as fB(!)\u0019kBT=!, the reduction of\nthe gap min( !e\u000b) due to the self-energy contributions is lin-\near in temperature. Then, because the magnon instability is a\nprecursor of the skyrmion nucleation and it occurs when the\nmagnon excitation gap collapses, we conclude that the criti-\ncal field linearly decreases with T. Note that this expansion\nis valid when a single-particle magnon excitation gap, !gap,\nis smaller than kBT. In typical magnetically-doped TIs, the\nperpendicular magnetic anisotropy is \u001810\u00006eV [48], which\ncorresponds to\u00180:1 K. Because the condition !gap<kBTis\nalways met within the temperature range of our simulations,\nthe linear- Tdependence observed in the inset of Fig. 4 can\nbe explained analytically based on the magnon excitations.\nNote that the instability and the nucleation of magnetic texture\nis a highly nonequilibrium process, and equilibrium analysis\nbased on the self-energy corrections may fail to completely\nexplain dynamical properties. We also neglected the magnon\nexcitation due to the electric field, which may a \u000bect the be-\nhavior in a low-temperature regime. However, it is still capa-\nble of qualitatively describing statistical properties such as the\naverage nucleation time and critical field.\nDiscussion.– Let us review potential candidate materials\nfor magnetic TIs. In this study, we used material param-\neters for a typical magnetic TI, such as Cr- and V-doped\n(Bi;Sb) 2Te3[47, 48]. The Curie temperature of this group\nof materials, however, is Tc\u001915\u001830K, and therefore the\nskyrmion state is only stable at the cryogenic temperatures,\nthus hindering some device applications. In order to overcome\nthis limitation, achieving high Tcis particularly desirable. In\na similar group of materials, such as MnSb 2Te4[55], a higher\nCurie temperature Tc\u001850K is observed. Furthermore, the\nincrease of Tcby 60K due to the exchange bias e \u000bect is ob-\nserved in a superlattice structure of Dy-doped Bi 2Te3and Cr-\ndoped Sb 2Te3[56]. As another approach to realizing high Tc5\nmagnetic TIs, bilayer systems consisting of a TI and magnetic\ninsulator with a higher Tcare expected to be more commer-\ncially realizable candidates. It is reported that a TI grown on\nYIG by molecular-beam epitaxy exhibits the anomalous Hall\ne\u000bect, i.e., a signature of magnetically-coupled surface states,\nup to 300K [57, 58]. Thus, with recent advances in nanotech-\nnology, there is a real prospect that room-temperature mag-\nnetic TIs will be within the reach in the near future.\nIn conclusion, we have comprehensively studied the\ncurrent-induced skyrmion nucleation on a surface of a topo-\nlogical insulator. As a system, we have considered a het-\nerostructure consisting of a TI and an ultrathin ferromag-\nnetic insulator (applicable to the recent experimental realiza-\ntions [21–24]) and employed the specific spin-transfer torque\nrealized on a TI surface coupling to magnetic local moments.\nBy solving the stochastic LLG equation, we have determined\nthe critical field Ecfor skyrmion nucleation and found that the\nnucleation time tn/(Ex\u0000Ec)\u00001, where Echas been found to\nbe proportional to the Gilbert damping \u000b. These results sug-\ngest that with the advances in nanotechnology, when much\ncleaner samples with lower damping will be available, faster\nskyrmion nucleation with lower fields may be achieved. Fur-\nthermore, we have investigated the temperature dependence\nof the nucleation and have observed the temperature e \u000bects on\nthe critical field, which linearly decreases as temperature rises.\nWe have given a quantum microscopic description for this lin-\near dependence based on magnon excitations. The self-energy\ncorrections due to the Dzyaloshinskii-Moriya interaction pro-\nvide linear in temperature reduction of the magnon excitation\ngap and the critical field. Our results give a phenomenologi-\ncal understanding of the skyrmion nucleation on a surface of a\ntopological insulator and open doors for developing TI-based\nskyrmionic memory and logic nanodevices.\nACKNOWLEDGMENTS\nThe authors are grateful to N. Nagaosa for insightful discus-\nsions. O.A.T. acknowledges the support from the Australian\nResearch Council (Grant No. DP200101027), the Cooperative\nResearch Project Program at the Research Institute of Electri-\ncal Communication, Tohoku University (Japan), and the NC-\nMAS grant.\nAppendix: Spin-transfer torque on the surface of a TI\nIn the appendix, we present a detailed derivation of spin-\ntransfer torques on the surface of a topological insulator. TheHamiltonian describing the surface states of a topological in-\nsulator is given by\nH=H0\u0000evFX\nk y\nk[A(t)\u0002\u001b]z k\u0000JX\nrn(r)\u0001 y\nr\u001b r;\n(A.1)\nwhere H0=P\nk y\nkh\n\u0000vF\u0010\nkx\u001by\u0000ky\u001bx\u0011\n\u0000J\u001bzi\n kdescribes a\nmassive Dirac Fermions with a helical spin texture [47],  kis\nthe Fermionic annihilation operator at momentum k,vFis the\nFermi velocity, \u001bis the Pauli matrix describing real spin de-\ngrees of freedom, A=(Ax;Ay) is the electromagnetic vector\npotential, Jis as-dexchange constant, and n(r)=ˆz+\u000en(r) is\nmagnetization of local moments. Here we take the saturated\nmagnetization direction along ˆzaxis and assume \u000en(r) is a\nsmall magnetic fluctuation around ˆzaxis, namely ˆz\u0001\u000en(r)=0\nandj\u000en(r)j<<1.\nBy treating the electromagnetic field and the magnetic fluc-\ntuation,\u000en(r), as perturbations, a non-equilibrium spin ac-\ncumulation induced by an electric field applied along the x-\ndirection is evaluated as\nh\u001b(r;t)i=\u0000iTr\u0002\u001bG<(r;r0;t;t)\u0003\f\f\fr0!r; (A.2)\nwhere G<(r;r;t;t)=ih y\nr0(t0) r(t)iis the lesser Green’s func-\ntion on the Keldysh contour and the trace is over the spin in-\ndices [59]. Turning into momentum space, the contour or-\ndered Green’s function obeys the Dyson equation given as\nGk;k0(t;t0)=gk(t\u0000t0)\u000ek;k0\n\u0000evFX\nkZ\nCdt1gk(t\u0000t1)Ax(t1)\u001byGk;k0(t1;t0)\n\u0000JX\nk;qZ\nCdt1gk(t\u0000t1)\u000en(q)\u0001\u001bGk\u0000q;k0(t1;t0);\n(A.3)\nwhere Gk;k0=P\nk;k0exp(\u0000ik\u0001r+ik0\u0001r0)G(r;r0) and gk=\n[i@t\u0000H0]\u00001is an unperturbed Green’s function. By expanding\nthe contour-ordered Green’s function up to the first order in\nthe external field and the magnetic fluctuation, \u000en(r), one can\nobtain that\nGk;k0(t;t0)\u0019\u0000evFJ\u000ek\u0000q;k0X\nk;qZ\nCdt1Z\nCdt2\u000en(q)\u0001h\ngk(t\u0000t1)Ax(t1)\u001bygk(t1\u0000t2)\u001bgk\u0000q(t2\u0000t0)\n+gk+q(t\u0000t1)\u001bgk(t1\u0000t2)Ax(t2)\u001bygk(t2\u0000t0)i\n: (A.4)\nNote that, in Eq. (A.4), we have neglected the terms irrele- vant to the non-equilibrium spin accumulation. By substitut-6\ning Eq. (A.4) into Eq. (A.2), the spin accumulation is given by\nh\u001b\u000b(r;t)i=ievFJX\nk;qeiq\u0001rAx(t1)\u000en\f(q)Tr\"Z\nCdt1Z\nCdt2\u001b\u000bgk(t\u0000t1)\u001bygk(t1\u0000t2)\u001b\fgk\u0000q(t2\u0000t)\n+Z\nCdt1Z\nCdt2\u001b\u000bgk+q(t\u0000t2)\u001b\fgk(t2\u0000t1)\u001bygk(t1\u0000t)#<\n: (A.5)\nWith use of the Langreth theorem,\n\"Z\nCdt1Z\nCdt2A(t\u0000t1)B(t1\u0000t2)C(t2\u0000t0)#<\n=Z1\n\u00001dt1Z1\n\u00001dt2\u0002Ar(t\u0000t1)Br(t1\u0000t2)C<(t2\u0000t0)\n+Ar(t\u0000t1)B<(t1\u0000t2)Ca(t2\u0000t0)+A<(t\u0000t1)Ba(t1\u0000t2)Ca(t2\u0000t0)\u0003;(A.6)\nwhere superscripts ( a;r) denote the advanced and retarded\nfunctions, the non-equilibrium spin accumulation is obtainedas\nh\u001b\u000b(r;t)i=X\nqZd\n2\u0019eiq\u0001r\u0000i\nt\u001f\u000b\f(q;\n)Ax(\n)\u000en\f(q):(A.7)\nHere\u001f\u000b\f(q;\n) is a spin susceptibility defined by\n\u001f\u000b\f(q;\n)=ievFJZd!\n2\u0019X\nkTrh\n\u001b\u000bgk(!+ \n)\u001bygk(!)\u001b\fgk\u0000q(!)+\u001b\u000bgk+q(!)\u001b\fgk(!)\u001bygk(!\u0000\n)i<\n\u0019i\n2\u0019evFJX\nkh\n\u001b\u000bgr\nk\u001byga\nk\u001b\fga\nk\u0000q+\u001b\u000bgr\nk+q\u001b\fgr\nk\u001byga\nki\n; (A.8)\nwhere gr;a\nk\u0011gr;a\nk(!=0)=\u0002\u0000H0\u0006i\u0011\u0003\u00001,\u0011is the damping\nrate induced by impurity scattering, and we have expanded\nthe function with respect to the frequency up to the first order.\nHere we have considered the small scattering regime ( \u0011 <<\nEF;J) and only retained the dominant contributions which\ninclude both the retarded and advanced Green’s functions. By\nexpanding the Green’s function with respect to the external\nmomentum, q, asgk+q\u0019gk\u0000vF\u000f\r\u001aq\rgk\u001b\u001agk+O(q2), one\ncan obtain that\n\u001f\u000b\f(q;\n)\u0019i\n2\u0019ev2\nFJX\nk\u000f\r\u001ah\n\u001b\u000bgr\nk\u001byga\nk\u001b\fga\nk\u001b\u001aga\nk\n\u0000\u001b\u000bgr\nk\u001b\u001agr\nk\u001b\fgr\nk\u001byga\nki\nq\r: (A.9)\nNote that we have dropped the O(q0)-terms because they are\nthe uniform contributions, namely corresponding to the spin-\norbit torque, which is well known and there is no need to\nreproduce in this Appendix. After momentum integration in\nEq. (A.9), the non-equilibrium spin accumulation correspond-\ning to the spin-transfer torques is obtained by\nh\u001b?(r;t)i=C(EF)r\u0001\u000en(r)Ex; (A.10)\nwhere C(EF)=eJ\n8\u0019E2\nFh\n\u001c2(E2\nF\u0000J2) ˆx\u0000\u001cJM zˆyi\nsgn(E F)\u0002(E2\nF\u0000J2),\u001c=1\n2\u0011is the electron scattering lifetime, sgn( x) is the\nsign function, and \u0002(x) is the Heaviside function. Finally, the\nspin-transfer torques are given by\nT=\u0000\r0J\nMS\u0018lˆz\u0002h\u001b?(r;t)i; (A.11)\nTx=\u0000e\r0\u000be\nMS\u0018l(r\u0001\u000en)Ex; (A.12)\nTy=\u0000e\r0\fe\nMS\u0018l(r\u0001\u000en)Ex; (A.13)\nwhere\u000be=\u001cJ3\n8\u0019E2\nFsgn(EF)\u0002(E2\nF\u0000J2) and\fe=\n\u001c2J2(E2\nF\u0000J2)\n8\u0019E2\nFsgn(EF)\u0002(E2\nF\u0000J2) are dimensionless coe \u000ecients\ncharacterizing the spin-transfer torque mediated by the Dirac\nelectrons [29, 30]. Note that Eq. 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Jpn. 75,\n113706 (2006)."    },    {        "title": "1903.05415v2.Higher_order_linearly_implicit_full_discretization_of_the_Landau__Lifshitz__Gilbert_equation.pdf",        "content": "arXiv:1903.05415v2  [math.NA]  20 Mar 2020HIGHER-ORDER LINEARLY IMPLICIT FULL DISCRETIZATION\nOF THE LANDAU–LIFSHITZ–GILBERT EQUATION\nGEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nAbstract. For the Landau–Lifshitz–Gilbert (LLG) equation of micromagnetics\nwe study linearly implicit backward difference formula (BDF) time discre tizations\nup to order 5 combined with higher-order non-conforming finite elem ent space\ndiscretizations, which are based on the weak formulation due to Alou ges but use\napproximate tangent spaces that are defined by L2-averaged instead of nodal or-\nthogonality constraints. We prove stability and optimal-order erro r bounds in the\nsituation of a sufficiently regular solution. For the BDF methods of or ders 3 to 5,\nthis requires that the damping parameter in the LLG equations be ab ove a posi-\ntive threshold; this condition is not needed for the A-stable method s of orders 1\nand 2, for which furthermore a discrete energy inequality irrespec tive of solution\nregularity is proved.\n1.Introduction\n1.1.Scope.In this paper we study the convergence of higher-order time and s pace\ndiscretizations of the Landau–Lifshitz–Gilbert (LLG) equation, wh ich is the basic\nmodel for phenomena in micromagnetism, such as in recording media [ 26, 36].\nThe main novelty of the paper lies in the construction and analysis of w hat is\napparently the first numerical method for the LLG equation that is second-order\nconvergent in both space and time to sufficiently regular solutions an d that satisfies,\nas an important robustness property irrespective of regularity, a discrete energy\ninequality analogous to that of the continuous problem.\nWe study discretization in time by linearly implicit backward difference fo rmu-\nlae (BDF) up to order 5 and discretization in space by finite elements o f arbitrary\npolynomial degree. For the BDF methods up to order 2 we prove opt imal-order\nerror bounds in the situation of a sufficiently regular solution and a dis crete energy\ninequality irrespective of solution regularity under very weak regula rity assumptions\non the data. For the BDF methods of orders 3 to 5, we prove optima l-order error\nbounds in the situation of a sufficiently regular solution under the add itional condi-\ntionthatthedampingparameter intheLLGequationbeaboveameth od-dependent\npositive threshold. However, no discrete energy inequality irrespe ctive of solution\nregularity is obtained for the BDF methods of orders 3 to 5.\nThe discretization in space is done by a higher-order non-conformin g finite ele-\nment method based on the approach of Alouges [4, 5], which uses a p rojection to\nDate: March 23, 2020.\n2010Mathematics Subject Classification. Primary 65M12, 65M15; Secondary 65L06.\nKey words and phrases. BDF methods, non-conforming finite element method, Landau–\nLifshitz–Gilbert equation, energy technique, stability.\n12 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nan approximate tangent space to the normality constraint. Contr ary to the point-\nwise orthogonality constraints in the nodes, which define the appro ximate tangent\nspace in those papers and yield only first-order convergence also f or finite elements\nwith higher-degree polynomials, we here enforce orthogonality ave raged over the\nfinite element basis functions. With these modified approximate tang ent spaces we\nproveH1-convergence of optimal order in space and time under the assump tion of\na sufficiently regular solution.\nKey issues in the error analysis are the properties of the orthogon al projection\nonto the approximate tangent space, the higher-order consiste ncy error analysis,\nand the proof of stable error propagation, which is based on non-s tandard energy\nestimates and uses both L2and maximum norm finite element analysis.\n1.2.The Landau–Lifshitz–Gilbert equation. The standard phenomenological\nmodel for micromagnetism is provided by the Landau–Lifshitz (LL) e quation\n(1.1) ∂tm=−m×Heff−αm×(m×Heff)\nwhere the unknown magnetization field m=m(x,t) takes values on the unit\nsphereS2,α >0 is a dimensionless damping parameter, and the effective mag-\nnetic fieldHeffdepends on the unknown m. The Landau–Lifshitz equation (1.1)\ncan be equivalently written in the Landau–Lifshitz–Gilbert form\n(1.2) α∂tm+m×∂tm= (1+α2)/bracketleftbig\nHeff−/parenleftbig\nm·Heff/parenrightbig\nm/bracketrightbig\n.\nIndeed, in view of the vector identity a×(b×c) = (a·c)b−(a·b)c,fora,b,c∈R3,\nwe have−m×/parenleftbig\nm×Heff/parenrightbig\n=Heff−/parenleftbig\nm·Heff/parenrightbig\nm,and taking the vector product of\n(1.1) withmand adding αtimes (1.1) then yields (1.2).\nSincem×ais orthogonal to m,for anya∈R3,it is obvious from (1.1) that\n∂tmis orthogonal to m:m·∂tm= 0; we infer that the Euclidean norm satisfies\n|m(x,t)|= 1 for allxand for allt, provided this is satisfied for the initial data.\nThe term in square brackets on the right-hand side in (1.2) can be re written as\nP(m)Heff, where (with Ithe 3×3 unit matrix)\nP(m) =I−mmT\nis the orthogonal projection onto the tangent plane to the unit sp hereS2atm.\nIn this paper we consider the situation\n(1.3) Heff=1\n1+α2/parenleftbig\n∆m+H/parenrightbig\n,\nwhereH=H(x,t) is a given external magnetic field. The factor 1 /(1 +α2) is\nchosen for convenience of presentation, but is inessential for th e theory; it can be\nreplaced by any positive constant factor.\nWiththischoiceof Heff, we arriveattheLandau–Lifshitz–Gilbert (LLG)equation\nin the form\n(1.4) α∂tm+m×∂tm=P(m)(∆m+H).\nWeconsiderthisequationasaninitial-boundaryvalueproblemonabou ndeddomain\nΩ⊂R3and a time interval 0 /lessorequalslantt/lessorequalslant¯t, with homogeneous Neumann boundaryHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 3\nconditions and initial data m0taking values on the unit sphere, i.e., the Euclidean\nnorm|m0(x)|equals 1 for all x∈Ω.\nWe consider the following weak formulation, first proposed by Alouge s [4, 5]:\nFind the solution m:Ω×[0,¯t]→S2withm(·,0) =m0by determining, at\nm(t)∈H1(Ω)3, the time derivative ∂tm(omitting here and in the following the\nargumentt) as that function in the tangent space\nT(m) :=/braceleftbig\nϕ∈L2(Ω)3:m·ϕ= 0 a.e./bracerightbig\n=/braceleftbig\nϕ∈L2(Ω)3:P(m)ϕ=ϕ}\nthat satisfies, for all ϕ∈T(m)∩H1(Ω)3,\n(1.5) α/parenleftbig\n∂tm,ϕ/parenrightbig\n+/parenleftbig\nm×∂tm,ϕ/parenrightbig\n+/parenleftbig\n∇m,∇ϕ/parenrightbig\n=/parenleftbig\nH,ϕ/parenrightbig\n,\nwhere the brackets ( ·,·) denote the L2inner product over the domain Ω. The\nnumerical methods studied in this paper are based on this weak form ulation.\n1.3.Previous work. There is a rich literature on numerical methods for Landau–\nLifshitz(–Gilbert) equations; for the numerical literature up to 20 07 see the review\nby Cimr´ ak [17].\nAlouges & Jaisson [4, 5] propose linear finite element discretizations in space and\nlinearly implicit backward Euler in time for the LLG equation in the weak fo rmula-\ntion (1.5) and prove convergence withoutrates towards nonsmooth weak solutions,\nusing a discrete energy inequality and compactness arguments. Co nvergence of this\ntype was previously shown by Bartels & Prohl [11] for fully implicit meth ods that\nare based on a different formulation of the Landau–Lifshitz equatio n (1.1). In [6],\nconvergence without rates towards weak solutions is shown for a m ethod that is\n(formally) of “almost” order 2 in time, based on the midpoint rule, for the LLG\nequation with an effective magnetic field of a more general type than (1.3).\nIn a complementary line of research, convergence withrates has been studied\nunder sufficiently strong regularity assumptions, which can, howev er, not be guar-\nanteed over a given time interval, since solutions of the LLG equation may develop\nsingularities. A first-order error bound for a linearly implicit time discr etization\nof the Landau–Lifshitz equation (1.1) was proved by Cimr´ ak [16]. Op timal-order\nerror bounds for linearly implicit time discretizations based on the bac kward Euler\nand Crank–Nicolson methods combined with finite element full discret izations for\na different version of the Landau–Lifshitz equation (1.1) were obta ined under suf-\nficient regularity assumptions by Gao [23] and An [7], respectively. In contrast to\n[4, 5, 6, 11], these methods do not satisfy an energy inequality irres pective of the\nsolution regularity.\nNumerical discretizations for the coupled system of the LLG equat ion (1.5) with\nthe eddy current approximation of the Maxwell equations are stud ied by Feischl &\nTran [21], with first-order error bounds in space and time under suffi cient regularity\nassumptions. This also yields thefirst result offirst-order conver gence ofthemethod\nof Alouges & Jaisson [4, 5].\nThere are several methods for the LLG equations that are of for mal order 2 in\ntime (thoughonlyoforder 1 inspace), e.g., [35, 31, 19], but noneof t hemcomes with\nan error analysis. Fully implicit BDF time discretizations for LLG equatio ns have4 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nbeen used successfully in the computational physics literature [37 ], though without\ngiving any error analysis.\nTo the authors’ knowledge, the second-order linearly implicit metho d proposed\nand studied here is thus the first numerical method for the LLG (or LL) equation\nthathasrigorousapriorierrorestimatesoforder2inbothspace andtimeunderhigh\nregularity assumptions and that satisfies a discrete energy inequa lity irrespective of\nregularity.\nWe conclude this brief survey of the literature with a remark: The ex isting con-\nvergence results either give convergence of a subsequence witho ut rates to a weak\nsolution(withoutimposingstrongregularityassumptions), orthey showconvergence\nwith rates towards sufficiently regular solutions (as we do here). Bo th approaches\nyield insight into the numerical methods and have their merits, and th ey comple-\nment each other. Clearly, neither approach is fully satisfactory, b ecause convergence\nwithout rates of some subsequence is nothing to observe inactual computations, and\non the other hand high regularity is at best provable for close to con stant initial con-\nditions [22] or over short time intervals. We regard the situation as a nalogous to\nthe development of numerical methods and their analysis in other fie lds such as\nnonlinear hyperbolic conservation laws: second-order methods ar e highly popular\nin that field, even though they can only be shown to converge with ve ry low order\n(1/2 or less or only without rates) for available regularity properties; s ee, e.g., [32,\nChapter 3]. Nevertheless, second-order methods arefavoredo ver first-order methods\nin many applications, especially if they enjoy some qualitative propert ies that give\nthem robustness in non-regular situations. A similar situation occur s with the LLG\nequation, where the most important qualitative property appears to be the energy\ninequality.\n1.4.Outline. InSection2we describe thenumerical methodsstudied inthispaper .\nThey use time discretization by linearly implicit BDF methods of orders u p to 5 and\nspace discretization by finite elements of arbitrary polynomial degr ee in a numerical\nscheme that is based on the weak formulation (1.5), with an approxim ate tangent\nspace that enforces the orthogonality constraint approximately in anL2-projected\nsense.\nIn Section 3 we state our main results:\n•For the full discretization of (1.5) by linearly implicit BDF methods of or ders 1\nand 2 and finite element methods of arbitrary polynomial degree we g ive optimal-\norder error bounds in the H1norm, under very mild mesh conditions, in the case\nof sufficiently regular solutions (Theorem 3.1). For these methods w e also show a\ndiscrete energy inequality that requires only very weak regularity a ssumptions on\nthe data (Proposition 3.1). This discrete energy inequality is of the s ame type as\nthe one used in [5, 11] for proving convergence without rates to a weak solution.\n•For the linearly implicit BDF methods of orders 3 to 5 and finite element m ethods\nwith polynomial degree at least 2, we have optimal-order error boun ds in theH1\nnorm only if the damping parameter αis larger than some positive threshold, which\ndepends on the order of the BDF method (Theorem 3.2). Moreover , a stronger (but\nstill mild) CFL condition τ/lessorequalslantchis required. A discrete energy inequality underHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 5\nvery weak regularity conditions is not available for the BDF methods o f orders 3\nto 5, in contrast to the A-stable BDF methods of orders 1 and 2.\nIn Section 4 we prove a perturbation result for the continuous pro blem by energy\ntechniques, as a preparation for the proofs of our error bounds for the discretization.\nIn Section 5 we study properties of the L2-orthogonal projection onto the discrete\ntangent space, which are needed to ensure consistency of the fu ll order and stability\nof the space discretization with the higher-order discrete tangen t space.\nIn Section 6 we study consistency properties of the methods and p resent the error\nequation.\nIn Sections 7 and 8 we prove Theorems 3.1 and 3.2, respectively. The higher-\norder convergence proofs are separated into consistency (Sec tion 6) and stability\nestimates. The stability proofs use the technique of energy estima tes, in an unusual\nversion where the error equation is tested with a projection of the discrete time\nderivative of the error onto the discrete tangent space. These p roofs are different\nfor the A-stable BDF methods of orders 1 and 2 and for the BDF met hods of orders\n3 to 5. For the control of nonlinearities, the stability proofs also re quire pointwise\nerror bounds, which are obtained with the help of finite element inver se inequalities\nfrom theH1error bounds of previous time steps.\nIn Section 9 we illustrate our results by numerical experiments.\nIn an Appendix we collect basic results on energy techniques for BDF methods\nthat are needed for our stability proofs.\n2.Discretization of the LLG equation\nWe now describe the time and space discretization that is proposed a nd studied\nin this paper.\n2.1.Time discretization by linearly implicit BDF methods. We shall dis-\ncretize the LLG equation (1.5) in time by the linearly implicit k-step BDF methods,\n1/lessorequalslantk/lessorequalslant5, described by the polynomials δandγ,\nδ(ζ) =k/summationdisplay\nℓ=11\nℓ(1−ζ)ℓ=k/summationdisplay\nj=0δjζj, γ(ζ) =1\nζ/bracketleftbig\n1−(1−ζ)k/bracketrightbig\n=k−1/summationdisplay\ni=0γiζi.\nWe lettn=nτ, n= 0,...,N,be a uniform partition of the interval [0 ,¯t] with\ntime stepτ=¯t/N.For thek-step method we require kstarting values mifor\ni= 0,...,k−1. Forn/greaterorequalslantk, we determine the approximation mntom(tn) as\nfollows. We first extrapolate the known values mn−k,...,mn−1to a preliminary\nnormalized approximation /hatwidermnattn,\n(2.1) /hatwidermn:=k−1/summationdisplay\nj=0γjmn−j−1/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1/vextendsingle/vextendsingle/vextendsingle.\nTo avoid potentially undefined quantities, we define /hatwidermnto be an arbitrary fixed\nunit vector if the denominator in the above formula is zero.6 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nThe derivative approximation ˙mnand the solution approximation mnare related\nby the backward difference formula\n(2.2) ˙mn=1\nτk/summationdisplay\nj=0δjmn−j,i.e.,mn=/parenleftig\n−k/summationdisplay\nj=1δjmn−j+τ˙mn/parenrightig\n/δ0.\nWe determine mnby requiring that for all ϕ∈T(/hatwidermn)∩H1(Ω)3,\n(2.3)α/parenleftbig\n˙mn,ϕ/parenrightbig\n+/parenleftbig\n/hatwidermn×˙mn,ϕ/parenrightbig\n+/parenleftbig\n∇mn,∇ϕ/parenrightbig\n=/parenleftbig\nH(tn),ϕ/parenrightbig\n˙mn∈T(/hatwidermn),i.e.,/hatwidermn·˙mn= 0.\nHere we note that on inserting the formula in (2.2) for mnin the third term of (2.3),\nwe obtain a linear constrained elliptic equation for ˙mn∈T(/hatwidermn)∩H1(Ω)3of the\nform\nα/parenleftbig\n˙mn,ϕ/parenrightbig\n+/parenleftbig\n/hatwidermn×˙mn,ϕ/parenrightbig\n+τ\nδ0/parenleftbig\n∇˙mn,∇ϕ/parenrightbig\n=/parenleftbig\nfn,ϕ/parenrightbig\n∀ϕ∈T(/hatwidermn)∩H1(Ω)3,\nwherefnconsistsofknownterms. Thebilinearformontheleft-handsideis H1(Ω)3-\ncoercive on T(/hatwidermn)∩H1(Ω)3, and hence the above linear equation has a unique\nsolution ˙mn∈T(/hatwidermn)∩H1(Ω)3by the Lax–Milgram lemma. Once this elliptic\nequation is solved for ˙mn, we obtain the approximation mn∈H1(Ω)3tom(tn)\nfrom the second formula in (2.2).\n2.2.Full discretization by BDF and higher-order finite elements .For a\nfamilyofregularandquasi-uniformfiniteelement triangulationsof Ωwithmaximum\nmeshwidth h >0 we form the Lagrange finite element spaces Vh⊂H1(Ω) with\npiecewise polynomials of degree r/greaterorequalslant1. We denote the L2-orthogonal projections\nonto the finite element space by Πh:L2(Ω)→VhandΠh=I⊗Πh:L2(Ω)3→V3\nh.\nWith a function m∈H1(Ω)3that vanishes nowhere on Ω, we associate the discrete\ntangent space\n(2.4)Th(m) ={ϕh∈V3\nh: (m·ϕh,vh) = 0∀vh∈Vh}\n={ϕh∈V3\nh:Πh(m·ϕh) = 0}.\nThis space is different from the discrete tangent space used in [4, 5 ], where the\northogonality constraint m·ϕh= 0 is required to hold pointwise at the finite\nelement nodes. Here, the constraint is enforced weakly on the finit e element space,\nas is done in various saddle point problems for partial differential equ ations, for\nexample forthedivergence-free constraint inthe Stokes problem [14, 25]. Incontrast\nto that example, here the bilinear form associated with the linear con straint, i.e.,\nb(m;ϕh,vh) = (m·ϕh,vh), depends on the state m. This dependence substantially\naffects both the implementation and the error analysis.\nFollowing the general approach of [4, 5] with this modified discrete ta ngent space,\nwe discretize (1.5) in space by determining the time derivative ∂tmh(t)∈Th(mh(t))\nsuch that (omitting the argument t)\n(2.5)α/parenleftbig\n∂tmh,ϕh/parenrightbig\n+/parenleftbig\nmh×∂tmh,ϕh/parenrightbig\n+/parenleftbig\n∇mh,∇ϕh/parenrightbig\n=/parenleftbig\nH,ϕh/parenrightbig\n∀ϕh∈Th(mh),\nwhere the brackets ( ·,·) denote again the L2inner product over the domain Ω.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 7\nThe full discretization with the linearly implicit BDF method is then readily\nobtained from (2.3): determine ˙mn\nh∈Th(/hatwidermn\nh) such that\n(2.6)α/parenleftbig˙mn\nh,ϕh/parenrightbig\n+/parenleftbig/hatwidermn\nh×˙mn\nh,ϕh/parenrightbig\n+/parenleftbig\n∇mn\nh,∇ϕh/parenrightbig\n=/parenleftbig\nHn,ϕh/parenrightbig\n∀ϕh∈Th(/hatwidermn\nh),\nwhere/hatwidermn\nhand˙mn\nhare related to mn−j\nhforj= 0,...,kin the same way as in (2.1)\nand (2.2) above with mn−j\nhin place ofmn−j, viz.,\n(2.7) ˙mn\nh=1\nτk/summationdisplay\nj=0δjmn−j\nh,/hatwidermn\nh=k−1/summationdisplay\nj=0γjmn−j−1\nh/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextendsingle.\nTo avoid potentially undefined quantities, we define /hatwidermn\nhto be an arbitrary fixed\nunit vector if the denominator in the above formula is zero. (We will, ho wever, show\nthat this does not occur in the situation of sufficient regularity.)\nTo implement the discrete tangent space Th(/hatwidermn\nh), there are at least two options:\nusing the constraints Πh(m·ϕh) = 0 or constructing a local basis of Th(m).\n(a)Constraints : Letφifori= 1,...,N:= dimVhdenote the nodal basis of\nVhand denote the basis functions of V3\nhbyφi=ek⊗φifori= (i,k), where\nekfork= 1,2,3 are the standard unit vectors of R3. We denote by MandA\nthe usual mass and stiffness matrices, respectively, with entries mij= (φi,φj)L2(Ω)\nandaij= (∇φi,∇φj)L2(Ω)3. We further introduce the sparse skew-symmetric matrix\nSn= (sn\ni,j)∈R3N×3Nwithentries sn\ni,j= (/hatwidermn\nh×φi,φj)L2(Ω)3andthesparseconstraint\nmatrixCn= (cn\ni,j)∈R3N×Nbycn\ni,j= (/hatwidermn\nh·φi,φj)L2(Ω). Finally, we denote the\nmatrix of the unconstrained time-discrete problem as\nKn=αI⊗M+τ\nδ0I⊗A+Sn.\nLet ˙mn∈R3Ndenote the nodal vector of ˙mn\nh∈Th(/hatwidermn\nh). In this setting, (2.6) yields\na system of linear equations of saddle point type\nKn˙mn+(Cn)Tλn=fn,\nCn˙mn= 0,\nwhereλn∈RNis the unknown vector of Lagrange multipliers and fn∈R3Nis a\nknown right-hand side.\n(b)Local basis : It is possible to compute a local basis of Th(m) by solving small\nlocal problems. To see that, let ω⊂Ωdenote a collection of elements of the mesh\nand letω⊃ωdenote the same set plus the layer of elements touching ω(the patch\nofω). A sufficient (and necessary) condition for ϕh∈V3\nhwith supp(ϕh)⊆ωto\nbelong toTh(m) is\n(2.8) ( m·ϕh,ψh) = 0 for all ψh∈Vhwith supp(ψh)⊆ω.\nIf we denote by # ωthe number of generalized hat functions of Vhsupported in ω,\nthe space of functions in V3\nhwith support in ωis 3#ω-dimensional. On the other\nhand, the space of test functions in (2.8) is # ω-dimensional. We may choose ω\nsufficiently large (depending only on shape regularity) such that 3# ω >#ωand\nhence (2.8) has at least one solution which is then a local basis functio n ofTh(m).\nChoosing different ωto coverΩyields a full basis of Th(m).8 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nLet us denote the so obtained basis of Th(/hatwidermn\nh) by (ψn\nℓ), given via ψn\nℓ=/summationtext\niφibn\niℓ,\nand the sparse basis matrix by Bn= (bn\niℓ). Then, the nodal vector ˙ mn=Bnxnis\nobtained by solving the linear system\n(Bn)TKnBnxn= (Bn)Tfn.\nAn advantage of this approach is that the dimension is roughly halved compared\nto the formulation with constraints. However, the efficiency of one approach versus\nthe other depends heavily on the numerical linear algebra used. Suc h comparisons\nare outside the scope of this paper.\nRemark 2.1. The algorithm described above does not enforce the norm constra int\n|m|= 1 at the nodes. The user might add a normalization step in the definit ion\nofmnin (2.2). However, here we do not consider this normalized variant of the\nmethod, whose convergence properties are not obvious to derive .\nRemark 2.2. Differently to [4], we do not use the pointwise discrete tangent space\nTpw\nh(m) ={ϕh∈V3\nh:m·ϕ= 0 in every node }\n={ϕh∈V3\nh:Ih(m·ϕh) = 0}=IhP(m)V3\nh,\nwhereIh:C(¯Ω)→Vhdenotesfiniteelementinterpolationand Ih=I⊗Ih:C(¯Ω)3→\nV3\nh. It is already reported in [4, Section 4] that an improvement of the o rder with\nhigher-degree finite elements could not be observed in numerical ex periments when\nusing the pointwise tangent spaces in the discretization (2.5). Our a nalysis shows\na lack of consistency of optimal order in the discretization with Tpw\nh(m), which\noriginates from the fact that IhP(m) is not self-adjoint. The order reduction can,\nhowever, be cured by adding a correction term: in the nth time step, determine\n˙mn\nh∈Tpw\nh(/hatwidermn\nh) such that for all ϕh∈Tpw\nh(/hatwidermn\nh),\n(2.9)α/parenleftbig˙mn\nh,ϕh/parenrightbig\n+/parenleftbig/hatwidermn\nh×˙mn\nh,ϕh/parenrightbig\n+/parenleftbig\n∇mn\nh,∇ϕh/parenrightbig\n−/parenleftbig\n∇/hatwidermn\nh,∇(I−P(/hatwidermn\nh))ϕh/parenrightbig\n=/parenleftbig\nP(/hatwidermn\nh)H(tn),ϕh/parenrightbig\n,\nwith notation /hatwidermn\nhand˙mn\nhas in (2.7). With the techniques of the present paper, it\ncan be shown that like (2.6), also this discretization converges with o ptimal order\nin theH1norm under sufficient regularity conditions. Since this paper is alread y\nrather long, we do not include the proof of this result. In contrast to (2.6) for the\nfirst- and second-order BDF methods, the method (2.9) does not admit anh- and\nτ-independent bound of the energy that is irrespective of the smoo thness of the\nsolution.\n3.Main results\n3.1.Error bound and energy inequality for BDF of orders 1 and 2. For\nthe full discretization with first- and second-order BDF methods a nd finite elements\nof arbitrary polynomial degree r/greaterorequalslant1 we will prove the following optimal-order error\nbound in Sections 5 to 7.\nTheorem 3.1 (Error bound for orders k= 1,2).Consider the full discretization\n(2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time discretiza-\ntion fork/lessorequalslant2and finite elements of polynomial degree r/greaterorequalslant1from a family ofHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 9\nregular and quasi-uniform triangulations of Ω. Suppose that the solution mof the\nLLG equation is sufficiently regular. Then, there exist ¯τ >0and¯h >0such that\nfor numerical solutions obtained with step sizes τ/lessorequalslant¯τand meshwidths h/lessorequalslant¯h, which\nare restricted by the very mild CFL-type condition\nτk/lessorequalslant¯ch1/2\nwith a sufficiently small constant ¯c(independent of handτ), the errors are bounded\nby\n(3.1) /ba∇dblmn\nh−m(tn)/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr)fortn=nτ/lessorequalslant¯t,\nwhereCis independent of h,τandn(but depends on αand exponentially on ¯t),\nprovided that the errors of the starting values also satisfy such a bound.\nThe precise regularity requirements are as follows:\n(3.2)m∈Ck+1([0,¯t],L∞(Ω)3)∩C1([0,¯t],Wr+1,∞(Ω)3),\n∆m+H∈C([0,¯t],Wr+1,∞(Ω)3).\nRemark 3.1 (Discrepancy from normality ).Sincem(x,tn) are unit vectors, an\nimmediate consequence of the error estimate (3.1) is that\n(3.3) /ba∇dbl1−|mn\nh|/ba∇dblL2(Ω)/lessorequalslantC(τk+hr) fortn=nτ/lessorequalslant¯t,\nwith a constant Cindependent of n,τandh. The proof of Theorem 3.1 also shows\nthat the denominator in the definition of the normalized extrapolate d value/hatwidermn\nh\nsatisfies\n/vextenddouble/vextenddouble/vextenddouble1−/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextenddouble/vextenddouble/vextenddouble\nL∞(Ω)/lessorequalslantCh−1/2(τk+hr)/lessorequalslant1\n2fortn=nτ/lessorequalslant¯t,\nwhich in particular ensures that /hatwidermn\nhis unambiguously defined.\nTesting with ϕ=∂tm∈T(m) in (1.5), we obtain (only formally, if ∂tmis not\ninH1(Ω)3)\nα(∂tm,∂tm)+(∇m,∂t∇m) = (H,∂tm),\nwhich, by integration in time and the Cauchy–Schwarz and Young ineq ualities, im-\nplies the energy inequality\n/ba∇dbl∇m(t)/ba∇dbl2\nL2+1\n2α/integraldisplayt\n0/ba∇dbl∂tm(s)/ba∇dbl2\nL2ds/lessorequalslant/ba∇dbl∇m(0)/ba∇dbl2\nL2+1\n2α/integraldisplayt\n0/ba∇dblH(s)/ba∇dbl2\nL2ds.\nSimilarly, wetestwith ϕh=˙mn\nh∈Th(/hatwidermn\nh)in(2.6). Thenwecanprovethefollow-\ning discrete energy inequality, which holds under very weak regularit y assumptions\non the data.\nProposition 3.1 (Energy inequality for orders k= 1,2).Consider the full dis-\ncretization (2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time\ndiscretization for k/lessorequalslant2and finite elements of polynomial degree r/greaterorequalslant1. Then, the10 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nnumerical solution satisfies the following discrete energy inequality :forn/greaterorequalslantkwith\nnτ/lessorequalslant¯t,\nγ−\nk/ba∇dbl∇mn\nh/ba∇dbl2\nL2+1\n2ατn/summationdisplay\nj=k/ba∇dbl˙mj\nh/ba∇dbl2\nL2/lessorequalslantγ+\nkk−1/summationdisplay\ni=0/ba∇dbl∇mi\nh/ba∇dbl2\nL2+τ\n2αn/summationdisplay\nj=k/ba∇dblH(tj)/ba∇dbl2\nL2,\nwhereγ±\n1= 1andγ±\n2= (3±2√\n2)/4.\nThis energy inequality is an important robustness indicator of the nu merical\nmethod. In [5, 11], such energy inequalitys are used to prove conve rgence with-\nout rates (for a subsequence τn→0 andhn→0) to a weak solution of the LLG\nequation for the numerical schemes considered there (which have γ±= 1, but this\nis inessential in the proofs).\nAs the proof of Proposition 3.1 is short, we give it here.\nProof.TheproofreliesontheA-stabilityofthefirst-andsecond-orderB DFmethods\nvia Dahlquist’s G-stabilitytheoryasexpressed inLemma 10.1ofthe Ap pendix, used\nwithδ(ζ) =/summationtextk\nℓ=1(1−ζ)ℓ/ℓandµ(ζ) = 1. The positive definite symmetric matrices\nG= (gij)k\ni,j=1are known to be G= 1 fork= 1 and (see [27, p.309])\nG=1\n4/parenleftbigg\n1−2\n−2 5/parenrightbigg\nfork= 2,\nwhich has the eigenvalues γ±= (3±2√\n2)/4.\nWe test with ϕh=˙mn\nh∈Th(/hatwidermn\nh) in (2.6) and note/parenleftbig\n/hatwidermn\nh×˙mn\nh,˙mn\nh/parenrightbig\n= 0, so that\nα/ba∇dbl˙mn\nh/ba∇dbl2\nL2+(∇mn\nh,∇˙mn\nh) = (Hn,˙mn\nh).\nThe right-hand side is bounded by\n(Hn,˙mn\nh)/lessorequalslantα\n2/ba∇dbl˙mn\nh/ba∇dbl2\nL2+1\n2α/ba∇dblHn/ba∇dbl2\nL2.\nRecalling the definition of ˙mn\nh, we have by Lemma 10.1\n(∇mn\nh,∇˙mn\nh)/greaterorequalslant1\nτk/summationdisplay\ni,j=1gij(∇mn−i+1\nh,∇mn−j+1\nh)−1\nτk/summationdisplay\ni,j=1gij(∇mn−i\nh,∇mn−j\nh).\nWe fix ¯nwithk/lessorequalslant¯n/lessorequalslant¯t/τand sum from n=kto ¯nto obtain\nk/summationdisplay\ni,j=1gij(∇m¯n−i+1\nh,∇m¯n−j+1\nh)+1\n2ατ¯n/summationdisplay\nn=k/ba∇dbl˙mn\nh/ba∇dbl2\nL2\n/lessorequalslantk/summationdisplay\ni,j=1gij(∇mk−i\nh,∇mk−j\nh)+τ\n2α¯n/summationdisplay\nn=k/ba∇dblHn/ba∇dbl2\nL2.\nNoting that\nγ−/ba∇dbl∇m¯n\nh/ba∇dbl2\nL2/lessorequalslantk/summationdisplay\ni,j=1gij(∇m¯n−i+1\nh,∇m¯n−j+1\nh),HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 11\nk/summationdisplay\ni,j=1gij(∇mk−i\nh,∇mk−j\nh)/lessorequalslantγ+k−1/summationdisplay\ni=0/ba∇dbl∇mi\nh/ba∇dbl2\nL2,\nwe obtain the stated result. /square\n3.2.Error bound for BDF of orders 3to5.For the BDF methods of orders\n3 to 5 we prove the following result in Section 8. Here we require a stro nger, but\nstill moderate stepsize restriction in terms of the meshwidth. More importantly, we\nmust impose a positive lower bound on the damping parameter αof (1.1).\nTheorem 3.2 (Error bound for orders k= 3,4,5).Consider the full discretization\n(2.6)of the LLG equation (1.4)by the linearly implicit k-step BDF time discretiza-\ntion for3/lessorequalslantk/lessorequalslant5and finite elements of polynomial degree r/greaterorequalslant2from a family of\nregular and quasi-uniform triangulations of Ω. Suppose that the solution mof the\nLLG equation has the regularity (3.2), and that the damping parameter αsatisfies\n(3.4)α>α kwith\nαk= 0.0913,0.4041,4.4348,fork= 3,4,5,respectively.\nThen, for an arbitrary constant ¯C >0, there exist ¯τ >0and¯h >0such that for\nnumerical solutions obtained with step sizes τ/lessorequalslant¯τand meshwidths h/lessorequalslant¯hthat are\nrestricted by\n(3.5) τ/lessorequalslant¯Ch,\nthe errors are bounded by\n/ba∇dblmn\nh−m(tn)/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr)fortn=nτ/lessorequalslant¯t,\nwhereCis independent of h,τandn(but depends on αand exponentially on ¯C¯t),\nprovided that the errors of the starting values also satisfy such a bound.\nTheorem 3.2 limits the use of the BDF methods of orders higher than 2 (and more\nseverely for orders higher than 3) to applications with a large dampin g parameter α,\nsuch as cases described in [24, 39]. We remark, however, that in man y situations\nαis of magnitude 10−2or even smaller [10]. A very small damping parameter α\naffects not only the methods considered here. To our knowledge, t he error analysis\nof any numerical method proposed in the literature breaks down as α→0, as does\nthe energy inequality.\nIt is not surprising that a positive lower bound on αarises for the methods of\nordersk/greaterorequalslant3, since they are not A-stable and a lower bound on αis required also for\nthe simplified linear problem ( α+i)∂tu=∆u, which arises from (1.4) by freezing m\nin the termm×∂tmand diagonalizing this skew-symmetric linear operator (with\neigenvalues ±i and 0) and by omitting the projection P(m) on the right-hand side\nof (1.4).\nThe proof of Theorem 3.2 uses a variant of the Nevanlinna–Odeh mult iplier tech-\nnique [34], which is described in the Appendix for the convenience of th e reader.\nWhile for sufficiently large αwe have an optimal-order error bound in the case of\na smooth solution, there is apparently no discrete energy inequality under weak\nregularity assumptions similar to Proposition 3.1 for the BDF methods of orders 3\nto 5.12 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nAs in Remark 3.1, the error bounds also allow us to bound the discrepa ncy from\nnormality.\n4.A continuous perturbation result\nIn this section we present a perturbation result for the continuou s problem, be-\ncause we will later transfer the arguments of its proof to the discr etizations to prove\nstability and convergence of the numerical methods.\nLetm(t) be a solution of (1.4) for 0 /lessorequalslantt/lessorequalslant¯t, and letm⋆(t), also of unit length,\nsolve the same equation up to a defect d(t) for 0/lessorequalslantt/lessorequalslant¯t:\n(4.1)α∂tm⋆+m⋆×∂tm⋆=P(m⋆)(∆m⋆+H)+d\n=P(m)(∆m⋆+H)+r,\nwith\nr=−/parenleftbig\nP(m)−P(m⋆)/parenrightbig\n(∆m⋆+H)+d.\nThen,m⋆also solves the perturbed weak formulation\nα(∂tm⋆,ϕ)+(m⋆×∂tm⋆,ϕ)+(∇m⋆,∇ϕ) = (r,ϕ)∀ϕ∈T(m)∩H1(Ω)3,\nand the error e=m−m⋆satisfies the error equation\n(4.2)α(∂te,ϕ)+(e×∂tm⋆,ϕ)+(m×∂te,ϕ)+(∇e,∇ϕ) =−(r,ϕ)\n∀ϕ∈T(m)∩H1(Ω)3.\nBefore we turn to the perturbation result, we need Lipschitz-typ e bounds for the\northogonal projection P(m) =I−mmTapplied to sufficiently regular functions.\nLemma 4.1. The projection P(·)satisfies the following estimates, for functions\nm,m⋆,v:Ω→R3, wheremandm⋆take values on the unit sphere and m⋆∈\nW1,∞(Ω)3:\n/ba∇dbl(P(m)−P(m⋆))v/ba∇dblL2(Ω)3/lessorequalslant2/ba∇dblv/ba∇dblL∞(Ω)3/ba∇dblm−m⋆/ba∇dblL2(Ω)3,/vextenddouble/vextenddouble∇/parenleftbig\n(P(m)−P(m⋆))v/parenrightbig/vextenddouble/vextenddouble\nL2(Ω)3×3/lessorequalslant2/ba∇dblm⋆/ba∇dblW1,∞(Ω)3/ba∇dblv/ba∇dblW1,∞(Ω)3/ba∇dblm−m⋆/ba∇dblL2(Ω)3\n+6/ba∇dblv/ba∇dblL∞(Ω)3/ba∇dbl∇(m−m⋆)/ba∇dblL2(Ω)3×3.\nProof.Settinge=m−m⋆, we start by rewriting\n(P(m)−P(m⋆))v=−(mmT−m⋆mT\n⋆)v=−(meT+emT\n⋆)v.\nThe first inequality then follows immediately by taking the L2norm of both sides\nof the above equality, using the fact that mandm⋆are of unit length. The second\ninequality is proved similarly, using the product rule\n∂i(P(m)−P(m⋆))v=−∂i(eeT+m⋆eT+emT\n⋆)v\n=−(∂ieeT+e∂ieT+∂im⋆eT+m⋆∂ieT+∂iemT\n⋆+e∂imT\n⋆)v\n+(meT+emT\n⋆)∂iv,\ntheL∞bound of∂im⋆, and the fact that /ba∇dble/ba∇dblL∞/lessorequalslant/ba∇dblm/ba∇dblL∞+/ba∇dblm⋆/ba∇dblL∞/lessorequalslant2./square\nWe have the following perturbation result.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 13\nLemma 4.2. Letm(t)andm⋆(t)be solutions of unit length of (1.5)and(4.1),\nrespectively, and suppose that, for 0/lessorequalslantt/lessorequalslant¯t, we have\n(4.3)/ba∇dblm⋆(t)/ba∇dblW1,∞(Ω)3+/ba∇dbl∂tm⋆(t)/ba∇dblW1,∞(Ω)3/lessorequalslantR\nand/ba∇dbl∆m⋆(t)+H(t)/ba∇dblL∞(Ω)3/lessorequalslantK.\nThen, the error e(t) =m(t)−m⋆(t)satisfies, for 0/lessorequalslantt/lessorequalslant¯t,\n(4.4) /ba∇dble(t)/ba∇dbl2\nH1(Ω)3/lessorequalslantC/parenleftig\n/ba∇dble(0)/ba∇dbl2\nH1(Ω)3+/integraldisplayt\n0/ba∇dbld(s)/ba∇dbl2\nL2(Ω)3ds/parenrightig\n,\nwhere the constant Cdepends only on α,R,K, and¯t.\nProof.Let us first assume that ∂tm(t)∈H1(Ω)3for allt. Following [21], we test in\nthe error equation (4.2) with ϕ=P(m)∂te∈T(m). By the following argument,\nthis test function is then indeed in H1(Ω)3and can be viewed as a perturbation\nof∂te:\nϕ=P(m)∂te=P(m)∂tm−P(m)∂tm⋆\n=P(m)∂tm−P(m⋆)∂tm⋆−(P(m)−P(m⋆))∂tm⋆\n=∂tm−∂tm⋆−(P(m)−P(m⋆))∂tm⋆,\nand so we have\n(4.5)ϕ=P(m)∂te=∂te+qwithq=−(P(m)−P(m⋆))∂tm⋆.\nBy Lemma 4.1 and using (4.3) we have\n(4.6) /ba∇dblq/ba∇dblL2/lessorequalslant2R/ba∇dble/ba∇dblL2and/ba∇dbl∇q/ba∇dblL2/lessorequalslantCR/ba∇dble/ba∇dblH1.\nTesting the error equation (4.2) with ϕ=∂te+q, we obtain\nα(∂te,∂te+q)+(e×∂tm⋆,∂te+q)+(m×∂te,∂te+q)\n+(∇e,∇(∂te+q)) =−(r,∂te+q),\nwhere, by (4.1) and Lemma 4.1 with (4.3), ris bounded as\n(4.7)/ba∇dblr/ba∇dblL2/lessorequalslant/ba∇dbl/parenleftbig\nP(m)−P(m⋆)/parenrightbig\n(∆m⋆+H)/ba∇dblL2+/ba∇dbld/ba∇dblL2\n/lessorequalslant2K/ba∇dble/ba∇dblL2+/ba∇dbld/ba∇dblL2.\nBy collecting terms, and using the fact that ( m×∂te,∂te) vanishes, we altogether\nobtain\nα/ba∇dbl∂te/ba∇dbl2\nL2+1\n2d\ndt/ba∇dbl∇e/ba∇dbl2\nL2=−α(∂te,q)−(e×∂tm⋆,∂te+q)−(m×∂te,q)\n−(∇e,∇q)−(r,∂te+q).\nFor the right-hand side, the Cauchy–Schwarz inequality and /ba∇dblm/ba∇dblL∞= 1 yield\nα/ba∇dbl∂te/ba∇dbl2\nL2+1\n2d\ndt/ba∇dbl∇e/ba∇dbl2\nL2/lessorequalslantα/ba∇dbl∂te/ba∇dblL2/ba∇dblq/ba∇dblL2+R/ba∇dble/ba∇dblL2(/ba∇dbl∂te/ba∇dblL2+/ba∇dblq/ba∇dblL2)\n+/ba∇dbl∂te/ba∇dblL2/ba∇dblq/ba∇dblL2+/ba∇dbl∇e/ba∇dblL2/ba∇dbl∇q/ba∇dblL2+/ba∇dblr/ba∇dblL2(/ba∇dbl∂te/ba∇dblL2+/ba∇dblq/ba∇dblL2).14 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nYoung’s inequality and absorptions, together with the bounds in (4.6 ) and (4.7),\nyield\nα1\n2/ba∇dbl∂te/ba∇dbl2\nL2+1\n2d\ndt/ba∇dbl∇e/ba∇dbl2\nL2/lessorequalslantc/ba∇dble/ba∇dbl2\nH1+c/ba∇dbld/ba∇dbl2\nL2.\nHere, we note that\n1\n2d\ndt/ba∇dble/ba∇dbl2\nL2= (∂te,e)/lessorequalslant1\n2/ba∇dbl∂te/ba∇dbl2\nL2+1\n2/ba∇dble/ba∇dbl2\nL2,so that /ba∇dbl∂te/ba∇dbl2\nL2/greaterorequalslantd\ndt/ba∇dble/ba∇dbl2\nL2−/ba∇dble/ba∇dbl2\nL2.\nCombining these inequalities and integrating in time, we obtain\n/ba∇dble(t)/ba∇dbl2\nH1/lessorequalslantc/ba∇dble(0)/ba∇dbl2\nH1+c/integraldisplayt\n0/ba∇dble(s)/ba∇dbl2\nH1ds+c/integraldisplayt\n0/ba∇dbld(s)/ba∇dbl2\nL2ds.\nBy Gronwall’s inequality, we then obtain the stated error bound.\nFinally, if∂tm(t) is not inH1(Ω)3for somet, then a regularization and density\nargument, which we do not present here, yields the result, since th e error bound\ndoes not depend on the H1norm of∂tm. /square\n5.Orthogonal projection onto the discrete tangent space\nFor consistency and stability of the full discretization, we need to s tudy properties\nof theL2(Ω)-orthogonal projection onto the discrete tangent space Th(m), which\nwe denote by\nPh(m):V3\nh→Th(m).\nWe do not have an explicit expression for this projection, but the pr operties stated\nin Lemmas 5.1 to 5.3 will be used for proving consistency and stability. W e recall\nthat we consider a quasi-uniform, shape-regular family Thof triangulations with\nLagrange finite elements of polynomial degree r.\nThe first lemma states that the projection Ph(m) approximates the orthogonal\nprojection P(m) =I−mmTonto the tangent space T(m) with optimal order. It\nwill be used in the consistency error analysis of Section 6.\nLemma 5.1. Form∈Wr+1,∞(Ω)3with|m|= 1almost everywhere we have\n/ba∇dbl(Ph(m)−P(m))v/ba∇dblL2(Ω)3/lessorequalslantChr+1/ba∇dblv/ba∇dblHr+1(Ω)3,\n/ba∇dbl(Ph(m)−P(m))v/ba∇dblH1(Ω)3/lessorequalslantChr/ba∇dblv/ba∇dblHr+1(Ω)3,\nfor allv∈Hr+1(Ω)3, whereCdepends on a bound of /ba∇dblm/ba∇dblWr+1,∞(Ω)3.\nThe second lemma states that the projection Ph(m) has Lipschitz bounds of the\nsame type as those of the orthogonal projection P(m) given in Lemma 4.1. It will\nbe used in the stability analysis of Sections 7 and 8.\nLemma 5.2. Letm∈W1,∞(Ω)3and/tildewiderm∈H1(Ω)3with|m|=|/tildewiderm|= 1almost\neverywhere and /ba∇dblm/ba∇dblW1,∞/lessorequalslantR. There exist CR>0andhR>0such that for\nh/lessorequalslanthR, for allvh∈V3\nh,\n(i)/ba∇dbl(Ph(m)−Ph(/tildewiderm))vh/ba∇dblL2(Ω)3/lessorequalslantCR/ba∇dblm−/tildewiderm/ba∇dblLp(Ω)3/ba∇dblvh/ba∇dblLq(Ω)3,\nfor(p,q)∈ {(2,∞),(∞,2)}, and\n(ii)/ba∇dbl(Ph(m)−Ph(/tildewiderm))vh/ba∇dblH1(Ω)3/lessorequalslantCR/ba∇dblm−/tildewiderm/ba∇dblH1(Ω)3/ba∇dblvh/ba∇dblL∞(Ω)3HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 15\n+CR/ba∇dblm−/tildewiderm/ba∇dblL2(Ω)3/ba∇dblvh/ba∇dblW1,∞(Ω)3.\nThe next lemma shows the Ws,p-stability of the projection. It is actually used for\np= 2 in the proof of Lemmas 5.1 and 5.2 and will be used for p= 2 in Section 6\nand forp=∞in Sections 7 and 8.\nLemma 5.3. There exists a constant depending only on p∈[1,∞]and the shape\nregularity of the mesh such that for all m∈W1,∞(Ω)3with|m|= 1almost every-\nwhere,\n/ba∇dblPh(m)vh/ba∇dblWs,p(Ω)3/lessorequalslantC/ba∇dblm/ba∇dbl2\nW1,∞(Ω)3/ba∇dblvh/ba∇dblWs,p(Ω)3\nfor allvh∈V3\nhands∈ {−1,0,1}.\nThese three lemmas will be proved in the course of this section, in whic h we\nformulate also three more lemmas that are of independent interest but will not be\nused in the following sections.\nIn the following, we use the dual norms\n/ba∇dblv/ba∇dblW−1,q:= sup\nw∈W1,p(v,w)\n/ba∇dblw/ba∇dblW1,pfor 1/p+1/q= 1.\nThe space W−1,1(Ω) is not the dual space of W1,∞(Ω) but rather defined as the\nclosure ofL2(Ω) withrespect to thenorm /ba∇dbl·/ba∇dblW−1,1. Wealso recall that Πh:Ws,p(Ω)\n→Ws,p(Ω) is uniformly bounded for s∈ {0,1}andp∈[1,∞] (see, e.g., [20]\nfor proofs in a much more general setting). By duality, we also obta in uniform\nboundedness for s=−1 andp∈[1,∞]. A useful consequence is that for vh∈Vh,\n/ba∇dblvh/ba∇dblW−1,q= sup\nw∈W1,p(vh,Πhw)\n/ba∇dblw/ba∇dblW1,p\n/lessorequalslantsup\nw∈W1,p(vh,Πhw)\n/ba∇dblΠhw/ba∇dblW1,psup\nw∈W1,p/ba∇dblΠhw/ba∇dblW1,p\n/ba∇dblw/ba∇dblW1,p/lessorsimilarsup\nwh∈Vh(vh,wh)\n/ba∇dblwh/ba∇dblW1,p.\nLemma 5.4. There holds /ba∇dblv/ba∇dblWs,p(Ω)≃supw∈W−s,q(Ω)(v,w)\n/bardblw/bardblW−s,q(Ω)with1/p+1/q= 1\nforp∈[1,∞]ands∈ {−1,0,1}.\nProof.The interesting case is ( s,p) = (1,∞) since all other cases follow by duality.\nForv∈W1,∞(Ω), thereexists asequence offunctions qn∈C∞\n0(Ω)3with/ba∇dblqn/ba∇dblL1= 1\nsuch that\n/ba∇dbl∇v/ba∇dblL∞= lim\nn→∞(∇v,qn) = lim\nn→∞−(v,divqn)/lessorequalslantsup\nq∈W1,1(v,divq)\n/ba∇dblq/ba∇dblL1.\nMoreover, there holds\n/ba∇dbldivq/ba∇dblW−1,1/lessorequalslantsup\nw∈W1,∞(q,∇w)\n/ba∇dbl∇w/ba∇dblL∞/lessorequalslant/ba∇dblq/ba∇dblL1.\nCombining the last two estimates shows\n/ba∇dbl∇v/ba∇dblL∞/lessorequalslantsup\nw∈W−1,1(v,w)\n/ba∇dblw/ba∇dblW−1,1.16 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nSince\n/ba∇dblv/ba∇dblL∞= sup\nw∈L1(v,w)\n/ba∇dblw/ba∇dblL1/lessorequalslantsup\nw∈W−1,1(v,w)\n/ba∇dblw/ba∇dblW−1,1,\nwe conclude the proof. /square\nLetthediscretenormalspace Nh(m) :=V3\nh⊖Th(m)begivenasthe L2-orthogonal\ncomplement of Th(m) inV3\nh. We note that\n(5.1) Nh(m) ={Πh(mψh) :ψh∈Vh}\nby the definition of Th(m). The functions in the discrete normal space are bounded\nfrom below as follows.\nLemma 5.5. For everyR >0, there exist hR>0andc >0such that for all\nm∈W1,∞(Ω)3with|m|= 1almost everywhere and /ba∇dblm/ba∇dblW1,∞(Ω)/lessorequalslantRand for all\nh/lessorequalslanthR,\n/ba∇dblΠh(mψh)/ba∇dblWs,p(Ω)3/greaterorequalslantc/ba∇dblψh/ba∇dblWs,p(Ω)\nfor allψh∈Vhand(s,p)∈ {−1,0,1}×[1,∞].\nProof.(a) We first prove the result for s∈ {−1,0}. LetIh:C(Ω)→V3\nhdenote the\nnodal interpolation operator and define mh:=Ihm∈V3\nh.\nThere holds\n/ba∇dblΠh(mhψh)/ba∇dblLp/greaterorequalslant/ba∇dblmhψh/ba∇dblLp−/ba∇dbl(I−Πh)(mhψh)/ba∇dblLp.\nMoreover, stability of ΠhinLp(Ω)3, for 1/lessorequalslantp/lessorequalslant∞, see [20], implies the estimate\n/ba∇dbl(I−Πh)(mhψh)/ba∇dblLp/lessorequalslant(1+C) inf\nvh∈V3\nh/ba∇dblmhψh−vh/ba∇dblLp.\nIn turn, this implies\n/ba∇dbl(I−Πh)(mhψh)/ba∇dblLp/lessorsimilar/ba∇dbl(I−Ih)(mhψh)/ba∇dblLp\n=/parenleftig/summationdisplay\nT∈Th/ba∇dbl(I−Ih)(mhψh)/ba∇dblp\nLp(T)3/parenrightig1/p\n.\nFor each element, the approximation properties of Ihshow\n/ba∇dbl(I−Ih)(mhψh)/ba∇dblLp(T)3/lessorsimilarhr+1/ba∇dbl∇r+1(mhψh)/ba∇dblLp(T)3\n/lessorequalslanthr+1/summationdisplay\ni+j=r+1/ba∇dbl∇min{i,r}mh/ba∇dblL∞(T)3/ba∇dbl∇min{j,r}ψh/ba∇dblLp(T)3.\nThus, multiple inverse estimates yield\n/ba∇dbl(I−Ih)(mhψh)/ba∇dblLp(T)3/lessorsimilarh/ba∇dblmh/ba∇dblW1,∞/ba∇dblψh/ba∇dblLp(T)3.\nMoreover, we have\n/ba∇dblmhψh/ba∇dblLp/greaterorequalslant/ba∇dblmψh/ba∇dblLp−/ba∇dbl(m−mh)ψh/ba∇dblLp/greaterorequalslant1\n2/ba∇dblψh/ba∇dblLp\nprovided that /ba∇dblm−mh/ba∇dblL∞/lessorequalslant1\n2, which in view of\n/ba∇dblm−mh/ba∇dblL∞=/ba∇dbl(I−Ih)m/ba∇dblL∞/lessorsimilarh/ba∇dbl∇m/ba∇dblL∞\nis satisfied for h/lessorequalslanthRwith a sufficiently small hR>0 that depends only on R.\nAltogether, this shows\n/ba∇dblΠh(mhψh)/ba∇dblLp/greaterorsimilar/ba∇dblψh/ba∇dblLpHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 17\nforh/lessorequalslanthR. Similarly we estimate\n/ba∇dblΠh((m−mh)ψh)/ba∇dblLp/lessorsimilar/ba∇dblm−mh/ba∇dblL∞/ba∇dblψh/ba∇dblLp/lessorsimilarh/ba∇dbl∇m/ba∇dblL∞/ba∇dblψh/ba∇dblLp.\nAltogether, we obtain\n/ba∇dblΠh(mψh)/ba∇dblLp/greaterorsimilar/ba∇dblΠh(mhψh)/ba∇dblLp−/ba∇dblΠh((mh−m)ψh)/ba∇dblLp/greaterorsimilar/ba∇dblψh/ba∇dblLp\nforh/lessorequalslanthR. This concludes the proof for s= 0. Finally, for s=−1 we note that by\nusing the result for s= 0 and an inverse inequality,\n/ba∇dbl(I−Πh)(mψh)/ba∇dblW−1,p/lessorsimilarh/ba∇dblψh/ba∇dblLp\n/lessorsimilarh/ba∇dblΠh(mψh)/ba∇dblLp/lessorsimilar/ba∇dblΠh(mψh)/ba∇dblW−1,p.\nSince/ba∇dblmψh/ba∇dblW−1,p/greaterorsimilar/ba∇dblm/ba∇dbl−1\nW1,∞/ba∇dblψh/ba∇dblW−1,p, this concludes the proof for s∈ {−1,0}.\n(b) It remains to prove the result for s= 1. Note that the result follows from\nduality if we show\n(5.2) /ba∇dblΠh(m·wh)/ba∇dblW−1,q/greaterorsimilar/ba∇dblwh/ba∇dblW−1,q\nfor allwh∈Nh(m). To see this, note that (5.2) implies\n/ba∇dblΠh(mψh)/ba∇dblW1,p/greaterorequalslantsup\nwh∈Nh(m)(ψh,Πh(m·wh))\n/ba∇dblwh/ba∇dblW−1,q\n/greaterorsimilarsup\nwh∈Nh(m)(ψh,Πh(m·wh))\n/ba∇dblΠh(m·wh)/ba∇dblW−1,q= sup\nωh∈Vh(ψh,ωh)\n/ba∇dblωh/ba∇dblW−1,q≃ /ba∇dblψh/ba∇dblW1,p,\nwhereweusedinthesecondtolastequalitythatpart(a)for s= 0alreadyshowsthat\ndim(Nh(m)) = dim(Vh) and since (5.2) implies that the map Nh(m)→Vh,wh/ma√sto→\nΠh(m·wh) is injective, it is already bijective. It remains to prove (5.2). To tha t\nend, we first show for wh=Πh(mωh)∈Nh(m) for someωh∈Vh, using the reverse\ntriangle inequality, that\n/ba∇dblm·wh/ba∇dblW−1,q/greaterorequalslant/ba∇dblωh/ba∇dblW−1,q−/ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q\n/greaterorsimilar/ba∇dblm/ba∇dbl−1\nW1,∞/ba∇dblwh/ba∇dblW−1,q−/ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q.\nWithmh:=Ih(m)∈V3\nh, the last term satisfies\n/ba∇dblm·(I−Πh)(mωh)/ba∇dblW−1,q/lessorsimilarh/ba∇dblm/ba∇dblW1,∞/ba∇dbl(I−Πh)(mωh)/ba∇dblLq\n/lessorsimilarh/ba∇dblm/ba∇dblW1,∞(/ba∇dblm−mh/ba∇dblL∞/ba∇dblωh/ba∇dblLq+h/ba∇dblmh/ba∇dblW1,∞/ba∇dblωh/ba∇dblLq),\nwhere we used the same arguments as in the proof of part (a) to ge t the estimate\n/ba∇dbl(I−Πh)(mhωh)/ba∇dblLq/lessorsimilarh/ba∇dblmh/ba∇dblW1,∞/ba∇dblωh/ba∇dblLq. The fact /ba∇dblmh/ba∇dblW1,∞/lessorsimilar/ba∇dblm/ba∇dblW1,∞, the\napproximation property /ba∇dblm−mh/ba∇dblL∞/lessorsimilarh/ba∇dblm/ba∇dblW1,∞, and an inverse inequality con-\nclude\n(5.3) /ba∇dblm·wh/ba∇dblW−1,q/greaterorsimilar/ba∇dblwh/ba∇dblW−1,q\nwith (hidden) constants depending only on /ba∇dblm/ba∇dblW1,∞and shape regularity of the\nmesh.18 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nToprove(5.2), itremainstoboundtheleft-handsideaboveby /ba∇dblΠh(m·wh)/ba∇dblW−1,q.\nTo that end, we note\n/ba∇dbl(I−Πh)(m·wh)/ba∇dblW−1,q/lessorsimilarh/ba∇dblwh/ba∇dblLq=hsup\nv∈Lp(wh,v)\n/ba∇dblv/ba∇dblLp\n/lessorsimilarhsup\nv∈Nh(m)(wh,v)\n/ba∇dblv/ba∇dblLp=hsup\nv∈Vh(Πh(m·wh),v)\n/ba∇dblΠh(mv)/ba∇dblLp/lessorsimilarh/ba∇dblΠh(m·wh)/ba∇dblLq,\nwhere we used part (a) for s= 0 for the last inequality. An inverse inequality\nand the combination with (5.3) imply (5.2) for h >0 sufficiently small in terms of\n/ba∇dblm/ba∇dbl−1\nW1,∞. This concludes the proof. /square\nLemma 5.6. Define the matrix M∈RN×N, whereNdenotes the dimension of Vh,\nbyMij:=h−3(Πh(mφj),Πh(mφi)). Under the assumptions of Lemma 5.5, there\nexistsC >0such that for h/lessorequalslanthR,\n/ba∇dblM/ba∇dblp+/ba∇dblM−1/ba∇dblp/lessorequalslantCfor1/lessorequalslantp/lessorequalslant∞,\nwhereCdepends only on the shape regularity.\nProof.Lemma 5.5 shows for x∈RN\n(5.4) Mx·x=h−3/ba∇dblΠh(mN/summationdisplay\ni=1xiφi)/ba∇dbl2\nL2/greaterorsimilarh−3/ba∇dblN/summationdisplay\ni=1xiφi/ba∇dblL2≃ |x|2,\nwhere|·|denotes the Euclidean norm on RN. Letd(i,j) := dist(zi,zj)h−3denote\nthe metric which (approximately) measures the number of elements between the\nsupports of φiandφj, corresponding to the nodes ziandzj, and letBd(z) denote\nthe corresponding ball. In the following, we use a localization propert y of theL2-\nprojection, i.e., there exist a,b>0 such that for all ℓ∈N,\n(5.5) /ba∇dblΠh(mφi)/ba∇dblL2(Ω\\Bℓ(zi))3/lessorequalslantae−bℓ/ba∇dblmφi/ba∇dblL2.\nThe proof of this bound is essentially contained in the proof of [9, Le mma 3.1].\nSince we use the very same arguments below, we briefly recall the st rategy: First,\none observes that the mass matrix /tildewiderM∈RN×Nwith entries /tildewiderMij:=h−3(φj,φi) is\nbandedinthesense that d(i,j)/greaterorsimilar1implies /tildewiderMij= 0, anditsatisfies /tildewiderMx·x/greaterorsimilar|x|2. As\nshown below, this implies that the inverse matrix /tildewiderM−1satisfies|(/tildewiderM−1)ij|/lessorsimilare−bd(i,j)\nfor someb >0 independent of h >0. Note that each entry of the vector field\nΠh(mφi)∈V3\nhcan be represented by/summationtextN\nj=1xk,jφj,k= 1,2,3,and is computed by\nsolving/tildewiderMxk=gk∈RNwithm= (m1,m2,m3)Tandgk,j:= (mkφi,φj). Hence, the\nexponential decay of /tildewiderM−1directly implies (5.5).\nFrom the decay property (5.5), we immediately obtain\n|Mij|/lessorequalslant/tildewideae−/tildewidebd(i,j)\nfor all 1/lessorequalslanti,j/lessorequalslantNand some /tildewidea,/tildewideb>0. This already proves /ba∇dblM/ba∇dblp/lessorequalslantC. We follow\nthe arguments from [28] to show that also M−1decays exponentially. To that end,HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 19\nnote that (5.4) implies the existence of c >0 such that /ba∇dblI−cM/ba∇dbl2=:q <1 and\nhence\n(5.6) M−1=c(I−(I−cM))−1=c∞/summationdisplay\nk=0(I−cM)k.\nClearly,I−cMinherits the decay properties from Mand therefore\n|((I−cM)k+1)ij|/lessorequalslant/tildewideak+1N/summationdisplay\nr1,...,rk=1e−/tildewideb(d(i,r1)+···+d(rk,j))\n/lessorequalslant/tildewideak+1/parenleftig\nmax\ns=1,...,NN/summationdisplay\nr=1e−/tildewidebd(s,r)/2/parenrightigk\ne−/tildewidebd(i,j)/2.\nThe value of max s=1,...,N/summationtextN\nr=1e−/tildewidebd(s,r)/2depends only on the shape regularity of the\ntriangulation and on /tildewideb, but is independent of h(it just depends on the number of\nelements contained in an annulus of thickness ≈h). This implies the existence of\n/tildewidec/greaterorequalslant1 such that\n|((I−cM)k+1)ij|/lessorequalslantmin{qk+1,/tildewideck+1e−/tildewidebd(i,j)/2}.\nThus, for /tildewideck+1/lessorequalslante/tildewidebd(i,j)/4, we have |((I−cM)k+1)ij|/lessorequalslante−/tildewidebd(i,j)/4, whereas for /tildewideck+1>\ne/tildewidebd(i,j)/4, we have |((I−cM)k+1)ij|/lessorequalslantqk+1<q/tildewidebd(i,j)/(4log(/tildewidec)). Altogether, we find some\n/tildewideb>0 (we reuse the symbol), independent of hsuch that\n|((I−cM)k+1)ij|/lessorequalslantq(k+1)/2|((I−cM)k+1)ij|1/2/lessorsimilarq(k+1)/2e−/tildewidebd(i,j).\nPlugging this into (5.6), we obtain\n|(M−1)ij|/lessorsimilar∞/summationdisplay\nk=0q(k+1)/2e−/tildewidebd(i,j)/lessorsimilare−/tildewidebd(i,j).\nThis yields the stated result. /square\nWe are now in a position to prove Lemma 5.3.\nProof of Lemma 5.3. (a) We first consider the case s= 0. In view of (5.1), we write\n(I−Ph(m))vh∈Nh(m) as\n(I−Ph(m))vh=h−3/2N/summationdisplay\ni=1xiΠh(mφi)\nfor some coefficient vector x∈RNand letbi:=h−3/2(vh,mφi) fori= 1,...,N.\nThen, there holds Mx=bwith the matrix Mfrom Lemma 5.6. This lemma and\ntheLp-stability of the L2-orthogonal projection Π h[20] imply that for p∈[1,∞],\n/ba∇dbl(I−Ph(m))vh/ba∇dblLp=/ba∇dblΠhh−3/2N/summationdisplay\ni=1ximφi/ba∇dblLp/lessorsimilar/ba∇dblh−3/2N/summationdisplay\ni=1ximφi/ba∇dblLp\n/lessorsimilarh−3/2/parenleftigN/summationdisplay\ni=1h3|xi|p/parenrightig1/p\n=h3/p−3/2|x|p=h3/p−3/2|M−1b|p/lessorsimilarh3/p−3/2|b|p.20 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nWith|bi|/lessorequalslanth−3/2/ba∇dblvh/ba∇dblLp(supp(φi))3h3(1−1/p)=/ba∇dblvh/ba∇dblLp(supp(φi))3h3/2−3/p, this shows\n/ba∇dblPh(m)vh/ba∇dblLp/lessorsimilar/ba∇dblvh/ba∇dblLp.\n(b) We now turn to the cases s=±1. Define the operator\n/tildewideP⊥\nh(m)vh:=Πh(mΠh(m·vh))\nandnotethat /tildewideP⊥\nh(m)vh∈Nh(m)aswellasker /tildewideP⊥\nh(m) =Th(m)(duetoLemma5.5).\nHowever, /tildewideP⊥\nh(m) is no projection. We observe for vh=Πh(mψh)∈Nh(m) that\n/ba∇dbl(I−/tildewideP⊥\nh(m))vh/ba∇dblW−1,p=/ba∇dblΠhmψh−Πh(mΠh(m·Πh(mψh)))/ba∇dblW−1,p\n/lessorsimilar/ba∇dblm/ba∇dblW1,∞/ba∇dblψh−m·Πh(mψh)/ba∇dblW−1,p\n=/ba∇dblm/ba∇dbl2\nW1,∞/ba∇dbl(I−Πh)(mψh)/ba∇dblW−1,p\n/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞h/ba∇dblψh/ba∇dblLp.\nWith Lemma 5.5 we conclude\n/ba∇dbl(I−/tildewideP⊥\nh(m))vh/ba∇dblW−1,p/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞h/ba∇dblvh/ba∇dblLp.\nSince/tildewideP⊥\nh(m)Ph(m) = 0 by definition of Th(m), we obtain with part (a) and an\ninverse inequality that for all vh∈V3\nh,\n/ba∇dbl(I−Ph(m)−/tildewideP⊥\nh(m))vh/ba∇dblW−1,p=/ba∇dbl(I−/tildewideP⊥\nh(m))(I−Ph(m))vh/ba∇dblW−1,p\n/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞h/ba∇dbl(I−Ph(m))vh/ba∇dblLp\n/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞h/ba∇dblvh/ba∇dblLp\n/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞/ba∇dblvh/ba∇dblW−1,p.\nTheW−1,p(Ω)-stability of Πhimplies/ba∇dbl/tildewideP⊥\nh(m)vh/ba∇dblW−1,p/lessorsimilar/ba∇dblm/ba∇dbl2\nW1,∞/ba∇dblvh/ba∇dblW−1,pand\nthe triangle inequality concludes the proof for s=−1. The case s= 1 follows by\nduality. /square\nProof of Lemma 5.2. (a) (s= 0) The projection vh:=Ph(m)vis given by the\nequation\n(vh,ϕh) = (v,ϕh)∀ϕh∈Th(m),\nwhich in view of the definition of Th(m) is equivalent to the solution of the saddle\npoint problem (with the Lagrange multiplier λh∈Vh)\n(vh,wh)+(m·wh,λh) = (v,wh)∀wh∈V3\nh,\n(m·vh,µh) = 0 ∀µh∈Vh.\nBy the first equation, we also obtain the identity Πh(mλh) = (I−Ph(m))vh, which\nwill be used below. Furthermore, /tildewidevh:=Ph(/tildewiderm)vis given by the same system with\n/tildewidermin place ofm, yielding a corresponding Lagrange multiplier /tildewideλh. Hence, the\ndifferenceseh:=vh−/tildewidevhandδh:=λh−/tildewideλhsatisfy\n(eh,wh)+(m·wh,δh) =−(wh,(m−/tildewiderm)/tildewideλh)∀wh∈V3\nh,\n(m·eh,µh) = −((m−/tildewiderm)·/tildewidevh,µh)∀µh∈Vh.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 21\nThe classical results on saddle-point problems (see [13, Proposition 2.1]) require two\ninf-sup conditions to be satisfied. First,\ninf\nqh∈Vhsup\nvh∈V3\nh(m·vh,qh)\n/ba∇dblvh/ba∇dblHs/ba∇dblqh/ba∇dblH−s>0\nholds uniformly in hdue to Lemma 5.5. Second,\ninf\nwh∈Th(m)sup\nvh∈Th(m)(vh,wh)\n/ba∇dblvh/ba∇dblHs/ba∇dblwh/ba∇dblH−s>0\nholdsuniformlyin hduetothestabilityestimatesfromLemma5.3(notingthat vh=\nPh(m)vhandwh=Ph(m)whforvh,wh∈Th(m)). For the above saddle-point\nproblems, these bounds for s= 0 give us an L2bound foreh=Ph(m)v−Ph(/tildewiderm)v:\nFrom [13] we obtain\n/ba∇dbl/tildewidevh/ba∇dblL2+/ba∇dbl/tildewideλh/ba∇dblL2/lessorsimilar/ba∇dblv/ba∇dblL2\nand\n/ba∇dbleh/ba∇dblL2+/ba∇dblδh/ba∇dblL2/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblL2+/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblL2.\nWith the stability from Lemma 5.3 and Lemma 5.5, we also obtain\n/ba∇dbl/tildewidevh/ba∇dblL∞+/ba∇dbl/tildewideλh/ba∇dblL∞/lessorsimilar/ba∇dblPh(/tildewiderm)v/ba∇dblL∞+/ba∇dbl(I−Ph(/tildewiderm))v/ba∇dblL∞/lessorsimilar/ba∇dblv/ba∇dblL∞.\nAltogether, this implies\n/ba∇dbleh/ba∇dblL2+/ba∇dblδh/ba∇dblL2/lessorsimilar/ba∇dblm−/tildewiderm/ba∇dblLp/ba∇dblv/ba∇dblLq\nfor (p,q)∈ {(2,∞),(∞,2)}.\n(b) (s= 1) For the H1(Ω)-estimate, we introduce the Riesz mapping Jhbetween\nVh⊂H1(Ω) and its dual Vh⊂H1(Ω)′, i.e., the isometry defined by\n(vh,Jhψh)H1=/a\\}b∇acketle{tvh,ψh/a\\}b∇acket∇i}ht ∀vh∈Vh, ψh∈Vh.\nByJh:=I⊗Jhwe denote the corresponding vector-valued mapping on V3\nh. We\nconsider the bilinear form on V3\nh×V3\nhdefined by\nah(vh,wh) =/a\\}b∇acketle{tvh,J−1\nhwh/a\\}b∇acket∇i}ht,vh,wh∈V3\nh,\nand reformulate the saddle-point problem for ( vh,λh)∈V3\nh×Vh⊂H1(Ω)3×H1(Ω)′\nas\nah(vh,wh)+/a\\}b∇acketle{tm·J−1\nhwh,λh/a\\}b∇acket∇i}ht=a(v,wh)∀wh∈V3\nh,\n/a\\}b∇acketle{tm·vh,J−1\nhµh/a\\}b∇acket∇i}ht = 0 ∀µh∈Vh.\nAs in the case s= 0 (algebraically it is the same system), we have vh=Ph(m)v\nandΠh(mλh) = (I−Ph(m))v. The system for eh=vh−/tildewidevhandδh=λh−/tildewideλh\nreads\nah(eh,wh)+/a\\}b∇acketle{tm·J−1\nhwh,δh/a\\}b∇acket∇i}ht=−/a\\}b∇acketle{t(m−/tildewiderm)·J−1\nhwh,/tildewideλh/a\\}b∇acket∇i}ht ∀wh∈V3\nh,\n/a\\}b∇acketle{tm·eh,J−1\nhµh/a\\}b∇acket∇i}ht =−/a\\}b∇acketle{t(m−/tildewiderm)·/tildewidevh,J−1\nhµh/a\\}b∇acket∇i}ht ∀µh∈Vh.\nThe above inf-sup bounds for s= 1 ands=−1 are precisely the inf-sup condi-\ntions that need to be satisfied for these generalized saddle-point p roblems (see [15,\nTheorem 2.1]), whose right-hand sides are bounded by\n|ah(v,wh)|/lessorequalslant/ba∇dblv/ba∇dblH1/ba∇dblJ−1\nhwh/ba∇dblH−1≃ /ba∇dblv/ba∇dblH1/ba∇dblwh/ba∇dblH122 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nand\n|/a\\}b∇acketle{t(m−/tildewiderm)·J−1\nhwh,/tildewideλh/a\\}b∇acket∇i}ht|/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblH1/ba∇dblwh/ba∇dblH1,\n|/a\\}b∇acketle{t(m−/tildewiderm)·/tildewidevh,J−1\nhµh/a\\}b∇acket∇i}ht|/lessorequalslant/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblH1/ba∇dblµh/ba∇dblH1.\nAs in the case s= 0, we obtain from Lemma 5.3 and Lemma 5.5 that\n/ba∇dbl/tildewidevh/ba∇dblW1,∞+/ba∇dbl/tildewideλh/ba∇dblW1,∞/lessorsimilar/ba∇dblPh(/tildewiderm)v/ba∇dblW1,∞+/ba∇dbl(I−Ph(/tildewiderm))v/ba∇dblW1,∞\n/lessorsimilar/ba∇dblv/ba∇dblW1,∞.\nHence, we obtain from [15, Theorem 2.1], for ( p,q)∈ {(2,∞),(∞,2)},\n/ba∇dbleh/ba∇dblH1/lessorsimilar/ba∇dbl(m−/tildewiderm)/tildewideλh/ba∇dblH1+/ba∇dbl(m−/tildewiderm)·/tildewidevh/ba∇dblH1\n/lessorsimilar1/summationdisplay\ns′=0/parenleftig\n/ba∇dblm−/tildewiderm/ba∇dblH1/ba∇dbl/tildewideλh/ba∇dblW1−s′,q+/ba∇dblm−/tildewiderm/ba∇dblWs′,p/ba∇dbl/tildewidevh/ba∇dblW1−s′,q/parenrightig\n/lessorsimilar1/summationdisplay\ns′=0/ba∇dblm−/tildewiderm/ba∇dblWs′,p/ba∇dblv/ba∇dblW1−s′,q.\nThis implies the H1(Ω)3estimate and hence concludes the proof. /square\nProof of Lemma 5.1. SincePh(m)vis the Galerkin approximation of the saddle\npoint problem for P(m)v(as in the previous proof), the C´ ea lemma for saddle-\npoint problems (see [13, Theorem 2.1]) shows in L2\n/ba∇dbl(Ph(m)−P(m))v/ba∇dblL2\n/lessorsimilarinf\n(wh,µh)∈V3\nh×Vh/parenleftig\n/ba∇dblP(m)v−wh/ba∇dblL2+/ba∇dblm·v−µh/ba∇dblL2/parenrightig\n/lessorsimilarhr+1/ba∇dblm/ba∇dblWr+1,∞/ba∇dblv/ba∇dblHr+1\nand similarly in H1, using [15, Theorem 2.1],\n/ba∇dbl(Ph(m)−P(m))v/ba∇dblH1\n/lessorsimilarinf\n(wh,µh)∈V3\nh×Vh/parenleftig\n/ba∇dblP(m)v−wh/ba∇dblH1+/ba∇dblm·v−µh/ba∇dblH1/parenrightig\n/lessorsimilarhr/ba∇dblm/ba∇dblWr+1,∞/ba∇dblv/ba∇dblHr+1.\nThis concludes the proof. /square\n6.Consistency error and error equation\nTo study the consistency errors, we find it instructive to separat e the issues of\nconsistency for the time and space discretizations. Therefore, w e first show defect\nestimates for the semidiscretization in time, and then turn to the fu ll discretization.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 23\n6.1.Consistency error of the semi-discretization in time. The order of both\nthe fully implicit k-step BDF method, described by the coefficients δ0,...,δ kand 1,\nandtheexplicit k-step BDFmethod, thatis themethoddescribed by thecoefficients\nδ0,...,δ kandγ0,...,γ k−1,isk,i.e.,\n(6.1)k/summationdisplay\ni=0(k−i)ℓδi=ℓkℓ−1=ℓk−1/summationdisplay\ni=0(k−i−1)ℓ−1γi, ℓ= 0,1,...,k.\nWe first rewrite the linearly implicit k-step BDF method (2.3) in strong form,\n(6.2) α˙mn+/hatwidermn×˙mn=P(/hatwidermn)(∆mn+Hn),\nwith Neumann boundary conditions.\nThe consistency error dnof the linearly implicit k-step BDF method (6.2) for the\nsolutionmis the defect by which the exact solution misses satisfying (6.2), and is\ngiven by\n(6.3) dn=α˙mn\n⋆+/hatwidermn\n⋆×˙mn\n⋆−P(/hatwidermn\n⋆)(∆mn\n⋆+Hn)\nforn=k,...,N, where we use the notation mn\n⋆=m(tn) and\n(6.4)/hatwidermn\n⋆=k−1/summationdisplay\nj=0γjmn−j−1\n⋆/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\n⋆/vextendsingle/vextendsingle/vextendsingle,\n˙mn\n⋆=P(/hatwidermn\n⋆)1\nτk/summationdisplay\nj=0δjmn−j\n⋆∈T(/hatwidermn\n⋆).\nNotethat thedefinition of ˙mn\n⋆contains theprojection P(/hatwidermn\n⋆), while ˙mnwasdefined\nwithout a projection (see the first formula in (2.2)), since ˙mn=P(/hatwidermn)˙mnis\nautomatically satisfied due to the constraint in (2.3).\nThe consistency error is bounded as follows.\nLemma 6.1. If the solution of the LLG equation (1.4)has the regularity\nm∈Ck+1([0,¯t],L2(Ω)3)∩C1([0,¯t],L∞(Ω)3)and∆m+H∈C([0,¯t],L∞(Ω)3),\nthen the consistency error (6.3)is bounded by\n/ba∇dbldn/ba∇dblL2(Ω)3/lessorequalslantCτk\nforn=k,...,N.\nProof.We begin by rewriting the equation for the defect as\n(6.5)dn=α˙mn\n⋆+/hatwidermn\n⋆×˙mn\n⋆−P(mn\n⋆)(∆mn\n⋆+Hn)\n−/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn).\nIn view of (1.4), we have\nP(mn\n⋆)(∆mn\n⋆+Hn) =α∂tm(tn)+mn\n⋆×∂tm(tn),\nand can rewrite (6.5) as\ndn=α/parenleftbig\n˙mn\n⋆−∂tm(tn)/parenrightbig\n+/parenleftbig\n/hatwidermn\n⋆×˙mn\n⋆−mn\n⋆×∂tm(tn)/parenrightbig\n−/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn),24 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\ni.e.,\ndn=α/parenleftbig˙mn\n⋆−∂tm(tn)/parenrightbig\n+(/hatwidermn\n⋆−mn\n⋆)×˙mn\n��+mn\n⋆×/parenleftbig˙mn\n⋆−∂tm(tn)/parenrightbig\n−/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn).\nTherefore,\n(6.6)dn=α˙dn+/hatwidedn×˙mn\n⋆+mn\n⋆×˙dn−/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn),\nwith\n(6.7) ˙dn:=˙mn\n⋆−∂tm(tn),/hatwidedn:=/hatwidermn\n⋆−mn\n⋆.\nNow, in view of the first estimate in Lemma 4.1, we have\n/ba∇dbl/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn)/ba∇dblL2/lessorequalslantC/ba∇dbl/hatwidermn\n⋆−mn\n⋆/ba∇dblL2,\ni.e.,\n(6.8) /ba∇dbl/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\n(∆mn\n⋆+Hn)/ba∇dblL2/lessorequalslantC/ba∇dbl/hatwidedn/ba∇dblL2.\nTherefore, it suffices to estimate ˙dnand/hatwidedn.\nTo estimate /hatwidedn, we shall proceed in two steps. First we shall estimate the extrap-\nolation error\n(6.9)k−1/summationdisplay\nj=0γjmn−j−1\n⋆−mn\n⋆\nand then /hatwidedn.\nBy Taylor expanding about tn−k,the leading terms of order up to k−1 cancel,\ndue to the second equality in (6.1), and we obtain\n(6.10)k−1/summationdisplay\ni=0γimn−i−1\n⋆−mn\n⋆=1\n(k−1)!/bracketleftiggk−1/summationdisplay\nj=0γj/integraldisplaytn−j−1\ntn−k(tn−j−1−s)k−1m(k)(s)ds\n−/integraldisplaytn\ntn−k(tn−s)k−1m(k)(s)ds/bracketrightigg\n,\nwithm(ℓ):=∂ℓm\n∂tℓ,whence\n(6.11)/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\ni=0γimn−i−1\n⋆−mn\n⋆/vextenddouble/vextenddouble/vextenddouble\nL2/lessorequalslantCτk.\nNow, for a normalized vector aand a non-zero vector b,we have\na−b\n|b|= (a−b)+1\n|b|(|b|−|a|)b,\nwhence/vextendsingle/vextendsinglea−b\n|b|/vextendsingle/vextendsingle/lessorequalslant2|a−b|.\nTherefore, (6.11) yields\n(6.12) /ba∇dbl/hatwidedn/ba∇dblL2/lessorequalslantCτk.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 25\nTo bound ˙dn,we use the fact that P(m(tn))∂tm(tn) =∂tm(tn)∈T(m(tn)), so\nthat we have\n˙dn=P(/hatwidermn\n⋆)1\nτk/summationdisplay\nj=0δjm(tn−j)−∂tm(tn)\n=P(/hatwidermn\n⋆)/parenleftig1\nτk/summationdisplay\nj=0δjm(tn−j)−∂tm(tn)/parenrightig\n+/parenleftbig\nP(/hatwidermn\n⋆)−P(m(tn))/parenrightbig\n∂tm(tn).\nBy Lemma 4.1 and (6.12), we have for the last term\n/ba∇dbl/parenleftbig\nP(/hatwidermn\n⋆)−P(m(tn))/parenrightbig\n∂tm(tn)/ba∇dblL2/lessorequalslantCτk.\nBy Taylor expanding the first term about tn−k,we see that, due to the order condi-\ntions of the implicit BDF method, i.e., the first equality in (6.1), the leadin g terms\nof order up to k−1 cancel, and we obtain\n(6.13)1\nτk/summationdisplay\nj=0δjm(tn−j)−∂tm(tn) =1\nk!/bracketleftigg\n1\nτk/summationdisplay\nj=0δj/integraldisplaytn−j\ntn−k(tn−j−s)km(k+1)(s)ds\n−k/integraldisplaytn\ntn−k(tn−s)k−1m(k+1)(s)ds/bracketrightigg\n,\nwhence\n(6.14) /ba∇dbl˙dn/ba∇dblL2/lessorequalslantCτk,\nprovided the solution mis sufficiently regular. Now, (6.6), (6.8), (6.14), and (6.12)\nyield\n(6.15) /ba∇dbldn/ba∇dblL2/lessorequalslantCτk.\nThis isthe desired consistency estimate, which isvalidfor BDFmethod s of arbitrary\norderk. /square\n6.2.Consistency error of the full discretization. Wedefine theRitzprojection\nRh:H1(Ω)→Vhcorresponding to the Poisson–Neumann problem via/parenleftbig\n∇Rhϕ,∇ψ/parenrightbig\n+/parenleftbig\nRhϕ,1/parenrightbig/parenleftbig\nψ,1/parenrightbig\n=/parenleftbig\n∇ϕ,∇ψ/parenrightbig\n+/parenleftbig\nϕ,1/parenrightbig/parenleftbig\nψ,1/parenrightbig\nfor allψ∈Vh, and we denote Rh=I⊗Rh:H1(Ω)3→V3\nh. We denote again\ntheL2-orthogonal projections onto the finite element space by Πh:L2(Ω)→Vh\nandΠh=I⊗Πh:L2(Ω)3→V3\nh. As in the previous section, we write Ph(m) for\ntheL2-orthogonal projection onto the discrete tangent space at m. We insert the\nfollowing quantities, which are related to the exact solution,\nmn\n⋆,h=Rhm(tn),\n/hatwidermn\n⋆,h=k−1/summationdisplay\nj=0γjmn−j−1\n⋆,h/slashig/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\n⋆,h/vextendsingle/vextendsingle/vextendsingle, (6.16)\n˙mn\n⋆,h=Ph(/hatwidermn\n⋆,h)1\nτk/summationdisplay\nj=0δjmn−j\n⋆,h∈Th(/hatwidermn\n⋆,h),26 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nintothelinearlyimplicit k-stepBDFmethod(2.6)andobtainadefect dn\nh∈Th(/hatwidermn\n⋆,h)\nfrom\n(6.17)α/parenleftbig\n˙mn\n⋆,h,ϕh/parenrightbig\n+/parenleftbig\n/hatwidermn\n⋆,h×˙mn\n⋆,h,ϕh/parenrightbig\n=−/parenleftbig\n∇mn\n⋆,h,∇ϕh/parenrightbig\n+/parenleftbig\nHn,ϕh/parenrightbig\n+/parenleftbig\ndn\nh,ϕh/parenrightbig\nfor allϕh∈Th(/hatwidermn\n⋆,h). By definition, there holds ( Rhϕ,1) = (ϕ,1) (this can be seen\nby testing with ψ= 1) and hence/parenleftbig\n∇mn\n⋆,h,∇ϕ/parenrightbig\n=/parenleftbig\n∇m(tn),∇ϕ/parenrightbig\n=−/parenleftbig\n∆m(tn),ϕ/parenrightbig\n.\nThus, we obtain the consistency error for the full discretization b y\n(6.18)dn\nh=Ph(/hatwidermn\n⋆,h)Dn\nhwithDn\nh=α˙mn\n⋆,h+/hatwidermn\n⋆,h×˙mn\n⋆,h−∆m(tn)−H(tn)\nforn=k,...,N. The consistency error is bounded as follows.\nLemma 6.2. If the solution of the LLG equation (1.4)has the regularity\nm∈Ck+1([0,¯t],L2(Ω)3)∩C1([0,¯t],Wr+1,∞(Ω)3)and\n∆m+H∈C([0,¯t],Wr+1,∞(Ω)3),\nthen the consistency error (6.18)is bounded by\n/ba∇dbldn\nh/ba∇dblL2(Ω)3/lessorequalslantC(τk+hr)\nfornwithkτ/lessorequalslantnτ/lessorequalslant¯t.\nProof.We begin by defining\nDn:=α∂tm(tn)+m(tn)×∂tm(tn)−∆m(tn)−H(tn)\nand note that P(mn\n⋆)Dn= 0. Here we denote again mn\n⋆=m(tn) and in the\nfollowing we use also the notations ˙mn\n⋆and/hatwidermn\n⋆as defined in (6.4). With this, we\nrewrite the equation for the defect as\ndn\nh=Ph(/hatwidermn\n⋆,h)Dn\nh−P(mn\n⋆)Dn\n=Ph(/hatwidermn\n⋆,h)/parenleftbig\nDn\nh−Dn/parenrightbig\n+/parenleftbig\nPh(/hatwidermn\n⋆,h)−Ph(/hatwidermn\n⋆)/parenrightbig\nDn\n+/parenleftbig\nPh(/hatwidermn\n⋆)−P(/hatwidermn\n⋆)/parenrightbig\nDn+/parenleftbig\nP(/hatwidermn\n⋆)−P(mn\n⋆)/parenrightbig\nDn\n≡I+II+III+IV.\nFor the term IVwe have by Lemma 4.1\n/ba∇dblIV/ba∇dblL2/lessorequalslant2/ba∇dbl/hatwidermn\n⋆−mn\n⋆/ba∇dblL2/ba∇dblDn/ba∇dblL∞,\nwhere the last term /hatwidermn\n⋆−mn\n⋆has been bounded in the L2norm byCτkin the proof\nof Lemma 6.1.\nThe term IIIis estimated using the first bound from Lemma 5.1, under our\nregularity assumptions, as\n/ba∇dblIII/ba∇dblL2/lessorequalslantChr.\nFor the bound on IIwe use Lemma 5.2 ( i) (withp= 2 andq=∞), to obtain\n/ba∇dblII/ba∇dblL2/lessorequalslantCR/ba∇dbl/hatwidermn\n⋆,h−/hatwidermn\n⋆/ba∇dblL2/ba∇dblDn/ba∇dblL∞,\nwhere, using (7.11), we obtain\n/ba∇dbl/hatwidermn\n⋆,h−/hatwidermn\n⋆/ba∇dblL2/lessorequalslant2/ba∇dbl/summationtextk\ni=1γi(Rh−I)mn−i\n∗/ba∇dblL2\nmin/vextendsingle/vextendsingle/summationtextk\ni=1γimn−i\n∗/vextendsingle/vextendsingle/lessorequalslantChr.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 27\nThe denominator is bounded from below by 1 −Cτk, because |mn\n∗|= 1 and\n|/summationtextk\ni=1γimn−i\n∗−mn\n∗|/lessorequalslantCτk. For the first term we have\n/ba∇dblI/ba∇dblL2/lessorequalslant/ba∇dblDn−Dn\nh/ba∇dblL2\n/lessorequalslantα/ba∇dbl∂tm(tn)−˙mn\n⋆,h/ba∇dblL2+/ba∇dblm(tn)×∂tm(tn)−/hatwidermn\n⋆,h×˙mn\n⋆,h/ba∇dblL2.\nThe terms /ba∇dbl∂tm(tn)−˙mn\n⋆/ba∇dblL2and/ba∇dblmn\n⋆×∂tm(tn)−/hatwidermn\n⋆×˙mn\n⋆/ba∇dblL2can be handled\nas in the proof of Lemma 6.1. Standard error estimates for the Ritz projectionRh\n(we do not exploit the Aubin–Nitsche duality here) imply\n/ba∇dbl(I−Rh)˙mn\n⋆/ba∇dblL2/lessorequalslantchr/ba∇dbl˙mn\n⋆/ba∇dblHr+1.\nTogether this yields, under the stated regularity assumption,\n/ba∇dblI/ba∇dblL2/lessorequalslantC(τk+hr),\nand the result follows. /square\n6.3.Error equation. We recall, from (2.6), the fully discrete problem with the\nlinearly implicit BDF method: find ˙mn\nh∈Th(/hatwidermn\nh) such that for all ϕh∈Th(/hatwidermn\nh),\n(6.19) α(˙mn\nh,ϕh)+(/hatwidermn\nh×˙mn\nh,ϕh)+(∇mn\nh,∇ϕh) = (H(tn),ϕh).\nThen, similarly as we have done in Section 4, we first rewrite (6.17): fo r all\nϕh∈Th(/hatwidermn\nh),\n(6.20) α(˙mn\n⋆,h,ϕh)+(/hatwidermn\n⋆,h×˙mn\n⋆,h,ϕh)+(∇mn\n⋆,h,∇ϕh) = (rn\nh,ϕh)\nwith\n(6.21) rn\nh=−(Ph(/hatwidermn\nh)−Ph(/hatwidermn\n⋆,h))(∆m⋆(tn)+H(tn))+dn\nh.\nThe erroren\nh=mn\nh−mn\n⋆,hsatisfies the error equation that is obtained by sub-\ntracting (6.20) from (6.19). We use the notations\n/hatwideen\nh=/hatwidermn\nh−/hatwidermn\n⋆,h, (6.22)\n˙en\nh=˙mn\nh−˙mn\n⋆,h=1\nτk/summationdisplay\nj=0δjen−j\nh+sn\nh, (6.23)\nwithsn\nh= (I−Ph(/hatwidermn\n⋆,h))1\nτk/summationdisplay\nj=0δjmn−j\n⋆,h.\nWe have the following bound for sn\nh.\nLemma 6.3. Under the regularity assumptions of Lemma 6.2, we have\n(6.24) /ba∇dblsn\nh/ba∇dblH1(Ω)3/lessorequalslantC(τk+hr).\nProof.We use Lemmas 5.1 and 5.3, and the bounds in the proof of Lemma 6.2.\nWe start by subtracting ( I−P(/hatwidermn\n⋆,h))∂tmn\n⋆= 0, and obtain (with ∂τmn\n⋆,h:=\n1\nτ/summationtextk\nj=0δjmn−j\n⋆,h)\nsn\nh= (I−Ph(/hatwidermn\n⋆,h))∂τmn\n⋆,h−(I−P(/hatwidermn\n⋆,h))∂tmn\n⋆\n= (∂τmn\n⋆,h−∂tmn\n⋆)−/parenleftbig\nPh(/hatwidermn\n⋆,h)∂τmn\n⋆,h−P(/hatwidermn\n⋆,h)∂tmn\n⋆/parenrightbig\n.28 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nThe first term above is bounded as O(τk+hr) via the techniques of the consistency\nproofs, Lemma 6.1 and 6.2. For the second term we have\nPh(/hatwidermn\n⋆,h)∂τmn\n⋆,h−P(/hatwidermn\n⋆,h)∂tmn\n⋆\n=Ph(/hatwidermn\n⋆,h)(∂τmn\n⋆,h−∂tmn\n⋆)+/parenleftbig\nPh(/hatwidermn\n⋆,h)−P(/hatwidermn\n⋆,h)/parenrightbig\n∂tmn\n⋆,\nwhere the first term is bounded as O(τk+hr), using Lemma 5.3 and the previous\nestimate, while the second term is bounded as O(hr) by theH1estimate from\nLemma 5.1. Altogether, we obtain the stated H1bound forsn\nh. /square\nWe then have the error equation\n(6.25)α(˙en\nh,ϕh)+(/hatwideen\nh×˙mn\n⋆,h,ϕh)+(/hatwidermn\nh×˙en\nh,ϕh)+(∇en\nh,∇ϕh) =−(rn\nh,ϕh),\nfor allϕh∈Th(/hatwidermn\nh), which is to be taken together with (6.21)–(6.23).\n7.Stability of the full discretization for BDF of orders 1 and 2\nFor the A-stable BDF methods (those of orders 1 and 2) we obtain t he follow-\ning stability estimate, which is analogous to the continuous perturba tion result\nLemma 4.2.\nLemma 7.1 (Stability for orders k= 1,2).Consider the linearly implicit k-step\nBDF discretization (2.6)fork/lessorequalslant2with finite elements of polynomial degree r/greaterorequalslant1.\nLetmn\nhandmn\n⋆,h=Rhm(tn)satisfy equations (2.6)and(6.17), respectively, and\nsuppose that the exact solution m(t)is bounded by (4.3)and/ba∇dblH(t)/ba∇dblL∞/lessorequalslantMfor\n0/lessorequalslantt/lessorequalslant¯t. Then, for sufficiently small h/lessorequalslant¯handτ/lessorequalslant¯τ, the erroren\nh=mn\nh−mn\n⋆,h\nsatisfies the following bound, for kτ/lessorequalslantnτ/lessorequalslant¯t,\n(7.1)/ba∇dblen\nh/ba∇dbl2\nH1(Ω)3/lessorequalslantC/parenleftigk−1/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nH1(Ω)3+τn/summationdisplay\nj=k/ba∇dbldj\nh/ba∇dbl2\nL2(Ω)3+τn/summationdisplay\nj=k/ba∇dblsj\nh/ba∇dbl2\nH1(Ω)3/parenrightig\n,\nwhere the constant Cis independent of h,τandn, but depends on α,R,K,M , and¯t.\nThis estimate holds under the smallness condition that the r ight-hand side is bounded\nbyˆchwith a sufficiently small constant ˆc(note that the right-hand side is of size\nO((τk+hr)2)in the case of a sufficiently regular solution ).\nCombining Lemmas 7.1, 6.2 and 6.3 yields the proof of Theorem 3.1 : These lem-\nmas imply the estimate\n/ba∇dblen\nh/ba∇dblH1(Ω)3/lessorequalslant/tildewideC(τk+hr)\nin the case of a sufficiently regular solution. Since then /ba∇dblRhm(tn)−m(tn)/ba∇dblH1(Ω)3/lessorequalslant\nChrand because of mn\nh−m(tn) =en\nh+(Rhm(tn)−m(tn)), this implies the error\nbound (3.1).\nThe smallness condition imposed in Lemma 7.1 is satisfied under the very mild\nCFL condition, for a sufficiently small ¯ c>0 (independent of h,τandn),\nτk/lessorequalslant¯ch1/2.\nTaken together, this proves Theorem 3.1.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 29\nProof.(a)Preparations. The proof of this lemma transfers the arguments of the\nproof of Lemma 4.2 to the fully discrete situation, using energy estim ates obtained\nby testing with (essentially) the discrete time derivative of the erro r, as presented\nin the Appendix, which is based on Dahlquist’s G-stability theory.\nHowever, testing the error equation (6.25) directly with ˙en\nhis not possible, since\n˙en\nhis not in the tangent space Th(/hatwidermn\nh). Therefore, as in the proof of Lemma 4.2, we\nagain start by showing that the test function ϕh=Ph(/hatwidermn\nh)˙en\nh∈Th(/hatwidermn\nh)∩H1(Ω)3\nis a perturbation of ˙en\nhitself:\nϕh=Ph(/hatwidermn\nh)˙en\nh=Ph(/hatwidermn\nh)˙mn\nh−Ph(/hatwidermn\nh)˙mn\n⋆,h\n=Ph(/hatwidermn\nh)˙mn\nh−Ph(/hatwidermn\n⋆,h)˙mn\n⋆,h+(Ph(/hatwidermn\n⋆,h)−Ph(/hatwidermn\nh))˙mn\n⋆,h.\nHere we note that Ph(/hatwidermn\nh)˙mn\nh=˙mn\nh∈Th(/hatwidermn\nh) by construction of the method(2.6),\nandPh(/hatwidermn\n⋆,h)˙mn\n⋆,h=˙mn\n⋆,h∈Th(/hatwidermn\n⋆,h) by the definition of ˙mn\n⋆,hin (6.4). So we have\nϕh=˙mn\nh−˙mn\n⋆,h−(Ph(/hatwidermn\nh)−P(/hatwidermn\n⋆,h))˙mn\n⋆,h,\nand hence\n(7.2)ϕh=Ph(/hatwidermn\nh)˙en=˙en\nh+qn\nhwithqn\nh=−(Ph(/hatwidermn\nh)−P(/hatwidermn\n⋆,h))˙mn\n⋆,h.\nTheproofnowtransferstheproofofthecontinuousperturbat ionresultLemma4.2\nto the discrete situation with some notable differences, which are em phasized here:\n(i) Instead of using the continuous quantities it uses their spatially d iscrete coun-\nterparts, in particular the discrete projections Ph(/hatwidermn\nh) andPh(/hatwidermn\n⋆,h), defined and\nstudied in Section 5. In view of the definition (2.1) and (6.16) of /hatwidermn\nhand/hatwidermn\n⋆,h, re-\nspectively, thisrequiresthat/summationtextk−1\nj=0γjmn−j−1\nh(x)and/summationtextk−1\nj=0γjmn−j−1\n⋆,h(x)arebounded\naway from zero uniformly for all x∈Ω.\n(ii) Instead of Lemma 4.1 we use Lemma 5.2 (with /hatwidermn\nhand/hatwidermn\n⋆,hin the role of /tildewiderm\nandm, respectively) to bound the quantity qn\nh. This requires that /hatwidermn\n⋆,hand˙mn\n⋆,h\nare bounded in W1,∞independently of h.\nAd(i): In order to show that |/summationtextk−1\nj=0γjmn−j−1\nh(x)|stays close to 1 for all x∈Ω,\nwe need to establish an L∞bound for the errors en−j−1\nh=mn−j−1\nh−mn−j−1\n⋆,h.\nWe use an induction argument and assume that for some time step nu mber ¯n\nwith ¯nτ/lessorequalslant¯twe have\n(7.3) /ba∇dblen\nh/ba∇dblL∞/lessorequalslantρ,for 0/lessorequalslantn<¯n,\nwhere we choose ρsu��ciently small independent of handτ. (In this proof it suffices\nto chooseρ/lessorequalslant1/(4Cγ), whereCγ=/summationtextk−1\nj=0|γj|= 2k−1.)\nNote that the smallness condition of the lemma implies that (7.3) is satis fied\nfor ¯n=k, because for the L∞errors of the starting values we have by an inverse\ninequality, for i= 0,...,k−1,\n/ba∇dblei\nh/ba∇dblL∞/lessorequalslantCh−1/2/ba∇dblei\nh/ba∇dblH1/lessorequalslantCh−1/2(ˆch)1/2=Cˆc1/2/lessorequalslantρ,\nprovided that ˆ cis sufficiently small (independent of τandh), as is assumed.\nWe will show in part (b) of the proof that with the induction hypothes is (7.3) we\nobtain also /ba∇dble¯n\nh/ba∇dblL∞/lessorequalslantρso that finally we obtain (7.3) for all¯nwith ¯nτ/lessorequalslant¯t.30 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nUsing reverse and ordinary triangle inequalities, the error bound of [12, Corol-\nlary 8.1.12] (noting that m(t)∈W2,∞(Ω) under our assumptions) and the L∞\nboundedness of ∂tm, and the bound (7.3), we estimate\n(7.4)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextendsingle−1/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextendsingle−|mn\n⋆|/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞/lessorequalslant/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\nj=0γjmn−j−1\nh−mn\n⋆/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞\n/lessorequalslant/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\nj=0γjen−j−1\nh/vextenddouble/vextenddouble/vextenddouble\nL∞+/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\nj=0γj(Rhmn−j−1\n⋆−mn−j−1\n⋆)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞+/vextenddouble/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\nj=0γj(mn−j−1\n⋆−mn\n⋆)/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL∞\n/lessorequalslant/vextenddouble/vextenddouble/vextenddoublek−1/summationdisplay\nj=0γjen−j−1\nh/vextenddouble/vextenddouble/vextenddouble\nL∞+Ch+Cτ/lessorequalslantk−1/summationdisplay\nj=0|γj| ·ρ+Ch+Cτ/lessorequalslant1\n2,\nprovided that handτare sufficiently small. The same argument also yields that/vextenddouble/vextenddouble|/summationtextk−1\nj=0γjmn−j−1\n⋆,h|−1/vextenddouble/vextenddouble\nL∞/lessorequalslant1\n2, and so we have\n(7.5)1\n2/lessorequalslant/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\nh(x)/vextendsingle/vextendsingle/vextendsingle/lessorequalslant3\n2and1\n2/lessorequalslant/vextendsingle/vextendsingle/vextendsinglek−1/summationdisplay\nj=0γjmn−j−1\n⋆,h(x)/vextendsingle/vextendsingle/vextendsingle/lessorequalslant3\n2\nfor allx∈Ω. In particular, it follows that /hatwidermn\nhand/hatwidermn\n⋆,hare unambiguously defined.\nAd(ii): The required W1,∞bound formn\n⋆,h=Rhm(tn) follows from the W1,∞-\nstability of the Ritz projection: by [12, Theorem 8.1.11] and by the as sumedW1,∞\nbound (4.3) for m(t),\n(7.6) /ba∇dblmn\n⋆,h/ba∇dblW1,∞/lessorequalslantC/ba∇dblm(tn)/ba∇dblW1,∞/lessorequalslantCR.\nThe bounds (7.5) and (7.6) for n/lessorequalslant¯nimply that also\n(7.7) /ba∇dbl/hatwidermn\n⋆,h/ba∇dblW1,∞/lessorequalslantCR\nforn/lessorequalslant¯n(with a different constant C). Using this bound in Lemma 5.3 and the\nassumedW1,∞bound (4.3) for ∂tm(t), we obtain with δ(ζ)/(1−ζ) =/summationtextk\nℓ=1(1−\nζ)ℓ−1/ℓ=:/summationtextk−1\nj=0µjζjthat\n/ba∇dbl˙mn\n⋆,h/ba∇dblW1,∞=/ba∇dblPh(/hatwidermn\n⋆,h)1\nτk/summationdisplay\nj=0δjmn−j\n⋆/ba∇dblW1,∞\n=/ba∇dblPh(/hatwidermn\n⋆,h)k−1/summationdisplay\nj=0µj1\nτ(mn−j\n⋆−mn−j−1\n⋆)/ba∇dblW1,∞\n=/ba∇dblPh(/hatwidermn\n⋆,h)k−1/summationdisplay\nj=0µj1\nτ/integraldisplaytn−j\ntn−j−1∂tm(t)dt/ba∇dblW1,∞\n/lessorequalslantCR/ba∇dblk−1/summationdisplay\nj=0µj1\nτ/integraldisplaytn−j\ntn−j−1∂tm(t)dt/ba∇dblW1,∞HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 31\n/lessorequalslantCRk−1/summationdisplay\nj=0|µj|R.\nWe can now establish a bound for qn\nhas defined in (7.2), using Lemma 5.2 together\nwith the above W1,∞bounds for /hatwidermn\n⋆,hand˙mn\n⋆,hto obtain\n(7.8) /ba∇dblqn\nh/ba∇dblL2/lessorequalslantc/ba∇dbl/hatwideen\nh/ba∇dblL2and/ba∇dbl∇qn\nh/ba∇dblL2/lessorequalslantc/ba∇dbl/hatwideen\nh/ba∇dblH1.\nWith theW1,∞bound of /hatwidermn\n⋆,hwe also obtain a bound of rn\nhdefined in (6.21). Using\nLemma 5.2 ( i) and recalling the L∞bound of∆m+Hof (4.3), we find that rn\nhis\nbounded by\n(7.9)/ba∇dblrn\nh/ba∇dblL2/lessorequalslant/ba∇dbl(Ph(/hatwidermn\nh)−Ph(/hatwidermn\n⋆,h))(∆mn\n⋆+Hn)/ba∇dblL2+/ba∇dbldn\nh/ba∇dblL2\n/lessorequalslantc/ba∇dbl/hatwideen\nh/ba∇dblL2+/ba∇dbldn\nh/ba∇dblL2.\n(b)Energy estimates. Forn/lessorequalslant¯nwith ¯nof (7.3), we test the error equation (6.25)\nwithϕh=˙en\nh+qn\nhand obtain\nα(˙en\nh,˙en\nh+qn\nh)+(/hatwideen\nh×˙mn\n⋆,h,˙en\nh+qn\nh)+(/hatwidermn\nh×˙en\nh,˙en\nh+qn\nh)\n+(∇en\nh,∇(˙en\nh+qn\nh)) =−(rn\nh,˙en\nh+qn\nh).\nBy collecting the terms, and using the fact that ( /hatwidermn\nh×˙en\nh,˙en\nh) = 0, we altogether\nobtain\nα/ba∇dbl˙en\nh/ba∇dbl2\nL2+(∇en\nh,∇˙en\nh) =−α(˙en\nh,qn\nh)−(/hatwideen\nh×˙mn\n⋆,h,˙en\nh+qn\nh)\n−(/hatwidermn\nh×˙en\nh,qn\nh)−(∇en\nh,∇qn\nh)−(rn\nh,˙en\nh+qn\nh).\nWe now estimate the term ( ∇en\nh,∇˙en\nh) on the left-hand side from below using\nDahlquist’s Lemma 10.1, so that the ensuing relation (10.2) yields\n(∇en\nh,∇˙en\nh)/greaterorequalslant1\nτ/parenleftig\n/ba∇dbl∇En\nh/ba∇dbl2\nG−/ba∇dbl∇En−1\nh/ba∇dbl2\nG/parenrightig\n+(∇en\nh,∇sn\nh),\nwhereEn\nh= (en−k+1\nh,...,en\nh) and theG-weighted semi-norm is given by\n/ba∇dbl∇En\nh/ba∇dbl2\nG=k/summationdisplay\ni,j=1gij(∇en−k+i\nh,∇en−k+j\nh).\nThis semi-norm satisfies the relation\n(7.10) γ−k/summationdisplay\nj=1/ba∇dbl∇en−k+j\nh/ba∇dbl2\nL2/lessorequalslant/ba∇dbl∇En\nh/ba∇dbl2\nG/lessorequalslantγ+k/summationdisplay\nj=1/ba∇dbl∇en−k+j\nh/ba∇dbl2\nL2,\nwhereγ−andγ+are the smallest and largest eigenvalues of the positive definite\nsymmetric matrix G= (gij) from Lemma 10.1.\nThe remaining terms are estimated using the Cauchy–Schwarz inequ ality and\n/ba∇dbl/hatwidermn\nh/ba∇dblL∞= 1; we altogether obtain\nα/ba∇dbl˙en\nh/ba∇dbl2\nL2+1\nτ/parenleftig\n/ba∇dbl∇En\nh/ba∇dbl2\nG−/ba∇dbl∇En−1\nh/ba∇dbl2\nG/parenrightig\n/lessorequalslantα/ba∇dbl˙en\nh/ba∇dblL2/ba∇dblqn\nh/ba∇dblL2+/ba∇dbl/hatwideen\nh/ba∇dblL2(/ba∇dbl˙en\nh/ba∇dblL2+/ba∇dblqn\nh/ba∇dblL2)\n+/ba∇dbl˙en\nh/ba∇dblL2/ba∇dblqn\nh/ba∇dblL2+/ba∇dbl∇en\nh/ba∇dblL2(/ba∇dbl∇qn/ba∇dblL2+/ba∇dbl∇sn\nh/ba∇dblL2)+/ba∇dblrn\nh/ba∇dblL2(/ba∇dbl˙en\nh/ba∇dblL2+/ba∇dblqn\nh/ba∇dblL2).32 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nWe now show an L2error bound for /hatwideen\nhin terms of (en−j−1\nh)k−1\nj=0. Using the fact that\nfora,b∈R3\\{0},\n(7.11)/vextendsingle/vextendsingle/vextendsingle/vextendsinglea\n|a|−b\n|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle(|b|−|a|)a+|a|(a−b)\n|a| |b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant2|a−b|\n|b|,\nand the lower bounds in (7.5) for both |/summationtextk−1\nj=0γjmn−j−1\nh|and|/summationtextk−1\nj=0γjmn−j−1\n⋆,h|, we\ncan estimate\n(7.12)/ba∇dbl/hatwideen\nh/ba∇dblL2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationtextk−1\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextendsingle/summationtextk−1\nj=0γjmn−j−1\nh/vextendsingle/vextendsingle/vextendsingle−/summationtextk−1\nj=0γjmn−j−1\n⋆,h/vextendsingle/vextendsingle/vextendsingle/summationtextk−1\nj=0γjmn−j−1\n⋆,h/vextendsingle/vextendsingle/vextendsingle/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2/lessorequalslantCk−1/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nL2.\nTo show a similar bound for /ba∇dbl∇/hatwideen\nh/ba∇dblL2we need the following two observations: First,\ntheW1,∞bounds formn−j−1\n⋆,hfrom (7.6) imply W1,∞boundedness for /hatwidermn\n⋆,hby\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j/parenleftbiggb\n|b|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂jb\n|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingleb(∂jb,b)\n|b|3/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nSecond, similarly we have/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂j/parenleftbigga\n|a|−b\n|b|/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja\n|a|−∂jb\n|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsinglea(∂ja,a)|b|3−b(∂jb,b)|a|3\n|a|3|b|3/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja\n|a|−∂jb\n|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+||a|3−|b|3||∂jb|\n|a|3|b|+|a(∂ja,a)−b(∂jb,b)|\n|b|3\n/lessorequalslant/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ja\n|a|−∂jb\n|b|/vextendsingle/vextendsingle/vextendsingle/vextendsingle+|a−b|(|b|2+|b||a|+|a|2)|∂jb|\n|a|3|b|\n+|a|2|∂ja−∂jb|\n|b|3+|a||∂jb||a−b|\n|b|3+|a−b||∂jb|\n|b|2.\nCombining these two observations, again with mhandm⋆,hin the role of aandb,\nrespectively, and the upper and lower bounds from (7.5) altogethe r yield\n(7.13) /ba∇dbl∇/hatwideen\nh/ba∇dbl2\nL2/lessorequalslantCk−1/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nH1.\nWe estimate further using Young’s inequality and absorptions into th e term\n/ba∇dbl˙en/ba∇dbl2\nL2, together with the bounds in (7.8) and (7.9), to obtain\nα1\n2/ba∇dbl˙en\nh/ba∇dbl2\nL2+1\nτ/parenleftig\n/ba∇dbl∇En\nh/ba∇dbl2\nG−/ba∇dbl∇En−1\nh/ba∇dbl2\nG/parenrightig\n/lessorequalslantck/summationdisplay\nj=0/ba∇dblen−j\nh/ba∇dbl2\nH1+c/ba∇dbldn\nh/ba∇dbl2\nL2+c/ba∇dbl∇sn\nh/ba∇dbl2\nL2.\nMultiplying both sides by τ, summing up from kton/lessorequalslant¯n, and using an absorption\nyield\nα1\n2τn/summationdisplay\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2+/ba∇dbl∇En\nh/ba∇dbl2\nG\n/lessorequalslant/ba∇dbl∇Ek−1\nh/ba∇dbl2\nG+cτn/summationdisplay\nj=k/ba∇dblej\nh/ba∇dbl2\nH1+cτn/summationdisplay\nj=k/parenleftbig\n/ba∇dbldj\nh/ba∇dbl2\nL2+/ba∇dblsj\nh/ba∇dbl2\nH1/parenrightbig\n+ck−1/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nL2.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 33\nWe then arrive, using (7.10), at\n(7.14)α1\n2τn/summationdisplay\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2+/ba∇dbl∇en\nh/ba∇dbl2\nL2/lessorequalslantcτn/summationdisplay\nj=k/ba∇dblej\nh/ba∇dbl2\nH1+cτn/summationdisplay\nj=k/parenleftbig\n/ba∇dbldj\nh/ba∇dbl2\nL2+/ba∇dblsj\nh/ba∇dbl2\nH1/parenrightbig\n+ck−1/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nL2,\nwithcdepending on α.\nSimilarly as in the time continuous case in the proof of Lemma 4.2, we con nect\n/ba∇dblen\nh/ba∇dbl2\nL2andτ/summationtextn\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2. We rewrite the identity\n1\nτk/summationdisplay\nj=0δjen−j\nh=˙en\nh−sn\nh, n/greaterorequalslantk,\nas\n1\nτn/summationdisplay\nj=kδn−jej\nh=˙ehn−sn\nh−gn\nh, n/greaterorequalslantk,\nwithδℓ= 0 forℓ>kand where\ngn\nh:=1\nτk−1/summationdisplay\ni=0δn−iei\nh\ndepends only on the starting errors and satisfies gn\nh= 0 forn/greaterorequalslant2k. With the inverse\npower series of δ(ζ),\nκ(ζ) =∞/summationdisplay\nn=0κnζn:=1\nδ(ζ),\nwe then have, for n/greaterorequalslantk,\nen\nh=τn/summationdisplay\nj=kκn−j(˙ej\nh−sj\nh−gj\nh).\nBy the zero-stability of the BDF method of order k/lessorequalslant6, the coefficients κnare\nuniformly bounded: |κn|/lessorequalslantcfor alln/greaterorequalslant0. Therefore we obtain via the Cauchy–\nSchwarz inequality\n/ba∇dblen\nh/ba∇dbl2\nL2/lessorequalslant2τ2/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\nj=kκn−j(˙ehj−sj\nh)/vextenddouble/vextenddouble/vextenddouble2\nL2+2τ2/vextenddouble/vextenddouble/vextenddouble2k−1/summationdisplay\nj=kκn−jgj\nh/vextenddouble/vextenddouble/vextenddouble2\nL2\n/lessorequalslant(2nτ)τc2n/summationdisplay\nj=k/ba∇dbl˙ehj−sj\nh/ba∇dbl2\nL2+2τ2c2k2k−1/summationdisplay\nj=k/ba∇dblgj\nh/ba∇dbl2\nL2\n/lessorequalslantCτn/summationdisplay\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2+Cτn/summationdisplay\nj=k/ba∇dblsj\nh/ba∇dbl2\nL2+Ck/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nL2.34 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nInserting this bound into (7.14) then yields\nα/ba∇dblen\nh/ba∇dbl2\nL2+/ba∇dbl∇en\nh/ba∇dbl2\nL2/lessorequalslantcτn/summationdisplay\nj=k/ba∇dblej\nh/ba∇dbl2\nH1+cτn/summationdisplay\nj=k/parenleftbig\n/ba∇dbldj\nh/ba∇dbl2\nL2+/ba∇dblsj\nh/ba∇dbl2\nH1/parenrightbig\n+ck−1/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nL2,\nand a discrete Gronwall inequality implies the stated stability result fo rn/lessorequalslant¯n. It\nthen follows from this stability bound, the smallness condition of the le mma and the\ninverse estimate from H1toL∞that (7.3) is satisfied also for ¯ n+1. This completes\nthe induction step for (7.3) and proves the stated error bound. /square\n8.Stability of the full discretization for BDF of orders 3 to 5\nStability for full discretizations using the BDF methods of orders 3 t o 5 can be\nshown under additional conditions on the damping parameter αand the stepsize τ.\nLemma 8.1 (Stability for orders k= 3,4,5).Consider the linearly implicit k-step\nBDF discretization (2.6)for3/lessorequalslantk/lessorequalslant5with finite elements of polynomial degree\nr/greaterorequalslant2. Letmn\nhandmn\n⋆,hsatisfy(2.6)and(6.17), respectively, and suppose that the\nregularity assumptions of Lemma 7.1 hold. Furthermore, ass ume that the damping\nparameterαsatisfies\n(8.1) α>α k:=ηk\n1−ηk\nwith the multiplier ηkof Lemma 10.2, and that τandhsatisfy the mild CFL-type\ncondition, for some ¯c>0,\n(8.2) τ/lessorequalslant¯ch.\nThen, for sufficiently small h/lessorequalslant¯handτ/lessorequalslant¯τ, the erroren\nh=mn\nh−mn\n⋆,hsatisfies\nthe following bound, for kτ/lessorequalslantnτ/lessorequalslant¯t,\n(8.3)/ba∇dblen\nh/ba∇dbl2\nH1(Ω)3/lessorequalslantC/parenleftigk−1/summationdisplay\ni=0/ba∇dblei\nh/ba∇dbl2\nH1(Ω)3+τn/summationdisplay\nj=k/ba∇dbldj\nh/ba∇dbl2\nL2(Ω)3+τn/summationdisplay\nj=k/ba∇dblsj\nh/ba∇dbl2\nH1(Ω)3/parenrightig\n,\nwhere the constant Cis independent of τ,handn, but depends on α,R,K,M , and\nexponentially on ¯c¯t. This estimate holds under the smallness condition that the\nright-hand side is bounded by ˆch3with a constant ˆc(note that the right-hand side is\nof sizeO((τk+hr)2)in the case of a sufficiently regular solution ).\nTogether with the defect bounds of Section 6, this stability lemma pr oves Theo-\nrem 3.2. We remark that the thresholds αk>0 defined here are the same as those\nappearing in Theorem 3.2.\nProof.The proof of this lemma combines the arguments of the proof of Lem ma 7.1\nwith a nonstandard variant of the multiplier technique of Nevanlinna a nd Odeh, as\noutlined in the Appendix. Since the size of the parameter αdetermines which BDF\nmethods satisfy the stability estimate, the dependence on αwill be carefully traced\nall along the proof.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 35\n(a)Preparations. As in the previous proof, we make again the induction hypoth-\nesis (7.3) for some ¯ nwith ¯nτ/lessorequalslant¯t, but this time with ρ=c0hfor some positive\nconstantc0:\n(8.4) /ba∇dblen\nh/ba∇dblL∞/lessorequalslantc0h, n< ¯n.\nBy an inverse inequality, this implies that /ba∇dblen\nh/ba∇dblW1,∞has anh- andτ-independent\nbound, and hence also /ba∇dblmn\nh/ba∇dblW1,∞forn<¯n. Together with (7.5), this implies\n(8.5) /ba∇dbl/hatwidermn\nh/ba∇dblW1,∞/lessorequalslantC\nand further\n(8.6) /ba∇dbl/hatwideen\nh/ba∇dblL∞/lessorequalslantCh.\nAs in the Appendix, we aim to subtract ηktimes the error equation for time\nstepn−1 from the error equation for time step n, and then to test with ϕh=\nPh(/hatwidermn\nh)˙en\nh∈Th(/hatwidermn\nh) (similarly as in the proof of Lemma 7.1). However, this is not\npossible directly due to the different test spaces at different time st eps:\nα(˙en\nh,ϕh)+(/hatwideen\nh×˙mn\n⋆,h,ϕh)\n+(/hatwidermn\nh×˙en\nh,ϕh)+(∇en\nh,∇ϕh) =−(rn\nh,ϕh),(8.7a)\nfor allϕh∈Th(/hatwidermn\nh), and\nα(˙en−1\nh,ψh)+(/hatwideen−1\nh×˙mn−1\n⋆,h,ψh)\n+(/hatwidermn−1\nh×˙en−1\nh,ψh)+(∇en−1\nh,∇ψh) =−(rn−1\nh,ψh),(8.7b)\nfor allψh∈Th(/hatwidermn−1\nh).\nAs in (7.2), we have\n(8.8)ϕh=Ph(/hatwidermn\nh)˙en\nh=˙en\nh+qn\nh,withqn\nh=−(Ph(/hatwidermn\nh)−Ph(/hatwidermn\n⋆,h))˙mn\n⋆,h,\nwhereqn\nhis bounded by (7.8).\nIn turn, the test function ψh=Ph(/hatwidermn−1\nh)˙en\nh∈Th(/hatwidermn−1\nh) is a perturbation of\nϕh=˙en\nh+qn\nh, since using (8.8) we obtain\nψh=Ph(/hatwidermn−1\nh)˙en\nh\n=Ph(/hatwidermn\nh)˙en\nh−(Ph(/hatwidermn\nh)−Ph(/hatwidermn−1\nh))˙en\nh\n=˙en\nh+qn\nh+pn\nhwithpn\nh=−(Ph(/hatwidermn\nh)−Ph(/hatwidermn−1\nh))˙en\nh.\nThe perturbation pn\nhis estimated using the second bound in Lemma 5.2 ( i) with\np=∞,q= 2, and noting (8.5). We obtain\n/ba∇dblpn\nh/ba∇dblL2/lessorequalslant/ba∇dbl(Ph(/hatwidermn\nh)−Ph(/hatwidermn−1\nh))˙en\nh/ba∇dblL2\n/lessorequalslantc/ba∇dbl˙en\nh/ba∇dblL2/ba∇dbl/hatwidermn\nh−/hatwidermn−1\nh/ba∇dblL∞\n/lessorequalslantc/ba∇dbl˙en\nh/ba∇dblL2/parenleftig\n/ba∇dbl/hatwideen\nh/ba∇dblL∞+/ba∇dbl/hatwidermn\n⋆,h−/hatwidermn−1\n⋆,h/ba∇dblL∞+/ba∇dbl/hatwideen−1\nh/ba∇dblL∞/parenrightig\n/lessorequalslantc/ba∇dbl˙en\nh/ba∇dblL2/parenleftig\n/ba∇dbl/hatwideen\nh/ba∇dblL∞+k−1/summationdisplay\nj=0|γj|/integraldisplaytn−j−1\ntn−j−2/ba∇dblRh∂tm(t)/ba∇dblL∞dt+/ba∇dbl/hatwideen−1\nh/ba∇dblL∞/parenrightig\n.36 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nWe have /ba∇dblRh∂tm(t)/ba∇dblL∞/lessorequalslantc/ba∇dbl∂tm(t)/ba∇dblW1,∞by [12, Theorem 8.1.11]. In view of (8.6)\nwe obtain, for τ/lessorequalslant¯Ch,\n(8.9) /ba∇dblpn\nh/ba∇dblL2/lessorequalslantCh/ba∇dbl˙en\nh/ba∇dblL2,\nand by an inverse estimate,\n(8.10) /ba∇dbl∇pn\nh/ba∇dblL2/lessorequalslantC/ba∇dbl˙en\nh/ba∇dblL2.\nWe also recall the bound (7.9) for /ba∇dblrn\nh/ba∇dblL2.\n(b)Energy estimates. By subtracting (8.7a) −ηk(8.7b) with the above choice of\ntest functions, we obtain\n(8.11)α(˙en\nh−ηk˙en−1\nh,˙en\nh+qn\nh)+(/hatwideen\nh×˙mn\n⋆,h−ηk/hatwideen−1\nh×˙mn−1\n⋆,h,˙en\nh+qn\nh)\n+(/hatwidermn\nh×˙en\nh−ηk/hatwidermn−1\nh×˙en−1\nh,˙en\nh+qn\nh)+(∇en\nh−ηk∇en−1\nh,∇(˙en\nh+qn\nh))\n−ηk/bracketleftbig\nα(˙en−1\nh,pn\nh)+(/hatwideen−1\nh×˙mn−1\n⋆,h,pn\nh)\n+(/hatwidermn−1\nh×˙en−1\nh,pn\nh)+(∇en−1\nh,∇pn\nh)/bracketrightbig\n=−(rn\nh−ηkrn−1\nh,˙en\nh+qn\nh)−ηk(rn−1\nh,pn\nh).\nWe estimate the terms of the error equation (8.11) separately and track carefully\nthe dependence on ηkandα.\nThe termα(˙en\nh−ηk˙en−1\nh,˙en\nh) is bounded from below, using Young’s inequality and\nabsorptions, by\nα(˙en\nh−ηk˙en−1\nh,˙en\nh)/greaterorequalslantα/parenleftbig\n1−1\n2ηk/parenrightbig\n/ba∇dbl˙en\nh/ba∇dbl2\nL2−α\n2ηk/ba∇dbl˙en−1\nh/ba∇dbl2\nL2,\nwhile the term ( ∇en\nh−ηk∇en−1\nh,∇˙en\nh) is bounded from below, via the relation (10.2)\nand (6.23), by\n(∇en\nh−ηk∇en−1\nh,∇˙en\nh)/greaterorequalslant1\nτ/parenleftig\n/ba∇dbl∇En\nh/ba∇dbl2\nG−/ba∇dbl∇En−1\nh/ba∇dbl2\nG/parenrightig\n+(∇en\nh−ηk∇en−1\nh,∇sn\nh),\nwithEn\nh= (en−k+1\nh,...,en\nh), and where the G-weighted semi-norm is generated by\nthe matrix G= (gij) from Lemma 10.1 for the rational function δ(ζ)/(1−ηkζ).\nThe remaining terms outside the rectangular bracket are estimate d using the\nCauchy–Schwarz and Young inequalities (the latter often with a suffi ciently small\nbut fixedh- andτ-independent weighting factor µ >0) and/ba∇dbl/hatwidermn\nh/ba∇dblL∞= 1 and\northogonality. We obtain, with varying constants c(which depend on αand are\ninversely proportional to µ)\nα(˙en\nh−ηk˙en−1\nh,qn\nh)+(/hatwideen\nh×˙mn\n⋆,h−ηk/hatwideen−1\nh×˙mn−1\n⋆,h,˙en\nh+qn\nh)\n+(/hatwidermn\nh×˙en\nh−ηk/hatwidermn−1\nh×˙en−1\nh,˙en\nh+qn\nh)+(∇en−ηk∇en−1\nh,∇qn\nh)\n/lessorequalslant/parenleftbig\nαµ+µ+1\n2ηk/parenrightbig\n/ba∇dbl˙en\nh/ba∇dbl2\nL2+/parenleftbig\nαµηk+1\n2ηk/parenrightbig\n/ba∇dbl˙en−1\nh/ba∇dbl2\nL2\n+c/parenleftbig\n/ba∇dblqn\nh/ba∇dblL2+/ba∇dbl/hatwideen\nh/ba∇dbl2\nL2+/ba∇dbl/hatwideen−1\nh/ba∇dbl2\nL2/parenrightbig\n+1\n2/parenleftbig\n/ba∇dbl∇en\nh/ba∇dbl2\nL2+η2\nk/ba∇dbl∇en−1\nh/ba∇dbl2\nL2+/ba∇dbl∇qn\nh/ba∇dblL2/parenrightbig\n/lessorequalslant/parenleftbig\nαµ+µ+1\n2ηk/parenrightbig\n/ba∇dbl˙en\nh/ba∇dbl2\nL2+/parenleftbig\nαµηk+1\n2ηk/parenrightbig\n/ba∇dbl˙en−1\nh/ba∇dbl2\nL2+ck/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nH1,\nwhere in the last inequality we used (7.12) and (7.13) to estimate /hatwideen\nh.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 37\nThe terms inside the rectangular bracket are bounded similarly, usin g (8.9) and\n(8.10) and the condition τ/lessorequalslant¯Ch, by\nα(˙en−1\nh,pn\nh)+(/hatwideen−1\nh×˙mn−1\n⋆,h,pn\nh)+(/hatwidermn−1\nh×˙en−1\nh,pn\nh)+(∇en−1\nh,∇pn\nh)\n/lessorequalslantµ/ba∇dbl˙en\nh/ba∇dbl2\nL2+ch/ba∇dbl˙en−1\nh/ba∇dbl2\nL2+c/parenleftbig\n/ba∇dbl/hatwideen−1\nh/ba∇dbl2\nL2+/ba∇dbl∇en−1\nh/ba∇dbl2\nL2/parenrightbig\n/lessorequalslantµ/ba∇dbl˙en\nh/ba∇dbl2\nL2+ck/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nH1.\nHereµis an arbitrarily small positive constant (independent of τandh), andc\ndepends on the choice of µ.\nIn view of (7.9), the terms with the defects rn\nhare bounded by\n−(rn\nh−ηkrn−1\nh,˙en\nh+qn\nh)−ηk(rn−1\nh,pn\nh)\n/lessorequalslantµ/ba∇dbl˙en\nh/ba∇dbl2\nL2+c/parenleftbig\n/ba∇dblrn\nh/ba∇dbl2\nL2+/ba∇dblrn−1\nh/ba∇dbl2\nL2+/ba∇dblqn\nh/ba∇dbl2\nL2/parenrightbig\n/lessorequalslantµ/ba∇dbl˙en\nh/ba∇dbl2\nL2+ck/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nL2+c1/summationdisplay\nj=0/ba∇dbldn−j\nh/ba∇dbl2\nL2.\nCombination of these inequalities yields\n/parenleftig\nα(1−1\n2ηk)−1\n2ηk−µ/parenrightig\n/ba∇dbl˙en\nh/ba∇dbl2\nL2−/parenleftig\nα\n2ηk+1\n2ηk+µαηk/parenrightig\n/ba∇dbl˙en−1\nh/ba∇dbl2\nL2\n+1\nτ/parenleftig\n/ba∇dbl∇En\nh/ba∇dbl2\nG−/ba∇dbl∇En−1\nh/ba∇dbl2\nG/parenrightig\n/lessorequalslantck/summationdisplay\nj=0/ba∇dblen−j−1\nh/ba∇dbl2\nH1+c1/summationdisplay\nj=0/ba∇dbldn−j\nh/ba∇dbl2\nL2+c/ba∇dbl∇sn\nh/ba∇dbl2\nL2.\nUnder condition (8.1) we have\nω:=α(1−ηk)−ηk>0.\nMultiplying both sides by τand summing up from ktonwithn/lessorequalslant¯nyields, for\nsufficiently small µ,\n1\n2ωτn/summationdisplay\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2+/ba∇dbl∇En\nh/ba∇dbl2\nG\n/lessorequalslantcτ/ba∇dbl˙ek−1\nh/ba∇dbl2\nL2+/ba∇dbl∇Ek−1\nh/ba∇dbl2\nG+cτn−1/summationdisplay\nj=0/ba∇dblej\nh/ba∇dbl2\nH1+cτn/summationdisplay\nj=k/ba∇dbldj\nh/ba∇dbl2\nL2+cτn/summationdisplay\nj=k/ba∇dbl∇sj\nh/ba∇dbl2\nL2.\nThe proof is then completed using exactly the same arguments as in t he last part of\ntheproofofLemma7.1,byestablishinganestimatebetween /ba∇dblen\nh/ba∇dbl2\nL2andτ/summationtextn\nj=k/ba∇dbl˙ej\nh/ba∇dbl2\nL2\nand using a discrete Gronwall inequality, and completing the induction step for\n(8.4). /square\n9.Numerical experiments\nTo obtain significant numerical results, we prescribe the exact solu tionmon\ngiven three-dimensional domains Ω:= [0,1]×[0,1]×[0,L] withL∈ {1/100,1/4}.38 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nThe discretizations of these domains will consist of a few layers of ele ments inz-\ndirection (one layer for L= 1/100 and ten layers for L= 1/4) and a later specified\nnumber of elements in xandydirections. This mimics the common case of thin\nfilm alloys as for example in the standard problems of the Micromagnet ic Modeling\nActivityGroupatNISTCenterforTheoreticalandComputational MaterialsScience\n(ctcms.nist.gov ). Moreover, this mesh structure helps to keep the computationa l\nrequirements reasonable and allow us to compute the experiments o n a desktop\nPC. We are aware that these experiments are only of preliminary nat ure and are\njust supposed to confirm the theoretical results. A more thorou gh investigation\nof the numerical properties of the developed method is needed. Th is will require\nus to incorporate preconditioning, parallelization of the computatio ns, as well as\nlower order energy contributions in the effective field (1.3) to be able to compare to\nbenchmark results from computational physics. This, however, is beyond the scope\nof this paper, and will be the topic of a subsequent work.\nWe consider the time interval [0 ,¯t] with¯t= 0.2 and define two different exact\nsolutions. Since within our computational budget either the time disc retization\nerror or the space discretization error dominates, we construct the solutions such\nthat the first oneis harder to approximate inspace, while thesecon d oneis harder to\napproximate in time. Both solutions are constant in z-direction as is often observed\nin thin-film applications.\n9.1.Implementation. The numerical experiments were conducted using the finite\nelement package FEniCS ( www.fenicsproject.org ) on a desktop computer. As al-\nreadydiscussed inSection2.2, thereareseveral ways toimplement thetangent space\nrestriction. We decided to solve a saddle point problem (variant (a) in Section 2.2)\nfor simplicity of implementation. For preconditioning, we used the blac k-box AMG\npreconditioner that comes with FEniCS. Although this might not be th e optimal\nsolution, it keeps the number of necessary iterative solver steps w ithin reasonable\nbounds. Assuming perfect preconditioning, the cost per time-ste p is then propor-\ntional to the number of mesh-elements. We observed this behavior approximately,\nalthough further research beyond the scope of this work is requir ed to give a definite\nconclusion.\n9.2.Exact solutions. We choose the damping parameter α= 0.2 and define\ng(t) := (¯t+0.1)/(¯t+0.1−t) as well as d(x) := (x1−1/2)2+(x2−1/2)2, which is\nthe squared distance of the projection of xto [0,1]×[0,1] and the point (1 /2,1/2).\nFor some constant C= 400 (a choice made to have pronounced effects), define\n(9.1)m(x,t) :=\nCe−g(t)\n1/4−d(x)(x1−1/2)\nCe−g(t)\n1/4−d(x)(x2−1/2)/radicalig\n1−C2e−2g(t)\n1/4−d(x)d(x)\nifd(x)/lessorequalslant1\n4andm(x,t) :=\n0\n0\n1\nelse.\nIt iseasy to check that |m(x,t)|= 1for all ( x,t)∈Ω×[0,¯t]. Moreover, ∂nm(x,t) =\n0 for allx∈∂Ω. We may calculate the time derivative of min a straightforwardHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 39\nfashion, i.e., ∂tm(x,t) = 0 ford(x)>1/4 and\n∂tm(x,t) =\n−g′(t)\n1/4−d(x)Ce−g(t)\n1/4−d(x)(x1−1/2)\n−g′(t)\n1/4−d(x)Ce−g(t)\n1/4−d(x)(x2−1/2)\ng′(t)\n1/4−d(x)C2e−2g(t)\n1/4−d(x)d(x)\nm3(x,t)\nifd(x)/lessorequalslant1\n4.\nHere,m3denotes the third component of mas defined above.\nThe second exact solution is defined via\n(9.2) /tildewiderm(x,t) :=\n−(x3\n1−3x2\n1/2+1/4)sin(3πt/¯t)/radicalbig\n1−(x3\n1−3x2\n1/2+1/4)2\n−(x3\n1−3x2\n1/2+1/4)cos(3πt/¯t)\n.\nDue to the polynomial nature in the first and the third component, a nd the well-\nbehaved square-root, the space approximation error does not d ominate the time\napproximation.\n9.3.The experiments. We now may compute the corresponding forcings Hresp.\n/tildewiderHto obtain the prescribed solutions by inserting into (1.4), i.e.,\nH=α∂tm+m×∂tm−∆m.\n(Note that we may disregard the projection P(m) from (1.4) since we solve in\nthe tangent space anyway.) We compute Hnumerically by first interpolating m\nand∂tmand then computing the derivatives. This introduces an additional e rror\nwhich is not accounted for in the theoretical analysis. However, th e examples below\nconfirm the expected convergence rates and hence conclude tha t this additional\nperturbation is negligible. Figure 9.1 shows slices of the exact solution at different\ntime steps. Figure 9.2 shows the convergence with respect to the t ime step size τ,\nwhile Figure 9.3 shows convergence with respect to the spatial mesh sizeh. All the\nexperiments confirm the expected rates for smooth solutions.\nFinally, we consider an example with nonsmooth initial data and consta nt right-\nhand side. The initial data are given by\n(9.3)m0(x) :=\nx1−1/2\nx2−1/2/radicalbig\n1−d(x)\nifd(x)/lessorequalslant1\n4andm0(x) :=\n0\n0\n1\nelse.\nWith the constant forcing field H:= (0,1,1)Twe compute a numerical approxima-\ntion to the unknown exact solution. Note that we do not expect any smoothness of\nthe solution (even the initial data is not smooth). Figure 9.4 neverth eless shows a\nphysically consistent decay of the energy /ba∇dbl∇m(t)/ba∇dblL2(Ω)3over time as well as a good\nagreement between different orders of approximation. Moreover , the computed ap-\nproximation shows little deviation from unit length as would be expecte d for smooth\nsolutions.40 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nFigure 9.1. Thefirstrowshowstheexactsolution m(x,t)from(9.1)\nforx∈[0,1]×[0,1]× {0}andt∈ {0,0.05,¯t}(from left to right),\nwhereas the second row shows the exact solution /tildewiderm(x,t) from (9.2)\nforx∈[0,1]×[0,1]×{0}andt∈ {0,0.2/6,0.2/3}(from left to right).\nWhile the problems are three-dimensional, the solutions are constan t\ninz-direction and we only show one slice of the solution.PSfrag replacements\n10−810−610−410−2100\n10−310−210−1k= 1\nk= 2\nk= 3\nk= 4\ntimestep τPSfrag replacements\n10−610−510−410−310−210−1100101\n10−310−210−1k= 1\nk= 2\nk= 3\nk= 4\ntimestep τ\nFigure 9.2. The plots show the error between computed solutions\nand exact solution /tildewidermfor a given time stepsize with a spatial poly-\nnomial degree of r= 2 and a spatial mesh size 1 /40 which results\nin≈6·104degrees of freedom per time step in the left plot. In the\nright plot we use a thicker domain D= [0,1]×[0,1]×[0,1/4] with 10\nelements in z-direction. This results in ≈4·105degrees of freedom\nper timestep. We use the k-step methods of order k∈ {1,2,3,4}and\nobserve the expected rates O(τk) indicated by the dashed lines. The\ncoarse levels of the higher order methods are missing because the kth\nstep is already beyond the final time ¯t.HIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 41\nPSfrag replacements\nr= 1\nr= 2\nr= 3\nr= 4\nmeshsize h10−410−310−210−1100\n10−1\nFigure 9.3. The plot shows convergence in meshsize hwith respect\nto the exact solution mfrom (9.1) on the domain D= [0,1]×[0,1]×\n[0,1/100]withonelayerofelementsin z-direction. Weusedthesecond\norder BDF method with τ= 10−3and spatial polynomial degrees\nr∈ {1,2,3,4}. The mesh sizes range from 1 /2 to 1/32. We observe\nthe expected rates O(hr) indicated by the dashed lines. The finest\nmesh-size for r= 4 does reach the expected error level. This is due to\nthe fact that the time-discretization errors start to dominate in t hat\nregion.\n10.Appendix: Energy estimates for backward difference formula e\nThe stability proofs of this paper rely on energy estimates, that is, on the use\nof positive definite bilinear forms to bound the error ein terms of the defect d.\nThis is, of course, a basic technique for studying the time-continuo us problem and\nalso for backward Euler and Crank–Nicolson time discretizations (se e, e.g., Thom´ ee\n[38]), but energy estimates still appear to be not well known for bac kward difference\nformula (BDF) time discretizations of order up to 5, which are widely u sed for\nsolving stiff ordinary differential equations. To illustrate the basic me chanism, we\nhere just consider the prototypical linear parabolic evolution equa tion in its weak\nformulation, given by two positive definite symmetric bilinear forms ( ·,·) anda(·,·)\non Hilbert spaces HandVwith induced norms |·|and/ba∇dbl·/ba∇dbl, respectively, and with\nVdensely and continuously embedded in H. The problem then is to find u(t)∈V\nsuch that\n(10.1) ( ∂tu,v)+a(u,v) = (f,v)∀v∈V,\nwith initial condition u(0) =u0. Ifu⋆is a function that satisfies the equation up to\na defectd, that is,\n(∂tu⋆,v)+a(u⋆,v) = (f,v)+(d,v)∀v∈V,42 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICHPSfrag replacements\n0.15\n0.14\n0.13\n0.12\n0.11\n0.1\n0.09\n0.08\n0.07\n0.06\n0.050 0.05 0.1 0.15 0.2\ntimer=k= 1\nr=k= 2\nr=k= 3\nr=k= 4r=k= 1\nr=k= 2\nr=k= 3\nr=k= 4PSfrag replacements10−1\n10−2\n10−3\n0 0 .05 0.1 0.15 0.2\ntime\nFigure 9.4. Left plot: Decay of energies /ba∇dbl∇m(t)/ba∇dblL2(Ω)3for the ap-\nproximations to the unknown solution with m0andHgiven in (9.3)\nand one line after (9.3). We plot four approximations of the k-step\nmethod with polynomial degree rforr=k∈ {1,2,3,4}. The spa-\ntial mesh-size is 1 /40 and the size of the timesteps is 10−3(blue) and\n10−2(red). Right plot: Deviation from unit length /ba∇dbl1−|m(t)|2/ba∇dblL∞(Ω)\nplotted over time for step sizes τ= 10−2(blue),τ= 10−3(red), and\nτ= 10−4(green). The solid lines indicate k= 1, whereas the dashed\nlines indicate k= 2. The spatial mesh-size is 1 /40 withr= 1.\nthen the error e=u−u⋆satisfies, in this linear case, an equation of the same form,\n(∂te,v)+a(e,v) = (d,v)∀v∈V,\nwith initial value e0=u0−u⋆\n0. Testing with v=eyields\n1\n2d\ndt|e|2+/ba∇dble/ba∇dbl2= (d,e).\nEstimating the right-hand side by ( d,e)/lessorequalslant/ba∇dbld/ba∇dbl⋆/ba∇dble/ba∇dbl/lessorequalslant1\n2/ba∇dbld/ba∇dbl2\n⋆+1\n2/ba∇dble/ba∇dbl2, with the dual\nnorm/ba∇dbl·/ba∇dbl⋆, and integrating from time 0 to tresults in the error bound\n|e(t)|2/lessorequalslant|e(0)|2+/integraldisplayt\n0/ba∇dbld(s)/ba∇dbl2\n⋆ds.\nOn the other hand, testing with v=∂teyields\n|∂te|2+1\n2d\ndt/ba∇dble/ba∇dbl2= (d,∂te),\nwhich leads similarly to the error bound\n/ba∇dble(t)/ba∇dbl2/lessorequalslant/ba∇dble(0)/ba∇dbl2+/integraldisplayt\n0|d(s)|2ds.\nThis procedure is all-familiar, but it is not obvious how to extend it to tim e dis-\ncretizations beyond the backward Euler and Crank–Nicolson metho ds. The use of\nenergy estimates for BDF methods relies on the following remarkable results.\nLemma 10.1. (Dahlquist [18]; see also [8] and [27, Section V.6]) Letδ(ζ) =δkζk+\n···+δ0andµ(ζ) =µkζk+···+µ0be polynomials of degree at most k(and at leastHIGHER-ORDER DISCRETIZATION OF THE LLG EQUATION 43\none of them of degree k)that have no common divisor. Let (·,·)be an inner product\nwith associated norm |·|.If\nReδ(ζ)\nµ(ζ)>0for|ζ|<1,\nthen there exists a positive definite symmetric matrix G= (gij)∈Rk×ksuch that\nforv0,...,v kin the real inner product space,\n/parenleftigk/summationdisplay\ni=0δivk−i,k/summationdisplay\nj=0µjvk−j/parenrightig\n/greaterorequalslantk/summationdisplay\ni,j=1gij(vi,vj)−k/summationdisplay\ni,j=1gij(vi−1,vj−1).\nIn combination with the preceding result for the multiplier µ(ζ) = 1−ηkζ,the\nfollowing property of BDF methods up to order 5 becomes important .\nLemma 10.2. (Nevanlinna & Odeh [34]) Fork/lessorequalslant5,there exists 0/lessorequalslantηk<1such\nthat forδ(ζ) =/summationtextk\nℓ=11\nℓ(1−ζ)ℓ,\nReδ(ζ)\n1−ηkζ>0for|ζ|<1.\nThe smallest possible values of ηkare\nη1=η2= 0, η3= 0.0836, η4= 0.2878, η5= 0.8160.\nPrecise expressions for the optimal multipliers for the BDF methods of orders 3,4\nand 5 are given by Akrivis & Katsoprinakis [1].\nAn immediate consequence of Lemma 10.2 and Lemma 10.1 is the relation\n(10.2)/parenleftigk/summationdisplay\ni=0δivk−i,vk−ηkvk−1/parenrightig\n/greaterorequalslantk/summationdisplay\ni,j=1gij(vi,vj)−k/summationdisplay\ni,j=1gij(vi−1,vj−1)\nwith a positive definite symmetric matrix G= (gij)∈Rk×k; it is this inequality that\nplays a crucial role in our energy estimates, and the same inequality f or the inner\nproducta(·,·).\nThe errorequationfortheBDFtimediscretization ofthelinear para bolicproblem\n(10.1) reads\n(˙en,v)+a(en,v) = (dn,v)∀v∈V,where ˙en=1\nτk/summationdisplay\nj=0δjen−j,\nwith starting errors e0,...,ek−1. When we test with v=en−ηken−1, the first term\ncan be estimated from below by (10.2), the second term is bounded f rom below by\n(1−1\n2ηk)/ba∇dblen/ba∇dbl2−1\n2ηk/ba∇dblen−1/ba∇dbl2, and the right-hand term is estimated from above by the\nCauchy-Schwarz inequality. Summing up from ktonthen yields the error bound\n(10.3) |en|2+τn/summationdisplay\nj=k/ba∇dblej/ba∇dbl2/lessorequalslantCk/parenleftigk−1/summationdisplay\ni=0/parenleftbig\n|ei|2+τ/ba∇dblei/ba∇dbl2/parenrightbig\n+τn/summationdisplay\nj=k/ba∇dbldj/ba∇dbl2\n⋆/parenrightig\n,\nwhereCkdepends only on the order kof the method. This kind of estimate for\nthe BDF error has recently been used for a variety of linear and non linear parabolic\nproblems [33, 3, 2, 30].44 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nOn the other hand, when we first subtract ηktimes the error equation for n−1\nfrom the error equation with nand then test with ˙ en, we obtain\n(˙en−ηk˙en−1,˙en)+a(en−ηken−1,˙en) = (dn−ηkdn−1,˙en).\nHere, thesecond termis boundedfrombelow by (10.2)withthe a(·,·)inner product,\nthe first term is bounded from below by (1 −1\n2ηk)|˙en|2−1\n2ηk|˙en−1|2, and the right-\nhand term is estimated from above by the Cauchy–Schwarz inequalit y. Summing\nup fromktonthen yields the error bound\n(10.4) /ba∇dblen/ba∇dbl2+τn/summationdisplay\nj=k|˙ej|2/lessorequalslantCk/parenleftigk−1/summationdisplay\ni=0/ba∇dblei/ba∇dbl2+τn/summationdisplay\nj=k|dj|2/parenrightig\n.\nIt is this type of estimate that we use in the present paper for the n onlinear problem\nconsidered here. It has previously been used in [29].\nAcknowledgment. The work of Michael Feischl, Bal´ azs Kov´ acs and Christian Lu-\nbichissupportedbyDeutscheForschungsgemeinschaft –projec t-id 258734477–SFB\n1173.\nReferences\n1. G. Akrivis and E. Katsoprinakis, Backward difference formulae :new multipliers and stability\nproperties for parabolic equations , Math. Comp. 85(2016) 2195–2216.\n2. G. Akrivis, B. Li, and C. Lubich, Combining maximal regularity and energy estimates for time\ndiscretizations of quasilinear parabolic equations , Math. Comp. 86(2017) 1527–1552.\n3. G. Akrivis and C. Lubich, Fully implicit, linearly implicit and implicit–explicit b ackward dif-\nference formulae for quasi-linear parabolic equations , Numer. Math. 131(2015) 713–735.\n4. F. Alouges, A new finite element scheme for Landau–Lifshitz equations , Disc. Cont. Dyn. Syst.\nSer. S.1(2008) 187–196.\n5. F. Alouges and P. 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Phys. 16(2014) 013032.46 GEORGIOS AKRIVIS, MICHAEL FEISCHL, BAL ´AZS KOV ´ACS, AND CHRISTIAN LUBICH\nDepartment of Computer Science & Engineering, University o f Ioannina, 45110\nIoannina, Greece, and Institute of Applied and Computation al Mathematics, FORTH,\n70013 Heraklion, Crete, Greece\nE-mail address :akrivis@cse.uoi.gr\nInstitute for Analysis and Scientific Computing (E 101), Te chnical University\nWien, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria\nE-mail address :michael.feischl @kit.edu\nE-mail address :michael.feischl @tuwien.ac.at\nMathematisches Institut, Universit ¨at T¨ubingen, Auf der Morgenstelle, D-72076\nT¨ubingen, Germany\nE-mail address :kovacs@na.uni-tuebingen.de\nMathematisches Institut, Universit ¨at T¨ubingen, Auf der Morgenstelle, D-72076\nT¨ubingen, Germany\nE-mail address :lubich@na.uni-tuebingen.de"    },    {        "title": "1809.11020v1.Isotropic_non_local_Gilbert_damping_driven_by_spin_currents_in_epitaxial_Pd_Fe_MgO_001__films.pdf",        "content": "Isotropic non -local Gilbert damping d riven by spin current s in epitaxial  \n Pd/Fe/MgO(001)  film s \nYan Li1,2,Yang Li1,2,Qian Liu3, Zhe Yuan3, Wei He1,Hao -Liang Liu1, Ke Xia3,Wei Yu1, \nXiang- Qun Zhang1, and Zhao- Hua Cheng1,2, * \n1State Key Laboratory of Magnetism  and Beijing National Laboratory for Condensed \nMatter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, \nChina \n2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing \n100049, China \n3The Center for Advanced Quantum Studies and Department of Physics, Beijing \nNormal University, 100875 China \nABSTRACT  \nAlthough both theoretical  predications and experimental observations \ndemonstrate d that the damping factor is anisotropic at ferromagne t/semiconductor \ninterface with robust interfacial  spin- orbit coupling , it is not well understood whether \nnon-local Gilbert damping driven by spin current s in heavy metal /ferromagnetic metal \n(HM/FM) bilayers is anisotropic or not. H ere, we investigated the in -plane angular - \nand frequenc y- dependen ce of magnetic relaxation of epitaxial Fe /MgO(001)  films  \nwith different capping layers of Pd and Cu. After disentangl ing the parasitic \ncontributions, such as two -magnon scattering  (TMS) , mosaicity,  and field-dragging \neffect, we unambiguously observed  that both local and non- local Gilbert damping are  \nisotropic  in Fe(001) plane , suggest ing that the pure spin current s absorption is \nindependent of Fe magnetization orientation  in the epitaxial  Pd/Fe heterostructure.  \nFirst principles calculation reveal s that the effective spin mixing conductance of Pd/Fe interface is nearly invariant for different magnetization directions in good \nagreement with the experimental observation s. These results offer a valuable insight \ninto the transmission and absorption of pure spin currents, and facilitate us to utilize  \nnext-generation spintronic devices.  \nPACS  number s: 72.25.Mk, 75.78.- n, 76.50.+g \n*Corresponding author  \nE-mail: zhcheng@iphy.ac.cn   \n \n  I. INTRODUCTION \nThe rapid development of spintronic devices inquires deeper understanding of \nthe magnetization relaxation mechanism1-3. The Gilbert damping factor, one of key \nparameter s in spin dynamics , characterizes the energy  transfer from the spin \nsubsystem to the lattice  and governs the magnetization  switching time and the critical \ncurrent density in spin transfer torque  devices4-6. Since the shape of Fermi surface \ndepends  on the orientation of the mag netization direction due to the spin- orbit \ninteraction, an anisotropic Gilbert damping  is expected  in single crystal ultrathin \nfilms7-10. Chen et al. discove red an anisotropic damping in the Fe/GaAs(001) ultrathin \nfilms where an robust interfacial spin -orbit field exists , due to GaAs substrate . The \nmagnitude of damping anisotropy , however, decreases with increasing Fe thickness , \nand disappears when the Fe thickness is larger than 1.9 nm11-13. \nBesides intrinsic Gilbert damping in ferromagnetic materials  (FM) , spin current s \nsink into heavy metal s (HM) or other magnetic layer s importing non -local Gilbert \ndamping in HM/FM bilayer s or spin valve structure  according to  spin pumping  \nmodel14-16. Although anisotropic magnetization relaxation in ferromagnetic \nmultilayers w as observed, it is debated whether the absorption of pure spin currents is \nanisotropic or isotropic  in ferromagnetic multilay ers17-21. This is because the \nfrequency - and angular -dependent ferromagnetic resonance (FMR) linewidth results \nare often  contaminated by parasitic contributions , such as two -magnon  scattering  \n(TMS), mosaicity, and field -dragging effect . Li et al. found that nearly isotropic \nabsorption of pure spin current in Co in Py1-xCux/Cu(5  nm)/Co(5 nm) trilayers using spin pumping technique22. Meanwhile, Baker et al. found an anisotropic absorption of \npure spin currents in Co 50Fe50/Cr/Ni 81Fe19 spin valves with variable Cr thickness, \nwhile the anisotropy is suppressed above the spin diffusion length23. Here, we \ninvestigated spin pumping and clarified the dependence of diverse  magnetic \nrelaxations on Fe  magnetic orientation using Vector Network Analyzer  ferromagnetic \nresonance  (VNA- FMR)  of epitaxial Fe /MgO(001)  films capped by Pd and Cu layers. \nSimple  FM/HM bilayers  would be a more convincing  candidate  to explore  the \nnon-local relaxation mechanism.  Exclu ding the misleading dragging effect and the \ndeceitful extrinsic terms,  we unambiguously observed that both local and non- local \nGilbert damping are isotropic  in Fe(001) plane . The i sotropic non- local Gilbert \ndamping suggest s that the pure spin current s abso rption is independent of Fe \nmagnetization orientation , which is supported by the first principle s calculation.  \nII. EXPERIMENTS  \nSample s were prepared in molecular beam epitaxy  chambers  with a basic \npressure-102 10× mbar24. Prior to deposition, MgO(001) substrate was annealed at 700 ℃  \nfor 2 hours, and then 6 nm Fe film was deposited on a MgO(001) substrate using \nelectron -beam gun , and finally 5 nm Pd w as covered on Fe films . The crystalline \nquality and epitaxial relationship was confirmed by high- resolution transmissio n \nelectron micro scopy (HRTEM), as shown in Fig . 1(a) and (b). It has been revealed \nthat the films were grown with the epitaxial relationship Pd(001)<110>||Fe(001)<100>||MgO(001)<110> (see the inset of Fig . 1(b)). For \ncomparison, Cu(3.5 nm)/Fe (6 nm)/MgO(001)  sample  was also  prepared. In-plane VNA- FMR measurements were performed by facing the sample down on employing \na co-planar waveguide (CPW) and recording the transmission coefficient S 2125-27. All \ndepositions and measurements were performed at room temperature.  \nIII. RESULTS AND DISCUSSI ON \nFig. 2(a) s hows schematically the  stacked sample and the measured \nconfigurat ion.  The representative FMR spectra at fixed frequency 13.4 GHz and \nvarious magnetic field angle s Hϕ are illustrated in Fig. 2(b). T he FMR signal  (the \ntransmission parameter S 21) is a superposition of symmetric and antisymmetric \nLorentzian  functions . The following  equation could be used to extract  the resonance \nfield H r and the resonance line width  H∆:  \n2\n21 0 22 22( / 2)( ) ( / 2)Re ( ) +( ) ( / 2) ( ) ( / 2)r\nrrH HH HSH SL DHH H HH H∆ − ∆= −− +∆ − +∆.            (1)  \nHere, Re S21, S0, H, L and D are the real part of transmission parameter,  the offset, the \nexternal  magnetic  field, the symmetric and antisymmetric magnitude , respectively25-27\n.  \nThe resonance frequency f  is given by Kittel formula28 \n0=2RR\nab f HHγµ\nπ                                                    (2)  \nwith2\n42 cos( ) (3 cos 4 ) / 4 sin ( 45 )R\nar MH M M d H H HH H ϕϕ ϕ ϕ = −++ + − −a, \n42 cos( ) cos 4 sin 2R\nb r MH M M HH H H ϕϕ ϕ ϕ= −+ − and \n02=out\nds\nsKHMMµ− . Here, γ and \n0µ are the gyromagnetic ratio and the vacuum permeability. H , H2, H4 and Ms are the \napplied  magnetic field, the uniaxial and  four-fold magnetic anisotropy field s and \nsaturation magnetization , respectively.  outK is the out -of-plane uniaxial magnetic \nanisotropy  constant. The equilibrium azimuthal angle of magnetization  Mϕis determined by the following equation:  \n42 sin( ) ( / 4)sin 4 ( / 2)cos 2 0r MH M M H HHϕϕ ϕ ϕ −+ + = .                    (3)  \nThe angular dependent FMR measurements were performed by rotating the \nsamples in plane while sweeping the applied magnetic field.  At a fixed frequen cy of \n13.4 GHz, the angular dependence of H r can be derived from Eq. (2) and plotted in \nFig. 2(c) and 2(d) for Fe/MgO(001) sample s capped by  Pd and Cu, respectively . It can \nbe seen clearly that the angular dependence of H r demonstrates a four -fold symmetry  \nand the values of 2=0H Oe, 4=625H Oe and 0 2.0dHµ=  T for Pd/ Fe/MgO(001)  \nand2=0H Oe, 4=625H Oe and 0 1.9dHµ= T for Cu /Fe/MgO(001) , respectivel y. \nCompar ing to the sample with Cu c apping l ayer, Pd/Fe interface modifies the \nout-of-plane uniaxial magnetic anisotropy, and has a negligible contribution to the \nin-plane  uniaxial magnetic anisotropy.  \nIn cont rast to the four -fold symmetry of H r, the angul ar dependence  of H∆for \nthe samples with Pd and Cu capping layers indicates  to be superposition of four-fold \nand quasi -eight -fold contributions , as shown in Fig. 3(a) and 3(b) , respectively . In fact, \nthe quasi -eight -fold broadening  also represent s a four-fold symmetry  with multiple \nextreme value point s. In the case of the sample with Pd capping layer, H∆exhibits \ntwo peaks around the  hard magnetization direction s Fe<1 10>, and the values of H∆ \nfor Fe<100>  and Fe<110 > direction s are almost the same (58 Oe). On the other hand, \na larger difference in the magnitude of  H∆ was observed along these two directions \nof Cu/Fe/MgO(001) sample , i.e. 71 Oe and 4 9 Oe for Fe<100> and Fe<110>  axes, \nrespectively . In order to understand the mechanism of anisotropic magnetic relaxation, we \nmust take both intrinsic and extrinsic contributions into account29-34. H∆ is follow ed \nby the expression32\n: \n_ =mosaicity TMS Gilbert dragging HH H H∆ ∆ +∆ +∆ .                              ( 4) \nThe first term denotes TMS, represent ing that a uniform prerecession magnon ( 0k=) \nis scattered into a degenerate magno n ( 0k≠) due to imperfect  crystal structure.  \nTherefore, the contribution of TMS to the linewidth reli es heavily on the symmetrical \ndistribution  of defects and manifest s anisotropic feature accordingly . The second  term \ndescribes the mo saicity contribution  in a film plane,  which is caused by a slightly  \nspread  of magnetic  parameters on a very large scale. The last term _ Gilbert draggingH∆ is \nthe Gilbert damping contribution with field -dragging . \nIn the case of Fe/ MgO( 001) epitaxial  film, the contribution of TMS to FMR \nlinewidth composes of numerous  two-fold and four -fold TMS channel s31-34, \nj,max 4 j,max j,max 2 j,maxcos ( ) cos 2( )TMS twofold M twofold fourfold M fourfold\njjH ϕϕ ϕϕ ∆ =Γ − +Γ − ∑∑ .          ( 5) \nHere, j,max\ntwofoldϕ  and j,max\nfourfoldϕ  represent angle of the maximum scattering rate  in \ntwo-fold and four -fold scatterings along the direction  j. However, the same values of \nH∆between Fe<100>  and Fe<110>  directions suggest that the TMS can be neglected \nin Pd/Fe/MgO(001 ) epitaxial film. On the other hand, the larger difference in the \nmagnitude of H∆ was observed along these two directions , suggesting  that either \nsignificant TMS contribution or anisotropic Gilbert damping exists in \nCu/Fe/MgO(001) s ample13, 32, 33. \nThe angular dependence of mosaicity contribution can be described as32, 34 =r\nmosaicity H\nHHH ϕϕ∂∆∆∂,                                                ( 6) \nwhere Hϕ∆  represents an in  plane variation  of mosaicity. 0mosaicityH∆=  Oe should \nbe hold along easy magnetization direction s and hard directions where =0r\nHH\nϕ∂\n∂. \nDue to magnetocrystalline anisotropy, magn etization would not always align at  \nthe direction of the applied field when the field  is weaker  than the saturation field. We \nevaluate the field -dragging effect during rotation of the sample or frequency -swept  \nbased on the numerical calculation  using Eq. ( 3). Fig. 4(a) shows  Hϕdependence on \nHϕ at 13.4 GHz. The relation  reveals a conspicuous  dragging effect with a  four-fold \nsymmetry. At 25Hϕ=a, HMϕϕ−  is as high as  12a. Fig. 4(b) sh ows Mϕ \ndependence on f  at various  Hϕ. When  the magnetic  field is applied along Fe<100> or \nFe<110>  directions , the magnetization is always aligned along the applied magnetic \nfield. However, there is a conspicuous  angle between  the magnetization and the \nmagnetic  field with the field along intermediate  axis. Owing to the angle between \nmagnetization and applied field , H∆ corresponding  to Gilbert contributio n with the \nfield-dragging could be disclose d according  to the following equations12, 13 \n_ = [Im( )]Gilbert draggingH χ ∆∆                                              ( 7) \nand 22 2[]Im( )( ) ()RR RR\ne f f ab a a ab s\nRR RR\na b ab e f f ab a bHH H H HH M\nH H HH HH H+ Haχa+=−+ ,                   (8) \nwhere aH and bH are R\naH  and R\nbH in non- resonance condition. The effective \nparameter  effa consist s of the intrinsic  Gilbert damping an d the non- local one driven \nby spin currents . \nGenerally , effa was obtained by the slope of the linear  dependence of H∆ on frequency f  along the directions without field -dragging  28: \n0\n04efffHHπa\nµγ∆ = +∆ ,                                                 (9)  \nwhere 0H∆ is inhomogeneous non- Gilbert linewidth at zero -frequency25-27. Fig. 5 \nshows  H∆  dependence on frequency at various Hϕ. Obviously , H∆ versus  f can \nbe fitted linear ly with  3\n/ 6.0 10Pd Fea−= ×  and 3\n/=4.2 10Cu Fea−×  for magnetic field \nalong easy axes Fe<1 00> or hard axes Fe<1 10> of the samples with Pd and Cu \ncapping layers,  respectively , indicating isotropic damping  (Fig. 5(f) and 5( j)). By \nusing the aforementioned isotropic damping factor s, the contributions  of TMS, \nmosaicity, and field -dragging effect  are separated from the angular  dependence of \nH∆ (Fig. 3 a-b). Table I summar izes the fitted parameters in the two  samples. \nCompared with Cu/Fe  sample , one observes  a significant reduction of mosaicity \nbroadening and a negligible  TMS term in Pd/Fe bilayers. In fact, due to high mobilit y, \nthe capping layer Cu forms nanocrystallites on Fe film, which causes interfacial \ndefects dependence on the crystallographic ax es35-38. The interfacial defects will \nimpact  a four-fold linewidth broadening  due to TM S. In contrast , the  excellent \nepitaxial quality  at Pd/Fe  interface not only ensures a sharp interfacial structure , but \nalso reduces defect density to decrease TMS contribution. Moreover, the mosaicity \ncontribution, indicat ing the fluctuation of the magne tic anisotropy  field, could be \nstrengthen  by the  interfacial  stacking faults. C onsequently , a fully epitaxial structure \ncould significantly decrease the extrinsic contributions, especially TMS and mosaicity  \nterms . \nTaking these contributions  to magnetizatio n relaxation into account, the frequency dependence of H∆ at various directions  can be well reproduced, as shown \nin Fig. 5(f)- (j). For other directions  rather than Fe<1 00> and Fe<1 10>, nonlinear \nrelationship between H∆ and f are evident  and illustrated in Fig . 5(g-i). At =20Hϕa, \nthe H∆ vs f curve brings out a slight bump comparing  to the linear  ones along hard \nor easy ax es. At =27Hϕa, H∆ has a rapid  decrease after H∆ experiencing an \nabrupt enhancement . At =33Hϕa, H∆ decreases more sharply after 11 GHz.  The \nnonlinearity can be ascrib ed to the parasitic contrib utions, such as  TMS, mosaicity, \nand field -dragging effect . It is virtually impossible  to stem from  TMS for the d istorted  \ncurves  because a nonlinear linewidth broadening due to TMS increases as frequency \nincreases, and approach es to saturation  at high frequency31. According to  the \ncalculation  in Fig. 4(b), there is a huge field -dragging effect except the applied \nmagnetic field H  along hard and  easy ax es. The field -dragging will make  H∆ vs f \ndeviate from the linear relationship . As expected , we could effectively fit  the \nexperimental data H∆ vs f using the following equation in association with the \noriginal formula s (7), \n0 [Im( )] HH χ ∆ =∆ +∆ .                                               ( 10) \nEq. (10) converges to the Eq. (9)  with the applied magnetic field along the directions \nwithout field -dragging, i.e. easy  axes Fe<1 00> or hard axes Fe<1 10>13. \nAfter distinguish ing the contributions  of extrinsic terms and field -dragging effect, \nthe Gilbert damping factors effa along various direc tions are  show n in Fig. 6(a). \nAccording to the classical spin pumping model14, precess ional magnetization in FM  \nlayer will pump spins into adjacent  nonmagnetic  metals  across interface. Cu with only s conduction band has a smaller spin- flip probability and a large r spin diffusion length \nthan 500 nm39, therefore, the reference sample Cu/Fe cannot  increase the Gilbert \ndamping due to a capp ing layer Cu. In contrast , Pd-layer  with stron g spin- orbit \ncoupling has a larger spin- flip probability , the injected spin currents are dissipated in \nPd-layer , and enhance the intrinsic Gilbert damping  of Fe film. The enhancement of \nthe Gilbert damping allows us to comprehend the non- local relaxation m echanism.  \nObviously, it can be seen from Fig. 6(a)  that there is no strong relation between the \nnon-local  Gilbert damping and the magnetization orientation in epitaxial film Pd/Fe.  \nThe parameters-3\n/=4.2 10Cu Fea ×  and -3\n/=6.0 10Pd Fea ×  are the Gilbert damping of \nPd/Fe and Cu/Fe , respectively. The non-local Gilbert damping could be evaluated \nusing  the effective spin mixing conductance effg↑↓14 \n// =4B\nPd Fe Cu Fe eff\ns FeggMtµaa aπ↑↓∆ −= .                                     ( 11) \nThe obtained isotropic value  19 2=1.23 10effgm↑↓ −×  is comparable to the literature s40-42.  \nIn order to theoretically  investigate  the dependence of the non- local Gilbert \ndamping  on the magnetization orientation, the first principles calculation was \nperformed  to calculate the  total Gilbert damping of the Pd /Fe/Pd multilayer  on the \nbasis of the scattering theory43-45. The electronic structure of the Pd/Fe  interface was \ncalculated self -consistently using the surface Green’s function technique implemented \nwith the tight- binding linearized muffin -tin orbitals method. Within the atomic sphere \napproximation, the charge and spin densities and the effective Kohn- Sham potentials \nwere evaluat ed inside atomic spheres. The total Gilbert damping was  then calculated \nusing the scattering theory of magnetization dissipation45. We simulate d the room tempe rature via introducing frozen thermal lattice disorder  into a 5x5 lateral \nsupercell43. The root -mean -squared displacement of the atoms is determined by the \nDebye model with the Debye temperature 470 K.  A 28x 28 k-mesh is used to sample \nthe two -dimensional Brillouin zone and five different configurations of disorder have \nbeen calculated for each Fe thickness.  The total Gilbert damping exhibits a linear \ndependence on the length of Fe and the intercept of the linear function can be \nextracted corresponding to the contribution of the spin pumping at the Pd/Fe  \ninterface44. The interfacial  contribution i s converted to the effective spin mixing \nconductance, plotted in Fig. 6( b) as a function of the magnetization orientation. It can \nbe seen that the effective spin mixing conductance  across  Pd/Fe  interface \n19 2=1.29 10effgm↑↓ −×  is independent  of the magnet ization direction , and  is in very \ngood agreement with the experimental value19 21.23 10 m−× . According to the \nElliott- Yafet mechanism  in a nonmagnetic metal , spins  relax indiscriminately energy \nand momentum along all orientation in Pd-layer  since a cubic metal  is expected to \npossess a weak anisotropy of the Elliott -Yafet parameter46. Incidentally , the fitting \nerror will  mislead  an aniso tropic Gilbert damping if ones  use Eq. ( 9) to fit the entire  \ncurves  H∆ vs f. Besides , an epitaxial  magnetic film integrated into a pseudo spin \nvalve could lead to an anisotropic absorption of spin current based on s pin transfer \ntorque mechanism  since it is demanding  to drag magnetization parallel ing to the \napplied field11. \n  I V.  CONCLUSIONS  \nIn summary, a  non-local  Gilbert damping is induced by the spin pumping  in \nPd/Fe bilayers  as spin currents transfer  angular momentum into the Pd- layer . Due to \nstrong magnetocrystalline anisotropy, the field -dragging effect makes the line width \nversus frequency deviate from the linear relationship except magnetic field along hard \nor easy ax es. Extrinsic relaxation , such as TMS  and mosaicity, relies heavil y on \nmagnetization orientation.  Howeve r, an epitaxial interface could significantly \ndecrease  and minimize  the extrinsic contributions, especially TMS and mosaicity. It is \nnoteworthy that an isotropic non- local Gilb ert damping factor is clarified after ruling \nout the misleading field-dragging effect and the deceitful extrinsic contributions. \nMagnetization  orientation has a negligible contribution to the non- local Gilbert \ndamping based on both theoretical and experimental results , manifesting  that the \nabsorption of pure spin currents across  interface Pd(100)[110]/Fe(001)[100]  is \nindependent of Fe magnetization orientation. Our works provide deeper i nsight  into \nthe non- local Gilbert damping mechanism. \n  ACKNOWLEDGMENTS  \nThis work is supported by the National Key Research Program of China (Grant Nos.  \n2015CB921403, 2016YFA0300701, and 2017YFB0702702), the National Natural \nSciences Foundation of China (Grant Nos.  51427801,1187411 ,51671212, and \n11504413) and the Key Research Program of Frontier Sciences, CAS (Grant Nos. \nQYZDJ -SSW -JSC023, KJZD -SW-M01 and ZDYZ2012- 2). The work at Beijing \nNormal University is partly supported by the National Natural Sciences Foundation of \nChina (Grant Nos. 61774017, 61704018, and 11734004), the R ecruitment Program of \nGlobal Youth Experts and the Fundamental Research Funds for the Central Universities (Grant No. 2018EYT03).  REFERENCES  \n1. I. Zutic, J. Fabian and S. Das Sarma, Rev. Mod. Phys. 76 (2), 323 -410 (2004).  \n2. K. Ando, S . Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa and E. Saitoh, Phys. Rev. Lett. 101  \n(3), 036601 (2008).  \n3. J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back and T. Jungwirth, Rev. Mod. Phys. 87 (4), \n1213- 1260 (2015).  \n4. V. Kambersky, Czech. J. P hys. 26 (12), 1366- 1383 (1976).  \n5. A. B. Cahaya, A. O. Leon and G. E. W. Bauer, Phys. Rev. B 96 (14) (2017).  \n6. D. Thonig, Y . Kvashnin, O. Eriksson and M. Pereiro, Phys. Rev. Materials 2  (1) (2018).  \n7. D. Steiauf and M. Fähnle, Phys. Rev. B 72 (6) (2005).  \n8. J. Seib, D. Steiauf and M. Fähnle, Phys. 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The sample is placed on the CPW for FMR measurement, and \ncould be rotated in plane . (b) Typical  real FMR spectra of Pd/Fe at fixed frequency \n13.4 GHz at various magnetic field angle sHϕ. Magnetic field angle Hϕ dependen ce \nof the resonanc e field H r at a fixed frequency 13.4 GHz  for Pd/Fe (c) and Cu /Fe (d) . \nThe red curves  are fit to Kittel’s formula  (2). (In order to show clearly the tendency , \nwe show the data at 45 225Hϕ−≤≤aa,  the same below ) \nFig. 3 ( Color online)  The measured linewidth H∆ as a function of Hϕat 13.4 GHz \nfor Pd /Fe (a) and Cu/Fe (b). The line width H∆ is superimposed by several  terms, \nsuch as TMS, mosaicity and Gilbert contribution with field- dragging.  \nFig. 4 (Color online)  Field -dragging effect for Pd/Fe. (a) The green line denotes the \nequilibrium direction of magnetization as a function of magnetic field angleHϕ at \n13.4 GHz. T he red line indicates the misalignment between  the magnetization and the \napplied magnetic field according ly. (b) The equilibrium  direction of the magnetization \nin the frequency -swept mode at variousHϕ. \nFig. 5 ( Color online)  Frequency dependence of the resonance  field Hr (a-e) and \nfrequency  dependence of the resonance line width H∆ (f-j) for Pd/Fe at  variousHϕ. The blue solid squares  and curves  in (f) and (j) corresponding  to frequency \ndependence of H∆ at 0Hϕ=a and 45Hϕ=a for Cu/Pd. \nFig. 6 (Color online)  Angular dependent Gilbert damping and first principles \ncalculation. ( a) The opened and solid green squares represent the obtained Gilbert \ndamping for Pd/Fe and Cu/Fe films, respectively. The red and blue lines are guide to \nthe eyes. ( b) The experimental and calculated spin mixing conductance as a function \nof the orientation of the equilibrium magnetization.  \nTable I The fitted  magnetic anisotropy parameters and magnetic relaxation \nparameters in Pd /Fe and Cu/Fe films . \n \n  Fig.1  \n \n \n  \nFig. 2  \n \n  \nFig. 3  \n \n  \nFig. 4  \n \n  \nFig. 5  \n \nFig. 6  \n \n  \n \nTable I  The fitted m agnetic anisotropy parameters and magnetic relaxation \nparameters in Pd /Fe and Cu/Fe films  in Fig. 3.  \nSample   4H(Oe)   2H(Oe)  0 dHµ (T)   effa   100γ<>Γ (710Hz)   ϕ∆(deg.) \nPd/Fe      625       0        2.0     0.0060       0           0.23 \nCu/Fe      625       0        1.9     0.0042      58           1.26 \n \n  "    },    {        "title": "2206.04899v1.Spin_Pumping_into_Anisotropic_Dirac_Electrons.pdf",        "content": "Spin Pumping into Anisotropic Dirac Electrons\nTakumi Funato1;2, Takeo Kato3, Mamoru Matsuo2;4;5;6\n1Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan\n2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n3Institute for Solid State Physics, The University of Tokyo, Kashiwa, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan and\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: June 13, 2022)\nWe study spin pumping into an anisotropic Dirac electron system induced by microwave irra-\ndiation to an adjacent ferromagnetic insulator theoretically. We formulate the Gilbert damping\nenhancement due to the spin current \rowing into the Dirac electron system using second-order\nperturbation with respect to the interfacial exchange coupling. As an illustration, we consider the\nanisotropic Dirac system realized in bismuth to show that the Gilbert damping varies according to\nthe magnetization direction in the ferromagnetic insulator. Our results indicate that this setup can\nprovide helpful information on the anisotropy of the Dirac electron system.\nI. INTRODUCTION\nIn spintronics, spin currents are crucial in using elec-\ntrons' charge and spin. Spin pumping, the spin current\ngeneration of conduction electrons from nonequilibrium\nmagnetization dynamics at magnetic interfaces, is a pop-\nular method for generating and manipulating spin cur-\nrents. In previous experimental reports on spin pumping,\nthe enhancement of Gilbert damping in ferromagnetic\nresonance (FMR) was observed due to the loss of angu-\nlar momentum associated with the spin current injection\ninto the nonmagnetic layer adjacent to the ferromagnetic\nlayer1{9. Mizukami et al. measured the enhancement of\nthe Gilbert damping associated with the adjacent non-\nmagnetic metal. They reported that the strong spin-orbit\ncoupling in the nonmagnetic layer strictly a\u000bected the\nenhancement of the Gilbert damping3{5. Consequently,\nelectric detection by inverse spin Hall e\u000bect, in which the\ncharge current is converted from the spin current, led to\nspin pumping being used as an essential technique for\nstudying spin-related phenomena in nonmagnetic mate-\nrials10{24. Saitoh et al. measured electric voltage in a\nbilayer of Py and Pt under microwave application. They\nobserved that charge current converted because of inverse\nspin Hall e\u000bect from spin current injected by spin pump-\ning11.\nIn the \frst theoretical report on spin pumping, Berger\npredicted an increase in Gilbert damping due to the spin\ncurrent \rowing interface between the ferromagnetic and\nnonmagnetic layers25,26. Tserkovnyak et al. calculated\nthe spin current \rowing through the interface27{29based\non the scattering-matrix theory and the picture of adi-\nabatic spin pumping30{32. They introduced a complex\nspin-mixing conductance that characterizes spin trans-\nport at the interfaces based on spin conservation and no\nspin loss. The spin mixing conductance can represent\nthe spin pumping-associated phenomena and is quanti-\ntatively evaluated using the \frst principle calculation33.\nNevertheless, microscopic analysis is necessary to under-stand the detailed mechanism of spin transport at the in-\nterface34{44. It was clari\fed that spin pumping depends\non the anisotropy of the electron band structure and spin\ntexture. Spin pumping is expected to be one of the probes\nof the electron states41{44.\nBismuth has been extensively studied because of its at-\ntractive physical properties, such as large diamagnetism,\nlargeg-factor, high e\u000ecient Seebeck e\u000bect, Subrikov-de\nHaas e\u000bect, and de Haas-van Alphen e\u000bect45,46. The\nelectrons in the conduction and valence bands near the\nL-point in bismuth, which contribute mainly to the vari-\nous physical phenomena, are expressed as e\u000bective Dirac\nelectrons. Thus, electrons in bismuth are called Dirac\nelectrons45{47. The doping antimony to bismuth is known\nto close the gap and makes it a topological insulator48,49.\nBecause of its strong spin-orbit interaction, bismuth has\nattracted broad attention in spintronics as a high e\u000ecient\ncharge-to-spin conversion material50{55. The spin current\ngeneration at the interface between the bismuth oxide\nand metal has been studied since a signi\fcant Rashba\nMicrowaveDirac electron\nsystem\nInterfacial\nexchange\nFerromagnetic\ninsulator\nFIG. 1. Schematic illustration of a bilayer system composed\nof the Dirac electron system and ferromagnetic insulator. The\napplied microwave excited precession of the localized spin in\nthe ferromagnetic insulator and spin current is injected into\nthe Dirac electron system.arXiv:2206.04899v1  [cond-mat.mes-hall]  10 Jun 20222\nspin-orbit interaction appears at the interface56. The\nspin injection into bismuth was observed due to spin\npumping from yttrium iron garnet or permalloy57{59.\nNevertheless, microscopic analysis of spin pumping into\nbismuth has not been performed. The dependence of the\nspin pumping on the crystal and band structure of bis-\nmuth remains unclear.\nThis study aims at a microscopic analysis of spin in-\njection due to spin pumping into an anisotropic Dirac\nelectron system, such as bismuth, and investigates the\ndependence of spin pumping on the band structure. We\nconsider a bilayer system comprising an anisotropic Dirac\nelectron system and a ferromagnetic insulator where a\nmicrowave is applied (see Fig. 1). The e\u000bect of the inter-\nface is treated by proximity exchange coupling between\nthe Dirac electron spins and the localized spins of the\nferromagnetic insulator34{44. We calculate the Gilbert\ndamping enhancement due to spin pumping from the fer-\nromagnetic insulator into the Dirac electron system up to\nthe second perturbation of the interfacial exchange cou-\npling. For illustarion, we calculate the enhancement of\nthe Gilbert damping for an anisotropic Dirac system in\nbismuth.\nThis paper is organized as follows: Sec. II describes\nthe model. Sec. III shows the formulation of the Gilbert\ndamping enhancement and discuss the e\u000bect of the inter-\nfacial randomness on spin pumping. Sec. IV summarizes\nthe results and demonstration of the Gilbert damping\nenhancement in bismuth. Sec. V presents the conclu-\nsion. The Appendices show the details of the calcula-\ntion. Appendix A de\fnes the magnetic moment of elec-\ntrons in a Dirac electron system. Appendix B provides\nthe detailed formulation of the Gilbert damping modu-\nlation, and Appendix C presents the detailed derivation\nof Gilbert damping modulation.\nII. MODEL\nWe consider a bilayer system composed of an\nanisotropic Dirac electron system and a ferromagnetic\ninsulator under a static magnetic \feld. We evaluate a\nmicroscopic model whose Hamiltonian is given as\n^HT=^HD+^HFI+^Hex; (1)\nwhere ^HD,^HFI, and ^Hexrepresent an anisotropic Dirac\nelectron system, a ferromagnetic insulator, and an inter-\nfacial exchange interaction, respectively.\nA. Anisotropic Dirac system\nThe following Wol\u000b Hamiltonian models the\nanisotropic Dirac electron system46,47,50:\n^HD=X\nkcy\nk(\u0000~k\u0001v\u001a2+ \u0001\u001a3)ck; (2)where 2\u0001 (6= 0) is the band gap, cy\nk(ck) is the electrons'\nfour-component creation (annihilation) operator, and v\nis the velocity operator given by vi=P\n\u000bwi\u000b\u001b\u000bwith\nwi\u000bbeing the matrix element of the velocity operator.\n\u001b= (\u001bx;\u001by;\u001bz) are the Pauli matrices in the spin space\nand\u001a= (\u001a1;\u001a2;\u001a3) are the Pauli matrices specifying the\nconduction and valence bands.\nFor this anisotropic Dirac system, the Matsubara\nGreen function of the electrons is given by\ngk(i\u000fn) =i\u000fn+\u0016\u0000~~k\u0001\u001b\u001a2+ \u0001\u001a3\n(i\u000fn+\u0016)2\u0000\u000f2\nk; (3)\nwhere\u000fn= (2n+ 1)\u0019=\fis the fermionic Matsubara fre-\nquencies with nbeing integers, \u0016(>\u0001) is the chem-\nical potential in the conduction band ~kis de\fned by\n~k\u0001\u001b=~k\u000b\u001b\u000b=k\u0001v, and\u000fkis the eigenenergy given\nby\n\u000fk=p\n\u00012+ (~kiwi\u000b)2=q\n\u00012+~2~k2: (4)\nThe density of state of the Dirac electrons per unit cell\nper band and spin is givcen by\n\u0017(\u000f) =n\u00001\nDX\nk;\u0015\u000e(\u000f\u0000\u0015\u000fk); (5)\n=j\u000fj\n2\u00192~3s\n\u000f2\u0000\u00012\n\u00013det\u000bij\u0012(j\u000fj\u0000\u0001); (6)\nwherenDis the number of unit cells in the system and \u000bij\nis the inverse mass tensor near the bottom of the band,\nwhich characterize the band structure of the anisotropic\nDirac electron system:\n\u000bij=1\n~2@2\u000fk\n@ki@kj\f\f\f\f\nk=0=1\n\u0001X\n\u000bwi\u000bwj\u000b: (7)\nThe spin operator can be de\fned as\n^sq=X\nkcy\nk\u0000q=2sck+q=2; (8)\nsi=m\n\u0001Mi\u000b\u001a3\u001b\u000b;(i=x;y;z ); (9)\nwhereMi\u000bare the matrix elements of the spin magnetic\nmoment given as50,51\nMi\u000b=\u000f\u000b\f\r\u000fijkwi\fwj\r=2: (10)\nThe detailed derivation of the spin magnetic moment can\nbe found in Appendix A.\nB. Ferromagnetic insulator\nThe bulk ferromagnetic insulator under a static mag-\nnetic \feld is described by the quantum Heisenberg model\nas\n^HFI=\u00002JX\nhi;jiSi\u0001Sj\u0000g\u0016BhdcX\niSX\ni; (11)3\nFIG. 2. Relation between the original coordinates ( x;y;z ) and\nthe magnetization-\fxed coordinates ( X;Y;Z ). The direction\nof the ordered localized spin hSi0is \fxed to the X-axis.\u0012is\nthe polar angle and \u001eis the azimuthal angle.\nwhereJis an exchange interaction, gis g-factor of the\nelectrons,\u0016Bis the Bohr magnetization, and hi;jirepre-\nsents the pair of nearest neighbor sites. Here, we have in-\ntroduced a magnetization-\fxed coordinate ( X;Y;Z ), for\nwhich the direction of the ordered localized spin hSi0is\n\fxed to the X-axis. The localized spin operators for the\nmagnetization-\fxed coordinates are related to the ones\nfor the original coordinates ( x;y;z ) as\n0\n@Sx\nSy\nSz1\nA=R(\u0012;\u001e)0\n@SX\nSY\nSZ1\nA; (12)\nwhereR(\u0012;\u001e) =Rz(\u001e)Ry(\u0012) is the rotation matrix com-\nbining the polar angle \u0012rotation around the y-axisRy(\u0012)\nand the azimuthal angle \u001erotation around the z-axis\nRz(\u001e), given by\nR(\u0012;\u001e) =0\n@cos\u0012cos\u001e\u0000sin\u001esin\u0012cos\u001e\ncos\u0012sin\u001ecos\u001esin\u0012sin\u001e\n\u0000sin\u0012 0 cos \u00121\nA:(13)\nBy applying the spin-wave approximation, the spin op-\nerators are written as S\u0006\nk=SY\nk\u0006iSZ\nk=p\n2Sbk(by\nk) and\nSX\nk=S\u0000by\nkbkusing magnon creation/annihilation op-\nerators,by\nkandbk. Then, the Hamiltonian is rewritten\nas\n^HFI=X\nk~!kby\nkbk; (14)\nwhere ~!k=Dk2+~!0withD=zJSa2being the spin\nsti\u000bness and zbeing the number of the nearest neighbor\nsites, and ~!0=g\u0016Bhdcis the Zeeman energy.C. Interfacial exchange interaction\nThe proximity exchange coupling between the electron\nspin in the anisotropic Dirac system and the localized\nspin in the ferromagnetic insulator is modeled by\n^Hex=X\nq;k(Tq;k^s+\nqS\u0000\nk+ h.c.); (15)\nwhereTq;kis a matrix element for spin transfer through\nthe interface and ^ s\u0006\nq= ^sY\nq\u0006i^sZ\nqare the spin ladder\noperators of the Dirac electrons. According to the re-\nlation between the original coordinate ( x;y;z ) and the\nmagnetization-\fxed coordinate ( X;Y;Z ), the spin oper-\nators of the Dirac electrons are expressed as\n0\n@sX\nsY\nsZ1\nA=R\u00001(\u0012;\u001e)0\n@sx\nsy\nsz1\nA; (16)\nwhereR\u00001(\u0012;\u001e) =Ry(\u0012)Rz(\u0000\u001e) is given by\nR\u00001(\u0012;\u001e) =0\n@cos\u0012cos\u001ecos\u0012sin\u001e\u0000sin\u0012\n\u0000sin\u001e cos\u001e 0\nsin\u0012cos\u001esin\u0012sin\u001ecos\u00121\nA:(17)\nThe spin ladder operators are given by\ns+=m\n\u0001aiMi\u000b\u001b\u000b; s\u0000=m\n\u0001a\u0003\niMi\u000b\u001b\u000b; (18)\nwhereai(i=x;y;z ) are de\fned by\n0\n@ax\nay\naz1\nA=0\n@\u0000sin\u001e+isin\u0012cos\u001e\ncos\u001e+isin\u0012sin\u001e\nicos\u00121\nA: (19)\nIII. FORMULATION\nApplying a microwave to the ferromagnetic insulator\nincludes the localized spin's precession. The Gilbert\ndamping constant can be read from the retarded magnon\nGreen function de\fned by\nGR\nk(!) =\u0000i\n~Z1\n0dtei(!+i\u000e)th[S+\nk(t);S\u0000\nk]i; (20)\nwithS+\nk(t) =ei^HT=~S+\nke\u0000i^HT=~being the Heisenberg\nrepresentation of the localized spin, since one can prove\nthat the absorption rate of the microwave is proportional\nto ImGR\nk=0(!) (see also Appendix B). By considering the\nsecond-order perturbation with respect to the matrix el-\nement for the spin transfer Tq;k, the magnon Green func-\ntion is given by34{44\nGR\n0(!) =2S=~\n(!\u0000!0) +i(\u000b+\u000e\u000b)!: (21)\nHere, we introduced a term, i\u000b!, in the denominator\nto express the spin relaxation within a bulk FI, where4\n\u000bindicates the strength of the Gilbert damping. The\nenhancement of the damping, \u000e\u000b, is due to the adjacent\nDirac electron system, calculated by\n\u000e\u000b=2S\n~!X\nqjTq;0j2Im\u001fR\nq(!); (22)\nwhere\u001fR\nq(!) is the retarded component of the spin sus-\nceptibility (de\fned below). We assume that the FMR\npeak described by Im GR\nk=0(!) is su\u000eciently sharp, i.e.,\n\u000b+\u000e\u000b\u001c1. Then, the enhancement of the Gilbert damp-\ning can be regarded as almost constant around the peak\n(!'!0), allowing us to replace !in\u000e\u000bwith!0.\nThe retarded component of the spin susceptibility for\nthe Dirac electrons:\n\u001fR\nq(!) =i\n~Z1\n\u00001dtei(!+i\u000e)t\u0012(t)h[s+\nq(t);s\u0000\n\u0000q]i: (23)\nThe retarded component of the spin susceptibility is\nderived from the following Matsubara Green function\nthrough analytic continuation i!l!~!+i\u000e:\n\u001fq(i!l) =Z\f\n0d\u001cei!l\u001ch^s+\nq(\u001c)^s\u0000\n\u0000qi; (24)\nwhere!l= 2\u0019l=\f is the bosonic Matsubara frequency\nwithlbeing integers. According to Wick's theorem,\nthe Matsubara representation of the spin susceptibility\nis given by\n\u001fq(i!l)\n=\u0000\f\u00001X\nk;i\u000fntr[s+gk+q(i\u000fn+i!l)s\u0000gk(i\u000fn)];(25)\nwhereP\ni\u000fnindicates the sum with respect to the\nfermionic Matsubara frequency, \u000fn= (2\u0019+ 1)n=\f. The\nimaginary part of the spin susceptibility is given by\nIm\u001fR\nq(!) =\u0000\u0019F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015\n\u0002h\nf(\u00150\u000fk+q)\u0000f(\u0015\u000fk)i\n\u000e(~!\u0000\u00150\u000fk+q+\u0015\u000fk);(26)\nwheref(\u000f) = (e\f(\u000f\u0000\u0016)+ 1)\u00001is the Fermi distribution\nfunction,\u0015=\u0006is a band index (see Fig. 3), and F(\u0012;\u001e)\nis the dimensionless function which depends on the di-\nrection of the ordered localized spin, de\fned by\nF(\u0012;\u001e) =\u00122m\n\u0001\u00132X\n\u000baiMi\u000ba\u0003\njMj\u000b: (27)\nFor detailed derivation, see Appendix C.\nIn this paper, we model the interfacial spin transfer as a\ncombination of the clean and dirty processes. The former\ncorresponds to the momentum-conserved spin transfer\nand the latter to the momentum-nonconserved one41,44.\nBy averaging over the position of the localized spin at\nFIG. 3. Schematic illustration of the band structure of the\nanisotropic Dirac electron system. The red band represents\nthe conduction band with \u0015= +, and the blue band repre-\nsents the valence band with \u0015=\u0000. The chemical potential\nis in the conduction band.\nthe interface, we can derive the matrix elements of the\ninterfacial spin-transfer process as\njTq;0j2=T2\n1\u000eq;0+T2\n2; (28)\nwhereT1andT2are the averaged matrix elements con-\ntributing to the clean and dirty processes, respectively.\nThen, the enhancement of the Gilbert damping is given\nby\n\u000e\u000b=2S\n~!F(\u0012;\u001e)n\nT1Im ~\u001fR\nuni(!0) +T2Im ~\u001fR\nloc(!0)o\n;\n(29)\nwhere\u001fR\nuni(!) and\u001fR\nuni(!) are the local and uniform spin\nsusceptibilities de\fned by\n~\u001fR\nloc(!0) =F\u00001(\u0012;\u001e)X\nq\u001fR\nq(!0); (30)\n~\u001fR\nuni(!0) =F\u00001(\u0012;\u001e)\u001fR\n0(!0); (31)\nrespectively. From Eq. (26), their imaginary parts are\ncalculated as\nIm ~\u001fR\nloc(!0) =\u0000\u0019n2\nDZ\nd\u000f\u0017(\u000f)\u0017(\u000f+~!0)\n\u0002\u00141\n2+2\u00012+\u000f2\n6\u000f(\u000f+~!0)\u0015h\nf(\u000f+~!0)\u0000f(\u000f)i\n;\n(32)\nIm ~\u001fR\nuni(!0) =\u0000\u0019nD\u0017\u0000~!0\n2\u0001~2!2\n0\u00004\u00012\n3~2!2\n0\n\u0002h\nf(~!0\n2)\u0000f(\u0000~!0\n2)i\n: (33)\nThe enhancement of the Gilbert damping, \u000e\u000b, depends\non the direction of the ordered localized spin through the5\nFIG. 4. FMR frequency dependence of the (a) local\nand (b) uniform spin susceptibilities. The local spin sus-\nceptibility is normalized by \u0019n2\nD\u00172\n0and scaled by 106, and\nthe uniform spin susceptibility is normalized by \u0019nD\u00170with\n\u00170\u00111=2\u00192~3p\ndet\u000bij. Note that kBis the Boltzmann con-\nstant. The line with kBT=\u0001 = 0:001 is absent in (a) because\nthe local spin susceptibility approaches zero at low tempera-\nture.\ndimensionless function F(\u0012;\u001e) regardless of the interfa-\ncial condition.\nBy contrast, the FMR frequency dependence of \u000e\u000bre-\n\rects the interfacial condition; for a clean interface, it is\ndetermined mainly by Im \u001fR\nuni(!0), whereas for a dirty\ninterface, it is determined by Im \u001fR\nloc(!0). The FMR fre-\nquency dependence of the local and uniform spin sus-\nceptibilities, Im \u001fR\nloc(!0) and Im\u001fR\nuni(!0), are plotted in\nFigs. 4 (a) and (b), respectively. The local and uniform\nspin susceptibilities are normalized by \u0019n2\nD\u00172\n0and\u0019nD\u00170,\nrespectively, where \u00170\u00111=2\u00192~3p\ndet\u000bijis de\fned. In\nthe calculation, the ratio of the chemical potential to the\nenergy gap was set to \u0016=\u0001'4:61, which is the value in\nthe bismuth46. According to Fig. 4 (a), the local spin sus-\nceptibility increases linearly with the frequency !in the\nlow-frequency region. This !-linear behavior can be re-\nproduced analytically for low temperatures and ~!\u001c\u0016:\nIm ~\u001floc(!0)'~!0\u0019\n2n2\nD[\u0017(\u0016)]2\u0014\n1 +2\u00012+\u00162\n3\u00162\u0015\n:(34)Fig. 4 (b) indicates a strong suppression of the uniform\nspin susceptibility below a spin-excitation gap ( !0<2\u0016).\nThis feature can be checked by its analytic form at zero\ntemperature:\nIm ~\u001fR\nuni(!0) =\u0019nD\u0017\u0000~!0\n2\u0001~2!2\n0\u00004\u00012\n3~2!2\n0\u0012(~!0\u00002\u0016):\n(35)\nThus, the FMR frequency dependence of the enhance-\nment of the Gilbert damping depends on the interfacial\ncondition. This indicates that the measurement of the\nFMR frequency dependence may provide helpful infor-\nmation on the randomness of the junction.\nIV. RESULT\nWe consider bismuth, which is one of the anisotropic\nDirac electron systems45,46,52,60,61. The crystalline struc-\nture of pure bismuth is a rhombohedral lattice with the\nspace group of R\u00163msymmetry, see Figs. 5 (a) and (b).\nIt is reasonable to determine the Cartesian coordinate\nsystem in the rhombohedral structure using the trigonal\naxis withC3symmetry, the binary axis with C2symme-\ntry, and the bisectrix axis, which is perpendicular to the\ntrigonal and binary axes. Hereafter, we choose the x-axis\nas the binary axis, the y-axis as the bisectrix axis, and\nthez-axis as the trigonal axis. Note that the trigonal, bi-\nnary, and bisectrix axes are denoted as [0001], [1 \u0016210], and\n[10\u001610], respectively, where the Miller-Bravais indices are\nused. The bismuth's band structure around the Fermi\nsurface consists of three electron ellipsoids at L-points\nand one hole ellipsoid at the T-point. It is well known\nthat the electron ellipsoids are the dominant contribu-\ntion to the transport phenomena since electron's mass\nis much smaller than that of the hole, see Fig. 5 (c).\nTherefore, the present study considers only the electron\nsystems at the L-points. The electron ellipsoids are sig-\nni\fcantly elongated, with the ratio of the major to minor\naxes being approximately 15 : 1. Each of the three elec-\ntron ellipsoids can be converted to one another with 2 \u0019=3\nrotation around the trigonal axis. The electron ellipsoid\nalong the bisectrix axis is labeled as e1, and the other\ntwo-electron ellipsoids are labeled e2 ande3. The in-\nverse mass tensor for the e1 electron ellipsoids is given\nby\n\u000b$\ne1=0\nB@\u000b10 0\n0\u000b2\u000b4\n0\u000b4\u000b31\nCA: (36)\nThe inverse mass tensor of the electron ellipsoids e2 and\ne3 are obtained by rotating that of e1 by 2\u0019=3 rotation6\nas below:\n\u000b$\ne2;e3=1\n40\nBB@\u000b1+ 3\u000b2\u0006p\n3(\u000b1\u0000\u000b2)\u00062p\n3\u000b4\n\u0006p\n3(\u000b1\u0000\u000b2) 3\u000b1+\u000b2\u00002\u000b4\n\u00062p\n3\u000b4\u00002\u000b4 4\u000b31\nCCA:\n(37)\nLet us express the dimensionless function F(\u0012;\u001e) rep-\nresenting the localized spin direction dependence of the\ndamping enhancement on the inverse mass tensors.\nF(\u0012;\u001e) =\u00122m\n\u0001\u00132X\n\u000bh\n(sin2\u001e+ sin2\u0012cos2\u001e)M2\nx\u000b\n+(cos2\u001e+ sin2\u0012sin2\u001e)M2\ny\u000b\n+ cos2\u0012(M2\nz\u000b\u0000sin 2\u001eMx\u000bMy\u000b)\n+ sin 2\u0012Mz\u000b(Mx\u000bcos\u001e+My\u000bsin\u001e)i\n: (38)\nHere, we use the following calculations:\nX\n\u000bM2\nx\u000b=\u00012\n4(\u000byy\u000bzz\u0000\u000b2\nyz)total=\u00012\n4m2\u0016\u0014?;(39)\nX\n\u000bM2\ny\u000b=\u00012\n4(\u000bzz\u000bxx\u0000\u000b2\nzx)total=\u00012\n4m2\u0016\u0014?;(40)\nX\n\u000bM2\nz\u000b=\u00012\n4(\u000bxx\u000byy\u0000\u000b2\nxy)total=\u00012\n4m2\u0016\u0014k;(41)\nX\n\u000bMi\u000bMj\u000b=\u00012\n4(\u000bik\u000bjk\u0000\u000bij\u000bkk)total= 0;(42)\nwherei;j;k are cyclic. (\u0001\u0001\u0001)totalrepresents the summa-\ntion of the contributions of the three electron ellipsoids,\nand \u0016\u0014k, \u0016\u0014?(>0) are the total Gaussian curvature of the\nthree electron ellipsoids normalized by the electron mass\nm, given by\n\u0016\u0014k= 3m2\u000b1\u000b2; (43)\n\u0016\u0014?=3\n2m2[(\u000b1+\u000b2)\u000b3\u0000\u000b2\n4]: (44)\nHence, the dimensionless function Fis given by\nF(\u0012) = (1 + sin2\u0012)\u0016\u0014?+ cos2\u0012\u0016\u0014k: (45)\nThe results suggest that the variation of the damping\nenhancement depends only on the polar angle \u0012, which is\nthe angle between the direction of the ordered localized\nspinhSi0and the trigonal axis. It is also found that the \u0012\ndependence of the damping enhancement originates from\nthe anisotropy of the band structure. The dimensionless\nfunctionF(\u0012) is plotted in Fig. 6 by varying the ratio\nof the total Gaussian curvatures x= \u0016\u0014?=\u0016\u0014k, which cor-\nresponds to the anisotropy of the band structure. Fig-\nure 6 shows that the \u0012-dependence of the damping en-\nhancement decreases with smaller xand the angular de-\npendence vanishes in an isotropic Dirac electron system\nBinaryBisectrixTrigonal\ne�e�e�(c)\nBinary(x)(a)\nBisectrix(y)Trigonal(z)\nBinaryBisectrixTrigonal(b)FIG. 5. (a) The rhombohedral lattice structure of bismuth.\nThex-axis,y-axis, andz-axis are chosen as the binary axis\nwithC2symmetry, the bisectrix axis, and the trigonal axis\nwithC3symmetry, respectively. The yellow lines represents\nthe unit cell of the rhombohedral lattice. (b) The rhombohe-\ndral structure viewed from the trigonal axis. (c) Schematic\nillustration of the band structure at the Fermi surface. The\nthree electron ellispoids at L-points are dominant contribu-\ntion to the spin transport.\nx= 1. Bismuth is known to have a strongly anisotropic\nband structure. The magnitude of the matrix elements of\nthe inverse mass \u000b1-\u000b4was experimentally determined as\nm\u000b1= 806,m\u000b2= 7:95,m\u000b3= 349, and m\u000b4= 37:6.\nThe total Gaussian curvatures are evaluated as46\n\u0016\u0014k'1:92\u0002104; (46)\n\u0016\u0014?'4:24\u0002105: (47)\nThe ratio of the total Gaussian curvature is estimated\nasx'22:1. Therefore, the damping enhancement is\nexpected to depend strongly on the polar angle \u0012in a bi-\nlayer system composed of single-crystalline bismuth and\nferromagnetic insulator. Conversely, the \u0012-dependence of\nthe damping enhancement is considered to be suppressed\nfor polycrystalline bismuth.\nThe damping enhancement is independent of the az-\nimuthal angle \u001e. Therefore, it is invariant even on ro-\ntating the spin orientation around the trigonal axis. The\nreason is that the azimuthal angular dependence of the\ndamping enhancement cancels out when the contribu-\ntions of the three electron ellipsoids are summed over,\nalthough each contribution depends on the azimuthal an-\ngle. The azimuthal angular dependence of the damping\nenhancement is expected to remain when strain breaks\nthe in-plane symmetry. Additionally, suppose the spin\ncan be injected into each electron ellipsoid separately,\ne.g., by interfacial manipulation of the bismuth atoms.\nIn that case, the damping enhancement depends on the\nazimuthal angle of the spin orientation of the ferromag-\nnetic insulator39. This may be one of the probes of the\nelectron ellipsoidal selective transport phenomena.7\n- /2\n0 /2\ntheta1.01.52.0damping_modulation\nFIG. 6. The \u0012-dependence of the damping enhancement\nfor di\u000berent x. The ratio of the total Gaussian curvatures\nx= \u0016\u0014?=\u0016\u0014krepresents the anisotropy of the band structure.\nThe blue line with x= 22:1 corresponds to the damping en-\nhancement in single-crystalline bismuth, and the other lines\ncorrespond to that in the weakly anisotropic band structure.\nAs can be seen from the graph, the \u0012-dependence of the damp-\ning enhancement decreases as the more weakly anisotropic\nband structure, and the angular dependence turns out to van-\nish in an isotropic Dirac electron system with x= 1.\nIt is also noteworthy that the damping enhancement\nvaries according to the ordered localized spin direction\nwith both clean and dirty interfaces; that is independent\nof whether momentum is conserved in interfacial spin\ntransport. Conversely, it was reported that the spin ori-\nentation dependence of the damping enhancement due to\nthe Rashba and Dresselhaus spin-orbit interaction turned\nout to vanish by interfacial inhomogeneity42,43.\nV. CONCLUSION\nWe theoretically studied spin pumping from a ferro-\nmagnetic insulator to an anisotropic Dirac electron sys-\ntem. We calculated the enhancement of the Gilbert\ndamping in the second perturbation concerning the prox-\nimity interfacial exchange interaction by considering\nthe interfacial randomness. For illustration, we calcu-\nlated the enhancement of the Gilbert damping for an\nanisotropic Dirac system realized in bismuth. We showed\nthat the Gilbert damping varies according to the polar\nangle between the ordered spin hSi0and the trigonal axis\nof the Dirac electron system whereas it is invariant in its\nrotation around the trigonal axis. Our results indicate\nthat the spin pumping experiment can provide helpful in-\nformation on the anisotropic band structure of the Dirac\nelectron system.\nThe Gilbert damping is invariant in the rotation\naround the trigonal axis because the contributions of each\nelectron ellipsoid depend on the in-plane direction of theordered spinhSi0. Nevertheless, the total contribution\nbecomes independent of the rotation of the trigonal axis\nafter summing up the contributions from the three elec-\ntron ellipsoids that are related to each other by the C3\nsymmetry of the bismuth crystalline structure. If the spin\ncould be injected into each electron ellipsoid separately,\nit is expected that the in-plane direction of the ordered\nlocalized spin would in\ruence the damping enhancement.\nThis may be one of the electron ellipsoid selective spin in-\njection probes. The in-plane direction's dependence will\nalso appear when a static strain is applied. A detailed\ndiscussion of these e\u000bects is left as a future problem.\nACKNOWLEDGMENTS\nThe authors would like to thank A. Yamakage and Y.\nOminato for helpful and enlightening discussions. The\ncontinued support of Y. Nozaki is greatly appreciated.\nWe also thank H, Nakayama for the daily discussions.\nThis work was partially supported by JST CREST Grant\nNo. JPMJCR19J4, Japan. This work was supported by\nJSPS KAKENHI for Grants (Nos. 20H01863, 20K03831,\n21H04565, 21H01800, and 21K20356). MM was sup-\nported by the Priority Program of the Chinese Academy\nof Sciences, Grant No. XDB28000000.\nAppendix A: Magnetic moment of electrons in Dirac\nelectron system\nIn this section, we de\fne the spin operators in the\nDirac electron systems. The Wol\u000b Hamiltonian around\nthe L point is given by HD=\u001a3\u0001\u0000\u001a2\u0019\u0001v, where\nvi=P\n\u000bwi\u000b\u001b\u000bwithwi\u000bbeing the matrix component\nof the velocity vectors and \u0019=p+e\ncAis the momen-\ntum operator including the vector potential. It is rea-\nsonable to determine the magnetic moment of electrons\nin an e\u000bective Dirac system as the coe\u000ecient of the Zee-\nman term. The Wol\u000b Hamiltonian is diagonalized by the\nSchrie\u000ber-Wol\u000b transformation up to v=\u0001 as below:\nei\u0018HDe\u0000i\u0018'\u0014\n\u0001 +1\n2\u0001(\u0019\u0001v)2\u0015\n\u001a3; (A1)\nwhere\u0018=\u001a1\n2\u0001\u0019\u0001vis chosen to erase the o\u000b-diagonal\nmatrix for the particle-hole space. We can proceed cal-\nculation as follows:\n(\u0019\u0001v)2=\u0019i\u0019jwi\u000bwj\f(\u000e\u000b\f+i\u000f\u000b\f\r\u001b\r);\n= (\u0019iwi\u000b)2+i\n2\u000f\u000b\f\r\u001b\r[\u0019\u0002\u0019]i\u000fijkwj\u000bwk\f;\n= \u0001\u0012\n\u0019\u0001\u000b\u0001\u0019+~e\nc\u0001Mi\u000b\u001b\u000bBi\u0013\n; (A2)\nwhere we used ( \u0019\u0002\u0019) =e~\ncir\u0002AandMi\u000bis de\fned as\nMi\u000b=1\n2\u000f\u000b\f\r\u000fijkwj\fwk\r: (A3)8\nFinally, we obtain\nei\u0018HDe\u0000i\u0018'\u0014\n\u0001 +\u0019\u0001\u000b$\u0001\u0019\n2\u0015\n\u0000Bi\u0016s;i; (A4)\nwhere\u0016s;iis a magnetic moment of the Dirac electrons\nde\fned as\n\u0016s;i=\u0000~e\n2c\u0001Mi\u000b\u001a3\u001b\u000b=\u0000~e\n2c\u0001Mi\u000b\u0012\n\u001b\u000b0\n0\u0000\u001b\u000b\u0013\n:\n(A5)\nIn the main text, we de\fned the spin operator sas the\nmagnetic moment \u0016sdivided by the Bohr magnetization\n\u0016B=~e=2mc, i.e.,\nsi=\u0000\u0016s;i\n\u0016B=m\n\u0001Mi\u000b\u0012\n\u001b\u000b0\n0\u0000\u001b\u000b\u0013\n: (A6)\nFor an isotropic Dirac system, the matrix component is\ngiven bywi\u000b=v\u000ei\u000band Eq. (A6) reproduces the well-\nknown form of the spin operator\ns=g\u0003\n2\u0012\n\u001b0\n0\u0000\u001b\u0013\n; (A7)\nwhereg\u0003= 2m=m\u0003is the e\u000bective g-factor with m\u0003=\n\u0001=v2being e\u000bective mass.\nAppendix B: Linear Response Theory\nIn this section, we brie\ry explain how the microwave\nabsorption rate is written in terms of the uniform spincorrelation function. The Hamiltonian of an external\ncircular-polarized microwave is written as\n^Hrf=\u0000g\u0016Bhrf\n2X\ni(S\u0000\nie\u0000i!t+S+\niei!t)\n=\u0000g\u0016BhrfpnF\n2(S\u0000\n0e\u0000i!t+S+\n0ei!t); (B1)\nwherehrfis an amplitude of the magnetic \feld of the\nmicrowave, S\u0006\nkare the Fourier transformations de\fned\nas\nS\u0006\nk=1pnFX\niS\u0006\nie\u0000ik\u0001Ri; (B2)\nandRiis the position of the locazed spin i. Using the lin-\near response theory with respect to ^Hrf, the expectation\nvalue of the local spin is calculated as\nhS+\n0i!=GR\n0(!)\u0002g\u0016BhrfpnF\n2; (B3)\nwhereGR\nk(!) is the spin correlation function de\fned in\nEq. (20). Since the microwave absorption is determined\nby the dissipative part of the response function, it is\nproportional to Im GR\n0(!), that reproduces a Lorentzian-\ntype FMR lineshape. As explained in the main text, the\nchange of the linewidth of the microwave absorption, \u000e\u000b,\ngives information on spin excitation in the Dirac system\nvia the spin susceptibility as shown in Eq. (22).\nAppendix C: Spin susceptibility of Dirac electrons\nIn this section, we give detailed derivation of Eq. (26). The trace part in Eq. (25) is calculated as\ntr[s+gk+q(i\u000fn+i!l)s\u0000gk(i\u000fn)] =[(i\u000fn+i!l+\u0016)(i\u000fn+\u0016) + \u00012]tr[s+s\u0000]\u0000tr[s+~(~k+~q)\u0001\u001bs\u0000~~k\u0001\u001b]\n[(i\u000fn+i!l+\u0016)2\u0000\u000f2\nk+q][(i\u000fn+\u0016)2\u0000\u000f2\nk]; (C1)\nwhere ( ~k+~q)\u0001\u001b= (k+q)\u0001v. Using the following relations\ntr[s+s\u0000] =\u00122m\n\u0001\u00132X\n\u000baiMi\u000ba\u0003\njMj\u000b; (C2)\ntr[s+~(~k+~q)\u0001\u001bs\u0000~~k\u0001\u001b] =\u00122m\n\u0001\u00132X\n\u000b(2aiMi\u000b~~k\u000ba\u0003\njMj\f~~k\f\u0000~2~k2aiMi\u000ba\u0003\njMj\u000b); (C3)\nthe spin susceptibility is given by\n\u001fq(i!l) =\u00002F(\u0012;\u001e)X\nk\f\u00001X\ni\u000fn(i\u000fn+i!l+\u0016)(i\u000fn+\u0016) + \u00012+~2~k2=3\n[(i\u000fn+i!l+\u0016)2\u0000\u000f2\nk+q][(i\u000fn+\u0016)2\u0000\u000f2\nk]; (C4)\nwhere we dropped the terms proportional to ~k\u000b~k\f(\u000b6=\f) because they vanish after the summation with respect\nto the wavenumber k. Here, we introduced a dimensionless function, F(\u0012;\u001e) = (2m=\u0001)2P\n\u000baiMi\u000ba\u0003\njMj\u000b, which9\ndepends on the direction of the magnetization of the FI. Representing the Matsubara summation as the following\ncontour integral, we derive\n\u001fq(i!l) =\u00002F(\u0012;\u001e)X\nkIdz\n4\u0019itanh\u0012\f(z\u0000\u0016)\n2\u0013z(z+i!l) + \u00012+~2~k2=3\n[(z+i!l)2\u0000\u000f2\nk+q][z2\u0000\u000f2\nk]; (C5)\n= 2F(\u0012;\u001e)X\nkIdz\n2\u0019if(z)z(z+i!l) + \u00012+~2~k2=3\n[(z+i!l)2\u0000\u000f2\nk+q][z2\u0000\u000f2\nk]; (C6)\nWe note that tanh( \f(z\u0000\u0016)=2) has poles at z=i\u000fn+\u0016and is related to the Fermi distribution function f(z) as\ntanh[\f(z\u0000\u0016)=2] = 1\u00002f(z). Using the following identities\n1\nz2\u0000\u000f2\nk=1\n2\u000fkX\n\u0015=\u0006\u0015\nz\u0000\u0015\u000fk; (C7)\nz\nz2\u0000\u000f2\nk=1\n2X\n\u0015=\u00061\nz\u0000\u0015\u000fk; (C8)\nthe spin susceptibility is given by\n\u001fq(i!l) =F(\u0012;\u001e)X\nkIdz\n2\u0019if(z)X\n\u0015;\u00150=\u0006\"\n1\n2+(\u00012+~2~k2=3)\u0015\u00150\n2\u000fk\u000fk+q#\n1\nz\u0000\u0015\u000fk1\nz+i!l\u0000\u00150\u000fk+q; (C9)\n=F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015f(\u00150\u000fk+q)\u0000f(\u0015\u000fk)\ni!l\u0000\u00150\u000fk+q+\u0015\u000fk: (C10)\nBy the analytic continuation i!l=~!+i\u000e, we derive the retarded spin susceptibility as below:\n\u001fR\nq(!) =F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015f(\u00150\u000fk+q)\u0000f(\u0015\u000fk)\n~!+i\u000e\u0000\u00150\u000fk+q+\u0015\u000fk: (C11)\nThe imaginary part of the spin susceptibility is given by\nIm\u001fR\nq(!) =\u0000\u0019F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015h\nf(\u00150\u000fk+q)\u0000f(\u0015\u000fk)i\n\u000e(~!\u0000\u00150\u000fk+q+\u0015\u000fk): (C12)\nFrom this expression, Eqs. 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The Landau-Lifshitz-Gilbert-Slonczewski (LLGS ) equation\nisexactlysolvedfortheonespinexcitationintheabsenceofonsitea nisotropy\nfor the excitations of spin with fields perpendicular and parallel to th e chain.\nWe show the removal of damping in the spin excitations by appropriat ely\nintroducing current and also the enhancement of angular frequen cy of the\noscillations due to field-like torque in the case of both perpendicular a nd par-\nallel field. The exactness of the analytical results is verified by matc hing with\nnumerical counterparts. Further, we numerically confirm the exis tence of in-\nphase and anti-phase stable synchronized oscillations for two spin- excitations\nin the presence of current with perpendicular field and field-like torq ue. We\nalso show that the one-spin excitation is stable against thermal nois e and\ngets only slightly modified against thermal fluctuations.\nKeywords: Spin torque, Spin transfer nano-oscillator, LLGS equation,\nPT-symmetry, Spin excitations.\n1. Introduction\nBeing an important aspect in applied magnetism [1, 2], both from theo-\nretical and application points of view [3, 4], the study of dynamics of c lassical\nEmail address: avadh@lanl.gov (Avadh Saxena)\nPreprint submitted to Physica A August 22, 2021Heisenberg ferromagnetic spin chain with anisotropic interaction is o f funda-\nmental interest in the context of spin waves in arbitrarily shaped ma gnetic\nstructures [5] and spin-transfer torque in ferromagnetic layers [28]. Although\nthere are continuum cases which are completely integrable with solito n so-\nlutions, no discrete integrable case has been studied except for th e Ishimori\nlattice with a modified version [7]. Two of the present authors have ide ntified\na number of special solutions in the discrete spin chain case under va rious\nsituations such as with external magnetic field and onsite anisotrop y [8]. Un-\nderstanding such classes of solutions in these classical spin chain sy stems is\none of the important areas of investigations in spin dynamics.\nDue to its practical relevance [9, 10] the occurrence of localized b reathers\noroscillationsinferromagneticspinchainswithsuitableonsiteanisotr opyhas\nbeen studied for several years and recently two ofus andSubash have studied\ntheexcitationsofone, twoandthreespinsalongwiththeirlinearsta bilityand\nhave obtainedexplicit analytical solutions fortheHeisenberg anisot ropicspin\nchain in the presence of external magnetic field with onsite anisotro py [11].\nAlso, the two of us have studied the dynamics of one/many spin excit ations\nand the impact of spin current torque in an anisotropic Heisenberg f erromag-\nnetic spin chain in a constant/variable external magnetic field analyt ically\nand numerically. We have also proved analytically that the spin curren t can\nbalance the damping effect and have extended the study of such mo del to\nshow that in (parity-time reversal or) PT-symmetric magnetic nan ostruc-\ntures the gain/loss terms are canceled by the ferromagnetic coup ling which\nleads to spin oscillations [12].\nOn the other hand the spin transfer torque has been a promising ca n-\ndidate, starting from its discovery by Berger [13] and Slonczewski [14] the-\noretically as well as experimentally, for magnetization switching and m ag-\nnetization oscillations with their corresponding applications in the writeop-\neration for nanomagnetic memory storage [15] and microwave gene ration\n[16]. The dynamics of magnetization driven by spin transfer torque c an be\ninvestigated numerically and analytically by solving the governing Land au-\nLifshitz-Gilbert-Slonczewski (LLGS) equation [2, 17]. The role of an addi-\ntional torque, known as field-like torque [18, 19, 20], in magnetizat ion has\nbeen examined recently for its fruitful outcomes such as zero field oscillations\n[21, 22] and elimination of steady state motion in coupled spin torque o scilla-\ntors [23], respectively. While the spin transfer torque transfers s pin angular\nmomentum from a pinned layer to a free layer of the spin valve system , the\nfield-like torque arises due to the precession of spin polarized electr ons, from\n2the pinned layer, around the free layer’s magnetic moment. The field -like\ntorque finds technological applications in domain wall reflection [24] a s well\nas in magnetization uniformity in heavy metal/ferromagnetic metal/h eavy\nmetal layer structures [25].\nThe dynamics of the Heisenberg one dimensional discrete ferromag netic\nspin chain for the localized excitations of the one or more spins driven by\nspin-transfer torque in the presence of field-like torque has not b een studied\nyet to the best of our knowledge.\nInthispaperweanalyticallysolve theLLGSequationalongwithfield-like\ntorque for the components of spin for the one-spin excitation and numerically\nsolve for the two-spin excitations in the presence of field both para llel and\nperpendicular to the direction of spin chain. Also, we show the enhan cement\nof the angular frequency of the oscillations due to the field-like torq ue and\nanisotropy. Further, we identify the conditions among field, curre nt and\nthe magnitude of field-like torque to obtain the undamped oscillations for\ndifferent cases. We also confirm the stable nature of the one-spin e xcitations\nagainst thermal fluctuations.\nThe organization of the paper is as follows. In Sec. 2 we introduce th e\nHamiltonianmodelofthespinchainsystem. Wesolvetheone-spinexc itation\nin the presence of field in Sec. 3 and current and field-like torque in Se c. 4.\nIn Sec. 5 we solve the general case with perpendicular field, curren t and\nfield-like torque. In Sec. 6 we deduce the one-spin excitation for th e case\nof parallel field, current and field-like torque. We briefly study the t wo-spin\nexcitationsinthepresence ofperpendicular field, current andfield -like torque\nin Sec. 7. In Sec. 8, we discuss the influence of thermal noise on one -spin\nexcitation. Finally, in Sec. 9 we present our main conclusions.\n2. Model for spin chain\nThe Hamiltonian corresponding to the evolution of Nnumber of spins of\na one-dimensional anisotropic Heisenberg ferromagnetic spin chain is given\nby\nH=−N/summationdisplay\n{n}(ASx\nnSx\nn+1+BSy\nnSy\nn+1+CSz\nnSz\nn+1)−D/summationdisplay\nn(Sz\nn)2−H./summationdisplay\nnSn,(1)\n3whereSx\nn,Sy\nnandSz\nnare the spin components of the classical unit spin vector\n/vectorSn, satisfying the condition\n(Sx\nn)2+(Sy\nn)2+(Sz\nn)2= 1, n= 1,2,...,N. (2)\nHereA,BandCare the exchange interaction parameters, Dis the onsite\nanisotropy parameter and His the external magnetic field. The Landau-\nLifshitz equation of motion for the nthspin of the chain, specified by the\nHamiltonian (1), is deduced by introducing appropriate spin Poisson b racket\nrelations as[26]\ndSn\ndt=Sn×Heff+αSn×(Sn×Heff), n= 1,2,...,N, (3)\nwhereHeff=−δH/δSnis the effective field and αis the Gilbert damping\nparameter.\n3. One-spin excitation in the presence of perpendicular fiel d\nConsider a one-dimensional spin chain with the excitation of one spin S0\nas follows:\n.....(1,0,0),(1,0,0),(Sx\n0,Sy\n0,Sz\n0),(1,0,0),(1,0,0).... (4)\nThe Hamiltonian for this system, with external field H= (0,0,H) alongz\ndirection, is written from Eq.(1) as\nH=−[(N−3)A+2ASx\n0]−D(Sz\n0)2−HSz\n0. (5)\nThe effective field is derived from the Hamiltonian given in Eq.(5) as\nHeff= 2Aˆi+[2DSz\n0+H]ˆk, (6)\nwhereˆiandˆkare unit vectors along positive xandzdirections, respectively.\nBy substituting Eq.(6) in Eq.(3), the equations of motion for the exc ited spin\nare obtained as\ndSx\n0\ndt= 2DSy\n0Sz\n0+HSy\n0+α/bracketleftbig\n−2A(1−(Sx\n0)2)+2DSx\n0(Sz\n0)2+HSx\n0Sz\n0/bracketrightbig\n,(7)\ndSy\n0\ndt= 2ASz\n0−2DSx\n0Sz\n0−HSx\n0+α/bracketleftbig\n2ASx\n0Sy\n0+2DSy\n0(Sz\n0)2+HSy\n0Sz\n0/bracketrightbig\n,(8)\ndSz\n0\ndt=−2ASy\n0+α/bracketleftbig\n2ASx\n0Sz\n0−2DSz\n0(1−(Sz\n0)2)−H(1−(Sz\n0)2)/bracketrightbig\n.(9)\n4From Eqs.(7),(8) and (9), one can verify that\nSx\n0dSx\n0\ndt+Sy\n0dSy\n0\ndt+Sz\n0dSz\n0\ndt= 0, (10)\nto confirm that S2= (Sx\n0)2+(Sy\n0)2+(Sz\n0)2= constant = 1 is conserved. By\nconsidering the case where the onsite anisotropy is zero, we can wr ite the\ndynamical equations from Eqs.(7),(8) and (9), for the one-spin ex citation as\ndSx\n0\ndt= HSy\n0+α/bracketleftbig\n−2A(1−(Sx\n0)2)+HSx\n0Sz\n0/bracketrightbig\n, (11a)\ndSy\n0\ndt= 2 ASz\n0−HSx\n0+α[2ASx\n0Sy\n0+HSy\n0Sz\n0], (11b)\ndSz\n0\ndt= −2ASy\n0+α/bracketleftbig\n2ASx\n0Sz\n0−H(1−(Sz\n0)2)/bracketrightbig\n. (11c)\nFigure 1: (Color online) Undamped oscillations of (a) Sx\n0, (b)Sy\n0and (c)Sz\n0forA=\n0.1,D= 0,H= 0.1 andα= 0. Here the red lines and black dots are plotted from\nanalytical (Eq.(19)) and numerical (Eq.(11)) results, respective ly. The initial conditions\nare (0.10,0.00,0.99). (d) The three-dimensional trajectory of S0.\n5Figure 2: (Color online) Damped oscillations of (a) Sx\n0, (b)Sy\n0and (c)Sz\n0forA= 0.1,D=\n0,H= 0.1 andα= 0.005 plotted from Eq.(19). In the insets for the intermediate range\nof time, the red lines and black dots are plotted from analytical (Eq.( 19)) and numer-\nical (Eq.(11)) results, respectively. The initial conditions are (0.10 ,0.00,0.99). (d) The\nthree-dimensional trajectory of S0. The insets show the oscillations of the corresponding\ncomponents between t= 4000 and t= 4250.\nSince the coupled system of Eqs.(11) is difficult to solve as such, they are\ntransformedintermsofthestereographic complex variable ωanditscomplex\nconjugate ω∗as\nω=Sx\n0+iSy\n0\n1+Sz\n0, ω∗=Sx\n0−iSy\n0\n1+Sz\n0, (12)\nusing the following transformations,\nSx\n0=ω+ω∗\n1+ωω∗, Sy\n0=−iω−ω∗\n1+ωω∗, Sz\n0=1−ωω∗\n1+ωω∗, i=√\n−1,(13)\nas follows:\ndω\ndt=A(α−i)/bracketleftbigg\nω2+H\nAω−1/bracketrightbigg\n=A(α−i)(ω−ω+)(ω−ω−),(14)\n6whereω±= (1/2A)(−H±Ω), Ω =√\nH2+4A2. Eq.(14) can be exactly\nsolved as\nω(t) =ω+−C ω−eΩ(α−i)t\n1−C eΩ(α−i)t, ω∗(t) =ω+−C∗ω−eΩ(α+i)t\n1−C∗eΩ(α+i)t,(15)\nwhereCandC∗are arbitrary complex constants which can be obtained by\nusing Eq.(15) as\nC=Cr+iCi=ω+−ω(0)\nω−−ω(0), C∗=Cr−iCi=ω+−ω∗(0)\nω−−ω∗(0).(16)\nThe constants CrandCican be reexpressed using Eq.(12) as follows:\nCr=(H2−Ω2)(1+Sz\n0(0))2+4AH(1+Sz\n0(0))Sx\n0(0)+1−(Sz\n0(0))2\n4A2(ω−(1+Sz\n0(0)−Sx\n0(0))2+(Sy\n0(0))2),(17)\nCi=Ω(1+Sz\n0(0))Sy\n0(0)\nA(ω−(1+Sz\n0(0)−Sx\n0(0))2+(Sy\n0(0))2). (18)\nThe components of S0can be obtained by substituting Eqs.(15) and (16) in\nEqs.(13) as\nSx\n0=2/braceleftbig\nω+e−2αΩt+(H/A)e−αΩt[Crcos(Ωt)+Cisin(Ωt)]+ω−(C2\nr+C2\ni)/bracerightbig\n(1+ω2\n+)e−2αΩt−2(1+(H2−Ω2)/4A2)e−αΩt[Crcos(Ωt)+Cisin(Ωt)]+(1+ ω2\n−)(C2r+C2\ni),\n(19a)\nSy\n0=2(Ω/A)e−αΩt[Crsin(Ωt)−Cicos(Ωt)]\n(1+ω2\n+)e−2αΩt−2(1+(H2−Ω2)/4A2)e−αΩt[Crcos(Ωt)+Cisin(Ωt)]+(1+ ω2\n−)(C2r+C2\ni),\n(19b)\nSz\n0=(1−ω2\n+)e−2αΩt−2(1−(H2−Ω2)/4A2)e−αΩt[Crcos(Ωt)+Cisin(Ωt)]+(1−ω2\n−)(C2\nr+C2\ni)\n(1+ω2\n+)e−2αΩt−2(1+(H2−Ω2)/4A2)e−αΩt[Crcos(Ωt)+Cisin(Ωt)]+(1+ ω2\n−)(C2\nr+C2\ni).\n(19c)\nFrom Eqs.(19), we can observe that continuous oscillations are pos sible only\nin the absence of damping ( α= 0). Also, in the presence of damping, when\nt→ ∞\nSx\n0(∞) =2ω−\n1+ω2\n−, Sy\n0(∞) = 0, Sz\n0(∞) =1−ω2\n−\n1+ω2\n−.(20)\nEqs.(20) clearly show that in the presence of damping the external field\nenablesS0to reach the steady state in the xz-plane. The time period of the\n7oscillations can be determined as T= 2π/Ω. The spin excitations of Sx\n0,Sy\n0\nandSz\n0in the absence and presence of damping are plotted in Figs.1 and 2\nrespectively by using the expressions given in Eqs.(19). The dots co rrespond\nto the numerical results plotted from Eqs.(11). Fig.1(d) confirms t he closed\nperiodic oscillations of the components in the absence of damping.\n4. One-spin excitation in the presence of spin-transfer tor que and\nfield-like torque\nBy considering the one-dimensional spin chain in the free layer of a sp in-\nvalve (tri-layer) structure, the dynamics of the nthspin in the presence of\ncurrent is governed by the following LLGS equation[21, 27],\ndSn\ndt=Sn×Heff+αSn×(Sn×Heff)+jSn×(Sn×Sp)+j βSn×Sp,(21)\nwherejis the magnitude of the spin-transfer torque [28] which can also be\nequivalently called the damping-like torque [29] and βis the magnitude of\nthe field-like torque. Sp= (1,0,0) is the polarization vector of the pinned\nlayer. The equations of motion of S0in the absence of onsite anisotropy and\nperpendicular field are obtained by substituting Eq.(6) in Eq.(21) as\ndSx\n0\ndt=−2αA2(1−(Sx\n0)2), (22a)\ndSy\n0\ndt= 2A1Sz\n0+α[2A2Sx\n0Sy\n0], (22b)\ndSz\n0\ndt=−2A1Sy\n0+α2A2Sx\n0Sz\n0, (22c)\nwhereA1=A+(jβ/2) andA2=A+(j/2α). Eqs.(22) can be transformed\ninto stereographic form using Eqs.(13) as follows:\ndω\ndt=−i(A1+iαA2)(ω2−1). (23)\nEq.(23) can be solved as\nω=1+C e2(αA2−iA1)t\n1−Ce2(αA2−iA1)t, ω∗=1+C∗e2(αA2+iA1)t\n1−C∗e2(αA2+iA1)t, (24)\n8whereC=Cr+iCi=ω(0)−1\nω(0)+1andC∗=Cr−iCi=ω∗(0)−1\nω∗(0)+1are arbitrary\nconstants. CrandCiare obtained as\nCr=(Sx\n0(0)−Sz\n0(0)−1)(Sx\n0(0)+Sz\n0(0)+1)+( Sy\n0(0))2\n(Sx\n0(0)+Sz\n0(0)+1)2+(Sy\n0(0))2,(25)\nCi=2Sy\n0(0)(1+Sz\n0(0))\n(Sx\n0(0)+Sz\n0(0)+1)2+(Sy\n0(0))2. (26)\nBy substituting Eqs.(24) in Eqs.(13) we can get the components of S0in the\npresence of spin-transfer and field-like torques as\nSx\n0=e−2(2αA+j)t−(C2\nr+C2\ni)\ne−2(2αA+j)t+(C2r+C2\ni), (27a)\nSy\n0=2e−(2αA+j)t(Cicos[(2A+jβ)t]−Crsin[(2A+jβ)t])\ne−2(2αA+j)t+(C2\nr+C2\ni),(27b)\nSz\n0=−2e−(2αA+j)t(Crcos[(2A+jβ)t]+Cisin[(2A+jβ)t])\ne−2(2αA+j)t+(C2r+C2\ni).(27c)\nFromEq.(27a)onecanverifythatirrespective ofthefield-like torq ue(asymp-\ntotically for large t)S0approaches (-1,0,0) or (1,0,0) when 2 αA+j >0 or\n2αA+j <0, respectively. Also, thecurrent damps outthesystem even inth e\nabsence of damping. Further, the periodic oscillations appear when the con-\ndition 2αA+j= 0 is satisfied. From Eqs.(27b) and (27c) it can be observed\nthat the angular frequency of the oscillations is 2 A+jβ, which interestingly\nimplies that the current can enhance the frequency of the oscillatio ns only in\nthe presence of field-like torque.\n5. Dynamics of one-spin excitation in the combined presence of\nperpendicular field, spin-transfer torque and field-like to rque\nEquations of motion in the presence of perpendicular magnetic field H=\n(0,0,H), spin-transfer torque and field-like torque without onsite anisot ropy\ncan be written from Eq.(21) as\ndSx\n0\ndt=HSy\n0+α/bracketleftbig\n−2A2(1−(Sx\n0)2)+HSx\n0Sz\n0/bracketrightbig\n, (28a)\ndSy\n0\ndt= 2A1Sz\n0−HSx\n0+α[2A2Sx\n0Sy\n0+HSy\n0Sz\n0], (28b)\ndSz\n0\ndt=−2A1Sy\n0+α/bracketleftbig\n2A2Sx\n0Sz\n0−H(1−(Sz\n0)2)/bracketrightbig\n, (28c)\n9Figure 3: (Color online) Undamped oscillations for (a) Sx\n0, (b)Sy\n0and (c) Sz\n0when\nA= 0.1,D= 0, H= 0.1,α= 0.005, j=−0.00125016 and β= 0.1. Here the red\nlines and black dots are plotted from analytical (Eq.(34)) and numer ical (Eq.(28)) re-\nsults, respectively. The initial conditions are (0.10,0.40,0.91). (d) Th e three-dimensional\ntrajectory of S0.\nwhereA1=A+(jβ/2) andA2=A+(j/2α). Eqs.(28) are transformed into\na stereographic equation using Eqs.(13) as follows:\ndω\ndt= (A2α−iA1)/bracketleftbigg\nω2+H(1+iα)\n(A1+iαA2)ω−1/bracketrightbigg\n= (A2α−iA1)(ω−ω+)(ω−ω−), (29)\nwhere now\nω±=/braceleftBigg\nPr±/radicalBigg\n(Q2r+Q2\ni)1/2\n1+(Qi/2Qr)2/bracerightBigg\n+i/braceleftBigg\nPi±(Qi/2Qr)/radicalBigg\n(Q2r+Q2\ni)1/2\n1+(Qi/2Qr)2/bracerightBigg\n,\n(30)\n10and\nPr=−H(A2α2+A1)\n2(A2\n2α2+A2\n1), Pi=−αH(A1−A2)\n2(A2\n2α2+A2\n1),\nQr=H2(A2α2+A1)2−H2α2(A1−A2)2+4(A2\n2α2+A2\n1)2\n4(A2\n2α2+A2\n1),\nQi=2αH2(A2α2+A1)(A1−A2)\n4(A2\n2α2+A2\n1)2.\nBy solving Eq.(29), we get\nω(t) =ω+−C ω−e(K+iΩ)t\n1−C e(K+iΩ)t, ω∗(t) =ω∗\n+−C∗ω∗\n−e(K−iΩ)t\n1−C∗e(K−iΩ)t,(31)\nwhereω∗\n+andω∗\n−are complex conjugates of ω+andω−, respectively. Cand\nC∗can be derived from Eq.(31) as C=Cr+i Ci=ω+−ω(0)\nω−−ω(0),C∗=Cr−i Ci=\nω∗\n+−ω(0)∗\nω∗\n−−ω(0)∗, whereω(0) =Sx\n0(0)+i Sy\n0\n1+Sz\n0(0)andω(0)∗=Sx\n0(0)−i Sy\n0\n1+Sz\n0(0). HereKand Ω are\ngiven by\nK= 2/radicalBigg\n(Q2\nr+Q2\ni)1/2\n1+(Qi/2Qr)2[A2α+A1(Qi/2Qr)], (32)\nΩ = 2/radicalBigg\n(Q2\nr+Q2\ni)1/2\n1+(Qi/2Qr)2[A2α(Qi/2Qr)−A1]. (33)\nFrom Eqs.(31), the components of S0can be determined using Eq.(13) as\nSx\n0= 2/braceleftbiggT1+T2e−2Kt+2[T3sin(Ωt)−T4cos(Ωt)]e−Kt\nT5+T6e−2Kt+2[T7sin(Ωt)+T8cos(Ωt)]e−Kt/bracerightbigg\n,(34a)\nSy\n0= 2/braceleftbiggT9+T10e−2Kt+2[T11sin(Ωt)+T12cos(Ωt)]e−Kt\nT5+T6e−2Kt+2[T7sin(Ωt)+T8cos(Ωt)]e−Kt/bracerightbigg\n,(34b)\nSz\n0=/braceleftbiggT13+T14e−2Kt+2[T15sin(Ωt)+T16cos(Ωt)]e−Kt\nT5+T6e−2Kt+2[T7sin(Ωt)+T8cos(Ωt)]e−Kt/bracerightbigg\n,(34c)\nwhere the explicit forms of T1,T2,...,T16are given in the Appendix.\nFrom Eqs.(34) we can identify that S0damps out and reaches steady\nstate when K/negationslash= 0. The steady state values of Sx\n0,Sy\n0andSz\n0are given by\nSx\n0=T1\nT5, Sy\n0=T9\nT5, Sz\n0=T13\nT5,asttends to ∞whenK >0,\nSx\n0=T2\nT6, Sy\n0=T10\nT6, Sz\n0=T14\nT6,asttends to ∞whenK <0.\n11 0 0.6 1.2\n-1 -0.5  0  0.5  1Ω\nβ\nFigure 4: (Color online) Enhancement of angular frequency of unda mped oscillations by\nthe field-like torque when A= 0.1,D= 0,α= 0.005 and j= 0.1.\nWhenK= 0, the undamped oscillations appear. The values of current,\nfield and field-like torque for which undamped oscillations are possible c an\nbe obtained from Eq.(32) as follows:\nA2α+A1(Qi/2Qr) = 0,\nH2+4A2/parenleftbig\nA12+α2A22/parenrightbig2\nA13+3α2A1A22+α4A23−α2A23= 0. (35)\nThe angular frequency of the undamped oscillations is derived by usin g\nEq.(35) in Eq.(33) as\nΩ =−2/bracketleftBigg\n(A1−A2)(A1+α2A2)/parenleftbig\nA12+α2A22/parenrightbig/radicalbig\nA12+4α2A22\nA13+3α2A1A22+α2(α2−1)A23/bracketrightBigg1/2\n.(36)\nThe existence of undamped oscillations is confirmed by plotting Sx\n0,Sy\n0and\nSz\n0in Figs.3(a), (b) and (c) respectively when H= 0.1, β= 0.1 andj=\n−0.00125. Also, Fig.3(d) shows the three-dimensional trajectory of S0of the\nundamped oscillations. The enhancement of angular frequency of u ndamped\noscillations by field-like torque, plotted from Eq.(36), is shown in Fig.4 w hen\nA= 0.1,D= 0,α= 0.005 and j= 0.1. Also, the enhancement of angular\nfrequency by the introduction of onsite anisotropy Din Eqs.(28) is shown\nappropriately in Fig.5 when A= 0.1,H= 0.1,α= 0.005,β= 0.1 and\nj=−0.01, from appropriate numerical analysis.\n12 0.4 0.8 1.2 1.6\n 0  0.4  0.8  1.2  1.6Ω\nD\nFigure 5: (Color online) Enhancement of frequency of undamped os cillations by the\nanisotropy DwhenA= 0.1,α= 0.005,β= 0.1,H= 0.1 andj=−0.01.\n6. One-spin excitation in the combined presence of parallel field,\nspin-transfer torque and field-like torque\nWhentheexternalmagneticfieldisappliedparalleltothechain, i.e. alo ng\nx-axis, the dynamical equations can be obtained from Eq.(21) by co nsidering\nH= (H,0,0) as\ndSx\n0\ndt= 2DSy\n0Sz\n0−[j+α(2A+H)](1−(Sx\n0)2)+2αDSx\n0(Sz\n0)2,(37a)\ndSy\n0\ndt= (2A+H+jβ)Sz\n0+[j+α(2A+H)]Sx\n0Sy\n0+2αDSy\n0(Sz\n0)2,(37b)\ndSz\n0\ndt=−(2A+H+jβ)Sy\n0+[j+α(2A+H)]Sx\n0Sz\n0−2αDSz\n0(1−(Sz\n0)2).\n(37c)\nThe above Eqs.(37) for D= 0 are transformed into a stereographic equation\nusing Eq.(13) as\ndω\ndt=−1\n2[α(2A+H)+j−i(2A+H+jβ)](1−ω2).(38)\nEq.(38) is solved as\nω=1+Ce[j+α(2A+H)−i(2A+H+jβ)]t\n1−Ce[j+α(2A+H)−i(2A+H+jβ)]t, ω∗=1+C∗e[j+α(2A+H)+i(2A+H+jβ)]t\n1−C∗e[j+α(2A+H)+i(2A+H+jβ)]t,\n(39)\n13whereC=Cr+iCi=ω(0)−1\nω(0)+1andC∗=Cr−iCi=ω∗(0)−1\nω∗(0)+1are arbitrary\nconstants. Here CrandCican be obtained by using Eqs.(12) as in Eqs.(25)\nand (26). By substituting Eqs.(39) into Eqs.(13) we can derive\nSx\n0=1−(C2\nr+C2\ni)e2[j+α(2A+H)]t\n1+(C2r+C2\ni)e2[j+α(2A+H)]t, (40a)\nSy\n0= 2e[j+α(2A+H)]t/braceleftbiggCicos([2A+H+jβ]t)−Crsin([2A+H+jβ]t)\n1+(C2r+C2\ni)e2[j+α(2A+H)]t/bracerightbigg\n,\n(40b)\nSz\n0=−2e[j+α(2A+H)]t/braceleftbiggCrcos([2A+H+jβ]t)+Cisin([2A+H+jβ]t)\n1+(C2r+C2\ni)e2[j+α(2A+H)]t/bracerightbigg\n.\n(40c)\nFrom Eqs.(40), one can understand that the spin Sx\n0switches asymptotically\n(t→ ∞) to +1 or −1 whenj+α(2A+H)<0 orj+α(2A+H)>0,\nrespectively. Theangularfrequencyoftheoscillationsisgivenby2 A+H+jβ,\nwhich implies that the angular frequency is independent of the curre nt in the\nabsence of field-like torque. Further, it can be noticed that the un damped\noscillations in the presence of parallel field are possible when the cond ition\nj+α(2A+H) = 0 is satisfied.\nManipulation of single electron spin states in solids is receiving much\nattention for quantum computing [30, 31], mainly for localized electro n spins\nin solids which show long relaxation and coherence times and their stat es can\nbe easily manipulated via microwave or radio frequency pulses [32]. Also ,\nsingle spin dynamics in a Heisenberg XXZ spin chain has been studied for a\nquantum transistor [33] and coherent manipulation of a single spin st ate by\nmicrowave pulses has been investigated [34].\nThe formation of localized spin excitations in a magnetic layer is exper-\nimentally possible. It has been proved that by means of antiferroma gnetic\ncoupling a reference layer with fixed magnetization direction can be f ormed\nfrom an oppositely magnetized pinned layer. Thus, it is possible to for m a\nferromagnetic layer with fixed direction of magnetization [35]. It ha s been\nexperimentally proved that by placing a nano-contact in this fixed lay er, the\nlocalized region of magnetization beneath it can be excited by passing a cur-\nrent [36, 37]. These works demonstrate the possibility of exciting loc alized\nspins without altering the spins outside of the localized region and red ucing\nthe number of spins by reducing the cross-sectional area of the n ano-contact.\n14Thus, thespin transfer torquecannot affect thespins other tha nthe localized\nspins.\n7. Two-spin excitation in the presence of perpendicular fiel d, spin-\ntransfer torque and field-like torque\nThe studies onone-spinexcitation canbeextended intomulti-spin ex cita-\ntionsingeneral. Inthissection we numerically study thetwo-spin exc itations\nin the presence of perpendicular field, current and field-like torque . The case\nof parallel field can also be similarly analyzed. Considering the one dimen -\nsional spin chain with the excitation of two spins S0andS1as follows,\n.....(1,0,0),(1,0,0),(Sx\n0,Sy\n0,Sz\n0),(Sx\n1,Sy\n1,Sz\n1),(1,0,0),(1,0,0).....,(41)\nthe Hamiltonian for this system, with perpendicular external field H=\n(0,0,H) along positive zdirection, is written from Eq.(1) as\nH=−[(N−4)A+ASx\n0+ASx\n0Sx\n1+ASx\n1+BSy\n0Sy\n1+CSz\n0Sz\n1]\n−D(Sz\n0)2−D(Sz\n1)2−HSz\n0−HSz\n1. (42)\nThe corresponding effective fields for the two spins S0andS1can be derived\nas\nHeff,S0=A(1+Sx\n1)ˆi+BSy\n1ˆj+[CSz\n1+H+2DSz\n0]ˆk, (43)\nHeff,S1=A(1+Sx\n0)ˆi+BSy\n0ˆj+[CSz\n0+H+2DSz\n1]ˆk. (44)\nThe LLGS equations corresponding to the spins Sn, n= 0,1,in the presence\nof field and current are given by\ndSn\ndt=Sn×Heff,Sn+αSn×(Sn×Heff,Sn)+jSn×(Sn×Sp)+j βSn×Sp.(45)\nThe corresponding dynamical equations for the components of S0andS1\n15Figure 6: (Color online) Undamped oscillations of S0(black line) and S1(red dots). (a)\nSx\n0,Sx\n1(b)Sy\n0,Sy\n1, (c)Sz\n0,Sz\n1and (d) magnetization trajectory when A= 0.1,B=\n0.1,C= 0.1,D= 0, H= 0.1414,α= 0.005, j=−0.001, β= 0.1. The initial conditions\nare (0.6,0.8,0.0).\nwithSp= (1,0,0) can be derived as\ndSx\n0\ndt=CSy\n0Sz\n1+2DSy\n0Sz\n0+HSy\n0−BSz\n0Sy\n1\n+α[−A(1−(Sx\n0)2)(1+Sx\n1)+BSx\n0Sy\n0Sy\n1+CSx\n0Sz\n0Sz\n1+2DSx\n0(Sy\n0)2+HSx\n0Sz\n0]\n−j(1−(Sx\n0)2), (46a)\ndSy\n0\ndt=ASz\n0(1+Sx\n1)−CSx\n0Sz\n1−2DSx\n0Sz\n0−HSx\n0\n+α[ASx\n0Sy\n0(1+Sx\n1)−BSy\n1(1−(Sy\n0)2)+CSy\n0Sz\n0Sz\n1+2DSy\n0(Sz\n0)2+HSy\n0Sz\n0]\n+jSx\n0Sy\n0+jβSz\n0, (46b)\ndSz\n0\ndt=BSx\n0Sy\n1−ASy\n0(1+Sx\n1)\n+α[ASx\n0Sz\n0(1+Sx\n1)+BSy\n0Sz\n0Sy\n1−(CSz\n1+2DSz\n0+H)(1−(Sz\n0)2)]\n+jSx\n0Sz\n0−jβSy\n0, (46c)\n16-1 0 1\n 19500  19750  20000(a)\nS0x, S1x\nt-1 0 1\n 19500  19750  20000(b)\nS0y, S1y\nt-1 0 1\n 19500  19750  20000(c)\nS0z, S1z\nt\nFigure 7: (Color online) Anti-phase synchronized oscillations of S0(black line) and S1(red\nline). (a) Sx\n0,Sx\n1(b)Sy\n0,Sy\n1and (c) Sz\n0,Sz\n1whenA= 0.1,B= 0.1,C= 0.1,D=\n0, H= 0.1414,α= 0.005, j=−0.001, β= 0. The initial conditions are (0.6,0.8,0.0) and\n(0.61,0.79,0.0).\ndSx\n1\ndt=CSy\n1Sz\n0+2DSy\n1Sz\n1+HSy\n1−BSz\n1Sy\n0\n+α[−A(1−(Sx\n1)2)(1+Sx\n0)+BSx\n1Sy\n1Sy\n0+CSx\n1Sz\n1Sz\n0+2DSx\n1(Sy\n1)2+HSx\n1Sz\n1]\n−j(1−(Sx\n1)2), (47a)\ndSy\n1\ndt=ASz\n1(1+Sx\n0)−CSx\n1Sz\n0−2DSx\n1Sz\n1−HSx\n1\n+α[ASx\n1Sy\n1(1+Sx\n0)−BSy\n0(1−(Sy\n1)2)+CSy\n1Sz\n1Sz\n0+2DSy\n1(Sz\n1)2+HSy\n1Sz\n1]\n+jSx\n1Sy\n1+jβSz\n1, (47b)\ndSz\n1\ndt=BSx\n1Sy\n0−ASy\n1(1+Sx\n0)\n+α[ASx\n1Sz\n1(1+Sx\n0)+BSy\n1Sz\n1Sy\n0−(CSz\n0+2DSz\n1+H)(1−(Sz\n1)2)]\n+jSx\n1Sz\n1−jβSy\n1. (47c)\nEqs.(46) and (47) for the case D= 0 can be transformed into the stereo-\n17Figure 8: (Color online) Damped oscillations of (a) Sx\n0(b)Sy\n0and (c)Sz\n0forA= 0.1,B=\n0.1,C= 0.1,D= 0, H= 0.1414,α= 0.005, j=−0.001, β= 0. The initial conditions\nfor the two spins are (0.6,0.8,0.0). The insets for the intermediate ra nge of time show\nsynchronization of the respective components of the two spins.\ngraphic form as\ndω0\ndt=−A\n2(α−i)(1−ω2\n0)/parenleftbigg\n1+ω1+ω∗\n1\n1+ω1ω∗\n1/parenrightbigg\n−B\n2(α−i)(1+ω2\n0)/parenleftbiggω1−ω∗\n1\n1+ω1ω∗\n1/parenrightbigg\n+C(α−i)ω0/parenleftbigg1−ω1ω∗\n1\n1+ω1ω∗\n1/parenrightbigg\n+H(α−i)ω0, (48a)\ndω1\ndt=−A\n2(α−i)(1−ω2\n1)/parenleftbigg\n1+ω0+ω∗\n0\n1+ω0ω∗\n0/parenrightbigg\n−B\n2(α−i)(1+ω2\n1)/parenleftbiggω0−ω∗\n0\n1+ω0ω∗\n0/parenrightbigg\n+C(α−i)ω1/parenleftbigg1−ω0ω∗\n0\n1+ω0ω∗\n0/parenrightbigg\n+H(α−i)ω1. (48b)\nEqs.(46) and (47) are numerically solved and the undamped in-phase\nsynchronized oscillations of spins S0andS1are plotted in Figs.6 when A=\n0.1,B= 0.1,C= 0.1,D= 0, H= 0.1414,α= 0.005, j=−0.001, β=\n180.1 for the same initial conditions (0.6,0.8,0.0). Interestingly, the two- spin\nsystem shows anti-phase synchronized oscillations when the initial c onditions\nare slightly different. Figs.7 show the undamped anti-phase synchro nized\noscillations of spins S0andS1whenA= 0.1,B= 0.1,C= 0.1,D= 0, H=\n0.1414,α= 0.005, j=−0.001, β= 0.1 for the different initial conditions\n(0.6,0.8,0.0) and (0.61,0.79,0.0). Damped oscillations in the absence of fie ld-\nlike torque are shown in Figs.8. Same results are obtained by solving th e\nsystem (48) as well.\n8. Effect of thermal noise on one-spin excitation in the prese nce of\nperpendicular field\n-1 0 1\n 0  50  100(a)\nS0x\nt(ns) 0 1\n 99  99.5  100(b)S0x\nt(ns)\n-1 0 1\n 99  99.5  100(c)\nS0y\nt(ns)\nFigure 9: (Color online) Numerically plotted temporal evolutions of (a ) & (b)Sx\n0and (c)\nSy\n0whenA= 1000 Oe, D=0,H= 1000 Oe (perpendicular field), α=0.005,j=-12.5016\n(-0.1043 mA) and β= 0.1. Here the red and black lines are plotted in the presence ( T=\n300 K) and absence ( T= 0 K) of thermal noise, respectively. The initial conditions are\n(0.10,0.40,0.91). (d) The three-dimensional trajectory of S0with (red line) and without\n(black line) the thermal noise.\n19Wealso investigate nowtheeffectofthermalfluctuationsonthedy namics\nof one-spin excitation. It is carried out by including the thermal field due to\nthermal noise in the effective field as follows [23]:\nHeff= 2Aˆi+[2DSz\n0+H]ˆk+Hth, (49)\nwhere the thermal field is given by\nHth=√\nFG, F=2αkBT\n(1+α2)Msµ0V△t. (50)\nIn the above equation, Gis the Gaussian random number generator vector\nof the oscillator with components ( Gx,Gy,Gz), which satisfies the statistical\nproperties /angbracketleftGm(t)/angbracketright= 0 and/angbracketleftGm(t)Gn(t′)/angbracketright=δmnδ(t−t′) for allm,n=x,y,z.\nHerekBis the Boltzmann constant, Tis the temperature, Ms= 1448.4\nemu/cc is the saturation magnetization, µ0is the magnetic permeability in\nfree space, V = 2.5 ×64×64 nm3[17] is a typical volume of the free layer and\n△tis the step size of the time scale used in the simulation.\nThe temporal evolution of Sx\n0is plotted in Fig.9(a) for the time range t\n= 0 to 100 ns, where the black and red solid lines are plotted in the abse nce\n(T= 0 K) and presence ( T= 300 K) of thermal noise, respectively, for the\nparameters [21, 23] A= 1000 Oe, D=0,H= 1000 Oe (perpendicular field),\nα=0.005,j=-12.5016 (-0.1043 mA) and β= 0.1 (See Appendix B). In Fig.9\nwe observe that there is a slight variationin the oscillation boundary w ithout\nany change in the amplitude. The smooth oscillations even in the prese nce\nof thermal noise are confirmed by plotting the temporal evolutions ofSx\n0,Sy\n0\nandSz\n0in Figs.9(b), (c) and (d), respectively. From these figures we obse rve\nthat the thermal noise only very slightly affects the time evolution of the\nspin.\n9. Conclusions\nBy solving the LLGS equation along with field-like torque we have an-\nalytically deduced the expressions for one-spin excitation in the pre sence of\nperpendicular/parallel fields. It has been observed that the field- like torque\nis essential to enhance the frequency of the oscillations and it incre ases the\nfrequency of oscillations for both the cases of perpendicular and p arallel\nmagnetic fields. Relevant conditions have been obtained among the c urrent,\nmagnetic field and field-like torque to obtain the undamped oscillations . The\n20numerical study has been extended to the case of two-spin excita tions and\nthe possibility of undamped in-phase and anti-phase synchronized o scilla-\ntions has been shown between the two spins. The investigations on o ne-spin\nexcitation against thermal fluctuations show that the system is on ly slightly\naffected by thermal noise. Our results are potentially important fo r under-\nstanding the spin dynamics in relevant magnetic materials and struct ures\n[1, 5].\n21Appendix A\nIn this appendix, we provide the full expressions of various parame ters\nTi, i= 1, 2, ...,16, given in Eq.(34):\nT1= (C2\nr+C2\ni)(Pr/radicalbig\n1+(Qi/2Qr)2−(Q2\nr+Q2\ni)1/4),\nT2= (Q2\nr+Q2\ni)1/4+Pr/radicalbig\n1+(Qi/2Qr)2,\nT3=CiPr/radicalbig\n1+(Qi/2Qr)2−Cr(Q2\nr+Q2\ni)1/4(Qi/2Qr),\nT4=CrPr/radicalbig\n1+(Qi/2Qr)2+Ci(Q2\nr+Q2\ni)1/4(Qi/2Qr),\nT5= (C2\nr+C2\ni)/bracketleftbig\n−2(Q2\nr+Q2\ni)1/4(Pr+Pi(Qi/2Qr))\n+/radicalbig\n1+(Qi/2Qr)2(1+P2\nr+P2\ni+/radicalBig\nQ2r+Q2\ni)/bracketrightbigg\n,\nT6= 2(Q2\nr+Q2\ni)1/4(Pr+Pi(Qi/2Qr))\n+/radicalbig\n1+(Qi/2Qr)2(1+P2\nr+P2\ni+/radicalBig\nQ2\nr+Q2\ni),\nT7= 2Cr(Q2\nr+Q2\ni)1/4(Pi−Pr(Qi/2Qr))\n+2Ci/radicalbig\n1+(Qi/2Qr)2(1+P2\nr+P2\ni+/radicalBig\nQ2\nr+Q2\ni),\nT8= 2Ci(Q2\nr+Q2\ni)1/4(Pi−Pr(Qi/2Qr))\n−2Cr/radicalbig\n1+(Qi/2Qr)2(1+P2\nr+P2\ni+/radicalBig\nQ2r+Q2\ni),\nT9= (C2\nr+C2\ni)(Pi/radicalbig\n1+(Qi/2Qr)2−(Q2\nr+Q2\ni)1/4(Qi/2Qr)),\nT10= (Q2\nr+Q2\ni)1/4(Qi/2Qr)+Pi/radicalbig\n1+(Qi/2Qr)2,\nT11=Cr(Q2\nr+Q2\ni)1/4+CiPi/radicalbig\n1+(Qi/2Qr)2,\nT12=Ci(Q2\nr+Q2\ni)1/4−CrPi/radicalbig\n1+(Qi/2Qr)2,\nT13= (C2\nr+C2\ni)/bracketleftBig\n2(Q2\nr+Q2\ni)1/4(Pr+Pi/radicalbig\n1+(Qi/2Qr)2)\n−/radicalbig\n1+(Qi/2Qr)2(1−P2\ni−P2\nr−(Q2\nr+Q2\ni)1/4)/bracketrightBig\n,\nT14=−2(Q2\nr+Q2\ni)1/4(Pr+Pi/radicalbig\n1+(Qi/2Qr)2)\n+/radicalbig\n1+(Qi/2Qr)2(1−P2\ni−P2\nr−(Q2\nr+Q2\ni)1/4),\nT15= 2Cr(Q2\nr+Q2\ni)1/4(Pr(Qi/2Qr)−Pi)−Ci/radicalbig\n1+(Qi/2Qr)2(P2\ni+P2\nr−1−(Q2\nr+Q2\ni)),\nT16= 2Cr(Q2\nr+Q2\ni)1/4(Pr(Qi/2Qr)−Pi)+Ci/radicalbig\n1+(Qi/2Qr)2(P2\ni+P2\nr−1−(Q2\nr+Q2\ni)).\n22Appendix B: Comparison of numerical parameters with realis tic\nmaterial parameters\nHere we will briefly explain the procedure to deduce the expressions for\ncurrentjandthe coefficient of field-like torque βby comparing Eq. (21) with\nthe standard form of Landau-Lifshitz-Gilbert-Slonczewski (LLG S) equation\nutilized for the spin torque nano oscillator (STNO) that consists of a ferro-\nmagnetic free layer and pinned layer with a nonmagnetic conducting s pacer\nlayer which separates the ferromagnetic free and pinned layers.\nAs discussed in Sec. 4 the LLGS equation for a spin in the presence of\nperpendicular field, current and field-like torque is given by\ndS\ndt=S×Heff+αS×(S×Heff)+jS×(S×Sp)+j βS×Sp,(A.1)\nwhere\nHeff= 2Aˆi+Hˆk. (A.2)\nUsing the orthogonality relation S.dS\ndt= 0, one can deduce from Eq.(A.1) the\nfollowing equation:\nS×dS\ndt=S×(S×Heff)−αS×Heff−jS×Sp+jβS×(S×Sp).\n(A.3)\nFrom Eq.(A.3) we can derive,\nS×(S×Heff) =S×dS\ndt+αS×Heff+jS×Sp−jβS×(S×Sp).\n(A.4)\nBy substituting Eq.(A.4) in Eq.(A.1) we obtain,\ndS\ndt= (1+α2)S×Heff+αS×dS\ndt+j(1−αβ)S×(S×Sp)+j(α+β)S×Sp.\n(A.5)\nWith a rescaling of time t→ −γ\n1+α2t, we get\ndS\ndt=−γS×Heff+αS×dS\ndt−γj1−αβ\n1+α2S×(S×Sp)−γjα+β\n1+α2S×Sp,\n(A.6)\n23The standard form of LLGS equation used for studying the unit mag netiza-\ntion vector mof the free layer of the STNO is given by [21, 23]\ndm\ndt=−γm×H′\neff+αm×dm\ndt+γHsm×(m×mp)−γHsβ′S×mp,\n(A.7)\nwhere\nH′\neff= (Hx+Kxmx)ˆi+(Hy+Kymy)ˆj+[Hz+(Kz−Nz)mz]ˆk,(A.8)\nand\nHs=¯hηI\n2eMsV. (A.9)\nHereH′effis the effective field that includes the external fields Hx,Hyand\nHzalongx,yandzdirections, respectively, anisotropy fields Kx,Kyand\nKzalongx,yandzdirections, respectively, and demagnetization field Nz\nin the free layer, γis the gyromagnetic ratio, αis the damping constant,\nthe unit vector mp= (1,0,0) is along the polarization of the pinned layer\nandβ′is field-like torque, ¯ h(=h/2π) is the reduced Planck’s constant, η\nis the dimensionless parameter which determines the magnitude of th e spin\ntransfer torque, Iis the current flowing through the free layer, eis charge\nof the electron, Msis the saturation magnetization and Vis the volume of\nthe free layer. Here, the demagnetization field has been included on ly for\nz-direction since the normal of the free layer plane is along the z-direction.\nBy comparing Eqs.(A.6) and (A.7) we obtain the relations\nI=−2eMsVj\n¯hη/parenleftbigg1−αβ\n1+α2/parenrightbigg\n, β′=α+β\n1−αβ, (A.10)\nand similarly by comparing Eqs.(A.2) and (A.8) we get\nHx= 2A, Hy= 0, Hz=H, Kx= 0, Ky= 0, Kz−Nz= 0.(A.11)\nThe material parameters are adopted from Refs. [21, 23], and are given\nbyα= 0.005, |β′| ≤0.5 (which gives the condition -0.506 ≤β≤0.493),η\n= 0.54,Ms= 1448.3 emu/c.c., V= 2.5×64×64 nm3. To verify the impact\nof thermal fluctuations we have numerically plotted the temporal e volutions\nofSand spin trajectory using Eq.(A.7) in Figs.9 with ( T= 300 K) and\nwithout ( T= 0 K) the thermal noise for the choice of the parameters [21, 23]\nA= 1000Oe, D=0,H= 1000Oe(perpendicular field), α=0.005,j=-12.5016\n(I = -0.1043 mA) and β= 0.1.\n24Acknowledgements\nThe research work of ML and RA was supported by a DST-SERB Distin -\nguished Fellowship (No.: SERB/F/6717/2017-18). ML also wishes to t hank\nthe Center for Nonlinear Studies, Los Alamos National Laboratory , USA for\nits warm hospitality during his visit in the summer of 2019. This work was\nsupported in part by the U.S. Department of Energy.\nReferences\nReferences\n[1] B. Hillerbrands, K. Ounadjela, Spin Dynamics in Confined Magnetic\nStructures, vols. I & II, Springer, Berlin, 2002.\n[2] M. Lakshmanan, Philos. Trans. R. Soc. A 369 (2011) 1280.\n[3] B. Georges, V. Cros, and A. Fert, Phys. Rev. B 73 (2006) 0604 R.\n[4] Z. 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Zhou1,4\n1Surface Physics State Laboratory and Department of Physics , Fudan University, Shanghai 200433, China\n2Department of Applied Science, College of William and Mary, Williamsburg, Virginia 23185\n3Key Laboratory of Micro and Nano Photonic Structures (Minis try of Education) and\nDepartment of Optical Science and Engineering, Fudan Unive rsity, Shanghai 200433, China and\n4Shanghai Key Laboratory of Special Artificial Microstructu re Materials and\nTechnology &Physics Department, Tongji University, Shanghai 200092, C hina\n(Dated: November 6, 2018)\nWe have experimentally and theoretically investigated the dependence of the intrinsic Gilbert\ndamping parameter α0on the spin-orbital coupling strength ξby using L1 0ordered FePd 1−xPtx\nternary alloy films with perpendicular magnetic anisotropy . With the time-resolved magneto-optical\nKerr effect, α0is found to increase by more than a factor of ten when xvaries from 0 to 1.0. Since\nchanges of other leading parameters are found to be neglecte d, theα0has for the first time been\nproven to be proportional to ξ2.\nPACS numbers: 75.78.Jp; 75.50.Vv; 75.70.Tj; 76.50.+g\nMagnetization dynamics has currently become one\nof the most popular topic in modern magnetism due\nto its crucial importance in information storage. Real\nspace trajectory of magnetization processional switching\ntriggered by magnetic field pulses, fs laser pulses, and\nspin-polarized current1–6, can be well described by\nthe phenomenological Landau-Lifshitz-Gilbert (LLG)\nequation that incorporates the Gilbert damping term7\nwhich controls the dissipation of magnetic energy\ntowards the thermal bath. The time interval from the\nnon-equilibrium magnetization to the equilibrium state\nis governed by the Gilbert parameter α. It has very\nrecently been shown that the laser-induced ultrafast\ndemagnetization is also controlled by the α8.\nThe intrinsic Gilbert damping α0has been exten-\nsively studied in theory9–15, and in general believed\nto arise from the spin orbital coupling (SOC). In the\nSOC torque-correlation model proposed by Kambersk´ y,\ncontributions of intraband and interband transitions are\nthought toplay adominant rolein the α0at lowand high\ntemperatures Tand are predicted to be proportional to\nξ3(ξ=the SOC strength) and ξ2, respectively10,14. Up\nto date, however, no experiments have been reported\nto demonstrate the quantitative relationship between\nα0andξalthough many experimental attempts have\nbeen made to study the α0in various metallic and alloy\nfilms16–23. It is hard to rule out effects other than the\nSOC because α0is also strongly related to parameters\nsuch as the electron scattering time and density of state\nD(EF) at Fermi surface EF21,23,24which in turn change\namong various metals and alloys. In order to rigorously\naddress the ξdependence of α0in experiments, it is\ntherefore essential to find magnetic alloys in which the\nξcan be solely adjusted while other parameters almost\nkeep fixed.\nIn this Letter, we elucidate the ξdependence of α0by\nusing L1 0FePd1−xPtx(=FePdPt) ternary alloy films.\nHere, only ξis modulated artificially by the Pt/Pd\nconcentration ratio because heavier atoms are expectedto have a larger ξ27–29and parameters other than ξare\ntheoretically shown to be almost fixed. Experimental\nresults have shown that α0is proportional to ξ2. It is\ntherefore the first time to have given the experimental\nevidence of the ξ2scaling law. This work will also facili-\ntate exploration of new magnetic alloys with reasonably\nlarge perpendicular magnetic anisotropy (PMA) and low\nα.\nL10FePdPt ternary alloy films with 0 ≤x≤1.0 were\ndeposited on single crystal MgO (001) substrates by\nmagnetron sputtering. The FePdPt composite target\nwas formed by putting small Pt and Pd pieces on an\nFe target. During deposition, the substrates were kept\nat 500◦C. After deposition, the samples were annealed\nin situ at the same temperature for 2 hours. The base\npressure of the deposition system was 1 ×10−5Pa and\nthe Ar pressure was 0.35 Pa. Film thickness was deter-\nmined by X-ray reflectivity (XRR) to be 12 ±1 nm. In\norder to measure the Gilbert damping parameter α25,26,\ntime-resolved magneto-optical Kerr effect (TRMOKE)\nmeasurements were performed in a pump-probe setup\nusing a pulsed Ti:sapphire laser in the wavelength of\n400 nm (800 nm) for pump (probe) pulses with a pulse\nduration of 200 fs and a repetition rate of 250 kHz. An\nintense pump pulse beam with a fluence of 0.16 mJ/cm2\nwas normally incident to excite the sample, and the\ntransient Kerr signal was detected by a probe pulse\nbeam which is time-delayed with respect to the pump.\nThe intensity ratio of the pump to probe pulses was\nset to be about 6:1, and their respective focused spot\ndiameters were 1 mm and 0.7 mm. A variable magnetic\nfieldHup to 5 T was applied at an angle of 45 degrees\nwith respect to the film normal using a superconducting\nmagnet. TRMOKE measurements were performed at\n200 K and other measurements were performed at room\ntemperature.\nMicrostructural analysis was accomplished with the\naid of X-ray diffraction (XRD). Figures 1(a)-1(c) show\nthe XRD patterns for L1 0FePdPt films with x= 1,2\n/s50/s52 /s51/s50 /s52/s48 /s52/s56 /s53/s54 /s45/s51/s48 /s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48 /s51/s48/s70/s101/s80/s116\n/s32/s32\n/s40/s97/s41\n/s70/s101/s80/s116/s40/s100/s41\n/s32/s32\n/s32/s79/s85/s84\n/s32/s73/s78\n/s40/s98/s41 /s70/s101/s80/s100\n/s48/s46/s53/s80/s116\n/s48/s46/s53\n/s32\n/s70/s101/s80/s100\n/s48/s46/s53/s80/s116\n/s48/s46/s53/s40/s101/s41\n/s32\n/s70/s101/s80/s100\n/s48/s46/s55/s53/s80/s116\n/s48/s46/s50/s53/s40/s99/s41\n/s50 /s32/s40/s100/s101/s103/s114/s101/s101/s41\n/s32/s73/s110/s116/s101/s114/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41\n/s109/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41\n/s40/s102/s41\n/s70/s101/s80/s100\n/s48/s46/s55/s53/s80/s116\n/s48/s46/s50/s53\n/s70/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41\n/s32\nFIG. 1: XRD patterns(a, b, c), out-of-plane and in-plane\nhysteresis loops (d,e,f)for L1 0FePd1−xPtxfilms with x= 1\n(a,d),x= 0.5 (b,e) and x= 0.25 (c,f).\nx= 0.5, andx= 0.25, respectively. The films are\nof the L1 0ordered structure in the presence of (001)\nsuperlattice peak. The chemical ordering degree Scan\nbe calculated with the intensity of the (001) and (002)\npeaks and found to be 0 .7±0.1 for all samples. Since\nno other diffraction peaks exist except for (001) and\n(002) ones, all samples are of L1 0single phase with c\naxis perpendicular to the film plane. Here, c= 3.694˚A.\nMagnetization hysteresis loops were measured by vibrat-\ning sample magnetometer. Figures 1(d)-1(f) display the\ncorresponding out-of-plane and in-plane magnetization\nhysteresis loops. As shown in Fig.1(d), for x= 1\n(L10FePt) the out-of-plane hysteresis loop is almost\nsquare-shaped with coercivity HC= 3.8 kOe, indicating\nthe establishment of high PMA. With decreasing x, the\nHCdecreases. For x= 0.25 in Fig. 1(f), HCapproaches\nzero and the out-of-plane and in-plane loops almost\noverlap with each other, indicating a weak PMA. Ap-\nparently, the PMA increases with increasing x. Similar\nphenomena have been reported elsewhere28,29.\nFigure 2(b) displays the typical TRMOKE results\nfor L1 0FePdPt films with x= 0.25 under θH= 45oas\nshown in Fig.2(a). For the time delay longer than 5.0\nps, damped oscillatory Kerr signals are clearly seen due\nto the magnetization precession. The precession period\nbecomes short significantly with increasing H. In order\nto extract the precession frequency, the Kerr signal was\nfitted by following exponentially damped sine function,\na+bexp(−t/t0) +Aexp(−t/τ)sin(2πft+ϕ), where\nparameters A,τ,fandϕare the amplitude, relaxation\ntime, frequency, and phase of damped magnetization\nprecession, respectively30. Here,a,b, andt0correspond\nto the background signal owing to the slow recovery\nprocess. The experimental data are well fitted by the\nabove equation, as shown in Fig.2(b).\nFigure 3(a) shows that for all samples studied\nhere, the extracted precession frequency fincreases\nmonotonically as Hincreases. Moreover, fshows an\nFIG. 2: Schematic illustration of the TRMOKE geometry (a)\nand TRMOKE results for x= 0.25 under various magnetic\nfields (b). Here θH= 45◦. Curves are shifted for clarity. The\nsolid lines are fit results.\nincreasing tendency with increasing xat fixed H. For\nx= 1 (L1 0FePt),fis in a very high frequency range of\n180-260 GHz due to the high PMA. Figure 3(b) shows\nthat the relaxation time τdisplays a decreasing trend\nwith increasing H. Moreover, τincreases by two orders\nof magnitude when Pd atoms are replaced by Pt ones.\nIn particular, we observed the short relaxation time\nτ= 3 ps for x= 1 (L1 0FePt). When the oscillation\nperiod is longer than the relaxation time for low Hthe\nprecession cannot be excited for x= 131.\nWithα≪1.0, one can obtain the follow-\ning dispersion equation, 2 πf=γ√H1H2, where\nH1=Hcos(θH−θ) +HKcos2θandH2=\nHcos(θH−θ)+HKcos2θ, whereHK= 2KU/MS−4πMS\nwith uniaxial anisotropy constant KU. The equilibrium\nmagnetization angle θis calculated from the following\nequation sin2 θ= (2H/HK)sin(θH−θ), which is derived\nby taking the minimum of the total free energy. The\nmeasured Hdependence of fcan be well fitted, as shown\nin Fig.3(a). With the measured MSof 1100 emu/cm3,\ntheKUcan be calculated. The gfactor is equal to 2.16\nforx= 1, 0.7, and 0.5, and to 2.10 and 2.03 for x= 0.25\nand 0.15, respectively. A small fraction of the orbital3\n/s48/s56/s48/s49/s54/s48/s50/s52/s48/s51/s50/s48/s52/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s50/s48/s46/s52/s120/s61/s49/s32/s32/s32/s32 /s120/s61/s48/s46/s55/s32/s32\n/s120/s61/s48/s46/s53/s32 /s120/s61/s48/s46/s50/s53/s32/s32\n/s120/s61/s48/s46/s49/s53/s102/s32/s40/s71/s72/s122/s41/s40/s97/s41\n/s40/s98/s41/s32/s40/s110/s115/s41\n/s72/s32/s40/s107/s79/s101/s41\nFIG. 3: Uniform magnetization precession frequency f(a)\nand relaxation time τ(b) as a function of Hfor all samples\nstudied here. Solid lines refer to fit results.\n/s48/s46/s48/s48/s46/s49/s48/s46/s50/s49/s50/s51/s52\n/s32/s77/s101/s97/s115/s117/s114/s101/s100\n/s32/s67/s97/s108/s99/s117/s108/s97/s116/s101/s100/s40/s98/s41\n/s32/s32/s40/s97/s41/s75\n/s85 /s32/s40/s49/s48/s55\n/s32/s101/s114/s103/s47/s99/s109/s51\n/s41\n/s32\n/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s40/s99/s41\n/s32/s32 /s32/s40/s101/s86/s41\n/s120\nFIG. 4: Measured KU(a), measured (solid box) and calcu-\nlated (solid circles) α0(b),ξcalculated in this work (solid\ncircles) and elsewhere38(open ones) (c) as a function of x.\nThe lines serve as a visual guide in (b) and refer to the fit\nresults in (c).\nangular momentum is therefore restored by the SOC10\nand close to results reported elsewhere32.\nThe measured Hdependence of τcan be well fitted\nbyτ= 2/αγ(H1+H2) with the fitted values of g\nandHKforα≪1.0. Here, the Gilbert damping αis\nan adjustable parameter. As shown in Fig.3(b), the\nexperimental and fitted data coincide with each other\nat highHand exhibit significant deviation from each\nother at low H. It is therefore illustrated that the\nextrinsic magnetic relaxation contributes to the αat\nlowHand becomes weak at high H. This is because\nthe extrinsic magnetic relaxation may arise from the\ninhomogeneous PMA distribution and the interfacial\neffect and is greatly suppressed under high H33–35. The\nintrinsic α0therefore plays a dominant role at high H,\nthat is to say, α0is fitted in Fig. 3(b).\nTo determine the SOC strength ξand intrinsicdamping parameter α0in L1 0ordered FePd 1−xPtx\nternary alloys, we perform spin dependent first prin-\nciples calculations based on linear muffin-tin orbital\ndensity functional theorem, where the lattice constants\narea= 3.86˚Aandc= 3.79˚Afor L1 0ordered FePt. The\nD(EF) is 2.55, 2.47, 2.43, and 2.39 per atom per eV for\nxvarying from 0, 0.5, 0.75, to 1.0, respectively. The\nα0was achieved by using spin-orbital torque-correlation\nmodel based on spin dependent electron band structures\nobtained above9,13.\nIt is significant to compare variations of the PMA\nandα0. Figures 4(a) & 4(b) show the KUandα0both\ndecrease with decreasing x. Similar variation trends of\nKUandα0have been observed for perpendicularly mag-\nnetized Pt/Co/Pt multilayers30. When the ξis smaller\nthan the exchange splitting, the magnetic anisotropy is\nthought to come from the second order energy correction\nof the SOC in the perturbation treatment and is roughly\nproportional to both the ξand the orbital angular\nmomentum. The orbital momentum in 3 dmagnetic\nmetallic films restored by the SOC is also proportional\nto theξand the PMA therefore is proportional to ξ2/W\nwith the bandwidth of 3 delectrons W36. Since the W\ndoes not change much with x, the enhanced PMA at\nhighxis attributed to a larger ξof Pt atoms compared\nwith that of Pd atoms27,37. Our calculations show ξ\nchange from 0 .20, 0.26, 0.41 to 0.58 (eV) when xvarying\nfrom 0, 0 .5, 0.75, to 1.0, as shown in Fig. 4(c). This is\nbecause the ξis 0.6, 0.20, and 0.06 (eV) for Pt, Pd, and\nFe atoms, respectively27,38and the effect of Fe atoms is\nnegligible compared with those of Pd and Pt atoms. The\npresent results of ξare in good agreement with previous\nab initio calculations38. Apparently, the PMA behavior\narises from the increase of ξat highx. As shown in\nFig. 4(b), measured and calculated results of α0are in\na good agreement. Since the lattice constant, D(EF),\nthe Curie temperature, the gyromagnetic ratio, and\nthe averaged spin are experimentally and theoretically\nshown to almost not change with x, the enhanced α0is\nmainly attributed to the ξincrease with increasing x.\nFigure 5 showsthat the α0is approximatelyproportional\ntoξ2, where the ξvalues at other xare exploited from\nthe fitted curve in Fig. 4(c). Since for the present\nL10ordered FePd 1−xPtxternary alloy films only ξis\ntuned with x, the present work has rigorously proven\nthe theoretical prediction about the ξ2scaling of α09. It\nis indicated that the α0at 200 K is mainly contributed\nby the interband contribution10,12,14. The electronic-\nscattering-based model of ferromagnetic relaxation is\ntherefore proved to be applicable for the α0in L10\nFePdPt ternary alloys9. In order to further verify the\nξ3dependence of α014, measurements of magnetization\nprecession at low temperatures need to be accomplished.\nIn summary, we have investigated the magneti-\nzation dynamics in L1 0FePdPt ternary alloy films\nusing TRMOKE. The intrinsic α0can be continuously\ntuned, showing a decrease with increasing Pd content\ndue to smaller ξcompared with that of Pt atoms. In4\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s32/s32/s48/s32\n/s32/s40/s101/s86/s50\n/s41\nFIG. 5: The measured (solid square) and calculated (solid\ncircles)α0versusξ2as a function of x. The dashed curve\nrefers to the linear fit results.particular, the ξ2dependence of α0has been rigorously\ndemonstrated in experiments. 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Zhou, Chem-\nical Composition Tuning of the Anomalous Hall Ef-\nfect in Isoelectronic L1(0) FePd 1−xPtxAlloy Films ,\narXiv:1112.0834v1"    },    {        "title": "0805.1320v2.Spin_dynamics_in__III_Mn_V_ferromagnetic_semiconductors__the_role_of_correlations.pdf",        "content": "arXiv:0805.1320v2  [cond-mat.str-el]  25 Aug 2008Spin dynamics in (III,Mn)V ferromagnetic semiconductors: the role of correlations\nM. D. Kapetanakis and I. E. Perakis\nDepartment of Physics, University of Crete, and Institute o f Electronic Structure & Laser,\nFoundation for Research and Technology-Hellas, Heraklion , Crete, Greece\n(Dated: November 6, 2018)\nWe address the role of correlations between spin and charge d egrees of freedom on the dynamical\nproperties of ferromagnetic systems governed by the magnet ic exchange interaction between itiner-\nant and localized spins. For this we introduce a general theo ry that treats quantum fluctuations\nbeyond the Random Phase Approximation based on a correlatio n expansion of the Green’s function\nequations of motion. We calculate the spin susceptibility, spin–wave excitation spectrum, and mag-\nnetization precession damping. We find that correlations st rongly affect the magnitude and carrier\nconcentration dependence of the spin stiffness and magnetiz ation Gilbert damping.\nPACS numbers: 75.30.Ds, 75.50.Pp, 78.47.J-\nIntroduction— Semiconductors displaying carrier–\ninduced ferromagnetic order, such as Mn–doped III-V\nsemiconductors, manganites, chalcogenides, etc, have re-\nceived a lot of attention due to their combined magnetic\nand semiconducting properties [1, 2]. A strong response\nof their magnetic properties to carrier density tuning via\nlight, electrical gates, or current[3, 4, 5] canlead to novel\nspintronics applications [6] and multifunctional magnetic\ndevices combining information processing and storage on\na single chip. One of the challenges facing such magnetic\ndevices concerns the speed of the basic processing unit,\ndetermined by the dynamics of the collective spin.\nTwo key parameters characterize the spin dynam-\nics in ferromagnets: the spin stiffness, D, and the\nGilbert damping coefficient, α.Ddetermines the long–\nwavelength spin–wave excitation energies, ωQ∼DQ2,\nwhereQis the momentum, and other magnetic prop-\nerties.Dalso sets an upper limit to the ferromagnetic\ntransition temperature: Tc∝D[1]. So far, the Tcof\n(Ga,Mn)As has increased from ∼110 K [2] to ∼173 K\n[1, 7]. It is important for potential room temperature\nferromagnetism to consider the theoretical limits of Tc.\nTheGilbertcoefficient, α, characterizesthedampingof\nthe magnetization precession described by the Landau–\nLifshitz–Gilbert (LLG) equation [1, 8]. A microscopic\nexpression can be obtained by relating the spin suscepti-\nbility of the LLG equation to the Green’s function [9]\n≪A≫=−iθ(t)<[A(t),S−\nQ(0)]> (1)\nwithA=S+\n−Q,S+=Sx+iSy.∝angbracketleft···∝angbracketrightdenotes the\naverage over a grand canonical ensemble and SQ=\n1/√\nN/summationtext\njSje−iQRj, whereSjare spins localized at N\nrandomly distributed positions Rj. The microscopic ori-\ngin ofαisstill notfully understood[9]. Amean–fieldcal-\nculation of the magnetization damping due to the inter-\nplay between spin–spin interactions and carrier spin de-\nphasingwasdevelopedin Refs.[9, 10]. Themagnetization\ndynamics can be probed with, e.g., ferromagnetic res-\nonance [11] and ultrafast magneto–optical pump–probe\nspectroscopy experiments [5, 12, 13, 14]. The interpre-tation of such experiments requires a better theoretical\nunderstanding of dynamical magnetic properties.\nIn this Letter we discuss the effects of spin–charge cor-\nrelations, due to the p–d exchange coupling of local and\nitinerant spins, on the spin stiffness and Gilbert damp-\ningcoefficient. Wedescribequantumfluctuationsbeyond\nthe Random Phase Approximation (RPA) [15, 16] with a\ncorrelationexpansion[17]ofhigherGreen’sfunctionsand\na 1/S expansion of the spin self–energy. To O(1/S2), we\nobtain a strong enhancement, as compared to the RPA,\nof the spin stiffness and the magnetization damping and\na different dependence on carrier concentration.\nEquations of motion— The magnetic propertiescan be\ndescribedby the Hamiltonian [1] H=HMF+Hcorr, where\nthe mean field Hamiltonian HMF=/summationtext\nknεkna†\nknaknde-\nscribes valence holes created by a†\nkn, wherekis the mo-\nmentum, nis the band index, and εknthe band disper-\nsion in the presenceof the mean field created by the mag-\nnetic exchangeinteraction[16]. The Mn impurities act as\nacceptors, creating a hole Fermi sea with concentration\nch, and also provide S= 5/2 local spins.\nHcorr=βc/summationdisplay\nq∆Sz\nq∆sz\n−q+βc\n2/summationdisplay\nq(∆S+\nq∆s−\n−q+h.c.),(2)\nwhereβ∼50–150meV nm3in (III,Mn)V semiconductors\n[1] is the magnetic exchane interaction. cis the Mn spin\nconcentration and sq= 1/√\nN/summationtext\nnn′kσnn′a†\nk+qnakn′the\nhole spin operator. ∆ A=A− ∝angbracketleftA∝angbracketrightdescribes the quan-\ntum fluctuations of A. The ground state and thermo-\ndynamic properties of (III,Mn)V semiconductors in the\nmetallic regime ( ch∼1020cm−3) are described to first\napproximation by the mean field virtual crystal approxi-\nmation,HMF, justified for S→ ∞[1]. Most sensitive to\nthe quantum fluctuations induced by Hcorrare the dy-\nnamical properties. Refs.[9, 15] treated quantum effects\ntoO(1/S) (RPA). Here we study correlations that first\narise atO(1/S2). By choosing the z–axis parallel to the\nground state local spin S, we have S±= 0 and Sz=S.\nThe mean hole spin, s, is antiparallel to S,s±= 0 [1].2\nThe spin Green’s function is given by the equation\n∂t≪S+\n−Q≫=−2iSδ(t)+βc≪(s×S−Q)+≫\n−i∆≪s+\n−Q≫+βc\nN×\n/summationdisplay\nkpnn′≪(σnn′×∆Sp−k−Q)+∆[a†\nknapn′]≫,(3)\nwhere ∆ = βcSis the mean field spin–flip energy gap\nands= 1/N/summationtext\nknσnnfknis the ground state hole spin.\nfkn=∝angbracketlefta†\nknakn∝angbracketrightis the hole population. The first line on\nthe right hand side (rhs) describes the mean field pre-\ncession of the Mn spin around the mean hole spin. The\nsecond line on the rhs describes the RPA coupling to the\nitinerant hole spin [10], while the last line is due to the\ncorrelations. The hole spin dynamics is described by\n(i∂t−εkn′+εk−Qn)≪a†\nk−Q↑ak↓≫\n=βc\n2√\nN/bracketleftbigg\n(fk−Qn−fkn′)≪S+\n−Q≫\n+/summationdisplay\nqm≪(σn′m·∆Sq)∆[a†\nk−Qnak+qm]≫\n−/summationdisplay\nqm≪(σmn·∆Sq)∆[a†\nk−Q−qmakn′]≫/bracketrightbigg\n.(4)\nThe firstterm on the rhsgivesthe RPAcontribution[10],\nwhile the last two terms describe correlations.\nThe correlation contributions to Eqs.(3) and (4) are\ndetermined by the dynamics of the interactions be-\ntween a carrier excitation and a local spin fluctuation.\nThis dynamics is described by the Green’s functions\n≪∆Sp−k−Q∆[a†\nknapn′]≫, whose equations of motion\ncouple to higher Green’s functions, ≪Sa†aa†a≫and\n≪SSa†a≫, describingdynamicsof threeelementaryex-\ncitations. To truncate the infinite hierarchy, we apply a\ncorrelation expansion [17] and decompose ≪Sa†aa†a≫\ninto all possible products of the form ∝angbracketlefta†aa†a∝angbracketright ≪S≫,\n∝angbracketleftS∝angbracketright∝angbracketlefta†a∝angbracketright ≪a†a≫,∝angbracketlefta†a∝angbracketright ≪∆S∆[a†a]≫, and∝angbracketleftS∝angbracketright ≪\na†aa†a≫c, where≪a†aa†a≫cis obtained after sub-\ntracting all uncorrelated contributions, ∝angbracketlefta†a∝angbracketright ≪a†a≫,\nfrom≪a†aa†a≫(we include all permutations of mo-\nmentum and band indices) [18]. Similarly, we decompose\n≪SSa†a≫into products of the form ∝angbracketleftSS∝angbracketright ≪a†a≫,\n∝angbracketleftS∝angbracketright∝angbracketlefta†a∝angbracketright ≪S≫,∝angbracketleftS∝angbracketright ≪∆S∆[a†a]≫, and∝angbracketlefta†a∝angbracketright ≪\n∆S∆S≫. This corresponds to decomposing all opera-\ntorsAinto average and quantum fluctuation parts and\nneglecting products of three fluctuations. We thus de-\nscribe all correlations between any twospin and charge\nexcitations and neglect correlations among threeor more\nelementary excitations (which contribute to O(1/S3))\n[18]. In the case of ferromagnetic β, as in the mangan-\nites, we recover the variational results of Ref.[19] and\nthus obtain very good agreement with exact diagonaliza-\ntionresultswhilereproducingexactlysolvablelimits (one\nelectron, half filling, and atomic limits, see Refs.[18, 19]).When treating correlations in the realistic (III,Mn)V\nsystem, the numerical solution of the above closed sys-\ntem of equations of motion is complicated by the cou-\npling of many momenta and bands and by unsettled is-\nsues regarding the role on the dynamical and magnetic\nanisotropy properties of impurity bands, strain, localized\nstates, and sp–d hybridization [1, 20, 21, 22, 23]. In the\nsimpler RPA case, which neglects inelastic effects, a six–\nband effective mass approximation [16] revealed an order\nof magnitude enhancement of D. The single���band RPA\nmodel [15] also predicts maximum Dat very small hole\nconcentrations, while in the six–band model Dincreases\nand then saturates with hole doping. Here we illustrate\nthe main qualitative features due to ubiquitous corre-\nlations important in different ferromagnets [19, 24] by\nadopting the single–band Hamiltonian [15]. We then dis-\ncuss the role of the multi–band structure of (III,Mn)V\nsemiconductors by using a heavy and light hole band\nmodel.\nIn the case of two bands of spin– ↑and spin– ↓states\n[15], we obtain by Fourier transformation\n≪S+\n−Q≫ω=−2S\nω+δ+ΣRPA(Q,ω)+Σcorr(Q,ω),(5)\nwhereδ=βcsgives the energy splitting of the local spin\nlevels. Σ RPAis the RPA self energy [15, 16].\nΣcorr=βc\n2N/summationdisplay\nkp/bracketleftBigg\n(Gpk↑+Fpk)ω+εk−εk+Q\nω+εk−εk+Q+∆+iΓ\n−(Gpk↓−Fpk)ω+εp−Q−εp\nω+εp−Q−εp+∆+iΓ/bracketrightBigg\n(6)\nis the correlated contribution, where\nGσ=≪S+∆[a†\nσaσ]≫\n≪S+≫, F=≪∆Sza†\n↑a↓≫\n≪S+≫.(7)\nΓ∼10-100meV is the hole spin dephasing rate [25]. Sim-\nilar to Ref.[10] and the Lindblad method calculation of\nRef.[14], we describe such elastic effects by substituting\nthe spin–flip excitation energy∆ by ∆+ iΓ. We obtained\nGandFbysolvingthecorrespondingequationstolowest\norder in 1/S, with βSkept constant, which gives Σcorrto\nO(1/S2). More details will be presented elsewhere [18].\nResults— Firstwestudythe spinstiffness D=DRPA+\nDcorr\n++Dcorr\n−. The RPA contribution DRPAreproduces\nRef.[15]. The correlated cotributions Dcorr\n+>0 and3\n0 0.1 0.2 0.3 0.4 0.5\np00.020.040.060.08D/D0 D\nDRPA\nDRPA+D(-)\n0 0.2 0.4\np00.020.040.06\n50 100 150\nβc  (meV)00.010.02D/D0\n50 100 150\nβc  (meV)00.020.040.06a) βc =70meV b) βc =150meV\nc) p =0.1 d) p =0.5\nFIG. 1: (Color online) Spin stiffness Das function of hole\ndoping and interaction strength for the single–band model.\nc= 1nm−3, Γ=0,D0=/planckover2pi12/2mhh,mhh= 0.5me.\nDcorr\n−<0 were obtained to O(1/S2) from Eq.(6) [18]:\nDcorr\n−=−/planckover2pi12\n2mhS2N2/summationdisplay\nkp/bracketleftBigg\nfk↓(1−fp↓)εp(ˆp·ˆQ)2\nεp−εk\n+fk↑(1−fp↑)εk(ˆk·ˆQ)2\nεp−εk/bracketrightBigg\n, (8)\nDcorr\n+=/planckover2pi12\n2mhS2N2/summationdisplay\nkpfk↓(1−fp↑)×\n/bracketleftBig\nεk(ˆk·ˆQ)2+εp(ˆp·ˆQ)2/bracketrightBig\n×\n/bracketleftbigg2\nεp−εk+1\nεp−εk+∆−∆\n(εp−εk)2/bracketrightbigg\n,(9)\nwhereˆQ,ˆk, andˆ pdenote the unit vectors.\nFor ferromagnetic interaction, as in the manganites\n[19, 24], the Mn and carrier spins align in parallel. The\nHartree–Fock is then the state of maximum spin and\nan exact eigenstate of the many–body Hamiltonian (Na-\ngaoka state). For anti–ferromagnetic β, as in (III,Mn)V\nsemiconductors, the ground state carrier spin is anti–\nparallel to the Mn spin and can increase via the scat-\ntering of a spin– ↓hole to an empty spin– ↑state (which\ndecreases Szby 1). Such quantum fluctuations give rise\ntoDcorr\n+, Eq.(9), which vanishes for fk↓= 0.Dcorr\n−comes\nfrom magnon scattering accompanied by the creation of\naFermi seapair. In the caseofaspin– ↑Fermi sea, Eq.(8)\nrecovers the results of Refs.[19, 24].\nWe evaluated Eqs.(8) and (9) for zero temperature\nafter introducing an upper energy cutoff corresponding\nto the Debye momentum, k3\nD= 6π2c, that ensures the\ncorrect number of magnetic ion degrees of freedom [15].0 0.1 0.2 0.3 0.4 0.5\np00.20.4D/D0\n0 0.1 0.2 0.3 0.4 0.5\np00.20.4\n0 0.1 0.2 0.3\nεF  (eV)00.010.02D/D0\n0 0.1 0.2 0.3 0.4 0.5\nεF (eV)00.020.04a) βc =70meV b) βc =150meV\nc) βc =70meV d) βc =150meV\nFIG. 2: (Color online) Spin stiffness Dfor the parameters of\nFig. 1. (a)and(b): two–bandmodel, (c)and(d): dependence\non the Fermi energy within the single–band model.\nFigs. 1(a) and (b) show the dependence of Don hole\ndoping, characterized by p=ch/c, for two couplings β,\nwhile Figs. 1(c) and (d) show its dependence on βfor\ntwo dopings p. Figure 1 also compares our full result, D,\nwithDRPAandDRPA+Dcorr\n−. It is clear that the cor-\nrelations beyond RPA have a pronounced effect on the\nspin stiffness, and therefore on Tc∝D[1, 7] and other\nmagnetic properties. Similar to the manganites [19, 24],\nDcorr\n−<0 destabilizes the ferromagneticphase. However,\nDcorr\n+stronglyenhances Das comparedto DRPA[15] and\nalso changes its p–dependence.\nThe ferromagnetic order and Tcvalues observed in\n(III,Mn)V semiconductors cannot be explained with the\nsingle–band RPA approximation [15], which predicts a\nsmallDthat decreases with increasing p. Figure 1\nshows that the correlations change these RPA results in\na profound way. Even within the single–band model,\nthe correlations strongly enhance Dand change its p–\ndependence: Dnow increases with p. Within the RPA,\nsuch behavior can be obtained only by including multiple\nvalence bands [16]. As discussed e.g. in Refs.[1, 7], the\nmain bandstructure effects can be understood by con-\nsidering two bands of heavy ( mhh=0.5me) and light (\nmlh=0.086me) holes. Dis dominated and enhanced by\nthe more dispersive light hole band. On the other hand,\nthe heavily populated heavy hole states dominate the\nstatic properties and EF. By adopting such a two–band\nmodel, we obtain the results of Figs. 2(a) and (b). The\nmain difference from Fig. 1 is the order of magnitude en-\nhancement of all contributions, due to mlh/mhh= 0.17.\nImportantly,thedifferencesbetween DandDRPAremain\nstrong. Regarding the upper limit of Tcdue to collective\neffects, we note from Ref.[7] that is is proportional to D\nand the mean field Mn spin. We thus expect an enhance-\nment, as compared to the RPA result, comparable to the4\n0 0.5 100.020.04αα\nαRPA\n0 0.5 100.020.04\n0 0.5 1\np00.020.04α\n0 0.5 1\np00.020.04a) βc =70meV b) βc =100meV\nc) βc =120meV d) βc =150meV\nFIG. 3: (Color online) Gilbert damping as function of hole\ndoping for different interactions β.c= 1nm−3,Γ = 20meV.\ndifference between DandDRPA.\nThe dopingdependence of Dmainlycomesfromits de-\npendence on EF, shown in Figs. 2(c) and (d), which dif-\nfers strongly from the RPA result. Even though the two\nband model captures these differences, it fails to describe\naccuratelythe dependence of EFonp, determined by the\nsuccessive population of multiple anisotropicbands. Fur-\nthermore, thespin–orbitinteractionreducesthe holespin\nmatrix elements [22]. For example, |σ+\nnn′|2is maximum\nwhen the bandstates arealsospin eigenstates. The spin–\norbit interaction mixes the spin– ↑and spin– ↓states and\nreduces|σ+\nnn′|2. From Eq.(3) we see that the deviations\nfromthe meanfield resultaredetermined bythe coupling\nto the Green’s functions ≪σ+\nnn′∆[a†\nnan′]≫(RPA),≪\n∆Szσ+\nnn′∆[a†\nnan′]≫(correctiontoRPAdueto Szfluctu-\nationsleadingto Dcorr\n+>0), and≪∆S+σz\nnn′∆[a†\nnan′]≫\n(magnon–Fermi sea pair scattering leading to Dcorr\n−<0).\nBoth the RPA and the correlation contribution arising\nfrom ∆Szare proportional to σ+\nnn′. Our main result, i.e.\ntherelativeimportance of the correlation as compared to\nthe RPA contribution, should thus also hold in the real-\nistic system. The full solution will be pursued elsewhere.\nWe now turn to the Gilbert damping coefficient, α=\n2S/ω×Im≪S+\n0≫−1atω→0 [9]. We obtain to\nO(1/S2) thatα=αRPA+αcorr, where αRPArecovers\nthe mean–field result of Refs [9, 10] and predicts a linear\ndependence on the hole doping p, while\nαcorr=∆2\n2N2S2/summationdisplay\nkpIm/bracketleftBigg\nfk↓(1−fp↑)\n∆+iΓ×\n/parenleftbigg1\nεp−εk−δ+1\nεp−εk+∆+iΓ/parenrightbigg/bracketrightBigg\n(10)\narises from the carrier spin–flip quantum fluctuations.Fig.(3) compares αwith the RPA result as function of\np. The correlations enhance αand lead to a nonlinear\ndependence on p, which suggests the possibility of con-\ntrolling the magnetization relaxation by tuning the hole\ndensity. A nonlinear dependence of αon photoexcitation\nintensity was reported in Ref.[13] (see also Refs.[12, 21]).\nWe conclude that spin–charge correlations play an im-\nportant role on the dynamical properties of ferromag-\nnetic semiconductors. For quantitative statements, they\nmust be addressed together with the bandstructure ef-\nfects particular to the individual systems. The correla-\ntions studied here should play an important role in the\nultrafast magnetization dynamics observed with pump–\nprobe magneto–optical spectroscopy [12, 13, 14, 21, 22].\nThis work was supported by the EU STREP program\nHYSWITCH.\n[1] T. Jungwirth et al., Rev. Mod. Phys. 78, 2006.\n[2] H. Ohno, Science 281, 951 (1998).\n[3] S. Koshihara et al., Phys. Rev. Lett. 78, 4617 (1997).\n[4] H. Ohno et al., Nature 408, 944 (2000).\n[5] J. Wang et al., Phys. Rev. Lett. 98, 217401 (2007).\n[6] S. A. Wolf et al., Science 294, 1488 (2001).\n[7] T. K. Jungwirth et al., Phys. Rev. B 72, 165204 (2005).\n[8] L. D. Landau, E. M. Lifshitz, and L. P. Pitaeviski, Sta-\ntistical Physics, Part 2 (Pergamon, Oxford, 1980).\n[9] J. Sinova et. al., Phys. Rev. B69, 085209 (2004); Y.\nTserkovnyak, G.A.Fiete, andB. I.Halperin, Appl.Phys.\nLett.84, 25 (2004).\n[10] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y, Phys. Sta t.\nSol.23, 501 (1967).\n[11] S. T. B. Goennenwein et al., Appl. Phys. Lett. 82, 730\n(2003).\n[12] J. Wang et al., J. Phys: Cond. Matt. 18, R501 (2006).\n[13] J. Qi et al., Appl. Phys. Lett. 91, 112506 (2007).\n[14] J. Chovan, E. G. Kavousanaki, and I. E. Perakis, Phys.\nRev. Lett. 96, 057402 (2006); J. Chovan and I. E. Per-\nakis, Phys. Rev. B 77, 085321 (2008).\n[15] J. K¨ onig, H–H Lin and A. H. MacDonald, Phys. Rev.\nLett.84, 5628, (2000); M. Berciu and R. N. Bhatt, Phys.\nRev. B66, 085207 (2002).\n[16] J. K¨ onig, T. Jungwirth, and A. H. MacDonald, Phys.\nRev. B64, 184423 (2001).\n[17] J. Fricke, Ann. Phys. 252, 479 (1996).\n[18] M. D. Kapetanakis and I. E. Perakis, arXiv:0806.0938v1 .\n[19] M. D. Kapetanakis, A. Manousaki, and I. E. Perakis,\nPhys. Rev. B 73, 174424 (2006); M. D. Kapetanakis and\nI. E. Perakis, Phys. Rev. B 75, 140401(R) (2007).\n[20] K. S. Burch et. al., Phys. Rev. Lett. 97, 087208 (2006).\n[21] J. Wang et. al., arXiv:0804.3456; K. S. Burch at. al.,\nPhys. Rev. B 70, 205208 (2004).\n[22] L. Cywi´ nski and L. J. Sham, Phys. Rev. B 76, 045205\n(2007).\n[23] X. Liu et. al., Phys. Rev. B 71, 035307 (2005); K.\nHamaya et. al., Phys. Rev. B 74, 045201 (2006).\n[24] D. I. Golosov, Phys. Rev. Lett. 84, 3974 (2000); N.\nShannon and A. V. Chubukov, Phys. Rev. B 65, 104418\n(2002).5\n[25] T. Jungwirth et. al., Appl. Phys. Lett. 81, 4029 (2002)."    },    {        "title": "1712.03550v1.Magnetic_field_gradient_driven_dynamics_of_isolated_skyrmions_and_antiskyrmions_in_frustrated_magnets.pdf",        "content": " \nMagnetic  field gradient  driven dynamics of isolated skyrmions and antiskyrmions  in \nfrustrated magnets  \n \nJ. J. Liang1, J. H. Yu1, J. C hen1, M. H. Qin1,*, M. Zeng1, X. B. Lu1, X. S. Gao1,  \nand J. –M. Liu2,† \n1Institute for Advanced Materials , South China Academy of Advanced Optoelectronics  and \nGuangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, \nSouth China Normal University, Guangzhou 510006, China  \n2Laboratory of Solid State Microstr uctures  and Innovative Center for Advanced \nMicrostructures , Nanjing University, Nanjing 210093, China  \n \n[Abstract]  The study of skyrmion/antiskyrmion motion in magnetic materials is very \nimportant in particular for the spintronics applications.  In this work , we stud y the dynamics of \nisolated skyrmions and antiskyrmions  in frustrated magnets  driven by magnetic field gradient , \nusing the Landau -Lifshitz -Gilbert simulations on the frustrated classical Heisenberg  model  on \nthe triangular lattice . A Hall-like motio n induced  by the gradient is revealed in bulk system, \nsimilar to that in the well -studied chiral magnets.  More interestingly, our work suggest s that \nthe lateral confinement in nano -stripes of  the frustrated system  can completely suppress the \nHall motion an d significantly speed up the motion along the gradient direction. The simulated \nresults  are well explained by the Thiele theory . It is demonstrated that t he acceleration of the \nmotion is mainly determined by the Gilbert damping constant , which provides use ful \ninformation  for finding potential materials for skyrmion -based spintronics .  \n \nKeywords: skyrmion dynamics, field gradient, frustrated magnets   \nPACS numbers: 12.39.Dc, 66.30.Lw .Kw, 75.10.Jm \n \n                                                        \nEmail: *qinmh@scnu.edu.cn , † liujm@nju.edu.cn  I. INTRODUCTION  \nMagnetic skyrmions  which are topological defects with vortex -like spin structures  have  \nattracted extensive  attention  since their discovery in chiral magnets  due to their interesting \nphysics and potential applications in spintronic devices.1-4 Specifically, the interesting  \ncharacters of skyrmions such  as the topological protection5, the ultralow critical currents \nrequired to drive skyrmions  (~105 Am-2, several orders of smaller than that for domain -wall \nmanipulation )3,6, and th eir nanoscale size  make s them proposed to be promising candidates \nfor low po wer consumption magnetic memories  and high -density data processing devices. \nTheoretically, the cooperation of the energy competition among the ferromagnetic, \nDzyaloshinskii -Moriya ( DM), and the Zeeman couplings and the thermal fluctuations is \nsuggested to stabilize the skyrmions .7,8 Moreover, the significant effects of the uniaxial stress \non the stabilization of the skyrmion lattice have been revealed in earlier works.9-12 On the \nskyrmion  dynamics , it has been suggested that the skyrmions in chiral magnets can be \neffectively modulated by spin-polarized current,13-16 microwave fields,17 magnetic field \ngradients,18,19 electric field gradients,20,21 temperature gradient s22 etc. So far, some of these \nmanipulations have been realized  in experiments.23  \nDefinitely , finding new magnetic systems with skyrmions is essential both in application \npotential and in basic physical research.24 More recently,  frustrated magnets  have been \nsuggested theoretically to host skyrmion lattice  phase . For example, skyrmion crystals an d \nisolated skyrmions have been reported in the frustrated Heisenberg model on the triangular \nlattice.25,26 In this system, i t is suggested that the skyrmion crystals  are stabilized by the \ncompeting ferromagnetic nearest -neighbor (NN) and antifer romagnetic next-nearest -neighbo r \n(NNN) interaction s and thermal fluctuations at finite  temperatures (T) under applied magnetic \nfield h. Furthermore, the uniaxial anisotropy strongly affect s the spin orders in triangular \nantiferromagnets  and stabilize s the isolated sk yrmions even at zero T.26,27  \nCompared with the skyrmions in chiral magnets, those in frustrated magnets  hold two \nadditional merits.  On the one hand, the skyrmion lattice constant is typically an order of \nmagnitude smaller than that of chiral magnets, and higher -density data processing devices  are \nexpected.  On the other hand, the skyrmions are with  two additional degrees -of-freedomvorticity and helicity ) due to the fact that the exchange interactions are \ninsensitive to the direction of spin rotation . As a  result, both skyrmion and anti skyrmion \nlattices are possible in frustrated magnets  which keep the Z2 mirror symmetry in the xy spin \ncomponent.  Furthermore , the dynamics of skyrmions /antiskyrmions  is probably  different from \nthat of chiral magnets , as revea led in earlier work which  studied the current -induced dynamics \nin nanostripes of frustrated magnets.28 It has been  demonstrated that the spin states formed at \nthe edges  create multiple edge channels and guide the skyrmion /antiskyrmion  motion.  \nIt is noted that spin-polarized  current  may not drive the skyrmion well for insulating \nmaterials , and other control parameters such as field gradient are preferred . In chiral magnets, \nfor example, the gradient  can induce a Hall -like motion of skyrmions, i. e., the mai n velocity \nv (perpendicular to the gradient direction ) is induced by the gradient, and a low velocity v|| \n(parallel to the gradient direction ) is induced by  the damping effect . Thus, the gradient -driven \nmotion of skyrmions and antiskyrmions in frustrated systems is also expected . Furthermore , it \nhas been suggested  that the confined geometry suppress es the current -induced Hall motion of \nskyrmions and speed s up the motion along the current  direction , which is instructive  for \nfuture application s.29 In some ex tent, the gradient -driven motion could also be strongly \naffected by confining potential in narrow constricted geometries.  Thus, as a first step, the \nfield-gradient -induced dynamics of skyrmions and antiskyrmions in bulk frustrated magnets \nas well as in constricted geometries urgently deserves to be revealed  theoretically . However, \nfew works on this subject have been reported, as far as we know.  \nIn this work , we stud y the skyrmion /antiskyrmion dynamics in frustrated magnets \ninduced by  magnetic field gradien ts using  Landau -Lifshitz -Gilbert  (LLG)  simulations and \nThiele approach  based on the frustrated classical Heisenberg model on two -dimensional \ntriangular lattice . A Hall-like motion is revealed in bulk system, similar to that in chiral \nmagnets. More interest ingly, our work demonstrates  that the edge  confinement in nanostripes \nof frustrated magnets c an completely suppress the Hall motion and significantly accelerate  the \nmotion along the gradient direction.  \nThe remainder of this manuscript is organized as foll ows: in Sec. II the model and the \ncalculation method will be described. Sec. III is attributed to the results and discussion, and \nthe conclusion is presented in Sec. IV .      \nII. MODEL AND METHODS   \nFollowing the earlier  work,28 we consider  the Hamiltonian  \n22'\n12\n, ,z z z\ni j i j i i i i\ni j i i i ijH J J h S D S D S          S  S S  S\n,     (1) \nwhere Si is the classical Heisenberg spin with unit length on site i. The first term is the \nferromagnetic NN interaction with J1 = 1 (we use J1 as the energy unit, for simplicity) , and t he \nsecond term is the antiferromagnetic NNN  interacti on with J2 = 0.5 , and the third term is the \nZeeman coupling with a linear gradient field h = h0 + g·r (h0 = 0.4, r is the coordinate , and g \nis the gradient vector with a strength  g) applied along the [001] direction ,28 and the fourth  \nterm is the bulk uniaxial anisotropy  energy with D = 0.2 , and the last term  is the easy plane \nanisotropy energy of  the edges  with D' = 2. D' is only consider ed at the edges for the  \nnanostripes  system , which may give  rise to several  types of  edge states, as uncovered in \nearlier work .28 However, it has been confirmed that the skyrmion s/antiskyrmions  in \nnanostripes move with the same speed when they are  captured by one of these  edge state s. In \nthis work, we mainly concern the gradient -driven moti on of isolated skyrmions /antiskyrmion.  \nWe study  the spin dynamics at zero T by numerically solving the LLG  equation: \nii\ni i idd\ndt dt   SSS f S\n,                                           (2) \nwith the local effective field fi = (∂H/∂ Si). Here, γ = 6 is the gyromagnetic ratio , α is the \nGilbert damping coefficient. We use the fourth -order Runge -Kutta method to solve the LLG \nequation.  The initial spin configurations are obtained by solving the LLG equation at  g = 0. \nSubsequently, t he spin dynamics  are investigated under gradient fields. Furthermore, the \nsimulated results are further explained using the approach proposed by Thiele.29 The \ndisplacement of the skyrmion/antiskyrmion  is characteri zed by the position of its  center  (X, \nY): (1 )d d (1 )d d\n,.\n(1 )d d (1 )d dzz\nzzx S x y y S x y\nXY\nS x y S x y\n\n\n                           (3) \nThen, the velocity v = (vx, vy) is numerically calculated by  \nd d , d d .xyv X t v Y t\n                                            (4) \nAt last, v and v|| are obtaine d through a s imple coordinate transformation .  \n \nIII. RESULTS AND DISCUSSION  \nFirst, we investigate  the spin configurations of possible isolated skyrmions  and \nantiskyrmions  with various vorticities and helicities obtained by LLG simulations of bulk \nsystem ( D' = 0) at zero g. Specifically, four typical isolated skyrmions with the topological  \ncharge Q = 1 have been observed in our simulations, as depicted  in Fig 1(a). The first two \nskyrmions are N éel-type ones with different helicities, and the remaining two sk yrmions are \nBloch -type ones. Furthermore, isolated antiskyrmoins are also possible in this system, and \ntheir spin configurations with Q = 1 are shown  in Fig. 1(b).  \nAfter the relaxation of the spin configurations  at g = 0, the magnetic field gradient is \napplied along the direction of θ = /6 (θ is the angle between the gradient vector  and the \npositive x axis, as shown in Fig. 2(a)) to study the dynamics of isolated skyrmions and \nantiskyrmions  in bulk system . The LLG simulation is performed on a 28 × 28 triangular \nlattice  with the periodic boundary condition  applied along  the y' direction  perpendicular  to the \ngradient . Furthermore , we constrain the spin directions  at the edge s along the x direction by Sz \n= 1 (red circles  in Fig. 2(a) ) to reduce  the finite lat tice size effect . Similar to that in chiral \nmagnets, the skyrmion /antiskyrmion motion can be also driven  by the magnetic field \ngradients  in frustrated magnets. Fig. 2 (b) and Fig. 2( c) give  respectively  the calculated v|| and \nv as functions of g at α = 0.04 . v|| of the skyrmion equals to that of the antiskyrmion , and \nboth v|| and v increase linearl y with g. For a fixed g, the value of v is nearly an order of \nmagnitude larger than that of v||, clearly exhibiting  a Hall -like motion of the skyrmions/a ntiskyrmions.  It is noted that  v is caused by the gyromagnetic force which \ndepends on the sign of the topological charge. Thus, along the y' direction, the skyrmion and \nantiskyrmion move oppositely  under the field gradient , the same as earlier report.18 Moreover, \nv|| is resulted  from the dissipative force which is  associated with the Gilbert damping. For \nexample , the linear dependence of v|| on the Gilbert damping constant α has been revealed in \nchiral magnets,18 which still hold s true for the frustrated magnets. The dependence of velocity \non α at g = 103 is depicted in Fig. 3 , which clearly demonstrates  that v|| increases linearly and \nv is almost invariant  with the increa se of α.  \nSubsequently , the simulated results are qualitatively  explained by Thiele equations:  \n|| || '', and ,HHv Gv Gv vXY        \n                        (5) \nwith the skyrmion/antiskyrmoin center ( X', Y') in the x'y' coordinate system.  Here,  \n2 2 2 S S S S S Sd S 4 , and d d . G r Q r rx y x x y y                   \n     (6) \nFor the  frustrated  bulk magnets with the magnetic field gradient applied along the x' direction, \nthere are   \n''1 , and  0.z\ni\niHHg S gqXY   \n                                (7)        \nFor α << 1, q is almost invariant and the velocities can be estimated from  \n|| 2,. v gq v gqGG \n \n                                            (8) \nThus, a proportional relation between the velocity and field gradient is clearly  demonstrated . \nFurthermore, v is inversely propor tional to G and/or the topological charge  Q, resulting in  \nthe fact that the skyrmion  and antiskyrmion  move along the y' direction oppositely, as \nrevealed in our simulations.  For current -induced motion of skyrmions, the lateral confine ment can suppress the Hall \nmotion and accelerate the motion  along the current direction.29-31 The confinement effects  on \nthe h-gradient driven  skyrmion/antiskyrmion motion  are also investigated in the nanostripes \nof frustrated magnets. For this case, the L LG simulation is performed on an 84 × 30 \ntriangular -lattice with an open boundary condition along the y direction. For convenience, the \nfield gradient is applied along the x direction.  The easy plane anisotropy  with D' = 2 is \nconsidered at the lateral edges , which gives rise to the edge state and in turn confines the \nskyrmions/antiskyrmions . Fig. 4(a) gives  the time dependence of the y coordinate of the \nskyrmion center for  α = 0.04 and g = 103. It is clearly shown that t he isolated skyrmion  \njumps into the channel at Y = 17 and then moves with a constant  speed  along the gradient  \n(negative x, for this case)  direction . Furthermore, the position of the channel changes only a \nlittle due to the small range of the gradient consi dered in this work , which never affects our \nmain conclusions.   \nMore interestingly, the skyrmion/antiskyrmion motion along the gradient direction can be \nsignificant accelerated  by the lateral confinement, as shown  in Fig. 4(b) which gives v|| (vx) as \na fun ction of g at α = 0.04. For a fixed g, v|| of the  nanostripes is almost two orders of \nmagnitude larger than that of bulk system. When the skyrmion/antiskyrmion is captured by \nthe edge state ( under which v = 0), the equation (5) gives  v|| = gqγ/αΓ. It is s hown that v|| is \ninversely proportional to α in this confined geometry, and small α result s in a high speed of \nmotion  of the skyrmion/antiskyrmion . The inversely proportional relation between v|| and α \nhas also been confirmed in our LLG simulations, as cle arly shown  in Fig. 4(c) which gives the \nsimulated v|| as a function  of 1/α at g = 103.  \nAt last, we study the effect of the reversed gradient g on the skyrmion/antiskyrmion \nmotion, and its trail is recorded  in Fig. 4(d). It is clearly shown that the rever sed gradient  \nmoves the skyrmion/antiskyrmion out of the former channel near one lateral edge and drives \nit to the new channel near the other lateral edge. Subsequently, the skyrmion/antiskyrmion is \ncaptured by the n ew channel and moves reversely, resulting  in the loop -like trail. As a result, \nour work suggests that one may modulate the moving channel by reversing the field gradient, \nwhich is meaningful for future applications such as  in data eras ing/restoring .  \nAnyway, it is suggested theoretically that the  confined geometry in nanostripes of frustrated magnets could  significantly speed up the field-driven motion of the isolated \nskyrmions/antiskyrmions, especially for system with small Gilbert damping constant. \nFurthermore , we would also like to point out th at th is acceleration  is probably available in \nother confined materials such as  chiral magnets, which deserves to be checked in future \nexperiments.  \n \nIV. CONCLUSION  \nIn conclusion, we have studied the magnetic -field-gradient -drive n motion of the isolated \nskyrmions and antiskyrmions in the frustrated triangular -lattice spin model using \nLandau -Lifshitz -Gilbert simulations and Thiele  theory . The Hall -like motion is revealed in \nbulk system, similar to that in chiral magnets. More interestingly, it is suggested  that the \nlateral confinement in the nanostripes of the frustrated system  can suppress the Hall motion \nand significantly speed up the motion along the gradient direction. The acceleration of the \nmotion is mainly determined by the Gilbert damping constant , whic h is helpful for finding \npotential materials for skyrmion -based spintronics .  \n \n \nAcknowledgement s: \nThe work is supported by the National Key Projects for Basic Research of China  (Grant \nNo. 2015CB921202 ), and the National Key Research Programme of China (Gra nt No. \n2016YFA0300101), and the Natural Science Foundation of China ( Grant No. 51332007 ), and \nthe Science and Technology Planning Project of Guangdong Province (Grant No. \n2015B090927006) . X. Lu also thanks for the support from the project for Guangdong \nProvince Universities and Colleges Pearl River Scholar Funded Scheme (2016).  \n \n References:  \n \n1. A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. N iklowitz, and P. Bö ni 2009 \nPhys. Rev. Lett. 102 186602  \n2. H. Wilhelm, M. Baenitz, M. Schmidt, U.K. Rö ß ler,  A. A. Leonov, and A. N. Bogdanov \n2011 Phys. Rev. Lett. 107 127203  \n3. F. Jonietz et al 2010  Science 330 1648  \n4. S. Mü hlbauer et al 2009 Science 323 915 \n5. J. Hagemeister, N. Romming, K. von Bergmann, E.Y . Vedmedenko , and R. Wiesendanger \n2015 Nat. Commun. 6 8455  \n6. X.Z. Yu, N. Kanazawa, W.Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y . Matsui, Y . Onose , \nand Y . Tokura 2012 Nat. Commun. 3 988 \n7. S. D. Yi, S . Onoda, N . Nagaosa, and J . H. Han 2009  Phy. Rev. B  80 054416  \n8. S. Buhrandt and L . Fritz 2013 Phy. Rev. B  88 195137  \n9. K. Shibata et al 2015 Nat. Nanotech.  10 589 \n10. Y . Nii et al 2015 Nat. 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Lett. 111 067203  23. W. Legrand et al 2017 Nano Lett. 17 2703 -2712  \n24. Joseph Barker, Oleg A . Tretiakov  2016 Phys. Rev. Lett. 116 147203  \n25. T. Okubo, S. Chung, and H. Kawamura 2012 Phys. Rev . Lett. 108 017206  \n26. A. O. Leonov and M. Mostovoy, 2015 Nat. Commun. 6 8275  \n27. S. Hayami, S. -Z. Lin, and C. D. Batista 2016 Phy. Rev. B 93 184413  \n28. A. O. Leonov and M.  Mostovoy 2017 Nat. Commun. 8 14394  \n29. J. Iwasaki, M. Mochizuki, and N. Nagaosa 2013 Nat. Nanotech. 8 742 \n30. I. Purnama, W. L. Gan, D. W. Wong,  and W. S. Lew 2015 Sci. Rep. 5 10620  \n31. X. Zhang, G. P. Zhao, H. Fangohr, J. P. Liu, W. X. Xia, J. Xia, and F. J. Morvan  2015 Sci. \nRep. 5 7643  \n \n \n \n \n \n FIGURE CAPTIONS  \n \nFig.1. Typical LLG snapshot  of the spin configuration s of skyrmion s and antiskyrmion s. (a) \nskyrmion  structure s and (b) antiskyrmion structure s with different helicit ies. \n \nFig.2. (a) Effective model on the triangular  lattice.  (b) v|| and ( c) v as functions of g at α = \n0.04 in bulk system . \n \nFig.3. (a) v|| and ( b) v as functions of  α at g = 103.  \n \nFig.4. (a) Time dependence of Y coordinate of the skyrmion center at  α = 0.01 and g = 103. v|| \nas a function of  (b) g at α = 0.04 , and (c) α at g = 103 in the nanostripes of frustrated magnets . \n(d) The trail of skyrmion /antiskyrmion . The red line records  the motion with gradient g along \nthe positive x direction , while  the blu e line records  the motion with the revers ed g.      \n \n  \n \n \nFig.1. Typical LLG snapshot  of the spin configuration s of skyrmion s and antiskyrmion s. (a) \nskyrmion  structure s and (b) antiskyrmion structures with different helicit ies.  \nFig.2. (a) Effective model on the triangular  lattice.  (b) v|| and (c) v as functions of g at α = \n0.04 in bulk system .  \nFig.3. (a) v|| and ( b) v as functions of α at g = 103.  \nFig.4. (a) Time dependence of Y coordinate of the skyrmion center at α = 0.01 and g = 103. v|| \nas a function of  (b) g at α = 0.04, and (c) α at g = 103 in the nanostr ipes of frustrated magnets . \n(d) The trail of skyrmion /antiskyrmion . The red line records  the motion with gradient g along \nthe positive x direction , while  the blu e line records  the motion with the reversed g. \n "    },    {        "title": "1609.08250v1.Anomalous_Feedback_and_Negative_Domain_Wall_Resistance.pdf",        "content": "Anomalous Feedback and Negative Domain Wall Resistance\nRan Cheng,1Jian-Gang Zhu,2and Di Xiao1\n1Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA\n2Department of Electrical and Computer Engineering,\nCarnegie Mellon University, Pittsburgh, PA 15213, USA\nMagnetic induction can be regarded as a negative feedback e\u000bect, where the motive-force opposes\nthe change of magnetic \rux that generates the motive-force. In arti\fcial electromagnetics emerging\nfrom spintronics, however, this is not necessarily the case. By studying the current-induced domain\nwall dynamics in a cylindrical nanowire, we show that the spin motive-force exerting on electrons\ncan either oppose or support the applied current that drives the domain wall. The switching into the\nanomalous feedback regime occurs when the strength of the dissipative torque \fis about twice the\nvalue of the Gilbert damping constant \u000b. The anomalous feedback manifests as a negative domain\nwall resistance, which has an analogy with the water turbine.\nI. INTRODUCTION\nMagnetization dynamics and electron transport are\ncoupled together in a reciprocal manner. Their interplay\nintroduces a variety of feedback phenomena [1{12]. For\nexample, when a background magnetization varies slowly\nover space and time, conduction electron spins will follow\nthe magnetization orientation. By doing so, the electron\nwave function acquires a geometric phase changing with\ntime, which behaves as a time-varying magnetic \rux and\nproduces a spin motive-force (SMF) according to Fara-\nday's e\u000bect [13, 14]. As a feedback, electrons driven by\nthe SMF react on the magnetization via the spin-transfer\ntorque (STT) [15{17]. This reaction leads to a modi\fed\nmagnetic damping, which hinders the magnetization dy-\nnamics that generates the SMF [8]. In parallel, when a\nmagnetic texture is driven into motion by a current, it in\nturn exerts SMFs on the electrons, resulting in a modi-\n\fed electrical resistivity that inhibits the growth of the\ndriving current [2, 3].\nSimilar feedback mechanisms also apply to magnetic\nheterostructures [11]. For example, spin current pumped\nfrom a precessing ferromagnet into an adjacent normal\nmetal experiences a back\row, which, in turn acts on the\nferromagnet through STT [18]. Because of the back\row-\ninduced STT, the e\u000bective spin-mixing conductance on\nthe interface is renormalized [19]. If the pumped spin\ncurrent is absorbed by a second ferromagnet instead of\n\rowing back, it will mediate a dynamical interlayer cou-\npling between the two ferromagnets [4, 10]. Recently, it\nhas also been shown that in the presence of the spin Hall\ne\u000bect, spin pumping and spin-back\row are connected\nthrough a feedback loop due to the combined e\u000bect of\nthe spin Hall and its reverse process [11, 12]. This novel\nfeedback mechanism, despite quadratic in the spin Hall\nangle, gives rise to a crucial nonlinear damping e\u000bect\nthat qualitatively changes the dynamical behavior of the\nmagnetization.\nIn electromagnetics, a negative feedback is ensured by\nthe Lenz law [20], which requires that the emf generated\nby Faraday's e\u000bect must oppose the change of magnetic\n\rux that causes the emf. For instance, an electric motorworks simultaneously as a dynamotor so that the induced\nemf counteracts the applied emf. As a result, the electric\ncurrent \rowing through its coil is attenuated and the\nresistance from I\u0000Vmeasurement is larger than the\nresistance of the coil. In the context of spintronics, the\ncurrent-induced magnetization dynamics plays the role\nof an electric motor, which in turn drives the current in\na similar fashion as a dynamotor. Regarding the Lenz\nlaw, one may expect an increased resistivity.\nIn this paper, however, we show that this naive expec-\ntation is not always correct. The feedback acting on the\ndriving current can also give rise to a reduced resistiv-\nity. As an example, we study the current-driven domain\nwall (DW) dynamics in a nanowire with cylindrical sym-\nmetry [21], and demonstrate that when the DW is set\ninto motion by an applied current, its reaction in the\nform of SMF can either propel or repel the electron mo-\ntion, creating either a negative or a positive DW resis-\ntance. The sign of the DW resistance re\rects the style\nof the feedback, which depends only on two phenomeno-\nlogical parameters|the Gilbert damping constant \u000band\nthe strength of the dissipative torque \f. To interpret\nsuch an anomalous feedback phenomenon, we make an\nanalogy to the working mechanism of a water turbine. It\nis observed that if a DW propels electrons along with its\nmotion, just like a rotating turbine wheel carriers water,\na negative DW resistance is produced.\nThe paper is organized as follows. In Sec. II, we es-\ntablish the general formalism. In Sec. III, we apply the\nformalism to a slowly-varying spin texture and derive the\nfeedback-induced change of dissipations. In Section IV,\nwe explore the current-driven DW dynamics in a cylindri-\ncally symmetric nanowire, and derive the DW resistance\nin terms of\u000band\f. In Section V, we provide an intuitive\ninterpretation of the anomalous feedback.\nII. DYNAMIC FEEDBACKS\nAs illustrated in Fig. 1, the interplay between local\nmagnetization and conduction electrons is resolved in\na dynamic feedback loop connecting energy dissipationarXiv:1609.08250v1  [cond-mat.mes-hall]  27 Sep 20162\nmagnetization\nelectrons\nfeedbackloopSpinmotive-forcespin-transfertorqueEJouleheatingGilbertdampingHeff\nFIG. 1. (Color online) The interplay between magnetization\nand conduction electrons generates a dynamic feedback loop\nthat connects the magnetic and electronic dissipations.\nchannels of each individual process. Under the adiabatic\nassumption [22], we regard the magnetic order parame-\nterm(r;t) as a slowly-varying vector in space and time\nso that conduction electron spins are able to adjust to\nthe magnetization direction. Given the magnetic free en-\nergyU[m(r;t)], we de\fne the e\u000bective magnetic \feld as\nHe\u000b=\u0000\u000eU=\u000em. In the di\u000busive region, nonlocal pro-\ncesses are suppressed, and the coupled dynamics of the\nsystem is described by\n(1\u0000^\u000bm\u0002)_m=\rHe\u000b\u0002m+\u001c(j); (1a)\nj=^G(m)E+\"(_m); (1b)\nwhere\ris the gyromagnetic ratio, ^ \u000bis the magnetic\ndamping tensor, ^G(m) is the conductivity tensor. The\nSTT\u001cand the motive force \"arelocal functions ofjand\n_m, respectively; they mix the dynamics of mwith that\nof electrons. Note that \u001cand\"may also depend on the\nspatial gradient of the magnetization rm. With proper\ninitial conditions, the evolution of mandjcan be solved\nby iterations of Eq. (1) on discretized spacetime grid. At\nany particular point ( r;t), one is allowed to eliminate j\n(or_m) by substituting Eq. (1b) into Eq. (1a) [or Eq. (1a)\ninto Eq. (1b)] if both \u001cand\"are local functions of the\nspace and time coordinates.\nSuch an elimination operation ful\flls the feedback loop\nillustrated in Fig. 1. For example, if E= 0, the current\njis only induced by the motion of mthrough\", which\nis simultaneously reacting on mby virtue of\u001c. In this\nregard, we can eliminate jby inserting Eq. (1b) into\nEq. (1a), which modi\fes the magnetic damping tensor\n^\u000b. In a parallel sense, if the magnetization dynamics is\nsolely driven by j(no magnetic \feld), it also generates a\nfeedback onjand renormalizes the conductivity tensor\n^G. The latter corresponds to the elimination of _mby\ninserting Eq. (1a) into Eq. (1b).\nThe dynamic feedback e\u000bects can be further elucidated\nby energy dissipations. Swapping the roles of the ther-\nmodynamic forces He\u000bandEwith the corresponding\ncurrents _mandj[23], we can rewrite Eq. (1) as\n\u0014\nHe\u000b\nE\u0015\n=\u0014\nL11L12\nL21L22\u0015\u0014\n_m\nj\u0015\n: (2)\nHere,L11is pertaining to the Gilbert damping, L12the\ncurrent-induced torque, L21the motive force, and L22the electrical resistivity. The Onsager's reciprocity rela-\ntion implies that LT\n12(m;He\u000b) =L21(\u0000m;\u0000He\u000b) [24].\nIf magnetization and current decouple, i.e.,L12= 0, the\nmagnetic free energy dissipates only through the Gilbert\ndamping _Um=\u0000He\u000b\u0001_m=\u0000L11_m2, while the elec-\ntron free energy dissipates only through the Joule heating\n_Ue=\u0000E\u0001j=\u0000L22j2. However, when the STT ( L12)\nand the motive force ( L21) are introduced, a feedback\nloop will connect the two channels of energy dissipation\nas shown in Fig. 1. For example, the magnetic dissipation\nis implemented by not only the Gilbert damping, but also\nthe Joule heating, since a magnetic precession inevitably\ndrives the electron motion that carries away the magnetic\nenergy and subsequently dissipates into heat. This mani-\nfests as a renormalization of the magnetic damping tensor\n^\u000b(thusL11). In a similar fashion, electron current can\nexcite magnetic precession, which takes away the elec-\ntron kinetic energy and damped into heat through the\nGilbert damping. As a result, the resistivity tensor L22\nis e\u000bectively modi\fed. The rates of free energy loss are\nthus\n_Um=\u0000_m\u0002\nL11\u0000L12L\u00001\n22L21\u0003_m\u0011\u0000L 11_m2;(3a)\n_Ue=\u0000j\u0002\nL22\u0000L21L\u00001\n11L12\u0003\nj\u0011\u0000L 22j2; (3b)\nwhereL11andL22are the response coe\u000ecients modi\fed\nby the dynamic feedback.\nIn general, if a system is driven by a set of Nthermody-\nnamic forces [or currents in the \\swapped\" convention,\nsee Eq. (2)] X1,X2,\u0001\u0001\u0001XN, there are Ncurrents (or\nforces)J1,J2,\u0001\u0001\u0001JNsatisfyingJa=LabXb, where the\nrepeated index is summed. By a straightforward deriva-\ntion elaborated in the Appendix, the renormalized energy\ndissipation rate through a particular channel kis\n_Uk=\u0000X2\nk\n[L\u00001]kk; (4)\nwhereL\u00001denotes the inverse of the response matrix.\nForN= 2, Eq. (4) reduces to Eq. (3). We mention\nthat Eq. (4) is quite general, where the thermodynamic\nforces (or currents) can be magnetic, electric, thermalic,\netc. However, to simplify the following discussions, we\ndo not include any thermoelectric e\u000bect, although they\nmay become important in many circumstances [9].\nIII. SPIN TEXTURE\nA. Damping\nAs mentioned earlier, a spacetime dependent magne-\ntizationm(r;t) drives local spin currents via the SMF.\nThe SMF that exerts on spin-up electrons is opposite\nto its counterpart that exerts on spin-down electrons:\n\"\"=\u0000\"#, where the spin direction is determined with\nrespect to the local and instantaneous m(r;t). Since the\nspin current is polarized along m, we only keep its \row3\ndirection in the subscript, so the i-component of the spin\ncurrent density is\njs\ni=\u0016B\ne(G\"\nik\"\"\nk\u0000G#\nik\"#\nk)\n=\u0016B\u0016hGc\nik\n2e2[(@tm\u0002@km)\u0001m+\f@tm\u0001@km];(5)\nwhereGc\nik=G\"\nik+G#\nikis theik-component of the conduc-\ntivity tensor, \u0016Bis the Bohr magneton, and the Land\u0013 e g-\nfactor of electrons is taken to be 2. The term proportional\nto\fis the dissipative SMF [25, 26], which is the recipro-\ncal e\u000bect of the dissipative STT; \fis a phenomenological\nconstant that characterizes the relative strengths of the\ndissipative contribution.\nAs a feedback e\u000bect, the locally pumped spin current\nacts on the magnetization through the STT. De\fne the\nelectron velocity \feld as u=js=Ms, whereMsis the\nsaturation magnetization. Then the STT consists of two\northogonal terms [17]\n\u001c= (ui@i)m\u0000\fm\u0002(ui@i)m: (6)\nInserting Eq. (5) into Eq. (6) yields a damping term that\nrenormalizes the original Gilbert damping. The Landau-\nLifshitz-Gilbert (LLG) equation becomes\n@tm=\rHe\u000b\u0002m+m\u0002(D\u0001@tm); (7)\nwhereDis the damping tensor that can be decomposed\ninto ^D=^D0+^D0, where ^D0=\u000b0^I is the original Gilbert\ndamping, and the feedback correction is\n^D0=\u0011[^S+^A] (8)\nwith\u0011=\u0016B\u0016h=(2e2Ms). In Eq. (8), the element of the\nsymmetric part is\nSab=Gc\nik[(m\u0002@im)a(m\u0002@km)b\n\u0000\f2(@im)a(@km)b\u0003\n; (9)\nand that of the antisymmetric part is\nAab=\fGc\nik[(@im)a(m\u0002@km)b\u0000(a*)b)];(10)\nwhere summations over repeated indices are assumed.\nIn matrix form, the feedback correction can be written\nas^D0=\u0011TSTT\nTSMF=\u0011Gc\nik[(m\u0002@im) +\f@im]\n[(m\u0002@km)\u0000\f@km]. This suggestive form interprets\nthe feedback loop as two combined processes: a dynamic\nmpumps a local spin current, which in turn acts on m\nitself, implementing the feedback e\u000bect. When \f!0,\nEq. (8) reduces to Eq. (11) in Ref. [8].\nHere is an important remark. Although equations (8){\n(10) are similar to the results derived in Ref. [6, 7], the un-\nderlying physics is fundamentally distinct. In Ref. [6, 7],\nthe damping renormalization is attributed to the current-\ninduced noise, and thermal \ructuation is the primary\nstimulus. Consequently, the coe\u000ecient of the damping\ntensor depends on temperature. By contrast, our results\nare valid even at zero temperature.B. Resistance\nWhen closing the feedback loop the other way around,\ni.e., currentSTT\u0000\u0000\u0000! LLGSMF\u0000\u0000\u0000! current, we will obtain the\nfeedback modi\fcation of the resistance. To perform this\ncalculation, we start with the LLG equation\n@tm=\rHe\u000b\u0002m+\u000bm\u0002@tm\n+ (ui@i)m\u0000\fm\u0002(ui@i)m;(11)\nthen combine all @tmterms so that\n@tm=\r\n1 +\u000b2[He\u000b\u0002m+\u000bm\u0002(He\u000b\u0002m)]\n+1 +\u000b\f\n1 +\u000b2(ui@i)m+\u000b\u0000\f\n1 +\u000b2m\u0002(ui@i)m;(12)\nwhereu=P\u0016Bjc=(eMs) withP= (nF\n\"\u0000nF\n#)=(nF\n\"+nF\n#)\nthe polarization of carrier density at the Fermi level. The\ncharge current density is now driven by both the SMF\nand an external electric \feld E,\njc\ni=jc(E)\ni+jc(smf)\ni =Gc\nikEk\n+Gs\nik\u0016h\n2e[(@tm\u0002@km)\u0001m+\f(@tm\u0001@km)];(13)\nwhereGs\nik=G\"\nik\u0000G#\nikis theik-component of the spin\nconductivity. It should not be confused that for the SMF-\ninduced electron \row, the spin current depends on the\ncharge conductivity [see Eq. (5)], whereas the charge cur-\nrent depends on the spin conductivity [8].\nWhen substituting the LLG equation into the SMF to\neliminate@tm, terms involving He\u000bresult in nonlinear\ndependence between jcandE, which in principle should\nbe solved numerically . Nevertheless, those terms can be\ndiscarded in many special cases. For instance, if the mag-\nnetic free energy is invariant under a particular motion\nofm, we haveHe\u000bkmat all times, thus those terms\nvanish identically. In such circumstances, Eis linear in\njc, and the feedback can be expressed analytically as a\nrenormalization of the resistivity tensor. We will restrict\nthe following discussion to this category.\nTo proceed, we insert Eq. (12) into Eq. (13) and make\nthe approximation that He\u000bkm. After some manipula-\ntions, we obtain\njc\ni+Gs\nikRk`jc\n`=Gc\nikEk; (14)\nwhere the element of the feedback matrix ^Ris\nRk`=P\u0016B\u0016h\n2e2Ms\u0014\u000b(1\u0000\f2)\u00002\f\n1 +\u000b2@km\u0001@`m\n+1 + 2\u000b\f\u0000\f2\n1 +\u000b2(@km\u0002@`m)\u0001m\u0015\n\u0011P\u0016B\u0016h\n2e2Ms[f(\u000b;\f)gk`+h(\u000b;\f)\nk`]: (15)\nThe symmetric part of ^Ris proportional to the quantum\nmetricgk`=@km\u0001@`m[27], while the antisymmetric4\npart is proportional to the Berry curvature \n k`= (@km\u0002\n@`m)\u0001m. To appreciate the physical meaning of ^R, we\nturn to the resistivity by multiplying [ ^Gc]\u00001on Eq. (14),\nwhich givesE= ^\u001ajc. The resistivity tensor is\n^\u001a= ^\u001a0(1 + ^Gs^R); (16)\nwhere ^\u001a0= [^Gc]\u00001is the bare resistivity tensor without\nfeedback, and ^Gs^Ris the feedback-induced renormaliza-\ntion. Depending on the spatial pattern of m(r;t) and\nthe relative ratio between \u000band\f, a particular element\nof^Rcan be either positive or negative.\nIV. DOMAIN WALL RESISTANCE\nTransverse DWs in thin cylindrical magnetic nanowires\nhave two salient features that arouse recent interest [21].\n(1) The inner structure of these DWs remain unchanged\nduring their propagation, thus our assumption He\u000bkm\nis respected at all times. (2) These DWs are massless and\nthe critical currents required to initiate their motions are\nzero. Because of the latter property, the DW resistance\npractically measurable from I-V curve solely stems from\nthe dynamic feedback e\u000bect, whereas the conventional\ntheory based on stationary DW con\fgurations [28, 29] is\nincomplete.\nSuch a DW is a one-dimensional soliton characterized\nby two spherical angles \u0012and\u001especifying the local ori-\nentation of the magnetization\n\u0012(x;t) = 2 arctan e[x\u0000xc(t)]=w; (17a)\n\u001e(x;t) =\u001e(t); (17b)\nwherexc(t) is the center of the DW, and wis the width\nof the DW (supposed to be much larger than the lat-\ntice spacing). In one dimensions, the antisymmetric part\nof Eq. (15) vanishes, ^\n = 0; ^Rhas only one compo-\nnent, andGs=PGc. In this case, Eq. (14) reduces\nto\u001ajc=E, where\u001a=\u0002\n\u001a0+P2\u0011f(\u000b;\f)j@xmj2\u0003\nwith\n\u0011=\u0016B\u0016h=(2e2Ms). The pro\fle function given by Eq. (17)\nyieldsj@xmj2= 1=[w2cosh2(x=w)]. By integrating \u001a\noverx2(\u00001;+1), we obtain the total resistance\nR=R0+\u000b(1\u0000\f2)\u00002\f\n1 +\u000b2\u0014P2\u0016B\u0016h\ne2Ms\u00151\nAw; (18)\nwhereAis the area of the cross section of the cylindrical\nnanowire. The second term of Eq. (18) is ascribed to the\ndynamic feedback e\u000bect, which scales inversely with w.\nSinceP2\u0016B\u0016h=(e2MsAw)>0, the sign of this correction\nis only determined by f(\u000b;\f) = [\u000b(1\u0000\f2)\u00002\f]=(1+\u000b2).\nConsider\u000b\u001c1 and\f\u001c1, thenf(\u000b;\f)\u0019\u000b\u00002\f. As\na result, the dynamical correction of the DW resistance\nis positive for \f <\u000b= 2, and negative for \f >\u000b= 2. Using\ntypical material parameters of permalloy, the feedback-\ninduced resistance of a 100nm wide DW with A\u001830nm2\nis in the range of 10\u00005to 10\u00004\n.A negative DW resistance indicates that the feedback\nexerting on the electrons by the DW is positive. To be\nspeci\fc, when the DW is set into motion by a current,\nit propels the electrons in their direction of motion, thus\nreducing the electrical resistance. In terms of the Lenz\nlaw, this means that the SMF induction enhances the\n\rux (geometric phase) change by making the electrons\nmore mobile, contrasting to the normal case where the\nSMF opposes the \rux change. It worths emphasizing that\nsuch an anomalous situation is unique to cylindrically\nsymmetric nanowires, while nanostrips are not applicable\nas the approximation He\u000bkmis invalid.\nV. DISCUSSION\nDi\u000berent from the static DW resistance [28, 29] that\nis absorbed by R0in our theory, the feedback-induced\nDW resistance is associated with the DW dynamics. The\npeculiarity of using a cylindrical nanowire is that the\nthreshold current to initiate the DW dynamics is tech-\nnically zero [21]. So, what we mean by DW resistance\nrefers to the di\u000berence in Rwhen comparing the results\nofI\u0000Vmeasurements between a freely moving DW and\na pinned DW on identical cylindrical nanowires under\nthe same voltage drop.\nThe key to understand why such di\u000berence is negative\nfor\f > 2\u000blies in the reaction SMF that propels the\nelectrons along the direction of the DW motion. It con-\ntradicts the case of an electric motor where the back emf\ninduction opposes the driving current and raises the sys-\ntem resistance. At the same time, we need to justify that\nsuch an anomalous feedback e\u000bect does not violate any\nfundamental physical law. To this end, we make a heuris-\ntic analogy between the current-induced DW dynamics in\ncylindrical nanowires and a water turbine with constant\npump, where the rotating wheel represents our moving\nDW. In fact, the linear velocity of the DW is proportional\nto its angular velocity, and their ratio is independent of\nthe current [21]. Therefore, it is equivalent to character-\nize the DW motion by its angular velocity, which is more\ncurrent without turbineI!terminal velocity\npumpI!!!\nFIG. 2. (Color online) Comparison between an electric motor\ndriven by a constant voltage and a water turbine driven by\na constant pump. The overall current Ias a function of the\nangular velocity !signals the nature of the feedback e\u000bect.5\ntransparent to compare with a turbine wheel. Drawing\nsuch an analogy is to show that a negative resistance is\nnot surprising, while the analogy itself is by no means\nexact.\nAs schematically illustrated in Fig. 2, the working\nmechanism of a water turbine is compared with an elec-\ntric motor. They have one thing in common: the steady-\nstate angular velocity !increases with decreasing load.\nSo by controlling the load, one can tune !in both cases.\nHowever, the feedback mechanisms in the two cases are\nremarkably di\u000berent. In an electric motor, if one raises !\nby reducing the load, the back emf induced by Faraday's\ne\u000bect will get larger, which counteracts the applied volt-\nage more strongly and reduces the overall current. Con-\nsequently, the resistance read o\u000b from the I\u0000Vcurve\nincreases. This realizes the usual negative feedback e\u000bect\nand respects the Lenz law since Idecreases when the mo-\ntor rotates faster. In sharp contrast, if one increases !of\na water turbine, the water \rows more easily in the pipe as\nthe turbine blades less block the water. As a result, the\n\\resistance\" of the entire turbine system appears to be\nsmaller. This feature marks an anomalous feedback: the\nwater current increases when the turbine rotates faster.\nIgnoring the mass and friction of the wheel, the max-\nimum achievable angular velocity (in the limit of zero\nload), hence the maximum water current, is set by the\nwater \row in the absence of the turbine. Now go back\nto our DW dynamics: reducing the DW pinning corre-\nsponds to reducing the load on a water turbine, which\nenhances the driving current in just a similar way as the\nenhancement of water \row.\nFinally, we comment on why the anomalous feedback\nis more likely to occur in one dimensions. Since \u000b;\f\u001c1,\nthe second term of Eq. (15) dominates the \frst term, andits coe\u000ecient is unlikely to \rip sign unless \fis greater\nthan unity. However, in higher dimensions, the second\nterm always exist, so the \frst term that could lead to the\nanomaly is suppressed. Although the second term only\nrefers to the transverse components of the transport, the\nboundary conditions on the edges can considerably com-\nplicate the e\u000bective value of the longitudinal component\nand obscure the observation.\nACKNOWLEDGMENTS\nThe authors are grateful to A. Brataas for insightful\ndiscussions. We also thank J. Xiao and M. W. Daniels\nfor useful comments. This study was supported by the\nU.S. Department of Energy, O\u000ece of BES, Division of\nMSE under Grant No. DE-SC0012509.\nAppendix: Derivation of Eq. (4)\nIf all channels are in open circuit conditions except for\na particular channel k, only the current Jkis nonzero\neven in the presence of all Nthermodynamic forces\nX1\u0001\u0001\u0001XN. The energy dissipation rate is then\n_Uk=\u0000JkXk=\u0000LkkX2\nk\u0000NX\ni6=kLkiXiXk; (A.1)\nwhere the \frst term is the usual dissipation term. We\nnow eliminate those cross terms XiXk(i6=k) in terms\nofX2\nk. Since all currents but Jkare zero, multiplying Xk\nonJi=LijXjwithi6=kgives:\n2\n6666666664L11L12\u0001\u0001\u0001L1;k\u00001L1;k+1\u0001\u0001\u0001L1N\n.....................\nLk\u00001;1Lk\u00001;2\u0001\u0001\u0001Lk\u00001;k\u00001Lk\u00001;k+1\u0001\u0001\u0001Lk\u00001;N\nLk+1;1Lk+1;2\u0001\u0001\u0001Lk+1;k\u00001Lk+1;k+1\u0001\u0001\u0001Lk+1;N\n.....................\nLN1LN2\u0001\u0001\u0001LN;k\u00001LN;k+1\u0001\u0001\u0001LNN3\n77777777752\n6666666664X1Xk\n...\nXk\u00001Xk\nXk+1Xk\n...\nXNXk3\n7777777775=\u0000X2\nk2\n6666666664L1k\n...\nLk\u00001;k\nLk+1;k\n...\nLNk3\n7777777775: (A.2)\nThe coe\u000ecient matrix consists of the remaining elements\nofLafter taking away the k-th row and the k-th column.\nRegarding the Cramer's rule, the cross term is solved as\nXiXk=X2\nkAki\nAkkfori6=k, whereAijis thei;j-th alge-\nbraic cofactor (minor) of L. Inserting this relation into\nEq. (A.1), and considering the identity of row expansiondet[L] =PN\ni=1LkiAki, we \fnally obtain\n_Uk=\u0000\u0014\nLkk+det[L]\u0000LkkAkk\nAkk\u0015\nX2\nk=\u0000X2\nk\n[L\u00001]kk;\nwhich proves Eq. (4).\n[1] G. E. Volovik, J. Phys. C 20, L83 (1987). [2] J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys.\nRev. Lett. 107, 136804 (2011).6\n[3] T. Schulz et al. , Nat. Phys. 8, 301 (2012).\n[4] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90,\n187601 (2003).\n[5] J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 70,\n172405 (2004).\n[6] J\u001crn Foros, A. Brataas, Y. Tserkovnyak, and G. E. W.\nBauer, Phys. Rev. B 78, 140402(R) (2008).\n[7] C. H. Wong and Y. Tserkovnyak, Phys. Rev. B 80,\n184411 (2009).\n[8] S. Zhang and S. S.-L. Zhang, Phys. Rev. Lett. 102,\n086601 (2009).\n[9] G. E. W. Bauer, S. Bretzel, A. Brataas, and Y.\nTserkovnyak, Phys. Rev. B 81, 024427 (2010).\n[10] H. Skarsv\u0017 ag, A. Kapelrud, and A. Brataas, Phys. Rev. B\n90, 094418 (2014).\n[11] R. Cheng, J.-G. Zhu, and D. Xiao, Phys. Rev. Lett., 117,\n097202 (2016).\n[12] R. Cheng, D. Xiao, and A. Brataas, Phys. Rev. Lett.\n116, 207603 (2016).\n[13] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98,\n246601 (2007).\n[14] S. A. Yang et al. , Phys. Rev. Lett. 102, 067201 (2009);\nS. A. Yang et al. , Phys. Rev. B 82, 054410 (2010).\n[15] J. Slonczewki, J. Magn. Magn. Mater. 159, L1 (1996);\nL. Berger, Phys. Rev. B 54, 9353 (1996).\n[16] Y. B. Bazaliy, B. A. Jones, and S. -C. Zhang, Phys. Rev.\nB57, R3213 (1998).\n[17] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).[18] Y. Tserkovnyak and A. Brataas, Phys. Rev. B 66, 224403\n(2002); X. Wang, G. E. W. Bauer, B. J. van Wees, A.\nBrataas, Y. Tserkovnyak, Phys. Rev. Lett 97, 216602\n(2006); H.-J. Jiao and G. E. W. Bauer, Phys. Rev. Lett.\n110, 217602 (2013).\n[19] Y. Zhou, H.-J. Jiao, Y.-T. Chen, G. E. W. Bauer, and J.\nXiao, Phys. Rev. B 88, 184403 (2013).\n[20] D. J. 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Lett. 83, 4401 (1999)."    },    {        "title": "1705.07489v2.Dynamical_depinning_of_chiral_domain_walls.pdf",        "content": "Dynamical depinning of chiral domain walls\nSimone Moretti,\u0003Michele Voto, and Eduardo Martinez\nDepartment of Applied Physics, University of Salamanca, Plaza de los Caidos, Salamanca 37008, Spain.\nThe domain wall depinning \feld represents the minimum magnetic \feld needed to move a domain\nwall, typically pinned by samples' disorder or patterned constrictions. Conventionally, such \feld\nis considered independent on the Gilbert damping since it is assumed to be the \feld at which the\nZeeman energy equals the pinning energy barrier (both damping independent). Here, we analyse\nnumerically the domain wall depinning \feld as function of the Gilbert damping in a system with per-\npendicular magnetic anisotropy and Dzyaloshinskii-Moriya interaction. Contrary to expectations,\nwe \fnd that the depinning \feld depends on the Gilbert damping and that it strongly decreases for\nsmall damping parameters. We explain this dependence with a simple one-dimensional model and\nwe show that the reduction of the depinning \feld is related to the \fnite size of the pinning barriers\nand to the domain wall internal dynamics, connected to the Dzyaloshinskii-Moriya interaction and\nthe shape anisotropy.\nI. INTRODUCTION\nMagnetic domain wall (DW) motion along ferromag-\nnetic (FM) nanostructures has been the subject of in-\ntense research over the last decade owing to its po-\ntential for new promising technological applications1,2\nand for the very rich physics involved. A consider-\nable e\u000bort is now focused on DW dynamics in systems\nwith perpendicular magnetic anisotropy (PMA) which\npresent narrower DWs and a better scalability. Typ-\nical PMA systems consist of ultrathin multi-layers of\nheavy metal/FM/metal oxide (or heavy metal), such as\nPt=Co=Pt3,4or Pt=Co=AlOx5{7, where the FM layer has\na thickness of typically 0 :6\u00001 nm. In these systems,\nPMA arises mainly from interfacial interactions between\nthe FM layer and the neighbouring layers (see Ref.8and\nreferences therein). Another important interfacial ef-\nfect is the Dzyaloshinskii-Moriya interaction (DMI)9,10,\npresent in systems with broken inversion symmetry such\nas Pt/Co/AlOx. This e\u000bect gives rise to an internal in-\nplane \feld that \fxes the DW chirality (the magnetization\nrotates always in the same direction when passing from\nup to down and from down to up domains) and it can\nlead to a considerably faster domain wall motion10and to\nnew magnetic patterns such as skyrmions11or helices12.\nNormally, DWs are pinned by samples' intrinsic disorder\nand a minimum propagation \feld is needed in order to\novercome such pinning energy barrier and move the DW.\nSuch \feld is the DW depinning \feld ( Hdep) and it repre-\nsents an important parameter from a technological point\nof view since a low depinning \feld implies less energy\nrequired to move the DW and, therefore, a energetically\ncheaper device.\nFrom a theoretical point of view, DW motion can be\ndescribed by the Landau-Lifshitz-Gilbert (LLG) equa-\ntion13which predicts, for a perfect sample without dis-\norder, the velocity vs\feld curve depicted in Fig. 1 and\nlabelled as Perfect . In a disordered system, experi-\nments have shown that a DW moves as a general one-\ndimensional (1D) elastic interface in a two-dimensional\ndisordered medium3,4and that it follows a theoreticalvelocityvsdriving force curve, predicted for such inter-\nfaces14,15(also shown in Fig. 1 for T= 0 andT= 300K).\nMoreover, this behaviour can be reproduced by including\ndisorder in the LLG equation16{18. At zero temperature\n(T= 0) the DW does not move as long as the applied\n\feld is lower than Hdep, while, at T6= 0, thermal ac-\ntivation leads to DW motion even if H < H dep(the so\ncalled creep regime). For high \felds ( H >> H dep) the\nDW moves as predicted by the LLG equation in a per-\nfect system. Within the creep theory, the DW is con-\nsidered as a simple elastic interface and all its internal\ndynamics are neglected. Conventionally, Hdepis consid-\nered independent of the Gilbert damping because it is as-\nsumed to be the \feld at which the Zeeman energy equals\nthe pinning energy barrier19,20(both damping indepen-\ndent). Such assumption, consistently with the creep the-\nory, neglects any e\u000bects related to the internal DW dy-\nnamics such as DW spins precession or vertical Bloch\nlines (VBL) formation21. The damping parameter, for\nits part, represents another important parameter, which\ncontrols the energy dissipation and a\u000bects the DW veloc-\nity and Walker Breakdown22. It can be modi\fed by dop-\ning the sample23or by a proper interface choice as a con-\nsequence of spin-pumping mechanism24. Modi\fcations of\nthe DW depinning \feld related to changes in the damping\nparameter were already observed in in-plane systems23,25\nand attributed to a non-rigid DW motion23,25. Oscilla-\ntions of the DW depinning \feld due to the internal DW\ndynamics were also experimentally observed in in-plane\nsimilar systems26. Additional dynamical e\u000bects in soft\nsamples, such as DW boosts in current induced motion,\nwere numerically predicted and explained in terms of DW\ninternal dynamics and DW transformations27,28.\nHere, we numerically analyse the DW depinning \feld\nin a system with PMA and DMI as function of the Gilbert\ndamping. We observe a reduction of Hdepfor low damp-\ning and we explain this behaviour by adopting a simple\n1D model. We show that the e\u000bect is due to the \fnite\nsize of pinning barriers and to the DW internal dynam-\nics, related to the DMI and shape anisotropy \felds. This\narticle is structured as follows: in Section II we present\nthe simulations method, the disorder implementation andarXiv:1705.07489v2  [cond-mat.mes-hall]  25 Aug 20172\ntheHdepcalculations. The main results are outlined and\ndiscussed in Section III, where we also present the 1D\nmodel. Finally, the main conclusions of our work are\nsummarized in Section IV.\n●●●●\n●\n●\n●\n●●●●●●●●●●�=��\n�=����\n●�������\n������� ������� ��������\n����★\nFIG. 1. DW velocity vsapplied \feld as predicted by the LLG\nequation in a perfect system and by the creep law atT= 0\nandT= 300K.\nII. MICROMAGNETIC SIMULATIONS\nWe consider a sample of dimensions\n(1024\u00021024\u00020:6) nm3with periodic bound-\nary conditions along the ydirection, in order to simulate\nan extended thin \flm. Magnetization dynamics is\nanalysed by means of the LLG equation13:\ndm\ndt=\u0000\r0\n1 +\u000b2(m\u0002He\u000b)\u0000\r0\u000b\n1 +\u000b2[m\u0002(m\u0002He\u000b)];\n(1)\nwhere m(r;t) =M(r;t)=Msis the normalized magneti-\nzation vector, with Msbeing the saturation magnetiza-\ntion.\r0is the gyromagnetic ratio and \u000bis the Gilbert\ndamping. He\u000b=Hexch+HDMI+Han+Hdmg+Hz^uz\nis the e\u000bective \feld, including the exchange, DMI, uni-\naxial anisotropy, demagnetizing and external \feld con-\ntributions13respectively. Typical PMA samples param-\neters are considered: A= 17\u000210\u000012J=m,Ms= 1:03\u0002\n106A=m,Ku= 1:3\u0002106J=m3andD= 0:9 mJ=m2,\nwhereAis the exchange constant, Dis the DMI constant\nandKuis the uniaxial anisotropy constant. Disorder is\ntaken into account by dividing the sample into grains\nby Voronoi tessellation29,30, as shown in Fig. 2(a). In\neach grain the micromagnetic parameters fMs;Dc;Kug\nchange in a correlated way in order to mimic a normally\ndistributed thickness31:\ntG=N(t0;\u000e)!8\n<\n:MG= (MstG)=t0\nKG= (Kut0)=tG\nDG= (Dct0)=tG; (2)\nwhere the subscript Gstands for grain, t0is the aver-\nage thickness ( t0= 0:6nm) and\u000eis the standard devi-\nation of the thickness normal distribution. The sample\nis discretized in cells of dimensions (2 \u00022\u00020:6)nm3,smaller than the exchange length lex\u00185nm. Grain size\nis GS=15 nm, reasonable for these materials, while the\nthickness \ructuation is \u000e= 7%. Eq. (1) is solved by the\n\fnite di\u000berence solver MuMax 3.9.329.\nA DW is placed and relaxed at the center of the sample\nas depicted in Fig. 2(b). Hdepis calculated by applying\na sequence of \felds and running the simulation, for each\n\feld, until the DW is expelled from the sample, or until\nthe system has reached an equilibrium state (i.e. the DW\nremains pinned): \u001cmax<\u000f(\u000b).\u001cmaxindicates the maxi-\nmum torque, which rapidly decreases when the system is\nat equilibrium. It only depends on the system parame-\nters and damping. For each value of \u000b, we choose a spe-\nci\fc threshold, \u000f(\u000b), in order to be sure that we reached\nan equilibrium state (see Supplementary Material32for\nmore details). The simulations are repeated for 20 dif-\nferent disorder realizations. Within this approach, Hdep\ncorresponds to the minimum \feld needed to let the DW\npropagate freely through the whole sample. In order to\navoid boundaries e\u000bects, the threshold for complete de-\npinning is set tohmzi>0:8, wherehmziis averaged over\nall the realizations, i.e. hmzi=PN\ni=1hmzii=N, where\nN= 20 is the number of realizations. We checked that,\nin our case, this de\fnition of Hdepcoincides with tak-\ningHdep= MaxfHi\ndepg, withHi\ndepbeing the depinning\n\feld of the single realization. In other words, Hdepcor-\nresponds to the minimum \feld needed to depin the DW\nfrom any possible pinning site considered in the 20 real-\nizations33.\nFollowing this strategy, the DW depinning \feld is nu-\nmerically computed with two di\u000berent approaches:\n(1) by Static simulations, which neglect any precessional\ndynamics by solving\ndm\ndt=\u0000\r0\u000b\n1 +\u000b2[m\u0002(m\u0002He\u000b)]: (3)\nThis is commonly done when one looks for a minimum\nof the system energy and it corresponds to the picture\nin whichHdepsimply depends on the balance between\nZeeman and pinning energies.34\n(2) by Dynamic simulations, which include precessional\ndynamics by solving the full Eq. (1). This latter method\ncorresponds to the most realistic case. Another way to\nestimate the depinning \feld is to calculate the DW veloc-\nityvs\feld curve at T= 0 and look for minimum \feld at\nwhich the DW velocity is di\u000berent from zero. For these\nsimulations we use a moving computational region and\nwe run the simulations for t= 80ns (checking that longer\nsimulations do not change the DW velocity, meaning that\nwe reached a stationary state). This second setup re-\nquires more time and the calculations are repeated for\nonly 3 disorder realizations.\nUsing these methods, the depinning \feld Hdepis cal-\nculated for di\u000berent damping parameters \u000b.3\n(a) (b)\nxy\n(c)\nFIG. 2. (a) Grains structure obtained by Voronoi tassellation.\n(b) Initial DW state. (c) Sketch of the internal DW angle \u001e.\nIII. RESULTS AND DISCUSSION\nA. Granular system\nOur \frst result is shown in Fig. 3(a)-(b), which depicts\nthe \fnal average magnetization hmzias function of the\napplied \feld for di\u000berent damping parameters. In the\nStatic simulations (Fig. 3(a)) Hdepdoes not depend on\ndamping, so that a static depinning \feld can be de\fned.\nConversely, in the Dynamic simulations (Fig. 3(b)), Hdep\ndecreases for low damping parameters. The depinning\n\feld is indicated by a star in each plot and the static\ndepinning \feld is labelled as Hs. The same result is ob-\ntained by calculating Hdepfrom the DW velocity vsap-\nplied \feld plot, shown in Fig. 3(c). The stars in Fig. 3(c)\ncorrespond to the depinning \felds calculated in the pre-\nvious simulations and they are in good agreement with\nthe values predicted by the velocity vs\feld curve. The\ndynamical depinning \feld \u00160Hd, normalized to the static\ndepinning \feld \u00160Hs= (87\u00061)mT, with \u00160being the\nvacuum permeability, is shown in Fig. 3(d) as function of\nthe damping parameter \u000b.Hdsaturates for high damp-\ning (in this case \u000b\u00150:5) while it decreases for low damp-\ning untilHd=Hs\u00180:4 at\u000b= 0:02. This reduction must\nbe related to the precessional term, neglected in the static\nsimulations. The same behaviour is observed with di\u000ber-\nent grain sizes (GS=5 and 30 nm) and with a di\u000berent\ndisorder model, consisting of a simple variation of the Ku\nmodule in di\u000berent grains. This means that the e\u000bect is\nnot related to the grains size or to the particular disorder\nmodel we used.\nAdditionally, Fig. 4 represents the DW energy35as\nfunction of DW position and damping parameter for\n\u00160Hz= 70 mT. At high damping, the average DW en-\nergy density converges to \u001b1\u001810 mJ=m2, in good agree-\nment with the analytical value \u001b0= 4pAK0\u0000\u0019D=\n10:4 mJ=m2, whereK0is the e\u000bective anisotropy K0=\nKu\u0000\u00160M2\ns=2. On the contrary, for low damping, the\nDW energy increases up to \u001b(0:02)\u001814 mJ=m2. This\nincrease, related to DW precessional dynamics, reduces\nthe e\u000bective energy barrier and helps the DW to over-\n●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■\n●α=����\n○α=���\n■α=���\n���������������������<��> ★(�) ������\n��\n●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○\n■■■■■■■■■■■■■■■■■\n�� �� �� ��������������������������\n������� �����(��)<��>★★★(�) �������\n●●●●●●●●●○○○○○○○○○■■■■■■■■■●α=����\n○α=���\n■α=���\n�� �� �� ��������������\n������� �����(��)�� ��������(�/�)★★★(�)\n●●●●● ● ●\n������������������������������������\n�������α��/��(�)FIG. 3. Average hmzias function of applied \feld for dif-\nferent damping parameters for the (a) Static simulations and\n(b)Dynamic simulations. (c) DW velocity vs applied \feld for\ndi\u000berent damping. (d) Dynamical depinning \feld, normalized\ntoHs, as function of damping.\ncome the pinning barriers. Fig. 4(c) shows the total en-\nergy of the system (including Zeeman). As expected36,\nthe energy decreases as the DW moves.\nFinally, Fig. 5 shows the DW motion as function of\ntime for\u000b= 0:02 and\u000b= 0:5, along the same grain\npattern (and therefore along the same pinning barriers).\nThe applied \feld is \u00160Hz= 70mT, which satis\fes\nHd(0:02)<Hz<Hd(0:5). The initial DW con\fguration\nis the same but, for \u000b= 0:02, VBL start to nucleate and\nthe DW motion is much more turbulent (see Supplemen-\ntary Material32for a movie of this process). At t= 4 ns\nthe DW has reached an equilibrium position for \u000b= 0:5,\nwhile it has passed through the (same) pinning barriers\nfor\u000b= 0:02. Thus, one might think that the reduction\nof the depinning \feld could be related to the presence4\n●●●●●●●● ●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●\n●\n●\n●○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n○○○○○○○○○○○■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ �� ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■\n●α=����\n○α=���\n■α=���\n��������������������������\n�� ��������(μ�)σ��(��/��)(�)\n●\n●●\n�����������������������������\n�������α<σ��> (��/��)(�)\n●●●●●●●●●●●●\n●●●●●●●●●●●●○○○○○○○○○○○○○○○○○○○○○○○■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■ ■ ■ ■ ■ ■ ■ ■ ■\n���������������-��-�������\n�� ��������(��)����� ������ �������(��/��)\n(�)\nFIG. 4. (a) DW energy density as function of DW posi-\ntion for di\u000berent damping. The \fnal drop corresponds to\nthe expulsion of the DW. (b) Average DW density as funci-\nton of damping. Dashed line represents the analytical value\n\u001b1\u001810 mJ=m2. (c) Total energy density of the system as\nfunction of DW position for di\u000berent damping parameters.\nof VBL and their complex dynamics21. Further insights\nabout this mechanism are given by analysing the DW\ndepinning at a single energy barrier as described in the\nnext subsection.\nB. Single barrier\nIn order to understand how the DW precessional dy-\nnamics reduces Hdep, we micromagnetically analysed the\nDW depinning from a single barrier as sketched in Fig. 6.\nWe considered a strip of dimensions (1024 \u0002256\u00020:6)nm3\nand we divided the strip into two regions, R1andR2,\nwhich are assumed to have a thickness of t1= 0:58 and\nt2= 0:62 nm respectively. Their parameters vary ac-\ncordingly (see Sec. II), generating the DW energy bar-\nrier (\u000e\u001b) shown in Fig. 6(b). A DW is placed and re-\nlaxed just before the barrier. The \fnite size of the DW\n(\u0019\u0001DW\u001815 nm, with \u0001 DWbeing the DW width pa-\nrameter) smooths the abrupt energy step and, in fact,\nthe energy pro\fle can be successfully \ftted by using theBloch pro\fle22\n\u001bDW=\u001b0+\n+\u0012\u000e\u001b\n2\u0013\u001a\n1 + cos\u0012\n2 arctan\u0014\nexp\u0012x0\u0000x\n\u0001DW\u0013\u0015\u0013\u001b\n;\n(4)\nwherex0= 20 nm is the step position, while \u001b0and\n\u001b1are the DW energies at the left and right side of the\nbarrier as represented in Fig. 6(b). This means that\nthe pinning energy barrier has a spatial extension which\nis comparable to the DW width. By performing the\nsame static and dynamic simulations, we obtain a static\ndepinning \feld of \u00160Hs= 120 mT and, when decreasing\nthe damping parameter, we observe the same reduction\nof the depinning \feld as in the granular system (see\nFig. 6(c)). In this case the DW behaves like a rigid\nobject whose spins precess coherently and no VBL\nnucleation is observed. Hence, Hdepreduction does not\ndepend directly on the presence of VBL but on the more\ngeneral mechanism of spins' precession already present\nin this simpli\fed case.\nNevertheless, an important characteristic of these single\nbarrier simulations is that the barrier is localized and it\nhas a \fnite size which is of the order of the DW width.\nNote that the same holds for the granular system:\ndespite a more complex barrier structure, the dimension\nof the single barrier between two grains has the size of\nthe DW width.\nThus, in order to understand the interplay between the\nDW precessional dynamics and the \fnite size of the bar-\nrier, we considered a 1D collective-coordinate model with\na localized barrier. The 1D model equations, describing\nthe dynamics of the DW position qand the internal angle\n\u001e(sketched in Fig. 2(c)), are given by16\n(1 +\u000b2)_\u001e=\r0[(Hz+Hp(q))\n\u0000\u000b\u0012\nHKsin 2\u001e\n2\u0000\u0019\n2HDMIsin\u001e\u0013\n|{z }\nHint(\u001e)];(5)\n(1 +\u000b2)_q\n\u0001DW=\r0[\u000b(Hz+Hp(q))\n+\u0012\nHKsin 2\u001e\n2\u0000\u0019\n2HDMIsin\u001e\u0013\u0015\n;(6)\nwhereHK=MsNxis the shape anisotropy \feld, favour-\ning Bloch walls, with Nx=t0log 2=(\u0019\u0001DW)37being the\nDW demagnetizing factor along the xaxis.HDMI =\nD=(\u00160Ms\u0001DW) is the DMI \feld. Hint(\u001e) represents\nthe internal DW \feld, which includes DMI and shape\nanisotropy. Hintfavours Bloch ( \u001e=\u0006\u0019=2) or N\u0013 eel wall\n(\u001e= 0 or\u001e=\u0019) depending on the relative strength\nofHKandHDMI. In our system, the DMI dominates\nover shape anisotropy since \u00160HDMI\u0018170 mT while\n\u00160HK\u001830 mT. Hence, the DW equilibrium angle is5\nOut[64]=\nOut[60]=\n�� (a) 𝜶=𝟎.𝟎𝟐time 0 0.1 ns 0.2 ns 0.3 ns 4 ns\ntime 0 0.1 ns 0.2 ns 0.3 ns 4 ns(b) 𝜶=𝟎.𝟓\n… \nOut[395]=mx\nOut[395]=mx\nOut[62]=\nOut[65]=\nFIG. 5. (a) Snapshots of the magnetization dynamics at subsequent instants under \u00160Hz= 70mT, for two di\u000berent damping:\n(a)\u000b= 0:02 and (b) \u000b= 0:5. The grains pattern, and therefore the energy barrier, is the same for both cases. In order to let\nthe DW move across more pinning sites, these simulations were performed on a larger sample with Lx= 2048 nm.\n\u001e=\u0019(\u001e= 0 or\u001e=\u0019additionally depends on the sign\nof the DMI). Hp(q) is the DW pinning \feld, obtained\nfrom the DW energy pro\fle (Eq. (4)) as follows: the max-\nimum pinning \feld is taken from the static simulations\nwhile the shape of the barrier is taken as the normalized\nDW energy gradient (see Supplementary Material32for\nmore details),\nHp(q) =Hs\u0012@\u001bDW(x)\n@x\u0013\nN=\n= 2Hsexp\u0010\nx0\u0000q\n\u0001DW\u0011\nsinh\n2 arctan\u0010\nexp\u0010\nx0\u0000q\n\u0001DW\u0011\u0011i\n1 + exp\u0010\n2(x0\u0000q)\n\u0001DW\u0011 :(7)\nThe corresponding pinning \feld is plotted in Fig. 7(a).38\nThe results for the dynamical Hdep, obtained with this\nmodi\fed 1D model, are plotted in Fig. 6(c) and they\nshow a remarkable agreement with the single barrier mi-\ncromagnetic simulations. This indicates that the main\nfactors responsible for the reduction of Hdepare already\nincluded in this simple 1D model. Therefore, additional\ninsights might come from analysing the DW dynamics\nwithin this 1D model. Fig. 7(b) and (c) represents the\nDW internal angle \u001eand the DW position qas function\nof time for di\u000berent damping. The plots are calculated\nwith\u00160Hz= 55 mT which satis\fes Hdep(0:02)< Hz<\nHdep(0:1)< H dep(0:5). As shown in Fig. 7(b) and (c),\nbelow the depinning \feld ( \u000b= 0:1,\u000b= 0:5), both the\ninternal angle and the DW position oscillate before reach-\ning the same \fnal equilibrium state. However, the am-plitude of these oscillations (the maximum displacement)\ndepends on the damping parameter. Fig. 7(d) shows the\n\fnal equilibrium position as function of the applied \feld\nfor di\u000berent damping. The equilibrium position is the\nsame for all damping and it coincides with the position\nat whichHz=Hp(q). Conversely, the maximum dis-\nplacement, shown in Fig. 7(e), strongly increases for low\ndamping parameters. For applied \feld slightly smaller\nthan the depinning \feld, the DW reaches the boundary\nof the pinning barrier, meaning that a further increase\nof the \feld is enough to have a maximum displacement\nhigher than the barrier size and depin the DW. In other\nwords, the decrease of the depinning \feld, observed in\nthe single barrier simulations, is due to DW oscillations\nthat depend on \u000band that can be larger than the bar-\nrier size, leading to DW depinning for lower \feld. The\nDW dynamics and the depinning mechanism are further\nclari\fed in Fig. 7(f) and Fig. 7(g). Fig. 7(f) represents\nthe DW coordinates fq;\u001egfor\u00160Hz= 55 mT and dif-\nferent damping. Before reaching the common equilib-\nrium state, the DW moves in orbits (in the fq;\u001egspace)\nwhose radius depends on the damping parameter. For\n\u000b= 0:5 (black line) the DW rapidly collapse into the \f-\nnal equilibrium state. Conversely, for \u000b= 0:1 (red open\ncircles), the DW orbits around the equilibrium state be-\nfore reaching it. If the radius of the orbit is larger than\nthe barrier size the DW gets depinned, as in the case\nof\u000b= 0:02 (blue full circles). This mechanism is also\nrepresented in Fig. 7(g), where the DW orbits are placed\nin the energy landscape. The energy is calculated as6\n●●●●●●●●●●●●●●●●●μ� ����������� ��●�� ��������������������������������������������\n�������α��/��○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○�� �������������(�)-���������������������������������������(��)σ��(��/��)��δσσ�σ�(b)\n(c)\nR1R2yx(a)\nFIG. 6. (a) Sketch of the two regions implemented for the\nsingle barrier (SB) micromagnetic simulations. (b) DW en-\nergy as function of DW position along the strip. Blue solid\nline represents the analytical value, red points the DW con-\nvoluted energy (due to the \fnite size of the DW) while black\ndashed line a \ft using Eq. 4. (c) Dynamical depinning \feld,\nnormalized to the static depinning \feld, for the single bar-\nrier simulations as function of damping, obtained from full\nmicromagnetic simulations and the 1D model.\n\u001b(q;\u001e) =\u001bDW(q;\u001e)\u00002\u00160MsHzq, where\u001bDWis given by\nEq. (4). Fig. 7(g) shows that the equilibrium state cor-\nresponds to the new minimum of the energy landscape.\nFurthermore, it con\frms that the applied \feld is below\nthe static depinning \feld, at which the pinning barrier\nwould have been completely lifted. Nevertheless, while\nreaching the equilibrium state, the DW moves inside the\nenergy potential and, if the radius of the orbit is larger\nthan the barrier size, the DW can overcome the pinning\nbarrier, as shown for \u000b= 0:02 in Fig. 7(g).\nAt this point we need to understand why the amplitude\nof the DW oscillations depends on damping. By solving\nEq. (5) and Eq.(6) for the equilibrium state ( _ q= 0, _\u001e=\n0) we obtain\n_q= 0)jHp(q)j=Hz+Hint(\u001e)\n\u000b\n\u0019Hz\u0000\u0019\n2HDMI\n\u000bsin\u001e; (8)\n_\u001e= 0)jHp(q)j=Hz\u0000\u000bHint(\u001e)\n\u0019Hz+\u000b\u0019\n2HDMIsin\u001e; (9)\nsince\u00160HDMI\u001d\u00160HKand, therefore, Hint\u0019\n\u0000(\u0019=2)HDMIsin\u001e. These equations have a single com-\nmon solution which corresponds to jHp(q)j=Hzand\n\u001e=\u001e0=\u0019(at whichHint(\u0019) = 0). However, at t= 0,the DW starts precessing under the e\u000bect of the applied\n\feld and, if \u001e6=\u0019whenjHp(q)j=Hz, the DW does not\nstop at the \fnal equilibrium position but it continues its\nmotion, as imposed by Eq. (8) and (9). In other words,\nthe DW oscillations in Fig. 7(b) are given by oscillations\nof the DW internal angle \u001e, around its equilibrium value\n\u001e0=\u0019. These oscillations lead to a modi\fcation of the\nDW equilibrium position due to the DW internal \feld\n(Hint(\u001e)), which exerts an additional torque on the DW\nin order to restore the equilibrium angle. As previously\ncommented, if the amplitude of these oscillations is large\nenough, the DW gets depinned. From Eq. (8) we see\nthat the new equilibrium position (and therefore the am-\nplitude of the oscillations) depends on the DMI \feld, the\nvalue of the DW angle \u001eand the damping parameter.\nIn particular, damping has a twofold in\ruence on this\ndynamics: one the one hand, it appears directly in\nEq. (8), dividing the internal \feld, meaning that for the\nsame deviation of \u001efrom equilibrium, we have a stronger\ninternal \feld for smaller damping. On the other hand,\nthe second in\ruence of damping is on the DW internal\nangle: once the DW angle has deviated from equilibrium,\nthe restoring torque due to DMI is proportional to the\ndamping parameter (see Eq. (9)). Hence, a lower damp-\ning leads to lower restoring torque and a larger deviation\nof\u001efrom equilibrium. The maximum deviation of \u001efrom\nequilibrium ( \u000e\u001e=\u001emax\u0000\u001e0) is plotted in Fig. 8(b) as\nfunction of damping for \u00160Hz= 40 mT. As expected, a\nlower damping leads to a larger deviation \u000e\u001e.\nIn this latter section, the DW was set at rest close to\nthe barrier and, therefore, the initial DW velocity is zero.\nNevertheless, one might wonder what happens when the\nDW reaches the barrier with a \fnite velocity. We simu-\nlated this case by placing the DW at an initial distance\nd1= 200 nm from the barrier. The depinning is further\nreduced in this case (see Supplementary Material32for\nmore details). However, in the static simulations, the de-\npinning \feld remains constant, independently from the\nvelocity at which the DW reaches the barrier, meaning\nthat the reduction of Hdepis again related to the DW\nprecession. When the DW starts from d1it reaches the\nbarrier precessing, thus with a higher deviation from its\nequilibrium angle, leading to a higher e\u000bect of the inter-\nnal \feld.\nC. Di\u000berent DMI and pinning barriers\nFinally, by using the 1D model it is possible to ex-\nplore the dependence of Hdepon the pinning potential\namplitudeHs(related to the disorder strength) and on\nthe DMI constant D. The depinning \feld as function of\ndamping for di\u000berent values of Hsis plotted in Fig. 9(a).\nThe reduction of Hdepis enhanced for larger values of\nHs(strong disorder). This is consistent with our expla-\nnation, since for strong disorder we need to apply larger\n\felds that lead to larger oscillations of \u001e.\nFig. 9(b) represents the dynamical Hdepas function of7\n●●●●●●●●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼�����������������������������������(��)�� ��������(��)●●●●●●●●■■■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆◆▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼○○○○○○○○○○○○○○○○○○○○○○●α=����■α=���◆α=���▲α=����▼α=���○α=�������������������������������������������������\n������� �����μ���(��)���������������(��)������������(��)●●●●●●●■■■■■■■■■◆◆◆◆◆◆◆◆◆◆◆◆▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼○○○○○○○○○○○○○○○○○○○○○●α=����■α=���◆α=���▲α=����▼α=���○α=�������������������������������������������������\n������� �����μ���(��)�����������(��)������������(��)-�������������-���-��-�������������(��)μ���(��)\nMax Displacement\nEq. Position(a)(b)(c)(d)\n(e)(f)\n(g)●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○○ 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▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼●α=����○α=���α=������������������������������������\n�(��)ϕ(°)\nFIG. 7. (a) Pinning \feld obtained from Eq. (7) as function of DW position. DW position internal angle \u001eas function of\ntime for di\u000berent damping parameter and \u00160Hz= 55 mT. (c) DW position qas function of time for di\u000berent damping and\n\u00160Hz= 55 mT. (d) Equilibrium position as function of applied \feld for di\u000berent damping. (e) Maximum DW displacement as\nfunction of the applied \feld for di\u000berent damping. (f) DW coordinates fq;\u001egfor\u00160Hz= 55 mT and di\u000berent damping. (g)\nDW coordinatesfq;\u001eginside the energy landscape: \u001b=\u001bDW(q;\u001e)\u00002\u00160MsHzq.\n������������������������������\n�������αδϕ(°)\nFIG. 8. Maximum deviation of \u001efrom its equilibrium posi-\ntion as function of damping.\ndamping for \u00160Hs= 120 mT and di\u000berent DMI con-\nstants (expressed in term of the critical DMI constant\nDc= 4pAK0=\u0019= 3:9 mJ=m2)39. In this case, the reduc-\ntion ofHdepis enhanced for low DMI, until D= 0:05Dc,\nbut a negligible reduction is observed for D= 0. This\nnon-monotonic behaviour can be explained by looking at\nthe dependence of \u000e\u001eandHinton the DMI constant.\nFig. 10(a) shows the maximum \ructuation \u000e\u001eas func-\ntion of DMI for \u00160Hz= 30 mT. \u000e\u001eincreases for low\nDMI and it has a maximum at \u0019HDMI =HK, which\nin our case corresponds to D= 0:014Dc. The increase\nof\u000e\u001efor small values of Dis due to the smaller restor-\ning torque in Eq. (9). This holds until \u0019HDMI =HK,\nwhere shape anisotropy and DMI are comparable and\nthey both a\u000bect the DW equilibrium con\fguration. As a\nconsequence, the reduction of Hdepis enhanced by de-creasingDuntilD\u00180:014Dc, while it is reduced if\n0< D < 0:014Dc. Another contribution is given by\nthe amplitude of the internal \feld, Hint. Fig. 10(b) de-\npicts\u00160Hintas function of \u000e\u001eandD. The maximum\n\u000e\u001e, obtained at \u00160Hz= 30 mT, is additionally marked\nin the plot. The internal \feld decreases with the DMI\nbut this reduction is compensated by an increase in \u000e\u001e,\nwhich leads to an overall increase of \u00160Hint, as discussed\nin the previous part. However, at very low DMI, the in-\nternal \feld is dominated by shape anisotropy and, inde-\npendently on the DW angle displacement, it is too small\nto have an e\u000bect on the depinning mechanism. Note,\nhowever, that the amplitude of Hintshould be compared\nwith the amplitude of the pinning barrier Hs. Fig. 9(b)\nis calculated with \u00160Hs= 120 mT and the internal \feld,\ngiven by shape anisotropy ( HK=2\u001815 mT), has indeed\na negligible e\u000bect. However, larger e\u000bects are observed,\nin the caseD= 0, for smaller Hs, with reduction of Hdep\nup toHd=Hs\u00180:6, as shown in Fig. 9(c), which is calcu-\nlated with\u00160Hs= 30 mT. In other words, the reduction\nof the depinning \feld depends on the ratio between the\npinning barrier and the internal DW \feld.\nFinally, it is interesting to see what happens for\nweaker disorder and di\u000berent DMI in the system with\ngrains. Fig. 11 shows the dynamical Hdep, for di\u000berent\npinning potential and di\u000berent DMI, obtained in the\ngranular system. The results are in good agreement with\nwhat predicted by the 1D model for di\u000berent disorder\nstrengths. However, we observe a smaller dependence\non the DMI parameter. This is due to two reasons:8\n●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ●\n■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■\n◆◆◆◆◆◆◆◆◆◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ▲ ▲\n●μ���=�� ��\n■μ���=�� ��\n◆μ���=�� ��\n▲μ���=��� ��\n������������������������������������\n�������α��/��(�)\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼\n○○○○○○○○○○○○○○○○○○○○\n□□□□□□□□□□□□□□□□□□□□\n●�=���� � �■�=��� � �\n◆�=��� � �▲�=��� � �\n▼�=��� � �○�=��� � � □�=���\n���������������������������������������\n�������α��/��(�) μ���=��� ��\n●●●●●●●●●●●●●●●●●●●●\n■■■■■■■■■■■■■■■■■■■■\n◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆\n▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲\n▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼\n○○○○○○○○○○○○○○○○○○○○\n□□□□□□□□□□□□□□\n□□□□□□\n●�=���� � �■�=���� � �\n◆�=���� � �▲�=���� � �\n▼�=���� � �○�=���� � � □�=���\n�����������������������������������������\n�������α��/��(�) μ���=�� ��\nFIG. 9. (a) Dynamical Hdepas function of damping for di\u000ber-\nentHs(disorder strength). (b) Dynamical Hdepas function\nof damping for di\u000berent DMI constant and \u00160Hs= 120 mT.\n(c) Dynamical Hdepas function of damping for di\u000berent DMI\nconstant and \u00160Hs= 30 mT.\n(1) in the system with grains the static pinning barrier\nis\u00160Hs= 87 mT and the dependence of the depinning\n\feld with DMI is smaller for smaller barriers, as shown\nin Fig. 9(c). (2) The DW motion in the granular\nsystem presents the formation of VBL which might also\ncontribute to the reduction of the depinning \feld. The\nmechanism is the same: a VBL is a non-equilibrium\ncon\fguration for the DW (as a deviation of \u001efrom\nequilibrium) that generates additional torques on the\nDW, which contribute to the DW depinning.\n����� ���� ��� �������������\n�/��δϕ(°)(�)\nπ����=��\nμ�����(��)\n� ��� ��� ��� ���\n������������������������������\n�/��δϕ(°)(�)FIG. 10. (a) Max DW angle \ructuation \u000e\u001e=\u001emax\u0000\u001eeq\nas function of DMI for \u00160Hz= 30 mT. (b) Internal DW\n\feld\u00160Hintas function of DMI and \u000e\u001e. The green points\ncorrespond the max \ructuation plotted in (a). Note that the\nscale is logarithmic in (a).\n●●●●● ● ●\n□□□□□ □\n●μ���=�� ��\n□μ���=�� ����������������/��(�)\n●●●●● ● ●\n◇◇◇◇◇ ◇\n○○○○○ ○\n●�=��� ��/��~���� �\n◇�=��� ��/��~���� �\n○�=�\n������������������������������\n�������α��/��(�)\nFIG. 11. (a) Dynamical Hdepas function of damping for dif-\nferentHs(disorder strength). (b) Dynamical Hdepas function\nof damping for di\u000berent DMI constants.9\nIV. CONCLUSIONS\nTo summarize, we have analysed the DW depinning\n\feld in a PMA sample with DMI and we found that Hdep\ndecreases with the damping parameter with reductions\nup to 50%. This decrease is related to the DW inter-\nnal dynamics and the \fnite size of the barrier: due to\nDW precession, the DW internal angle ( \u001e) deviates from\nequilibrium and triggers the internal DW \feld (DMI and\nshape anisotropy) which tries to restore its original value.\nAt the same time, the internal \feld pushes the DW above\nits equilibrium position within the energy barrier. This\nmechanism leads to DW oscillations and, if the ampli-\ntude of the oscillations is higher than the barrier size,\nthe DW gets depinned for a lower \feld. Deviations of \u001e\nfrom equilibrium and DW oscillations are both damping\ndependent and they are enhanced at low damping.\nIn the system with grains the mechanism is the same\nbut deviations from the internal DW equilibrium include\nthe formation of VBL with more complex dynamics.\nThe e\u000bect is enhanced for low DMI (providing that\u0019HDMI> H K) and for stronger disorder since we need\nto apply larger external \felds, which lead to larger DW\noscillations. These results are relevant both from a tech-\nnological and theoretical point of view, since they \frstly\nsuggest that a low damping parameter can lead to a\nlowerHdep. Furthermore, they show that micromagnetic\ncalculations of the depinning \feld, neglecting the DW\nprecessional dynamics can provide only an upper limit\nforHdep, which could actually be lower due to the DW\nprecessional dynamics.\nV. ACKNOWLEDGEMENT\nS.M. would like to thank K. Shahbazi, C.H. Mar-\nrows and J. Leliaert for helpful discussions. This work\nwas supported by Project WALL, FP7- PEOPLE-2013-\nITN 608031 from the European Commission, Project No.\nMAT2014-52477-C5-4-P from the Spanish government,\nand Project No. SA282U14 and SA090U16 from the\nJunta de Castilla y Leon.\n\u0003Corresponding author: simone.moretti@usal.es\n1D. Allwood, Science 309, 1688 (2005).\n2S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n3P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier,\nJ. Ferr\u0013 e, V. Baltz, B. Rodmacq, B. Dieny, and R. L.\nStamps, Physical Review Letters 99, 217208 (2007).\n4J. Gorchon, S. Bustingorry, J. Ferr\u0013 e, V. Jeudy, A. B.\nKolton, and T. Giamarchi, Physical Review Letters 113,\n027205 (2014), arXiv:1407.7781.\n5T. A. Moore, I. M. Miron, G. Gaudin, G. Serret, S. Auf-\nfret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and\nM. Bon\fm, Applied Physics Letters 93, 262504 (2008),\narXiv:0812.1515.\n6I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-\nPrejbeanu, S. Au\u000bret, B. 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In fact, by in-creasing the sample dimension along the xdirection, we\nincrease the probability of \fnding the highest possible hj\nin the single realization and the average of Hi\ndepwill tend\nto the maximum.\n34This is solved by the Relax solver of MuMax with the as-\nsumption\u000b=(1 +\u000b2) = 1.\n35The DW energy is calculated as the energy of the system\nwith the DW minus the energy of the system without the\nDW (uniform state). The pro\fle is obtained by moving the\nDW with an external applied \feld and then subtracting the\nZeeman energy.\n36X. Wang, P. Yan, J. Lu, and C. He, Annals of Physics\n324, 1815 (2009), arXiv:0809.4311.\n37S. Tarasenko, a. Stankiewicz, V. Tarasenko, and J. Ferr\u0013 e,\nJournal of Magnetism and Magnetic Materials 189, 19\n(1998).\n38The same results are obtained with a Gaussian barrier,\nmeaning that the key point is the \fnite size of the barrier\nrather than its shape.\n39ForD >D c, DW have negative energies and the systems\nspontaneously breaks into non-uniform spin textures.\nAppendix A: Maximum torque and equilibrium state\nIn this section we show in more detail how the maximum torque represents an indicator of the equilibrium state.\nMaximu torque is de\fned as\n\u001cmax\n\r0= Maxf\u00001\n1 +\u000b2mi\u0002He\u000b;i\u0000\u000b\n1 +\u000b2mi\u0002(mi\u0002He\u000b;i)g=1\n\r0Max\u0012dmi\ndt\u0013\n; (A1)\nover all cells with label i=f1;:::;N =Nx\u0001Nyg. MuMax3.9.329can provide this output automatically if selected.\nWe perform the same simulations as indicated in the main text, without any stopping condition, but simply running\nfort= 20 ns. Fig. 12(a) shows the average mzcomponent for \u000b= 0:2 andBz= 10 mT, while Fig. 12(b) depicts the\ncorresponding maximum torque. We can see that, once the system has reached equilibrium, the maximum torque has\ndropped to a minimum value. The same results is obtained for di\u000berent damping but the \fnal maximum torque is\ndi\u000berent. Numerically this value is never zero since it is limited by the code numerical precision and by the system\nparameters, in particular by damping.\nFig. 12(c) represents the maximum torque as function of applied \feld for di\u000berent damping. The maximum torque\nis clearly independent on the applied \feld but depends on the damping value. Finally, Fig. 12(d) shows the max\ntorque as function of damping. The maximum torque decreases with damping and it saturates for \u000b\u00150:5 since we\nhave reached the minimum numerical precision of the code29. For higher damping the maximum torque oscillates\naround this minimum sensibility value, as shown in the inset of Fig. 12(d). The value obtained with these preliminary\nsimulations is used to set a threshold \u000f(\u000b) for the depinning \feld simulations in order to identify when the system has\nreached an equilibrium. Furthermore, additional tests were performed, without putting any max torque condition,\nbut simply running the simulations for a longer time ( t= 80;160 ns) and calculating the depinning \feld in order\nto ensure that the results obtained with these two method were consistent, i.e., that we have actually reached an\nequilibrium state with the maximum torque condition.11\n������������������-���-���-���\n����(��)<��>⨯��-�α=���(�)\n���������������������������\n����(��)��� ������/γ �(��)α=���(�)\n○ ○ ○ ○ ○ ○ ○ ○\n□ □ □ □ □ □ □ □\n◇◇◇◇◇◇◇◇\n○α=����□α=���◇α=���\n� �� �� �� ��������������������\nμ���(��)��� ������/γ �(��)(�)\n●\n●●\n��������������������-�������������������\n�������α��� ������/γ �(��)(�)\n���������������������\n����(��)τ���/γ�(��)α=���\nFIG. 12. (a) average mzas function of time. (b) Max torque/ \r0(\u001cmax) as function of time. \u001cmaxrapidly decreases when the\nsystem is at equilibrium. (c) Max torque as function of applied \feld for di\u000berent damping. (d) Max torque at equilibrium as\nfunction of damping. The inset shows the max torque as function of time for \u000b= 0:5.\nAppendix B: 1D energy barrier\nAs commented in the main text, the pinning \feld implemented in the 1D model simulations is obtained by using the\nshape of the DW energy pro\fle derivative @\u001b(x)=@x(beingxthe DW position) and the amplitude of the depinning\n\feld obtained in the full micromagnetic simulations Hsfor the single barrier case. Namely\nHdep=Hs\u0012@\u001b(x)\n@x\u0013\nN; (B1)\nwhere we recall that Nstands for the normalized value. This choice might sound unusual and needs to be justi\fed.\nIn fact, having the DW energy pro\fle, the depinning \feld could be simply calculated as20\nHdep=1\n2\u00160Ms@\u001b(x)\n@x: (B2)\nThis expression is derived by imposing that the derivative of the total DW energy E(x) = 2\u00160MsHzx+\u001b(x) (Zeeman\n+ internal energy) must be always negative. However, in our case also Ms(x) depends on the DW position and the\nresults obtained with Eq. B2 is di\u000berent from the depinning \feld measured in the static single barrier simulations.\nFor this reason we use Eq. B1 which keep the correct barrier shape and has the measured static value.\nFinally, we recall that equivalent results are obtained by using a simple Gaussian shape for the pinning \feld, meaning\nthat the key point is the localized shape of the barrier, rather than its exact form.\nAppendix C: Dynamical depinning for a moving Domain Wall\nIn this section we show the results for the dynamical depinning \feld when the DW is placed at an initial distance of\nd1= 200 nm from the barrier. In this way the DW hits the pinning with an initial velocity. The d0case corresponds to\nthe DW at rest relaxed just before the barrier and extensively analysed in the main text. Also for this con\fguration\nwe performed static and dynamic simulations, neglecting or including the DW precessional dynamics respectively.\nThe depinning \feld for the d1case is further reduces at small damping, reaching Hd=Hs\u00180:08 (Hd= 9 mT and\nHs= 120 mT) at \u000b= 0:02. Nevertheless, the depinning \feld remains constant in the static simulations independently\non the velocity at which the DW hits the barrier. This suggests that, rather than related to the DW velocity, the\nreduction is again related to the DW precession. When the DW starts from d1it reaches the barrier precessing, thus\nwith a higher displacement from its equilibrium angle, leading to a higher e\u000bect of the internal \feld.12\n●●●●●●\n○○○○○○ ◆◆◆◆◆ ◆ □□□□□ □\n●��(�������)\n○��(�������)◆��(������)\n□��(������)\n��� ��� ��� ��� ������������������������\n�������α��/��\nFIG. 13. Dynamical depinning \feld as function of damping for static and dynamic simulations for the d0andd1cases."    },    {        "title": "2210.08429v1.Magnetic_damping_anisotropy_in_the_two_dimensional_van_der_Waals_material_Fe__3_GeTe__2__from_first_principles.pdf",        "content": "Magnetic damping anisotropy in the two-dimensional van der Waals material\nFe3GeTe 2from \frst principles\nPengtao Yang, Ruixi Liu, Zhe Yuan, and Yi Liu\u0003\nThe Center for Advanced Quantum Studies and Department of Physics,\nBeijing Normal University, 100875 Beijing, China\n(Dated: October 18, 2022)\nMagnetization relaxation in the two-dimensional itinerant ferromagnetic van der Waals ma-\nterial, Fe 3GeTe 2, below the Curie temperature is fundamentally important for applications to\nlow-dimensional spintronics devices. We use \frst-principles scattering theory to calculate the\ntemperature-dependent Gilbert damping for bulk and single-layer Fe 3GeTe 2. The calculated damp-\ning frequency of bulk Fe 3GeTe 2increases monotonically with temperature because of the dominance\nof resistivitylike behavior. By contrast, a very weak temperature dependence is found for the damp-\ning frequency of a single layer, which is attributed to strong surface scattering in this highly con\fned\ngeometry. A systematic study of the damping anisotropy reveals that orientational anisotropy is\npresent in both bulk and single-layer Fe 3GeTe 2. Rotational anisotropy is signi\fcant at low tem-\nperatures for both the bulk and a single layer and is gradually diminished by temperature-induced\ndisorder. The rotational anisotropy can be signi\fcantly enhanced by up to 430% in gated single-layer\nFe3GeTe 2.\nI. INTRODUCTION\nNewly emerged intrinsic two-dimensional (2D) ferro-\nmagnetic (FM) van der Waals (vdW) materials1{6have\nbecome the subject of intense research. Weak vdW\nbonding facilitates the extraction of thin layers down to\natomic thicknesses, whereas strong magnetocrystalline\nanisotropy protects long-range magnetic order. These\nmaterials provide an exciting arena to perform funda-\nmental investigations on 2D magnetism and promis-\ning applications of low-dimensional spintronics devices.\nAmong these materials, Fe 3GeTe 2(FGT) is especially\nattractive for its itinerant ferromagnetism and metal-\nlicity, such that both spin and charge degrees of free-\ndom can be exploited for designing functional devices.\nBulk FGT has a relatively high Curie temperature ( TC)\nof approximately 220-230 K.7{11Atomically thin lay-\ners of FGT have lower TCs, which, however, have been\nraised to room temperature (by ionic gating4) and be-\nyond (by patterning12). As a FM metal at reasonably\nhigh temperature, FGT opens up vast opportunities for\napplications.13{23\nThe dynamical properties of FGT critically a\u000bect the\napplicability and performance of these proposed low-\ndimensional spintronics devices. The most salient of\nthese properties is the dynamical dissipation of mag-\nnetization. It is usually described using a phenomeno-\nlogical parameter called Gilbert damping, which char-\nacterizes the e\u000eciency of the instantaneous magneti-\nzation to align eventually with the e\u000bective magnetic\n\feld during its precessional motion. Although this pa-\nrameter has been extensively studied in conventional\nFM materials, such as 3 dtransition metals and alloys,\ntwo key issues with the Gilbert parameter of FGT re-\nmain to be addressed: the temperature dependence and\nanisotropy (one naturally expects anisotropic damping\nin FGT because of its layered structure and the strong\nmagnetocrystalline anisotropy). Temperature-dependentGilbert damping was \frst observed in Fe24and later more\nsystematically in Fe, Co and Ni.25{27A nonmonotonic\ntemperature dependence has been found, for which a so-\ncalled \\conductivitylike\" component decreases with in-\ncreasing temperature, usually at low temperatures, and a\n\\resistivitylike\" component increases with temperature,\nusually at high temperatures. This nonmonotonic be-\nhavior has been successfully described by the torque-\ncorrelation model28and reproduced by \frst-principles\ncomputations.29{32Anisotropic damping was \frst theo-\nretically predicted in FM metals33and in noncollinear\nmagnetic textures.34With di\u000berent orientation of the\nequilibrium magnetization with respect to the crystal-\nlographic axes, the damping parameter can be quanti-\ntatively di\u000berent in general. This is referred to as the\norientational anisotropy. Even for the same equilibrium\nmagnetization orientation in a single crystalline lattice,\nthe magnetization may precess instantaneously along\ndi\u000berent directions resulting in the so-called rotational\nanisotropy.33The orientational anisotropy of damping\nhas been observed in recent experiments on single-crystal\nFM alloys,35{37but the underlying physical mechanism\nremains unclear.\nThe dimensionless Gilbert damping parameter \u000bcan\nbe expressed in terms of a frequency \u0015via\u0015=\u000b\rM ,38\nwhereM=jMjis the magnetization magnitude and \ris\nthe gyromagnetic ratio. Despite of the di\u000berent dimen-\nsions, these two parameters are equivalent39and both\npresent in literature for experimental24{27,35{37and the-\noretical studies.28,29,31,33,34,40{42\nIn this study, we systematically investigate\ntemperature-dependent Gilbert damping in single-\nlayer (SL) and bulk FGT using \frst-principles scattering\ntheory. Considering that the magnetization perpen-\ndicular to the 2D atomic planes is favored by the\nstrong magnetocrystalline anisotropy, we calculate the\ndamping as a function of temperature below TCand\n\fnd nearly temperature-independent damping in thearXiv:2210.08429v1  [cond-mat.mes-hall]  16 Oct 20222\n(a)\nFeⅠ\nFeⅡ\nGe\nTe(b)\nFIG. 1. (a) Side and (b) top view of the lattice structure\nfor bulk Fe 3GeTe 2. The black dashed frame delineates the\nin-plane unit cell.\nSL and damping dominated by resistivitylike behavior\nin the bulk. Varying the equilibrium direction of the\nFGT magnetization produces a twofold symmetry in\ndamping. When the magnetization is aligned inside\nthe 2D planes, a remarkable rotational anisotropy in\nthe Gilbert damping is present for in- and out-of-plane\nrotating magnetization.\nThis paper is organized as follows. The crystalline\nstructure of SL and bulk FGT is brie\ry introduced in Sec.\nII, followed by a description of our theoretical methods\nand computational details. The calculated temperature-\ndependent damping in SL and bulk FGT is presented in\nSec. III. The two types of damping anisotropy, i.e., orien-\ntational and rotational anisotropy, are analyzed in Sect.\nIV. Conclusions are drawn in Sec. V.\nII. GEOMETRIC STRUCTURE OF FGT AND\nCOMPUTATIONAL METHODS\nThe lattice structure of FGT is shown in Fig. 1. Two\ndi\u000berent types of Fe atoms occupy inequivalent Wycko\u000b\nsites and are denoted as FeI and FeII. Five atomic layers\nstack along the caxis to form an SL of FGT: Ge and\nFeII constitute the central atomic layer perpendicular to\nthecaxis, and two FeI layers and two Te layers are lo-\ncated symmetrically above and beneath the central layer,\nrespectively. Single layers with ABAB :::stacking form\nthe bulk FGT, where Layer A is translated in plane with\nrespect to Layer B, such that the Ge atoms in Layer A\nlie on top of the Te and FeII atoms in Layer B.\nThe electronic structure of bulk and SL FGT has\nbeen determined using the linear augmented plane wave\nmethod43within the local density approximation (LDA).\nDi\u000berent types of exchange-correlation functionals have\nbeen investigated in the literature, among which LDA\nwas found to yield satisfactory structural and magnetic\nproperties for FGT.44We employ experimentally ob-\ntained lattice constants7for bulk FGT calculations and\nobtain magnetic moments of 1.78 \u0016Band 1.13\u0016Bfor\nthe two types of Fe, respectively. The initial structure\nof a single layer is taken from the bulk lattice and fully\nrelaxed, resulting in an in-plane constant a= 3:92\u0017A.\nA vacuum spacing of 11.76 \u0017A is chosen to exclude theinterlayer interaction under periodic boundary condi-\ntions. The magnetic moments for the Fe atoms in SL\nFGT are obtained as 1.72 \u0016Band 1.01\u0016B. All the\ncalculated magnetic moments are in good agreement\nwith experimental7,8,45,46and calculated values44,47,48re-\nported in the literature.\nThe Gilbert damping calculation is performed using\nthe scattering theory of magnetization dissipation pro-\nposed by Brataas et al.49Within this theory, a single do-\nmain FM metal is sandwiched between two nonmagnetic\n(NM) metal leads. The Gilbert damping that charac-\nterizes the energy dissipation during magnetization dy-\nnamics can be expressed in terms of a scattering ma-\ntrix and its derivative with respect to the magnetiza-\ntion direction. We thus construct a two-terminal trans-\nport structure as Au jFGTjAu, where the Au lattice is\nslightly deformed to match that of FGT: we use 3 \u00021 and\n4\u00021 unit cells (UCs) of Au (001) to match the UCs of\nSL and bulk FGT, respectively. To investigate the ef-\nfect of temperature on Gilbert damping, we use a frozen\nthermal lattice and spin disorder31,40,50to mimic lattice\nvibration and spin \ructuation at \fnite temperatures in\nFGT. The measured Debye temperature \u0002 D= 232 K\nand temperature-dependent magnetization for the bulk8\nand SL3are employed to model the lattice and spin disor-\nder. In the scattering calculations, lateral supercells are\nemployed to satisfy periodic boundary conditions perpen-\ndicular to the transport direction. The electronic poten-\ntials required for the transport calculation are calculated\nself-consistently using a minimal basis of tight-binding\nlinear mu\u000en-tin orbitals (TB-LMTOs), and the result-\ning band structures for SL and bulk FGT e\u000bectively re-\nproduce those obtained using the linear augmented plane\nwave method. Then, the scattering matrices consisting\nof re\rection and transmission probability amplitudes for\nthe Bloch wave functions incident from the NM leads are\ndetermined by the so-called \\wave function matching\"\nmethod, which is also implemented using TB-LMTOs.40\nOther computational details can be found in our previous\npublications.31,40{42In this work, we focus on the damp-\ning with collective magnetization dynamics in the long-\nwave limit corresponding to the reported values in ex-\nperiment via ferromagnetic resonance and time-resolved\nmagneto-optical Kerr e\u000bect. The damping with a \f-\nnite wavelength can be determined in our framework of\nscattering calculation42or using the torque-correlation\nmodel,51but the wavelength dependence of damping is\nbeyond the scope of the current study.\nIII. TEMPERATURE-DEPENDENT DAMPING\nThe strong magnetocrystalline anisotropy of FGT re-\nsults in the equilibrium magnetization being naturally\nperpendicular to the atomic layers. Slightly excited mag-\nnetization deviates from the plane normal (denoted as ^ z)\nand relaxes back by dissipating energy and angular mo-\nmentum, as schematized in the inset of Fig. 2(a). The3\n𝛂∥𝛂∥𝑴(𝑡)𝑥𝑦𝑧\n5101520F|| (10-3)\n0 0.2 0.4 0.6 0.8 1T/TC468Q|| (108 Hz)Lattice disorder only(b)(a)\nBulkSingle layer\nBulkSingle layer\nFIG. 2. The calculated dimensionless Gilbert damping pa-\nrameter\u000bk(a) and corresponding damping frequency \u0015k(b)\nfor single-layer and bulk Fe 3GeTe 2as a function of tempera-\nture. The relaxation of the instantaneous magnetization M(t)\nresults in a change in the in-plane magnetization component,\nwhich is parallel to the atomic planes, as schematized in the\ninset of (a). The empty symbols in (b) denote the damping\nfrequencies that are calculated considering only thermal lat-\ntice disorder. The green line indicates the linear temperature\ndependence.\nGilbert damping parameter \u000bkdescribes the e\u000eciency\nof such a dissipative process. The calculated \u000bkof SL\nand bulk FGT is plotted in Fig. 2(a) as a function of\ntemperature. The damping for both increases monoton-\nically with the temperature. This behavior resembles\nthe so-called \\resistivitylike\" damping observed in many\nsingle-crystal FM metals.24{26However, the damping \u000bk\nfor the bulk tends to diverge as the temperature ap-\nproachesTC. This divergence originates from vanishing\nmagnetization, as has been found in three-dimensional\nFM alloys.42Therefore, as temperatures approaching TC,\nit is more appropriate to use the damping frequency pa-\nrameter\u0015=\u000b\rM .\nThe calculated damping frequencies are shown in\nFig. 2(b). The damping of a SL FGT, \u0015S\nk, is larger\nthan the damping of the bulk, \u0015B\nk, especially at low\ntemperatures. This di\u000berence can be attributed to the\nstrong surface e\u000bect of highly con\fned SL FGT. The\nlowered symmetry at the surface signi\fcantly enhances\nspin-orbit coupling (SOC),52which enables the dissipa-\ntion of angular momentum from electronic spins to the\norbital degree of freedom and then into the lattice reser-voir. In addition, as the thickness of a single layer is\nconsiderably smaller than the electronic mean free path,\nconduction electrons in FGT are strongly scattered by\nthe surface. Therefore, the two necessary ingredients for\nGilbert damping, namely, SOC and electronic scattering,\nare both enhanced in the SL compared with the bulk, re-\nsulting in a larger damping for the SL.\nThe calculated damping frequency \u0015S\nkremains nearly\nconstant with increasing temperature, except for a mi-\nnor increase at T > 0:6TC. To gain further insight into\nthe temperature e\u000bect, we perform the damping calcu-\nlation considering only lattice disorder, where the calcu-\nlated\u0015S\nlatare plotted as red empty circles in Fig. 2(b).\nLattice-disorder-induced damping in the SL FGT, \u0015S\nlat,\nexhibits a very weak temperature dependence, indicating\nthat increasing lattice vibration does not in\ruence the\ndamping frequency. The di\u000berence between \u0015S\nlatand\u0015S\nk\nincreases slightly only near TC, which can be attributed\nto the strong spin \ructuation. The overall weak tem-\nperature dependence in the damping for a single layer\nindicates that a non-thermal disorder scattering mecha-\nnism is dominant: the strong surface scattering in such\na thin layer (only a few \u0017A) combined with the enhanced\nSOC at the surfaces is the main channel for the magnetic\ndamping in the SL FGT instead of spin \ructuation and\nlattice vibration. Gilbert damping with a similarly weak\ntemperature dependence has also been found in a permal-\nloy,40,53where chemical disorder scattering overwhelms\nthermally induced disorder.\nThe temperature dependence of the bulk damping fre-\nquency is signi\fcantly di\u000berent from that of the SL. The\ncalculated bulk damping, \u0015B\nk, (shown by the black solid\ndiamonds in Fig. 2(b)) increases linearly with the temper-\nature. This typical resistivitylike behavior suggests that\nthe interband transition in bulk FGT is the dominant\ndamping mechanism.54We also calculate the damping\nfrequency\u0015B\nlatconsidering only lattice disorder, as shown\nas the black empty diamonds in Fig. 2(b). Comparing the\nresults corresponding to the solid and empty diamonds\nleads us to conclude that both lattice and spin disorder\nsubstantially contribute to damping in bulk FGT. As the\ntemperature approaches TC, the bulk damping is compa-\nrable with that in the single layer.\nIV. ANISOTROPIC DAMPING\nThe damping torque exerted on the magnetization\nin the Landau-Lifshitz-Gilbert equation has the general\nform of M(t)\u0002[~\u000b\u0001_M(t)], where the Gilbert damping\nparameter ~\u000bor the corresponding frequency is a tensor.\nThis tensor and its elements depend on both the instan-\ntaneous M(t) and its time derivative _M(t), where the\nanisotropy has been extensively analyzed using theoret-\nical models55and \frst-principles calculations.33,34Fol-\nlowing the de\fnition given by Gilmore et al. ,33we call\nthe anisotropic damping that depends on the equilibrium\norientation of Meqthe orientational anisotropy and that4\n𝑴𝐞𝐪𝑥𝑦𝑧𝜃\n-U/2 -U/40U/4U/2V81216F|| (10-3)Single layerBulk\nFIG. 3. The calculated Gilbert damping parameter \u000bkfor SL\n(red circles) and bulk FGT (black diamonds) as a function\nof the angle between the equilibrium magnetization Meqand\nthe atomic layer normal (^ z) of Fe 3GeTe 2. The lines are \ftted\nusingC0+C2cos 2\u0012.\ndepending on _M(t) the rotational anisotropy. Consider-\ning the layered structure of vdW materials, the lowered\nsymmetry should result in remarkable anisotropy for the\nmagnetization relaxation. Both the orientational and ro-\ntational anisotropy in bulk and SL FGT have been sys-\ntematically analyzed in this section. Notably, the damp-\ning tensor is reduced to a scalar for the con\fguration\nshown in Fig. 2.\nUnder a large in-plane magnetic \feld, the perpendicu-\nlar magnetization of FGT can be tilted toward the exter-\nnal \feld direction, which is de\fned as the y-axis without\nloss of generality. Thus, the angle between the equilib-\nrium magnetization Meqand the plane normal ^ zis re-\nferred to as \u0012, as shown in the inset of Fig. 3. At \u0012= 0,\nas studied in Sec. III, \u000bxx=\u000byy=\u000bk. For\u00126= 0,\n\u000bxx=\u000bkstill holds, whereas the other diagonal element\n\u000byydepends on speci\fc values of \u0012. Here, we focus on \u000bk\nto study the orientational anisotropy of damping. The\ncalculated in-plane damping \u000bkis plotted as a function\nof\u0012in Fig. 3 for a SL at 77 K and bulk FGT at 100\nK. The temperature is chosen in this way to obtain the\nsame relative magnetization for the two systems, namely,\nM=M s= 88%, according to the experimentally measured\nmagnetization as a function of temperature.3,8The same\ntwofold symmetry is found for the damping parameters\nof both SL and bulk FGT, which can be e\u000bectively \ftted\nusing a cos 2 \u0012term. As the magnetization rotates away\nfrom the easy axis, \u000bkincreases and reaches a maximum\nwhen the magnetization aligns inside the FGT layer. The\nchanges, [\u000b(\u0012=\u0006\u0019=2)\u0000\u000b(\u0012= 0)]=\u000b(\u0012= 0), are 62% for\nthe SL and 39% for the bulk. A similar dependence of\nthe damping on the magnetization orientation has been\nrecently observed in single-crystal CoFe alloys.35,36The\npredicted anisotropic damping of FGT shown in Fig. 3\nshould analogously be experimentally observable.\nThe rotational anisotropy of damping33in FGT is most\nsigni\fcant when the equilibrium magnetization lies in-\nside the atomic plane of FGT (along the hard axis), i.e.,\n0 0.20.40.6 0.81T/TC120150180QC/Q|| (%)\n0 0.2 0.4 0.6 0.8 1T/TC10152025Q (108 Hz)Bulk\nSingle layerBulkQCQ||Single layer\nBulkBulk\n𝛂∥𝑴(𝐭)\n𝑥𝑦𝑧(a)𝜶\"(b)FIG. 4. (a) Schematic of damping with the equilibrium mag-\nnetization Meqlying inside the atomic plane. Then, the in-\nstantaneous magnetization M(t) dissipates both the in- and\nout-of-plane spin angular momentum. The two types of dis-\nsipation are denoted as \u000bk(\u0015k) and\u000b?(\u0015?). (b) The calcu-\nlated Gilbert damping frequency \u0015k(?)as a function of tem-\nperature for single-layer and bulk Fe 3GeTe 2. The inset shows\nthe ratio of the two frequencies \u0015?=\u0015k.\n\u0012=\u0006\u0019=2. As schematized in Fig. 4(a), the magne-\ntization M(t) loses its in- or out-of-plane components\ndepending on the instantaneous precessional direction\n_M(t). In this case, one has \u000bxx=\u000bkand\u000bzz=\u000b?,\nwhereas the o\u000b-diagonal elements of the damping ten-\nsor are guaranteed to remain zero by symmetry.40The\ncalculated\u0015kand\u0015?for SL and bulk FGT are shown\nas a function of temperature in Fig. 4(b). For SL FGT,\n\u000bk(as shown by the circles with horizontal hatching) is\nnearly independent of temperature, which is the same as\nforMeqalong the easy axis. This result suggests that de-\nspite the sizable orientational anisotropy in the damping\nof SL FGT, the temperature has very little in\ruence on\nthe speci\fc values of the damping frequency. The calcu-\nlated\u0015?for the SL (shown by the red circles with vertical\nhatching) is considerably larger than \u0015kat low tempera-\ntures but decreases with increasing temperature. \u0015?be-\ncomes comparable with \u0015knear the Curie temperature,\nindicating that the rotational anisotropy is signi\fcantly\ndiminished by temperature.\nThe calculated \u0015kfor bulk FGT with Meqalong the\nhard axis (shown by the black diamonds with horizontal\nhatching) is temperature-independent, in sharp contrast\nto the linear temperature dependence of \u0015kwith Meq\nalong the easy axis shown in Fig. 2(b). This result sug-\ngests that the damping is already saturated in this case5\n-0.4 -0.2 0 0.2 0.4\nE-EF (eV)100200300400500λ⊥/λ|| (%)50 K\n77 K\n100 KSingle layer\nFIG. 5. The calculated rotational damping anisotropy for\nsingle-layer Fe 3GeTe 2as a function of the Fermi energy at\ndi\u000berent temperatures.\nat a su\u000eciently large scattering rate, where saturated\ndamping has also been found in FM Ni.25The calculated\n\u0015?of bulk FGT is also larger than \u0015kat low tempera-\ntures and slightly decreases with increasing temperature.\nWe summarize the results for the rotationally anisotropic\ndamping frequency by plotting the ratio between \u0015?and\n\u0015kin the inset of Fig. 4(b). The ratio for both SL and\nbulk FGT decreases with increasing temperature and\napproaches unity near TC. This behavior is consistent\nwith the results calculated using the torque-correlation\nmodel,33where rotationally anisotropic damping disap-\npears gradually as the scattering rate increases. In highly\ndisordered systems, the damping is more isotropic, as in-\ntuitively expected.\nWe emphasize that the calculated \u0015?values are dis-\ntinct from those reported in previous studies in the\nliterature,55that is,\u0015?was found to vanish in single-\ncrystal monoatomic FM layers based on the breathing\nFermi surface model.56{58Interband scattering is ne-\nglected in the breathing Fermi surface model. However,\nthe resistivitylike behavior of our calculated \u0015kfor bulk\nFGT shows that interband scattering plays an important\nrole in this vdW FM material.\nOne of the unique advantages of 2D vdW materials\nis the tunability of the electronic structure via electri-\ncal gating.4,59To simulate such a scenario, we slightly\nadjust the Fermi level EFof SL FGT without changing\nthe band structure for simplicity. The calculated rota-\ntional anisotropy in the damping \u0015?=\u0015kof SL FGT is\nshown as a function of the Fermi energy in Fig. 5. At all\nthe temperatures considered, the anisotropy ratio \u0015?=\u0015k\nincreases dramatically as EFis lowered by 0.3 eV, es-\npecially at low temperatures, and only exhibits minor\nchanges when EFis increased. At 50 K, the ratio \u0015?=\u0015k\nbecomes as high as 430%, which is almost three times\nlarger than that obtained without gating. This result\nsuggests that a small quantity of holes doped into SLFGT at low temperatures remarkably enhances the rota-\ntional damping anisotropy.\nV. CONCLUSIONS\nWe have systematically studied Gilbert damping in a\n2D vdW FM material Fe 3GeTe 2by using \frst-principles\nscattering calculations where the temperature-induced\nlattice vibration and spin \ructuation are modeled by\nfrozen thermal lattice and spin disorder. When the mag-\nnetization is perpendicular to the 2D atomic plane, the\ndamping frequency of bulk FGT increases linearly with\nthe temperature, whereas that of SL FGT exhibits a\nweak temperature dependence. The di\u000berence can be\nattributed to surface scattering (which is absent in the\nbulk) dominating scattering due to temperature-induced\ndisorder in SLs, which have a thickness smaller than the\nelectronic mean free path. The anisotropy of Gilbert\ndamping in this 2D vdW material has also been thor-\noughly investigated. The orientational anisotropy, which\ndepends on the direction of the equilibrium magnetiza-\ntion with respect to the atomic planes, exhibits twofold\nrotational symmetry in both the bulk and SL. When\nthe equilibrium magnetization is parallel to the atomic\nplane, the damping is signi\fcantly enhanced compared to\nthat with the magnetization perpendicular to the atomic\nplane. The rotational anisotropic damping depending on\nthe direction of motion of the instantaneous magnetiza-\ntion is remarkable with the equilibrium magnetization ly-\ning inside the atomic plane. With an out-of-plane compo-\nnent in the timederivative of the precessional magnetiza-\ntion, the damping frequency ( \u0015?) is much larger than the\none where only in-plane magnetization is varying ( \u0015k).\nThe ratio\u0015?=\u0015kis larger than unity for both the bulk\nand a single layer and decreases with increasing temper-\nature. In SL FGT, \u0015?=\u0015kcan be enhanced up to 430%\nby slight holedoping at 50 K.\nAntiferromagnetic order has recently been discovered\nin 2D vdW materials (as reviewed in Ref. 60 and the ref-\nerences therein) and some intriguing properties are found\nin their damping behaviors.61,62Owing to the more com-\nplex magnetic order, more than a single parameter is\nnecessary in describing the damping in antiferromagnetic\ndynamics.63,64It would be very interesting to study the\nmagnetization relaxation in these 2D materials with more\ncomplex magnetic order.\nACKNOWLEDGMENTS\nThe authors are grateful to Professor Xiangang Wan\nat Nanjing University for his support and helpful dis-\ncussions. Financial support for this study was provided\nby the National Natural Science Foundation of China\n(Grants No. 11734004 and No. 12174028).6\n\u0003yiliu@bnu.edu.cn\n1Cheng Gong, Lin Li, Zhenglu Li, Huiwen Ji, Alex Stern,\nYang Xia, Ting Cao, Wei Bao, Chenzhe Wang, Yuan\nWang, Z. Q. Qiu, R. J. 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The current-voltage character-\nistics of the junction demonstrate a pattern of subharmonic current steps which forms a devil’s\nstaircase structure. We show that a width of the steps become s maximal at ferromagnetic reso-\nnance. Moreover, we demonstrate that the structure of the st eps and their widths can be tuned\nby changing the frequency of the external magnetic field, rat io of Josephson to magnetic energy,\nGilbert damping and the junction size.\nThis paper is submitted to LTP Journal.\nI. INTRODUCTION\nJosephson junction with ferromagnet layer (F) is\nwidely considered to be the place where spintronics and\nsuperconductivity fields interact1. In these junctions\nthe supercurrent induces magnetization dynamics due\nto the coupling between the Josephson and magnetic\nsubsystems. The possibility of achieving electric con-\ntrol over the magnetic properties of the magnet via\nJosephson current and its counterpart, i.e., achieving\nmagnetic control over Josephson current, recently at-\ntracted a lot of attention1–7. The current-phase rela-\ntion in the superconductor-ferromagnet-superconductor\njunction (SFS) junctions is very sensitive to the mutual\norientation of the magnetizations in the F-layer8,9. In\nRef.[10] the authors demonstrate a unique magnetization\ndynamics with a series of specific phase trajectories. The\norigin of these trajectories is related to a direct coupling\nbetween the magnetic moment and the Josephson oscil-\nlations in these junctions.\nExternal electromagnetic field can also provide a cou-\npling between spin wave and Josephson phase in SFS\njunctions11–17. Spin waves are elementary spin excita-\ntions which considered to be as both spatial and time\ndependent variations in the magnetization18,19. The fer-\nromagnetic resonance(FMR) correspondsto the uniform\nprecession of the magnetization around an external ap-\nplied magnetic field18. This mode can be resonantly ex-\ncited by ac magnetic field that couples directly to the\nmagnetization dynamics as described by the Landau-\nLifshitz-Gilbert (LLG) equation18,19.\nIn Ref.[18] the authors show that spin wave resonance\nat frequency ωrin SFS implies a dissipation that is mani-\nfested as adepressionin the IV-characteristicofthe junc-\ntion when /planckover2pi1ωr= 2eV, where/planckover2pi1is the Planck’s constant,\ne is the electron charge and Vis the voltage across the\njunction. The ac Josephson current produces an oscil-\nlating magnetic field and when the Josephson frequencymatches the spin wave frequency, this resonantly excites\nthe magnetization dynamics M(t)18. Due to the non-\nlinearity of the Josephson effect, there is a rectification\nof current across the junction, resulting in a dip in the\naverage dc component of the suppercurrent18.\nIn Ref.[13] the authors neglect the effective field due\nto Josephson energy in LLG equation and the results re-\nveal that even steps appear in the IV-characteristic of\nSFS junction under external magnetic field. The ori-\ngin of these steps is due to the interaction of Cooper\npairs with even number of magnons. Inside the ferro-\nmagnet, if the Cooper pairs scattered by odd number of\nmagnons, no Josephson current flows due to the forma-\ntion of spin triplet state13. However, if the Cooper pairs\ninteract with even number of magnons, the Josephson\ncoupling between the s-wave superconductor is achieved\nand the spin singlet state is formed, resulting in flows of\nJosephsoncurrent13. In Ref.[20]weshowthat takinginto\naccount the effective field due to Josepshon energy and\nat FMR, additional subharmonic current steps appear in\nthe IV-characteristic for overdamped SFS junction with\nspin wave excitations (magnons). It is found that the po-\nsition of the current steps in the IV-characteristics form\ndevil’s staircase structure which follows continued frac-\ntion formula20. The positions of those fractional steps\nare given by\nV=\nN±1\nn±1\nm±1\np±..\nΩ, (1)\nwhere Ω = ω/ωc,ωis the frequency of the external ra-\ndiation, ωcis the is the characteristic frequency of the\nJosephson junction and N,n,m,pare positive integers.\nIn this paper, we present a detailed analysis for the\nIV-characteristics of SFS junction under external mag-\nnetic field, and show how we can control the position\nof the subharmonic steps and alter their widths. The\ncoupling between spin wave and Josephson phase in SFS\njunction is achieved through the Josephson energy and\ngauge invariant phase difference between the S-layers. In\nthe framework of our approach, the dynamics of the SFS2\njunction isfully describedbytheresistivelyshuntedjunc-\ntion (RSJ) model and LLG equation. These equations\nare solved numerically by the 4thorder Runge-Kutta\nmethod. The appearance and position of the observed\ncurrent steps depend directly on the magnetic field and\njunction parameters.\nII. MODEL AND METHODS\nF\nss\nHacxyz\nH0I\nI\nFIG. 1. SFS Josephson junction. The bias current is applied\nin x-direction, an external magnetic field with amplitude Hac\nand frequency ωis applied in xy-plane and an uniaxial con-\nstant magnetic field H0is applied in z-direction.\nIn Fig 1 we consider a current biased SFS junction\nwhere the two superconductors are separated by ferro-\nmagnet layer with thickness d. The area of the junction\nisLyLz. An uniaxial constant magnetic field H0is ap-\nplied in z-direction, while the magnetic field is applied in\nxy-plane Hac= (Haccosωt,Hacsinωt,0)withamplitude\nHacand frequency ω. The magnetic field is induced in\nthe F-layer through B(t) = 4πM(t), and the magnetic\nfluxes in z- and y-direction are Φ z(t) = 4πdLyMz(t),\nΦy(t) = 4πdLzMy(t), respectively. The gauge-invariant\nphase difference in the junction is given by21:\n∇y,zθ(y,z,t) =−2πd\nΦ0B(t)×n, (2)\nwhereθis the phase difference between superconducting\nelectrodes, and Φ 0=h/2eis the magnetic flux quantum\nandnis a unit vector normal to yz-plane. The gauge-\ninvariantphasedifference in terms ofmagnetizationcom-\nponents reads as\nθ(y,z,t) =θ(t)−8π2dMz(t)\nΦ0y+8π2dMy(t)\nΦ0z,(3)\nwhere Φ 0=h/(2e) is the magnetic flux quantum.\nAccordingtoRSJ model, the currentthroughthe junc-\ntion is given by13:\nI\nI0c= sinθ(y,z,t)+Φ0\n2πI0cRdθ(y,z,t)\ndt,(4)\nwhereI0\ncis the critical current, and R is the resistance\nin the Josephson junction. After taking into account thegaugeinvarianceincludingthemagnetizationoftheferro-\nmagnetandintegratingoverthejunction areatheelectric\ncurrent reads13:\nI\nI0c=Φ2\nosin(θ(t))sin/parenleftBig\n4π2dMz(t)Ly\nΦo/parenrightBig\nsin/parenleftBig\n4π2dMy(t)Lz\nΦo/parenrightBig\n16π4d2LzLyMz(t)My(t)\n+Φ0\n2πRI0cdθ(y,z,t)\ndt. (5)\nThe applied magnetic field in the xy-plane causes pre-\ncessionalmotionofthemagnetizationinthe F-layer. The\ndynamics of magnetization Min the F-layer is described\nby LLG equation\n(1+α2)dM\ndt=−γM×Heff−γ α\n|M|[M×(M×Heff)](6)\nThe total energy of junction in the proposed model is\ngivenby E=Es+EM+EacwhereEsistheenergystored\nin Josephson junction, EMis the energy of uniaxial dc\nmagnetic field (Zeeman energy) and Eacis the energy of\nac magnetic field:\nEs=−Φ0\n2πθ(y,z,t)I+EJ[1−cos(y,z,t)],\nEM=−VFH0Mz(t),\nEac=−VFMx(t)Haccos(ωt)−VFMy(t)Hacsin(ωt)(7)\nHere,EJ= Φ0I0\nc/2πis the the Josephson energy, H0=\nω0/γ,ω0is the FMR frequency, and VFis the volume of\nthe ferromagnet. We neglect the anisotropy energy due\nto demagnetizing effect for simplicity. The effective field\nin LLG equation is calculated by\nHeff=−1\nVF∇ME (8)\nThus, the effective field Hmdue to microwave radiation\nHacand uniaxial magnetic field H0is given by\nHm=Haccos(ωt)ˆex+Hacsin(ωt)ˆey+H0ˆez.(9)\nwhile the effective field ( Hs) due to superconducting part\nis found from\nHs=−EJ\nVFsin(θ(y,z,t))∇Mθ(y,z,t).(10)\nOne should take the integration of LLG on coordinates,\nhowever, the superconducting part is the only part which\ndepends on the coordinate so, we can integrate the ef-\nfective field due to the Josephson energy and insert the\nresult into LLG equation. Then, the y- and z-component\nare given by\nHsy=EJcos(θ(t))sin(πΦz(t)/Φ0)\nVFπMy(t)Φz(t)/bracketleftbigg\nΦ0cos(πΦy(t)/Φ0)\n−Φ2\n0sin(πΦy(t)/Φ0)\nπΦy(t)/bracketrightbigg\nˆey, (11)\nHsz=EJcos(θ(t))sin(πΦy(t)/Φ0)\nVFπMz(t)Φy(t)/bracketleftbigg\nΦ0cos(πΦz(t)/Φ0)\n−Φ2\n0sin(πΦz(t)/Φ0)\nπΦz(t)/bracketrightbigg\nˆez. (12)3\nAs a result, the total effective field is Heff=Hm+\nHs. In the dimensionless form we use t→tωc,ωc=\n2πI0\ncR/Φ0is the characteristic frequency, m=M/M0,\nM0=∝ba∇dblM∝ba∇dbl,heff=Heff/H0,ǫJ=EJ/VFM0H0,hac=\nHac/H0, Ω =ω/ωc, Ω0=ω0/ωc,φsy=4π2LydM0/Φo,\nφsz=4π2lzdM0/Φo. Finally, the voltage V(t) =dθ/dtis\nnormalized to /planckover2pi1ωc/(2e). The LLG and the effective field\nequations take the form\ndm\ndt=−Ω0\n(1+α2)/parenleftbigg\nm×heff+α[m×(m×heff)]/parenrightbigg\n(13)\nwith\nheff=haccos(Ωt)ˆex+(hacsin(Ωt)+ΓijǫJcosθ)ˆey\n+ (1+Γ jiǫJcosθ)ˆez, (14)\nΓij=sin(φsimj)\nmi(φsimj)/bracketleftbigg\ncos(φsjmi)−sin(φsjmi)\n(φsjmi)/bracketrightbigg\n,(15)\nwherei=y,j=z. The RSJ in the dimensionless form is\ngiven by\nI/I0\nc=sin(φsymz)sin(φszmy)\n(φsymz)(φszmy)sinθ+dθ\ndt.(16)\nThe magnetization and phase dynamics of the SFS\njunction can be described by solving Eq.(16) together\nwith Eq.(13). To solve this system of equations, we em-\nploy the fourth-order Runge-Kutta scheme. At each cur-\nrent step, we find the temporal dependence of the volt-\nageV(t), phase θ(t), andmi(i=x,y,z) in the (0 ,Tmax)\ninterval. Then the time-average voltage Vis given by\nV=1\nTf−Ti/integraltext\nV(t)dt, whereTiandTfdetermine the in-\nterval for the temporal averaging. The current value is\nincreased or decreased by a small amount of δI (the bias\ncurrent step) to calculate the voltage at the next point\nof the IV-characteristics. The phase, voltage and mag-\nnetization components achieved at the previous current\nstep are used as the initial conditions for the next cur-\nrent step. The one-loop IV-characteristic is obtained by\nsweeping the bias current from I= 0 toI= 3 and back\ndown to I= 0. The initial conditions for the magnetiza-\ntion components are assumed to be mx= 0,my= 0.01\nandmz=/radicalBig\n1−m2x−m2y, while for the voltage and\nphase we have Vini= 0,θini=0. The numerical param-\neters (if not mentioned) are taken as α= 0.1,hac= 1,\nφsy=φsz= 4,ǫJ= 0.2 and Ω 0= 0.5.\nIII. RESULTS AND DISCUSSIONS\nItiswell-knownthatJosephsonoscillationscanbesyn-\nchronized by external microwave radiation which leads\nto Shapiro steps in the IV-characteristic22. The position\nof the Shapiro step is determined by relation V=n\nmΩ,\nwheren,mare integers. The steps at m= 1 are calledharmonics, otherwise we deal with synchronized subhar-\nmonic (fractional) steps. We show below the appearance\nof subharmonics in our case.\nFirst we present the simulated IV-characteristics at\ndifferent frequencies of the magnetic field. The IV-\ncharacteristics at three different values of Ω are shown\nin Fig 2(a).\nFIG. 2. (a) IV-characteristic at three different values of Ω.\nFor clarity, the IV-characteristics for Ω = 0 .5 and Ω = 0 .7\nhave been shifted to the right, by ∆ I= 0.5 and ∆ I= 1,\nrespectively with respect to Ω = 0 .2; (b) An enlarged part\nof the IV-characteristic with Ω = 0 .7. To get step voltage\nmultiply the corresponding fraction with Ω = 0 .7.\nAs we see, the second harmonic has the largest step\nwidth at the ferromagnetic resonance frequency Ω = Ω 0,\ni.e., the FMR is manifested itself by the step’s width.\nThere are also many subharmonic current steps in the\nIV-characteristic. We have analyzed the steps position\nbetween V= 0 and V= 0.7 for Ω = 0 .7 and found dif-\nferent level continued fractions, which follow the formula\ngiven by Eq.(1) and demonstrated in Fig.2(b). We see4\nthe reflection of the second level continued fractions 1 /n\nand 1−1/nwithN= 1. In addition to this, steps with\nthird level continued fractions 1 /(n−1/m) withN= 1\nis manifested. In the inset we demonstrate part of the\nfourth level continued fraction 1 −1/(n+ 1/(m+1/p))\nwithn= 2 and m= 2.\nIn case of external electromagnetic field which leads to\nthe additional electric current Iac=AsinΩt, the width\nof the Shapiro step is proportional to ∝Jn(A/Ω), where\nJnis the Bessel function of first kind. The preliminary\nresults (not presented here) show that the width of the\nShapiro-like steps under external magnetic field has a\nmore complex frequency dependence20. This question\nwill be discussed in detail somewhere else.\nThe coupling between Josephson phase and magneti-\nzation manifests itself in the appearance of the Shapiro\nsteps in the IV-characteristics at fractional and odd mul-\ntiplies of Ω20. In Fig.3 we demonstrate the effect of the\nratio of the Josephson to magnetic energy ǫJon appear-\nance of the steps and their width for Ω = 0 .5 where the\nenlarged parts of the IV-characteristics at three differ-\nent values of ǫJare shown. As it is demonstrated in\nthe figures, at ǫJ= 0.05 only two subharmonic steps\nappear between V= 1 and V= 1.5 (see hollow ar-\nrows). An enhanced staircase structure appears by in-\ncreasing the value of ǫJ, which can be see at ǫJ= 0.3\nand 0.5. Moreover, an intense subharmonic steps appear\nbetween V= 1.75 andV= 2 forǫJ= 0.5. The posi-\ntions for these steps reflect third level continued fraction\n(N−1)+1/(n+1/m)withN=4 andn=1 [see Fig.3(b)].\nLet us now demonstrate the effect of Gilbert damping\non the devil’s staircase structure. The Gilbert damping\nαis introduced into LLG equation23?to describe the\nrelaxation of magnetization dynamics. To reflect effect\nof Gilbert damping, we show an enlarged part of the IV-\ncharacteristic at three different values of αin Fig.4.\nThewidthofcurrentstepat V= 2Ωisalmostthesame\nat different values of α(e.g., see upward inset V= 2Ω).\nThe subharmonic current step width for V= (n/m)Ω (n\nis odd,mis integer) is decreasing with increasing α. In\naddition a horizontal shift for the current steps occurs.\nWe see the intense current steps in the IV-characteristic\nfor small value of α= 0.03 (see black solid arrows). With\nincrease in Gilbert damping (see α= 0.1, 0.16 and 0 .3)\nthe higher level subharmonic steps disappear. It is well-\nknownthatatlargevalueof αtheFMRlinewidthbecome\nmore broadening and the resonance frequency is shifted\nfrom Ω 0. Accordingly, the subharmonic steps disappear\nat large value of α. Furthermore, using the formula pre-\nsented in Ref.[20] the width at Ω = Ω 0for the fractional\nand odd current steps is proportional to (4 α2+α4)−q/2\n×(12+3α2)−k/2, whereqandkare integers.\nFinally, we demonstrate the effect of the junction size\non the devil’s staircase in the IV-characteristic under ex-\nternalmagneticfield. Thejunction sizechangesthe value\nofφsyandφsz. In Fig.5(a) we demonstrate the effect of\nthe junction thickness by changing φsz(φsyis qualita-\nFIG. 3. (a) An enlarged part of the IV-characteristic at\ndifferent values of ǫJin the interval between V= 1 and V=\n1.5; (b)Thesameintheintervalbetween V= 1.75andV= 2.\nFor clarity, the IV-characteristics for ǫJ= 0.3, and 0 .5 have\nbeen shifted to right, by ∆ I= 0.07, and 0 .14, respectively\nwith respect to the case with ǫJ= 0.05.\ntively the same).\nWe observe an enhanced subharmonic structure with\nincrease of junction size or the thickness of the ferro-\nmagnet. In Ref.[13] the authors demonstrated that the\ncritical current and the width of the step at V= 2Ω as a\nfunction of Lz/Lyfollow Bessel function of first kind. In\nFig.5(b), we can see the parts of continued fraction se-\nquences for subharmonic steps between V= 1 andV= 2\natφsz=φsy= 6. Current steps between V= 1 and\nV= 1.5 reflect the two second level continued fractions\n(N−1)+ 1/nandN−1/nwithN= 3 in both cases,\nwhile for the steps between V= 1.5 andV= 2 follow\nthe second level continued fraction ( N−1) + 1/nwith\nN= 4.\nFinally, wediscussthepossibilityofexperimentallyob-5\nFIG. 4. An enlarged part of IV-characteristic for four differ -\nent values of Gilbert damping for Ω = 0 .5. The inset shows an\nenlargedpartofcurrentstepwithconstantvoltage at V= 2Ω.\nserving the effects presented in this paper. For junction\nsized= 5nm, Ly=Lz= 80nm, critical current I0\nc≈\n200µA, saturation magnetization M0≈5×105A/m,\nH0≈40mT and gyromagnetic ratio γ= 3πMHz/T,\nwe find the value of φsy(z)=4π2Ly(z)dM0/Φ0= 4.8 and\nǫJ= 0.1. With the same junction parameters one can\ncontrol the appearance of the subharmonic steps by tun-\ning the strength of the constant magnetic field H0. Esti-\nmations showthat for H0= 10mT, the value of ǫJ= 0.4,\nand the fractional subharmonic steps are enhanced. In\ngeneral, the subharmonic steps are sensitive to junction\nparameters, Gilbert damping and the frequency of the\nexternal magnetic field.\nIV. CONCLUSIONS\nIn this work, we have studied the IV-characteristics\nof superconductor-ferromagnet-superconductor Joseph-\nson junction under external magnetic field. We used a\nmodified RSJ model which hosts magnetization dynam-\nics in F-layer. Due to the external magnetic field, the\ncouplingbetweenmagneticmomentandJosephsonphase\nis achieved through the effective field taking into account\nthe Josephson energy and gauge invariant phase differ-\nence between the superconducting electrodes. We have\nsolvedasystemofequationswhichdescribethe dynamics\nof the Josephson phase by the RSJ equation and magne-\ntization dynamics by Landau-Lifshitz-Gilbert equation.\nThe IV-characteristic demonstrates subharmonic current\nsteps. The pattern of the subharmonic steps can be con-\ntrolled by tuning the frequency of the ac magnetic field.\nWe show that by increasing the ratio of the Josephson to\nmagneticenergyanenhancedstaircasestructureappears.\nFinally, we demonstrate that Gilbert damping and junc-\nFIG. 5. (a) IV-characteristic at three different values of\nφsz= 0.7,3,6 andφsy=φsz. (b) An enlarged part of the IV-\ncharacteristic at φsz=φsy=6. The hollow arrows represent\nthe starting point of the sequences. To get step voltage we\nmultiply the corresponding fraction by Ω = 0 .5.\ntion parameters can change the subharmonic step struc-\nture. The observed features might find an application in\nsuperconducting spintronics.\nV. ACKNOWLEDGMENT\nWe thank Dr. D. V. Kamanin and Egypt JINR col-\nlaboration for support this work. The reported study\nwas partially funded by the RFBR research Projects No.\n18-02-00318 and No. 18-52-45011-IND. Numerical cal-\nculations have been made in the framework of the RSF\nProject No. 18-71-10095.6\nREFERENCES\n∗majed@sci.cu.edu.eg\n†shukrinv@theor.jinr.ru\n1J. Linder and K. Halterman, Phys. Rev. B 90, 104502\n(2014).\n2Yu. M. Shukrinov, A. Mazanik, I. Rahmonov, A. Botha,\nand A. Buzdin, EPL122, 37001 (2018).\n3Yu. M. Shukrinov, I. Rahmonov, K. Sengupta, and A.\nBuzdin, Appl. Phys. Lett. 110, 182407 (2017).\n4A. Buzdin, Phys. Rev. Lett. 101, 107005 (2008).\n5A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n6F. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.\nPhys.77, 1321 (2005).\n7A. A. Golubov, M. Y. Kupriyanov, and E. IlIchev, Rev.\nMod. Phys. 76, 411 (2004).\n8M. A. Silaev, I. V. Tokatly, and F. S. Bergeret, Phys. Rev.\nB95, 184508 (2017).\n9I. Bobkova, A. Bobkov, and M. Silaev, Phys. Rev. B 96,\n094506 (2017).\n10Y. M. Shukrinov, I. Rahmonov, and K. Sengupta, Phys.\nRev. B99, 224513 (2019).\n11M. Weides, M. Kemmler, H. Kohlstedt, R. Waser, D.\nKoelle, R. Kleiner, and E. 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Rev.\nB 97, 224514 (2018).\n21K. K. Likharev, Dynamics of Josephson junctions and cir-\ncuits, Gordon and Breach science publishers -Switzerland\n(1986).\n22S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).\n23T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443-\n3449 (2004).\n24M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102,\n137601(2009)."    },    {        "title": "1812.08404v1.Laser_Controlled_Spin_Dynamics_of_Ferromagnetic_Thin_Film_from_Femtosecond_to_Nanosecond_Timescale.pdf",        "content": "1\n \n \nLaser Controlled Spin Dynamics \nof Ferromagnetic Thin Film \nfrom \nFemtosecond to Nanosecond Timescale\n \nSucheta Mondal and Anjan Barman\n*\n \nDepartment of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for \nBasic Sciences, Block JD, Secto\nr III, Salt Lake, Kolkata 700 106, India.\n \n*\nabarman@bose.res.in\n \nKey words: (\nThin Film Heterostructures, Ultraf\na\nst Demagnetization, Gilbert Damping, Time\n-\nresolved\n \nMagneto\n-\noptical Kerr Effect\n)\n \n \n \nLaser induced modulatio\nn of the \nmagnetization dynamics \noccurring over various time\n-\nscales \nhave been unified\n \nhere \nfor\n \na \nNi\n80\nFe\n20\n \nthin \nfilm excited \nby\n \namplified \nfemtosecond laser pulses. \nThe weak correlation between demagnetization time and pump fluence with substantial \nenhancemen\nt in remagnetization time is demo\nn\nstrated using three\n-\ntemperature model \nconsidering the temperatures of electron, spin\n \nand \nlattice.\n \nThe \npicosecond\n \nmagnetization \ndynamics is modeled using \nthe L\nandau\n-\nLifshitz\n-\nGilbert equation. \nW\nith increasing pump fluence \nth\ne Gilbert damping parameter shows significant enhancement\n \nfrom its intrinsic value \ndue to\n \nincrement in the \nratio\n \nof electronic temperature to Curie temperature within very short time \nscale. The precessional frequency experiences \nnoticeable \nred shift with i\nncreasing pump fluence. \nThe changes in the local magnetic properties due to accumulation and dissipation of thermal \nenergy within the probed volume are described by the \nevolution of \ntemporal chirp parameter in a \ncomprehensive manner.\n \nA unification of ultra\nfast magnetic processes and its control over broad \ntimescale would enable the integration of various magnetic processes in a single device and use \none effect to control another.\n \n \n \n \n \n \n 2\n \n \nI. INTRODUCTION\n \nRecent development in \nmagnetic \nstorage \n[1] \nand \nmemory\n \n[2\n] \ndevice\ns\n \nheavily relies\n \nup\non \nincreasing\n \nswitching \nspeed and \ncoherent \nswitching \nof \nmagnetic states \nin \nferromagnetic thin films\n \nand patterned structures\n.\n \nO\nperating speeds of information storage devices have progressed into \nthe \nsub\n-\ngig\nahertz \nregime\n \nand controlled switching in \nindividual \nlayers of magnetic \nmultilayers \nand hetero\nstructures \nhas become \nimportant\n. \nThe relaxation processes \ninvolved in magnetization \ndynamics set \nnatural limit\ns\n \nfor \nthese \nswitching times and data transfer rates.\n \nIn the context of \nprecessional \nmagnetization dynamics \nthe \nnatural \nrelaxation \nrate \nagainst the small perturbation is \nexpressed as Gilbert damping\n \n(\nα\n)\n \naccording to the Landau\n-\nLiftshiz\n-\nGilbert (LLG) equation\n \n[3, \n4]\n.\n \nThis \nis analogous to viscous damping of the mechanical frictional torque\n \nand\n \nleads to the \ndirect dissipation of energy from the uniform precessional mode to thermal bath in case of zero \nwav\ne\n-\nvector excitation.\n \nGilbert damping \noriginates from spin\n-\norbit coupling and depends on the \ncoupling strength and \nd\n-\nband width of the \n3\nd\n \nferromagnet\n \n[5]\n. Th\ne damping\n \ncan be \nvaried \nby\n \nvarious \nintrinsic and extrinsic \nmechanisms including\n \nphonon drag\n \n[6]\n, Edd\ny current\n \n[7],\n \ndoping\n \n[8]\n \nor\n \ncapping\n \n[9]\n \nwith other material\n, injection of spin current\n \n[10]\n, magnon\n-\nmagnon scattering\n \n[11]\n \nand\n \ncontrolling \ntemperature of\n \nthe system\n \n[12]\n.\n \nIntri\nn\nsic and extrinsic \nnature of Gilbert \ndamping\n \nwere primarily studied by using fe\nrromagnetic resonance (FMR) technique. When the \nmagnetization is aligned with either in\n-\nplane or \nout\n-\nof\n-\nplane\n \napplied magnetic field, the \nlinewidth is proportional to the frequency with a slope determined by\n \ndamping co\nef\nf\nicient. This \nis the homogeneous or \nintrinsic contribution to the FMR linewidth. However, experiments show \nan additional frequency\n-\nindependent contribution to the linewidth\n \ncorresponds to \ninhomogeneous line broadening\n \n[13, 14]\n.\n \n \n \nHowever\n,\n \nstate\n-\nof\n-\nthe\n-\nart technique based on \npump\n-\nprobe geomet\nry\n \nhas been developed and rigorously exploited for measuring ultrafast \nmagnetization dynamics of ferromagnetic thin films during last few decades\n \n[15, 16]\n.\n \nUsing \ntime\n-\nresolved magneto\n-\noptical Kerr effect (TR\n-\nMOKE) \ntechnique \none can directly address the \npro\ncesses which are responsible for the excitation and relaxation of a magnetic system on their \ncharacteristic time scales\n \n[17\n-\n19]\n.\n \nGenerally \nduring\n \nthe pump\n-\nprobe measurements pump fluence \nis kept low to avoid nonlinear effects and sample surface degradation\n. Some recent experiments \nreveal that nonlinear spin waves play a \ncrucial \nrole in high power thin film precession\nal\n \ndynamics by introducing spin\n-\nwave instability\n \n[20]\n \nsimilar to \nFMR\n \nexperiments \nby \nappl\nication \nof\n \nhigh rf power\n \n[21]\n. \nThe coercivity and aniso\ntropy of the ferromagnetic thin films \ncan \nalso\n \nbe\n \nlowered by pump fluence\n,\n \nwhich \nmay\n \nhave potential application in heat assisted \nmagnetic \nrecording\n \n(HAMR)\n \n[22]\n. \nRecent\n \nreport\ns\n \nreveal \nthat damping \ncoefficient\n \ncan be increased or \ndecreased noticeably in the \nhigher excitation regime due to opening of further energy dissipation\n \nchannels\n \nbeyond a threshold\n \npump power\n \n[23\n-\n25]\n. \nNot only relaxation parameters but also \nfrequency \nshift\n \ndue to enhancement in pump power\n \nhas been \nobserved\n \n[20]\n. \nHowever,\n \nthe \nexperimental\n \nevidence for \nlarge \nmodulation of Gilbert damping along with frequency \nshift \nand \ntemporal chirping of the uniform precessio\nn\nal \nmotion \nis absent \nin the literature\n. \nThis \ninvestigation demands suitable choice of material, \nand here we have chosen \nPermalloy\n \n(Ni\n80\nFe\n20\n 3\n \n \nor Py here\n \non)\n \nbecause \nof its\n \nhigh permeability, \nnegligible magneto\n-\ncrystalline anisotropy, \nvery \nlow coercivity, large anisotropic magnetoresistance with reasonably low damping. Also, \ndue to \nits\n \nnegligible\n \nmagnetostriction P\ny\n \nis less sensitive to st\nrain and stress exerted during the thermal \ntreatment in \nHAMR\n \n[22]\n. \n \nIn this \narticle\n,\n \nwe have \nused \nfemto\n-\nsecond amplified laser\n \npulses\n \nfor excitation and detection of \nultrafast magnetization\n \ndynamics in \na \nP\ny\n \nthin \nfilm. Pump fluence dependent ultrafast \ndemag\nnetization is \ninvestigated\n \nalong with fast \nand slow \nremagnetization\n. \nOur comprehensive \nstudy \nof \nthe \npicosecond\n \ndynamics \nreveals transient nature of enhanced Gilbert damping in \npresence of high pump fluence\n. Also\n,\n \nthe time\n-\nvarying precession is subjected to\n \ntemporal \nchirping \nwhich occurs due to enhancement of temperature of the probed volume within a very \nshort time scale \nbeing followed by\n \nsuccessive\n \nheat dissipation. \nThis fluence dependent \nmodulation of magnetization dynamics will undoubtedly found suitable\n \napplication\n \nin spintronic \nand magnonic devices\n.\n \nII. SAMPLE PREPARATION AND CHARACTERIZATION\n \n20\n-\nnm\n-\nthick Permalloy (Ni\n80\nFe\n20\n, Py hereafter) film was deposited by using electron\n-\nbeam \nevaporation technique (SVT Associates, model: Smart Nano Tool AVD\n-\nE01) (ba\nse pressure = 3 \n× 10\n−8\n \nTorr, deposition rate = 0.2 Å/S) on 8 × 8 mm\n2\n \nsilicon (001) wafer coated with 300\n-\nnm\n-\nthick SiO\n2\n. Subsequently, 5\n-\nnm\n-\nthick SiO\n2\n \nis deposited over the Ni\n80\nFe\n20\n \nusing rf sputter\n-\ndeposition technique (base pressure = 4.5 × 10\n−7\n \nTorr, Ar pressure = 0.5 mTorr\n, deposition r\nate = \n0.2 Å/S, rf power = 60 W). \nThis capping layer protects the surface from environmental \ndegradation, oxidation and laser ablation during the pump\n-\nprobe experiment using femtosecond \nlaser pulses. \nFrom the vibrating sample magnetometry (VSM\n) we have obtained the saturation \nmagnetization (M\ns\n) and Curie temperature (T\nc\n) to be 850 emu/cc and 86\n3\n \nK\n,\n \nrespectively\n \n[26]\n.\n \n \nTo study the ultrafast magnetization dynamics of this sample, we have used a custom\n-\nbuilt time \nresolved magneto optical Kerr eff\nect (TRMOKE) magnetometer based on optical pump\n-\nprobe \ntechnique as shown in Fig. 1 (a). Here, the second harmonic (λ = 400 nm, repetition rate = 1 kHz, \npulse width > 40 fs) of amplified femtosecond laser pulse generated from a regenerative \namplifier system\n \n(Libra, Coherent) is used to excite the dynamics while the fundamental laser \npulse (λ = 800 nm, repetition rate = 1 kHz, pulse width ≈ 40 fs) is used as probe to detect the \ntime\n-\nresolved polar Kerr signal from the sample. The temporal resolution of the me\nasurement is \nlimited by the cross\n-\ncorrelation between the pump and probe pulses (\n≈120 fs). The probe beam \nhaving diameter of about 100 µm is normally incident on the sample whereas the pump beam is \nkept slightly defocused (spot size is about 300 µm) and is\n \nobliquely (\n≈ 30\n◦\n \nwith normal to the \nsample plane) incident on the sample maintaining an excellent spatial overlap with the probe \nspot. Time\n-\nresolved Kerr signal is collected from the uniformly excited part of the sample and \nslight misalignment during the \ncourse of the experiment does not affect the pump\n-\nprobe signal \nsignificantly. A large magnetic field of 3.5 kOe is first applied at a small angle of about 10° to 4\n \n \nthe sample plane to saturate its magnetization. This is followed by reduction of the magnetic \nfield to the bias field value (\nH \n= in\n-\nplane component of the bias field), which ensures that the \nmagnetization remains saturated along the bias field direction. The tilt of magnetization from the \nsample plane ensures a finite demagnetizing field along the \ndirection of the pump pulse, which is \nfurther modified by the pump pulse to induce a precessional dynamics within the sample\n \n[17]\n. In \nour experiment a 2\n-\nns time window has been used, which gave a damped uniform precession of \nmagnetization. The pump beam is\n \nchopped at 373 Hz frequency and the dynamic signal in the \nprobe pulse is detected by using a lock\n-\nin amplifier in a phase sensitive manner. Simultaneous \ntime\n-\nresolved reflectivity and Kerr rotation data were measured \nand no significant breakthrough \nof one\n \ninto another has been found\n \n[26]\n.\n \nThe probe fluence is kept constant at 2 mJ\n/\ncm\n2\n \nduring \nthe measurement to avoid additional contribution to the modulation of spin dynamics via laser \nheating. Pump fluence (\nF\n) was varied from 10 to 55 mJ\n/\ncm\n2\n \nto study the fl\nuence dependent \nmodulation in magnetization dynamics. All the experiments were performed under ambient \ncondition and room temperature. \n \nIII. RESULTS AND DISCUSSIONS\n \nA.\n \nLaser\n \ninduced ultrafast demagnetization\n \n \nWhen a femtosecond laser pulse \ninteracts\n \nwith\n \na\n \nferromagnetic \nthin film in its saturation \ncondition\n, \nthe magnetization of the system is partially or fully lost within hundreds of \nfemtosecond as measured by the time\n-\nresolved Kerr rotation or ellipticity.\n \nThis is known as \nultrafast \ndemagnetization of the\n \nferromagnet\n \nand was first observed by Beaurepire et al. in 1996 \n[\n27\n]\n. \nThis is generally followed by a fast recovery of the magnetization within sub\n-\npicosecond to \nfew picosecond\ns\n \nand a slower recovery within tens to hundreds of picoseconds, known as the fa\nst \nand slow remagnetization\n.\n \nIn many cases the slower recovery is accompanied by a coherent \nmagnetization precession and damping [\n17\n]. \nIn\n \nour\n \npump\n-\nprobe experiment, \nthe sample \nmagnetization is maintained in \nthe saturated state by application of a magnetic \nfield \nH\n \n= \n2.4 kOe\n \nbefore zero delay\n. \nRight after the zero\n-\ndelay\n \nand the \ninteraction\n \nof the pump pulse\n \nwith the \nelectrons in the \nferromagnetic \nmetal, ultrafast demagnetization takes place\n.\n \nThe local \nmagnetization is immediately quenched within first few hun\ndreds of fs \nfollowed by a subsequent \nfast \nremagnetization \nin next\n \nfew ps\n \n[27]\n. \nFigure \n1\n(b) shows ultrafast demagnetization \nobtained \nfor\n \ndifferent pump fluences. \nSeveral models have been proposed over \ntwo decades to explain the \nultrafast demagnetization\n \n[16\n, 28\n-\n31]\n.\n \nOut of those \na phenomenological thermodynamic model\n, \ncalled three temperature m\nodel\n \n[\n27, \n32, 33]\n \nhas been most widely used\n,\n \nwhere the dynamics of \nthese spin fluctuations can be describes as:\n \n)\n(\n)\n)(\n(\n0\n)\n(\n2\n1\n)\n(\n1\n2\n1\nt\nM\ne\nA\nA\ne\nA\nA\nA\nt\nM\nlat\nel\nsp\nel\nt\nlat\nel\nsp\nel\nlat\nel\nt\nsp\nel\nlat\nel\nsp\nel\nlat\nel\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 \n(1)\n.\n \nThis is an approximated\n \nform based on the assumption that the electron temperature rises \ninstantaneously upon laser excitation\n \nand can be applied to fit time\n-\nresolved \nKerr rotation \ndata \ntaken within few picoseconds timescale\n.\n \nT\nhe whole system i\ns divided into three subsystems: 5\n \n \nel\nectron, spin and lattice system. On laser excitation the hot electrons are created above Fermi \nlevel. Then during energy rebalancing between the subsystems\n,\n \nquenched magnetization relaxes\n \nback to the initial state.\n \nThe two exponential functions \nin the abov\ne equation \nmirror the \ndemagnetization given by demagnetization time (τ\nel\n-\nsp\n) for energy transfer between electron\n-\nspin \nand the decay of electron temperature \n(\nτ\nel\n-\nlat\n) \nowing to the tr\nansfer of energy to the lattice. In \naddition to these characteristics time\n \nconstants, the spin\n-\nlattice relaxation time also can be \nextracted \nby including another exponential term in the above equation\n \nif the spin specific heat is \ntaken into account [\n34\n]\n.\n \nθ\n \nis the Heaviside step function and \nΓ(t) \nstands for \nthe Gaussian function \nto be convoluted\n \nwith the laser pulse envelope determining the temporal resolution\n \n(showing the \ncross c\norrelation between the probe an\nd pump pulse)\n.\n \nThe constant, \nA\n1\n \nindicates the ratio \nbetween amount of magnetization after equilibrium between electrons, s\npins, and lattice is \nrestored and the initial magnetization. A\n2 \nis proportional to the initial electronic temperature rise.\n \nWe have plotted A1 and A2, normalized with their values at the highest fluence, as a function of \npump fluence in Fig. \n3S\n \nof the supp\nlemental material\n \nwhich shows that magnitude of both \nparameters increases with \nlaser \nfluence\n \n[26]\n.\n \nWe have \nobserved\n \nthat with increasing fluence the \ndemagnetization time has been \nnegligibly varied within a range of 250\n±40\n \nfs.\n \nThe weak or no \ncorrelation bet\nween the pump fluence and the demagnetization rate describes the intrinsic nature \nof the spin scattering\n, governed by various mechanisms including Elliott\n-\nYafet mechanism\n \n[\n35\n]\n. \nAnother \nimportant\n \nobservation here is that \nthe delay of demagnetization process\nes \nwhich is the \ntime delay between pump pulse (full width at half maxima, FWHM \n≈ 130±20 fs) and starting \npoint of the ultrafast demagnetization,\n \nbecomes shorter due to increase in pump fluences. A \nplausible explanation for this is the dependence of \ndelay o\nf \ndemagnetization on the electron\n-\nthermaliz\nation\n \ntime which is eventually proportional to electron density or pump fluences\n \n[\n3\n6\n]\n. \nOn the other hand\n,\n \nfast \nremagnetization \ntime has been found to be increased noticeably from \n0.40\n \n±\n \n0.05 ps to 0.8\n0 \n±\n \n0.05 ps w\nithin the experimental fluence range\n \nof 10\n-\n55 mJ/cm\n2\n. The \nlarger is the pump fluence, the higher is the electron temperature or further the spin temperature. \nTherefore, it is reasonable that magnetization recovery time increases with the pump fluence.\n \n \nB. \nPump fluence dependent modulation in Gilbert damping\n \n \nF\nig\nure\n \n1 \n(c) shows the representat\nive Kerr rotation data for \nF\n \n= 25\n \nmJ/cm\n2\n \nconsisting of three \ntemporal\n \nregions\n,\n \ni.e.\n \nultrafast \ndemagnetization\n, fast remagnetization and slow\n \nremagnetization \nsuperposed \nwith \ndamped \nprecession within\n \nthe time window\n \nof 2 ns\n. We \nprocess\n \nthe \nmagnetization \nprecession part\n \nafter subtracting \na\n \nbi\n-\nexponential background \nto estimate the \ndamping and its modulation\n. \nThe slower remagnetization is \nmainly \ndue to heat diffusion from th\ne \nlattice to the substrate and surrounding. Within our experimental fluence range the slow \nremagnetization time has increased from \n≈0.4 ns to ≈1.0 ns. \nThe precessiona\nl dynamics is \ndescribed\n \nby phenomenological Landau\n-\nLifshitz\n-\nGilbert \n(LLG) \nequation, \n \ndt\nM\nd\nM\nM\nH\nM\ndt\nM\nd\ns\neff\n\n\n\n\n\n\n\n\n\n\n\n\n                                                                                  \n        \n    \n(2)\n 6\n \n \nwhere \nγ\n \nis the gyromagnetic ratio, \nM\n \nis magnetization\n, \nα\n \nis Gilbert damping constant\n \nand \nH\neff\n \nis \nthe effective magnetic field consisting of \nseveral field components. \nThe \nvariation \nof precessional \nfrequency with the angle between sample plane and bias \nmagnetic \nfield direction is plotted in \nF\nig. \n1 \n(d\n), which\n \nreveals that there is no uniaxial anisotropy present in this sample.\n \n \nThe energy deposit\ned by the pump pulse, in terms of heat within the probed volume, plays a very \ncrucial role in modification of local magnetic properties\n,\n \ni.e. magnetic moment, anisotropy, \ncoercivity, magnetic susceptibility\n,\n \netc. With increasing fluence the precessional fr\nequency \nexperienced a red shift\n \n[20, 25]. Thus, at the onset of the precessional dynamics (about 10 ps \nfrom zero delay), for relatively high fluence, the initial frequency (\nf\ni\n) will be smaller than its \nintrinsic value (in absence of any significant heat di\nssipation). As time progresses and the sample \nmagnetization gradually attains its equilibrium value, the precessional frequency continuously \nchanges, causing a temporal chirping of the damped oscillatory Kerr signal.\n \nThe frequency shift \ncan be \nestimated\n \nfr\nom the amount of temporal chirping\n \n[\n3\n7\n].\n \nFigure \n2 \n(a) shows the background \nsubtracted \ntime\n-\nresolved Kerr rotation data (\nprecessional \npart)\n \nfor different pump fluences fitted \nwith \na \ndamped sinusoidal function with added temporal chirping\n,\n \n \n)\n)\n(\n2\nsin(\n/\n\n\n\n\n\nt\nbt\nf\nAe\ni\nt\nk\n\n\n\n                                                                                         \n \nwhere \nA\n, \nτ\n, \nf\ni\n, b\n \nand \nΦ \nare the amplitude of the magnetization precession, the relaxation time, the \ninitial precessional frequency, \nchirp parameter \nand \ninitial \nphase, respectively. \n \nAt this point, we \nare unsure of the exact nature of the damping, \ni.e.\n \nit may consis\nt of both intrinsic and extrinsic \nmechanisms and hence we term it as\n \neffective damping parameter \n(\nα\neff\n)\n \nwhich can be\n \nextracted \nusing the following formula\n \n[3\n8\n]\n, \n \n)\n2\n4\n(\n1\neff\neff\nM\nH\n\n\n\n\n\n                                                                        \n                                        \n(\n3\n)\n \nγ\n \n= 1.\n83\n \n×10\n7\n \nHz/\nOe\n \nfor Py \nan\nd \nM\neff\n \nis the effecti\nve magnetization including pump\n-\ninduced \nchanges\n \nat \nH\n \n= 2.4 kOe\n. \nThis formula is exploited to extract effective damping parameter \nprecisely\n \nin the moderate bias fi\neld regime. \nThe variation of relaxation time and effective \ndamping \nare\n \nplotted with pump fluence in \nF\nig. \n2 \n(b) and (c). \nHere, \nτ \ndecreases with fluence while \ndamping increases \nsignificantly with respect to\n \nits \nintrinsic value within this fluence range. We \nh\nave repeated the experiment for two different field values (2.4 and 1.8 kOe). The slope of \nfluence dependent damping remains unaltered\n \nfor both the field values\n. \nWe have also observed \nincrease in relative amplitude\ns\n \nof precession with pump fluence as shown\n \nin the inset of Fig. 2 \n(c). \nTo verify the transient nature of damping we have performed another set of experiment \nwhere the probed area is exposed to different pump fluences \n(\nF\ni\n) \nfor several minutes. After the \nirradiation, the precessioanl dynamics is mea\nsured from that area\n \nwith fixed probe and pump \nfluences 2 and 10 mJ/cm\n2\n, respectively. We found that damping remains almost constant for all \nthe measurements\n \n(as shown in Fig.\n \n2 (d))\n. These results demonstrate that the enhancement of 7\n \n \ndamping is transient a\nnd only exists in the presence of high pump fluence but dropped to its \noriginal value when the pump laser was set to initial fluence.\n \n \nThe bias field dependence of precessional dynamics at four different pump fluence\ns\n \nis studied to \ngain more insight about \nthe origin of fluence dependent damping.\n \nFirst, we plotted the average \nfrequency \n(\nf\nFFT\n)\n \nwith bias field which is obtained from the fast Fourier transformation (FFT) of \nthe precessional data in \nF\nig. \n3\n \n(a). The experimental data points are fitted with the Ki\nttel formula, \n \n)\n4\n(\n2\neff\nFFT\nM\nH\nH\nf\n\n\n\n\n\n      \n                                                                                             \n      \n \n(\n4\n)\n \nM\neff\n \nis the effective magne\ntization of the sample. Figure \n3 \n(b) shows that effective magnetization \ndoes not v\nary much within the applied fluence range. So\n,\n \nwe infer that \nwith increasing fluence \nthere is no \ninduced \nanisotropy \ndeveloped in the system\n \nwhich can modify the effective damping\n \nup to this extent\n \n[23]\n. The variation of relaxation time with bias field for \nfour different pump \nfluences are plotted in \nF\nig. \n3 \n(c). Relaxation\n \ntime is increased with decreasing\n \nfield for each case \nbut for the higher fluence regime, those value\ns seem\n \nto be fluctuating. \nThis depend\nence of τ on \nfield\n \nwas fitted with eq\nuation\n \n3 to extract damping coefficient at different fluence values. We \nhave further plotted the damping coefficient as a function of precession frequency (\nf\nFFT\n)\n \n[see \nsupplementa\nl\n \nmaterial\n, \nF\nig. \n4\nS\n]\n \n[26]\n, which shows a\nn invariance of \nα\neff\n \nwith \nf\nFFT\n. From that we \ncan infer that the damping coefficient in our sample within the experimental field and fluence \nregime are intrinsic in nature and hence, we may\n \nnow\n \nterm it as the intrinsic damping coefficient \nα\n0\n.\n \nThe extrinsic \ncontributions to damping mainly come from magnetic anisotropy field, two\n-\nmagnon scattering, multimodal dephasing for excitation of several spin\n-\nwave modes, etc, which \nare negligible in our present case. \n \nF\nigure\n \n3\n(d)\n \nshows the variation of \nα\n0\n \nwith pump flue\nnce, which shows that even the intrinsic \ndamping is significantly increasing with pump fluence \n[20, \n3\n9\n]\n.\n \nFor generation of perpendicular \nstanding spin\n-\nwave modes the film needs to be thick enough. Though the film thickness is 20 nm \nhere, but within the app\nlied bias field range we have not found any other magnetic mode \nappearing with the uniform Kittel mode within the frequency window of our interest\n \n(as shown \nin \nF\nig. \n5\nS of suppleme\nn\ntal material\n)\n \n[26]\n.\n \nAlso\n,\n \nfor 20\n-\nnm\n-\nthick \nPy \nfilm,\n \nthe effect of eddy \ncurre\nnt will be negligible\n \n[\n40\n]\n.\n \nThe overlap between spatial profile of focused probe and pump \nlaser spot may lead to the generation of magnons that propagate away from the region that is \nbeing probed. \nGenerally\n,\n \nenhancement of \nnonlocal damping by spin\n-\nwave emi\nssion becomes \nsignificant\n \nwhen the excitation area is less than 1 \nµm\n. Recently \nJ. \nWu \net al.\n \nshowed that \npropagation of magnetostatic spin waves could be significant even for probed regions of tens of \nmicrons in size\n \n[\n4\n1\n]\n. \nAlso\n,\n \nby generating spin\n-\nwave trap\n \nin the pump\n-\nprobe experiment \nmodification of precessional frequency in ferromagnetic thin film due to accumulation and \ndissipation of thermal energy within the probed volume has been reported\n \n[\n4\n2\n]\n. D\nuring \nour\n \nexperiment\n \nthe \noverlap between \nprobe \nand pump \nspot \nis \nmaintai\nned carefully\n \nand\n \nKerr signal is 8\n \n \ncollected from the uniformly excited part of the sample so that slight misalignment during the \ncourse of experiment does not \nintroduce \nany \nnonlocal effects\n. \nWe will now substantiate our \nresults with some theo\nretical arguments which involve the calculation of electronic temperature \nrise in the system due to \napplication of higher \npump fluence. The electronic temperature (\nT\ne\n) is \nrelated to absorbed laser energy per unit volume (\nE\na\n) according to the following \nequa\ntion\n \n[\n4\n3\n]\n, \n \n2\n/\n)\n(\n2\n0\n2\nT\nT\nE\ne\na\n\n\n\n                                                                                                           \n \n(\n5\n)\n \nwhere, \nξ\n \nis the electronic specific heat of the system and \nT\n0\n \nis \nthe\n \ninitial electronic temperature \n(room temperature here). \nFirst, \nwe have estimated \nE\na\n \naccording to the optical parameters of the \nsample\n \nby using the following equation,\n \n \n]\n/\n)\n1\n(\n)\n1\n[(\nd\nR\nF\ne\nE\nd\na\n\n\n\n\n\n    \n                                                                                                 \n(\n6\n)\n \nwhere\n, \nd\n \nis sample thickness, \nΨ\n \nis optical penetration depth (\n~\n17 nm for \n400\n-\nnm pump\n \nlaser \nin \n20\n-\nnm\n-\nthick Py\n \nfilm\n), \nR\n \nis the reflectivity of the sample \n(0.\n5 \nmeasured \nfor \nthe \nPy\n \nfilm\n) \nand \nF\n \nis \napplied pump fluence. \nBy solving equations (\n5\n) and (\n6\n) \nwe have \nobserved \nthat \nT\ne\n \nincreases from \n≈\n1800\n \nto\n \n4\n5\n00 K within our experimental fluence range\n \nof \n10\n \nto \n55 mJ/cm\n2\n. \nDecay time of the \nelectron temperature and other r\nelevant parameters (i.e. \nE\na\n, T\ne\n \nat various flue\nnce\ns\n) are described \nin the \nsupplementa\nl\n \nmaterial\n \n[26]\n. \nThe sample remains in its magnetized state even if the \nelectronic temperature exceeds the \nCurie temperature \nT\nc\n \n. \nImportantly, ratio of the system \ntemperat\nure\n,\n \nT\n \n(as decay of electronic temperature is strongly correlated with rise of lattice \ntemperature)\n \nto \nT\nc\n \nis affecting the magnetization relaxation time\n \nwhich\n \nfundamentally depends \non susceptibility\n. Accordingly damping \nshould\n \nbe proportional to susceptibi\nlity\n \nwhich \nis\n \nstrongly \ntemperature dependent\n \n[\n40\n]\n.\n \nVarious procedures for exciting precessional dynamics in \nferromagnets show the different mechanisms to be responsible for exploration of different energy \ndissipation channels. The spin\n-\nphonon interaction m\nechanism, which historically has been \nthought to be the main contribution to magnetization damping, is important for picosecond\n-\nnanosecond applications at high temperatures such as spin caloritronics. But for laser\n-\ninduced \nmagnetization dynamics, where spi\nn\n-\nflips occur mainly due to electron scattering, quantum \nLandau\n-\nLifshitz\n-\nBloch equation is sometimes exploited to explain the temperature dependence \nof damping by considering a simple spin\n-\nelectron interaction as a source for magnetic relaxation\n \n[4\n4\n]\n. This\n \napproach suggests that increasing ratio between \nsystem \ntemperature and Curie \ntemperature\n \ninduces electron\n-\nimpurity \nled \nspin\n-\ndependent scattering. Even slightly below \nT\nc\n \na \npure change in the magnetization magnitude oc\nc\nurs\n \nwhich causes \nthe\n \nenhancement of \nda\nmping\n. \nAlso our experimental results revea\nl that the precession amplitude and\n \ndamping have been \nsubjected to a sudden change for F > 30 mJ/cm\n2\n. Energy density deposited in the probed volume \nis proportional to pump fluence. For higher fluence, the temperatu\nre dependence of the electronic \nspecific heat plays major role\n.\nThe \nincrease in the electronic \nspecific heat\n \nvalue\n \nwith temperature 9\n \n \nmay lead\n \nto\n \nlonger thermal\n-\nrelaxation time\n. We infer that relative balance between the energy \ndepo\nsited\n \ninto the lattice and electron system is \nalso \ndifferent for higher fluence regime \ncompared to that in the lower fluence regime. Thus\n,\n \nthe system temperature remains well above \nCurie temperature for F > 30 mJ/cm\n2\n, during the onset of precession for t \n≥\n \n10 ps. This may open \nup additional energy dissipation channel for the magnetization relaxation process over \nnanoseconds time scale.\n \nSometimes within very short time scale the spin temperature can go \nbeyond the Curie temperature leading towards formation o\nf paramagnetic state but that is a \nhighly non\n-\nequilibrium \ncase\n \n[\n45\n]\n. \nHowever\n \nwe believe that \nin our experiment, \neven for the \nhigh \nfluence limit and \nin \nlocal thermal \nequilibrium\n \nthe ferro\nmagnetic\n \nto paramagnetic tra\nn\nsition is not \nobserved\n.\n \nR\nepetitive measur\nements established \nthe \nreversibility of the damping parameter\n \nand \nbias\n-\nmagnetic\n-\nfield dependence of precessional frequency confirms ferromagnetic \nnature\n \nof the \nsample\n.\n \nC. \nFrequency modulation and temporal chirping\n \nPump fluence also eventua\nlly modulates the precessional frequency by introducing temporal \nchirping in the uniform precession. After immediate arrival of pump pulse, due to enhancement \nof the surface temperature, the net magnetization is reduced in \npicosecond \ntime scale which \nresul\nts in chirping of the precessional \noscillation\n. The initial frequency\n \n(\nf\ni\n)\n \nis reduced with \nrespect to its intrinsic value at a constant field. But when the probed volume cools with time, the \nspins try to retain their original precessional frequency. Thus\n,\n \nwithin a fixed time window, the \naverage frequency (\nf\nFFT\n) \nalso undergoes slight modification. In the high fluence regime, \nsignificant red shift is observed in both \nf\nFFT\n \nand \nf\ni\n. For \nH\n \n= 2.4 and 1.8 kOe, modulation of \nfrequency is found to be 0.020 GHz.cm\n2\n/mJ\n \nfor \nf\nFFT\n \nand 0.028 GHz.cm\n2\n/mJ for \nf\ni\n, from the slope \nof linear fit (as shown in \nF\nig. 4(a)). The \nf\nFFT\n \nis redu\nced by 7.2\n% of the extrapolated value at zero \npump fluence for both the fields.\n \nOn the other hand, \nf\ni\n \nis decreased \nby\n \n8.7% of its zero pump value\n \nf\nor the highest pump fluence\n. \nThe temporal chirp parameter\n, \nb \nshows giant enhancement within the experimental fluence range \n(\nF\nig. \n4\n(b)). \nFor \nH\n \n= 2.4 kOe, \nb\n \nhas increased up\n \nto \nten \ntimes (from 0.03 GHz/ns to 0.33 GHz/ns) \nin this fluence limit which implies a\nn increase in frequency of 0.66 GHz. Within our \nexperimental \nscan\n \nwindow (2 ns)\n, the maximum frequency \nshift\n \nis found to be 4.5% for \nF\n \n= 55 \nmJ/cm\n2\n.\n \nFor another bias field (\nH\n \n= 1.8 kOe), the enhancement of chirp parameter follows the \nsimilar\n \ntrend. \nThis \nult\nrafast \nmodulation is attributed to the \nthermal effect\n \non the\n \nlocal magnetic \nproperties within \nthe \nprobe\nd\n \nvolume\n \nand \nis inferred to be reversible\n \n[\n3\n7\n]\n.\n \nWe \nhave also \nplotted \nthe variation of \nb\n \nwith applied bias field for four different pump \nfluencies\n. \nI\nt see\nms to be almost \nconstant for all the \nfield values\n \nin moderate fluence regime \n(as shown in \nF\nig. \n4\n \n(c))\n. \nBut for \nF\n \n= \n40 mJ/cm\n2\n, data points are relatively scattered and large errors have been considered to take care \nof those fluctuations. \n \n 10\n \n \nIV. CONCLUSION\n \nIn\n \nessence, \nfluence dependent study \nof ultrafast magnetization dynamics \nin \nNi\n80\nFe\n20\n \nthin film\n \nreveals very weak correlation between ultrafast demagnetization time and Gilbert damping \nwithin our experimental fluence range. W\ne have reported \nlarge\n \nenhancement o\nf damping with \npump fluence. \nF\nrom the bias field \nas well as pump fluence \ndependence of \nexperimentally \nobtained \ndynamic\nal\n \nparameters we have excluded all the possible extrinsic contributions \nand \nobserved a pump\n-\ninduced modulation of intrinsic Gilbert dampin\ng. Also\n,\n \nfrom repetitive \nmeasurements with different pump irradiat\nion we have shown that the pump\n-\ninduced changes are \nreversible in nature. \nEnha\nn\ncement of the system temperature to Curie temperature ratio is \nbelieved to be responsible for increment in \nrema\ngnetization\n \ntimes and damping. \nThe temporal \nchirp parameter has been found to be increased \nby \nup to \nten \ntimes within the experimental \nfluence range\n,\n \nwhile the frequency experiences a significant red\n \nshift. \nF\nrom application point of \nview, \nas increasing dema\nnd for faster and efficient magnetic memory devices, has led the \nscientific community in the extensive research field of ultrafast magnetization dynamics, our \nresults will further enlighten the understanding of modulation of magnetization dynamics in \nferro\nmagnetic systems in presence of higher pump fluence.\n \nUsually l\now damping materials are \npreferred because it is easier to switch their magnetization in expense of smaller energy\n, lower \nwrite current in STT\n-\nMRAM devices and longer propagation length of spin \nwaves im magnonic \ndevices\n. \nOn the other hand,\n \nhigher damping is also required to \nstop \nthe post switching ringing of \nthe signal. \nThe results also have important implications on the emergent field of\n \nall\n-\noptical\n \nhelicity dependent\n \nswitching [4\n6\n-\n4\n8\n]. \nIn thi\ns context, the \ntransient \nmodulation of Gilbert \ndamping \nand other dynamical parameters \nin ferromagnetic materials is of fundamental interest \nfor characterizing and controlling ultrafast responses in magnetic structures. \n \n \nAcknowledgements:\n \nWe gratefully ack\nnowledge the financial support from S. N. Bose National \nCentre for Basic Sciences (grant no.: \nSNB/AB/12\n-\n13/96\n \nand SNB/AB/18\n-\n19/211\n) and \nDepartment of Science and Technology (DST), Government of India (grant no.: \nSR/NM/NS\n-\n09/2011\n).  We also gratefully ackno\nwledge the technical assistance of Dr. Jaivardhan Sinha and \nMr. Samiran Choudhury for preparation of the sample. SM acknowledges DST for INSPIRE \nfellowship.\n \n \nReferences\n:\n \n[\n1\n]\n \nO. Hellwig, A. Berger, T. Thomson, E. Dobisz, Z. Z Bandic, H. Yang, D. S. Kercher \nand E. \nE. Fullerton, \nSeparating dipolar broadening from the intrinsic switching field distribution in \nperpendicular patterned media\n,\n \nAppl. Phys. Lett.\n \n90\n,\n \n162516\n \n(2007)\n.\n 11\n \n \n[\n2\n]\n \nS. Tehrani, E. Chen, M. Durlam, M. DeHerrera, J. M. Slaughter, J. Shi and G. Kersz\nykowski,\n \nHigh density submicron magnetoresistive random access memory\n,\n \nJ. Appl. Phys. \n85\n,\n \n5822\n \n(1999)\n.\n \n[\n3\n] \nL. Landau and E. 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Phys.\n \nMaterials \nand devices for all\n-\noptical helicity dependent switching,\n \n50\n, 133002 (2017).\n \n 15\n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1\n: (a) Schematic of experimental geometry\n.\n \nIn the inset,\n \nφ is\n \nshown as\n \nin\n-\nplane rotational angle of\n \nH, \n(b) \npump fluence dependence of ultrafast demagnetization; \nSolid lines are fit\nting line\ns. P\nump fluences (\nF\n) having unit of \nmJ/cm\n2 \nare mentioned in numerical figure. The Gaussian envelope of laser pulse is presented to describe the \nconvolution. (c) Repre\nsentative time resolved Kerr rotation data with three distinguished temporal regions for \nF\n \n= \n25 mJ/cm\n2\n. (d) Angular variation of precessional frequency at \nH\n \n= 1.1 kOe for 20\n-\nnm\n-\nthick Py film. \nφ is presented \nin degree.\n \n \n \n 16\n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2\n: (a) Background subtracted time\n-\nresolved Kerr rotation data for different pump fluences at \nH\n \n= 2.4 kOe. \nF\n \nhaving unit of mJ/cm\n2 \nis mentioned in numerical figure. \nSolid lines are fit\nting lines\n. \nPump f\nluence\n \ndependen\nce of\n \n(b) \nrelaxation time (\nτ\n) and (c) effective damping (\nα\neff\n). \nBlack \nand \nblue \nsymbols represent the variation of these \nparameters at two different field values, \nH\n \n= 2.4 and 1.8 kOe, respectively.\n \nA\nmplitude of precession is also plotted \nwith pump \nfluence for \nH\n \n= 2.4 kOe\n,\n \n(d) Variation of effective damping with irradiation fluence \n(\nF\ni\n) \nat \nH\n \n= 2.4 kOe. \nIn order to check the possible damage in the sample as high fluence values the pump fluence was taken up to the \ntargeted value of \nF\ni\n \nfor several minut\nes followed by reduction of the pump fluence to a constant value of 10 mJ/cm\n2\n \nand the pump\n-\nprobe measurement was performed. The damping coefficient is found to be unaffected by the \nirradiation fluence as shown in (d). \n \n \n \n \n \n \n \n \n \n17\n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3\n: (a) Bias\n \nfield dependence of precessional frequency for \nF\n \n= 10 mJ/cm\n2\n. The red solid line indicates the \nKittel fit. (b) Pump fluence dependence of effective magnetization (M\neff\n) of the probed volume. (c) Bias field \ndependence of relaxation time (\nτ\n) for four differ\nent fluences. \nF\n \nhaving unit of mJ/cm\n2 \nis mentioned in numerical \nfigures. Solid lines are the fitted data. (d) Variation of intrinsic Gilbert damping (\nα\n0\n) with pump fluence.\n \n \n \n \n \n \n \n \n \n \n \n \n \n 18\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 4\n: (a) Pump\n-\nfluence dependence of precessional fre\nquencies for \nH\n \n= 2.4 and 1.8 kOe. Red and black symbols \nrepresent the variation of average frequency (\nf\nFFT\n) and initial frequency (\nf\ni\n) respectively. (b) Variation of temporal \nchirp parameter ‘\nb’\n \nwith pump fluence for two different magnetic field values. (c\n) Variation of temporal chirp \nparameter with bias field for four different pump fluences. \nF\n \nhaving unit of mJ/cm\n2 \nis mentioned in numerical figure.\n \nDotted lines are guide to eye.\n \n \n \n "    },    {        "title": "1512.00557v1.Bose_Einstein_Condensation_of_Magnons_Pumped_by_the_Bulk_Spin_Seebeck_Effect.pdf",        "content": "Bose-Einstein Condensation of Magnons Pumped by the Bulk Spin Seebeck E\u000bect\nYaroslav Tserkovnyak,1Scott A. Bender,1Rembert A. Duine,2and Benedetta Flebus2\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,\nUtrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands\nWe propose inducing Bose-Einstein condensation of magnons in a magnetic insulator by a heat\n\row oriented toward its boundary. At a critical heat \rux, the oversaturated thermal gas of magnons\naccumulated at the boundary precipitates the condensate, which then grows gradually as the thermal\nbias is dialed up further. The thermal magnons thus pumped by the magnonic bulk (spin) Seebeck\ne\u000bect must generally overcome both the local Gilbert damping associated with the coherent magnetic\ndynamics as well as the radiative spin-wave losses toward the magnetic bulk, in order to achieve\nthe threshold of condensation. We quantitatively estimate the requisite bias in the case of the\nferrimagnetic yttrium iron garnet, discuss di\u000berent physical regimes of condensation, and contrast\nit with the competing (so-called Doppler-shift) bulk instability.\nPACS numbers: 72.25.Mk,72.20.Pa,75.30.Ds,85.75.-d\nIntroduction .|The rapidly developing thermoelectric\ntransport capabilities to probe nonconducting materials\nare instigating a shift in the \feld of spintronics toward in-\nsulating magnets [1{3]. While allowing for seamless spin\ninjection and detection at their boundaries [4{6], insulat-\ning magnets (including ferromagnets, antiferromagnets,\nand spin liquids) may o\u000ber also e\u000ecient spin propaga-\ntion owing to the lack of the electronic channels for dis-\nsipation of angular momentum. Recent measurements of\nspin signals mediated by thick layers of antiferromagnetic\nnickel oxide [7] and, especially, long di\u000busion lengths of\nmagnons in ferrimagnetic yttrium iron garnet (YIG) [8{\n10], even at room temperature, bear this view out.\nThe bosonic nature of magnons, furthermore, naturally\nlends itself to condensation instabilities when driven by\nlarge biases into a nonlinear response [11{13]. While the\nelectric spin Hall driving of magnetic insulators [14, 15]\nclosely mimics the familiar spin-transfer torque insta-\nbilities of conducting ferromagnets [16], the possibility\nof inducing magnonic (Bose-Einstein) condensation also\nby a heat \rux [12] o\u000bers new exciting opportunities\nthat are unique to the insulating heterostructures. The\nkey physics here is played out in the framework of the\nspin Seebeck/Peltier phenomenology [17], according to\nwhich the heat and spin currents carried by magnons\nare intricately intertwined [18]. While the problem of\nthe thermoelectrically-driven magnon condensation has\nbeen systematically addressed previously in thin-layer\nheterostructures [12, 13], the more basic regime of an\ninterfacial condensation induced by a bulk heat \rux re-\nmains unexplored. This concerns the standard geometry\nof the (longitudinal) spin Seebeck e\u000bect, which is suit-\nable for complex lateral heterostructures that could ulti-\nmately give rise to useful devices [19].\nApplying a large heat \rux from a ferromagnet toward\nits interface with another material (either conducting or\ninsulating), which can carry heat but blocks spin \row,\nleads to a nonequilibrium pile up of magnons at the\nT(x)x0µ(x)µ0.B\u0000n(x, t)nonmagnetic substratemagnetic insulator\njx✓(!,k)zFIG. 1. A monodomain ferromagnet with uniform equilibrium\nspin density pointing in the \u0000zdirection (in the presence of\na magnetic \feld Bpointing up along z). A positive ther-\nmal gradient, @xT >0, induces magnonic \rux jxtowards the\ninterface, where an excess of thermal magnons is accumulat-\ning over their spin-di\u000busion length \u0015. When the correspond-\ning nonequilibrium interfacial chemical potential \u00160reaches a\ncritical value (in excess of the magnon gap), the magnetic or-\nder undergoes a Hopf bifurcation toward a steady precessional\nstate, whose Gilbert damping and radiative spin-wave losses\nare replenished by the thermal-magnon pumping /\u00160. The\ncoherent transverse magnetic dynamics decays away from the\ninterface as nx\u0000iny/ei(kx\u0000!t), where Imk>0.\nboundary. See Fig. 1 for a schematic. When the associ-\nated chemical potential of magnons exceeds the lowest-\nmode frequency of the magnet, the latter gets pumped\nby the magnonic thermal gas, leading to its condensation\nat a critical bias. The problem of \fnding the threshold\nfor this phenomenon as well as considering detrimental\nand competing e\u000bects are the main focus of this Letter.\nOnce experimentally established, such pumped conden-\nsates should provide a fertile platform for studying and\nexploiting spin super\ruidity [20].\nTwo-\ruid magnon hydrodynamics .|The interplay be-\ntween thermal-magnon transport and coherent order-\nparameter dynamics is naturally captured within thearXiv:1512.00557v1  [cond-mat.mes-hall]  2 Dec 20152\ntwo-\ruid formalism developed in Ref. [21]. Namely, we\nstart with a generic long-wavelength spin Hamiltonian\nH=Z\nd3r\u0012\n\u0000A\n2s^s\u0001r2^s+B^sz+K\n2s^s2\nz\u0013\n; (1)\nwhere ^sis the spin-density operator (in units of ~),A\nis the magnetic sti\u000bness, Bthe external \feld along the\nzaxis,Kthe quadratic anisotropy in the same direc-\ntion (with K > 0 corresponding to the easy xyplane\nandK < 0 easyzaxis), andsthe saturation spin den-\nsity. We then perform the Holstein-Primako\u000b transfor-\nmation [22] to the bosonic \feld ^\b\u0019(^sx\u0000i^sy)=p\n2s,\nwhich is composed of the super\ruid order parameter\n\b\u0011h^\biand the quantum-\ructuating piece ^\u001e:^\b = \b+ ^\u001e.\nThese relate to the original spin variables as s\u0011h^si\u0019\n(p\n2sRe\b;\u0000p\n2sIm\b;nc+nx\u0000s), where \b =pnce\u0000i'\nandnx=h^\u001ey^\u001ei, withncandnxbeing respectively the\ncondensed and thermal magnon densities. It is clear that\n'is the azimuthal angle of the coherent magnetic pre-\ncession in the xyplane.\nFollowing the Landau-Lifshitz-Gilbert (LLG) phe-\nnomenology [23] of long-wavelength spin-wave dynamics,\nthe following hydrodynamic equations are obtained [21]:\n_nx+r\u0001jx+\u001b\u0016=\u00152= 2\u0011(!\u0000\u0016=~)nc; (2)\nfor the normal dynamics, where jx=\u0000\u001br\u0016\u0000&rT(\u001b\nbeing the magnon conductivity, &the bulk Seebeck co-\ne\u000ecient, and \u0016the chemical potential) is the thermal\nmagnon \rux and \u0015is the magnon di\u000busion length, and\n_nc+r\u0001jc+ 2\u000b!nc= 2\u0011(\u0016=~\u0000!)nc; (3a)\n~(!\u0000\n)\u0000Knc\ns=A\u0014\n(r')2\u0000r2pncpnc\u0015\n;(3b)\nfor the condensate, where jc=\u0000(2A=~)ncr'and~\n =\nB\u0000K(1\u00002nx=s) is the magnon gap (where we take for nx\nto be the equilibrium cloud density at the ambient tem-\nperatureTand self-consistently suppose that \n >0, so\nthat the ferromagnet is in the normal state with n=\u0000z\nin equilibrium [21]). Furthermore, \u0011\u0018(K=T )2(T=Tc)3\nis the dimensionless constant parametrizing the rate of\nthe thermal-cloud|condensate scattering [21], in terms\nof the Curie temperature Tc.\nFor our present purposes, it will be convenient to re-\ncast the condensate dynamics (3) in the form of the LLG\nequation, as discussed in Ref. [13]:\n~(1 +\u000bn\u0002)_n\u0000[~\n +K(1 +n\u0001z)]z\u0002n\n=An\u0002r2n+\u0011n\u0002(\u0016z\u0002n\u0000~_n);(4)\nwhere the second term on the right-hand side is the local\nthermomagnonic torque parametrized by \u0011. Rewriting\nEq. (2) in the same spirit, we have:\n_nx+r\u0001jx+\u001b\u0016=\u00152=\u0011sz\u0001n\u0002(_n\u0000\u0016z\u0002n=~):(5)Spin Seebeck-driven instability .|For the boundary\nconditions at the interface, x= 0, we will take the sim-\nplest scenario of a hard wall, for which both the thermal\nand coherent spin currents vanish, leading to\n\u001b@x\u0016+&@xT= 0 and @xn= 0; (6)\nwith the latter corresponding to the usual exchange\nboundary condition for classical ferromagnetic dynam-\nics. Below or near the onset of magnetic instability (con-\ndensation in the language of Ref. [12]), we can neglect\nthe right-hand side of Eq. (5). This produces the spin-\ndi\u000busion equation, which is solved by\n\u0016(x) =\u00160e\u0000x=\u0015;where\u00160=\u0015&@xT=\u001b; (7)\nin the steady state (established in response to a uniform\nthermal gradient @xT) and subject to the boundary con-\ndition (6). The magnon chemical potential \u0016is, natu-\nrally, maximized at the interface.\nFor the remainder of this section, we analyze Eq. (4)\nsubject to the magnonic torque induced by \u0016(x) in\nEq. (7). Let us \frst solve the problem in the limit \u0015!1\n(relative to other relevant lengthscales, to be identi\fed\nbelow) resulting in homogeneous dynamics. Rewriting\nthe corresponding LLG equation (4) as\n~(_n\u0000~\nz\u0002n) =n\u0002(\u0011\u00160z\u0002n\u0000~\u000b~_n)\n\u0019(\u0011\u00160\u0000~\u000b~~\n)n\u0002z\u0002n;(8)\nwhere ~\u000b\u0011\u000b+\u0011and~~\n\u0011~\n+K(1+n\u0001z). Here, we as-\nsumed ~\u000b;\u0011\u001c1 and thus approximated _n\u0019~\nz\u0002nin the\nGilbert damping term in going to the second line. It is\nnow easy to see that when the antidamping torque /\u0011\novercomes net damping ~ \u000b, the static equilibrium state\nn=\u0000zbecomes unstable [16]. In the case of the easy-\naxis anisotropy, K < 0, this leads to magnetic switch-\ning toward the stable n=zstate when \u00160>(~\u000b=\u0011)~\n.\nIn the more interesting easy-plane case, K > 0 (corre-\nsponding to repulsive magnon-magnon interactions), the\nanisotropy stabilizes magnetic dynamics at a limit cycle\n(realizing a Hopf bifurcation). The corresponding pre-\ncession angle \u0012is then found to be\n\u0012= 2 sin\u00001r\n\u0011\u00160\u0000~\u000b~\n2~\u000bK; (9)\neventually saturating at \u0012!\u0019when\u00160\u0015(~\u000b=\u0011)(~\n +\n2K).\nLet us estimate the thermal gradient necessary to reach\nthe critical heat \rux for condensation, \u00160= (~\u000b=\u0011)~\n, in\nthe case of yttrium iron garnet. The critical thermal\ngradient is given by\n@xT(c)=~\u000b\n\u0011\u001b\n\u0015&~\n: (10)\nFollowing the magnon-transport theory of Ref. [21] (Sup-\nplemental Material), \u001b=&\u00181 [24]. Taking conservatively3\n~\u000b=\u0011\u0018100 [13] and \u0015\u001810\u0016m [9] at room temperature\n(which is consistent with theoretical estimates based on\nRef. [13]), we get @xT(c)\u00181 K/\u0016m, for \n=2\u0019\u00182 GHz\n(corresponding to a kG \feld). Achieving such thermal\ngradients should be experimentally feasible [2, 8].\nCondensate out\row .|More generally, for \fnite \u0015, the\ncondensate is driven near the interface (where \u00166= 0)\nand should eventually decay su\u000eciently deep into the\nferromagnet. This causes spin super\row away from the\ninterface, furnishing radiative spin-wave losses into the\nbulk, which should suppress condensation and raise the\nheat-\rux threshold. The corresponding instability is de-\nscribed by the LLG equation (4), which we rewrite more\ncompactly as\n~(_n\u0000~\nz\u0002n) =An\u0002@2\nxn+n\u0002(\u0011\u0016z\u0002n\u0000~\u000b~_n);(11)\nwhere both ~\n and\u0016become position dependent (both\ndecreasing away from the interface toward \n and 0, re-\nspectively, in the bulk). Supposing a smooth onset of\ninstability, we will look for the thermal threshold by set-\nting ~\n!\n.\nTaking, furthermore, the opposite extreme of \u0015!0\n(relative to the absolute value of the condensate wave\nnumber, to be checked for internal consistency later), we\ncan integrate Eq. (11) over a distance \u0019\u0015near the inter-\nface [noting that n\u0002@2\nxn\u0011@x(n\u0002@xn)] to obtain the\nboundary condition,\n~\u0015(_n\u0000\nz\u0002n)\u0019An\u0002@xn+\u0015n\u0002(\u0011\u00160z\u0002n\u0000~\u000b~_n);\n(12)\nfor the intrinsic bulk dynamics,\n~(_n\u0000\nz\u0002n) =An\u0002@2\nxn\u0000~\u000b~n\u0002_n; (13)\nin the ferromagnet. In order to \fnd the steady-state\nlimit-cycle solution at the onset of the condensation, we\nlinearize these equations with respect to small deviations\nmaway from equilibrium, n\u0011\u0000z+m, and solve for\nthe ansatz m\u0011mx\u0000imy/ei(kx\u0000!t)(requiring that\nImk>0 and!is real valued), to obtain\n~(!\u0000\n) =Ak2\u0000i~\u000b~!; (14)\nsubject to the boundary condition\ni~(!\u0000\n) =Ak=\u0015\u0000\u0011\u00160+ ~\u000b~!: (15)\nThe \frst term on the right-hand side of this equation\ndescribes coherent spin out\row into the bulk, the sec-\nond term magnonic pumping, and the last term Gilbert\ndamping. The critical chemical potential is correspond-\ningly raised as\n\u00160=~\u000b~!+ARek=\u0015\n\u0011: (16)\nThe spin Seebeck-induced magnonic pumping /\u0011thus\nneeds to overcome the condensate out\row /Ain addi-\ntion to the Gilbert damping /~\u000b. We proceed to solveEqs. (14), (15) supposing that Im k\u001c\u0015\u00001, for internal\nconsistency, and \fnd\nImk= \n~\u000bp\n\u0015\n2\u00152s!2=3\n;Rek=r\nImk\n\u0015=\u0012~\u000b\n2\u0015\u00152s\u00131=3\n;\n(17)\nwhere\u0015s\u0011p\nA=~\n (\u001810 nm, using \n =2\u0019\u00182 GHz\nand typical YIG parameters [25]). In deriving Eqs. (17),\nwe have assumed that ~ \u000b\u001c\u0015=\u0015s, which should not be\nan issue in practice. The \fnal internal consistency check\nis Imk\u001c\u0015\u00001, which thus boils down to ~ \u000b(\u0015=\u0015s)2\u001c1.\nFor YIG with ~ \u000b\u001810\u00004, this would be borderline when\n\u0015=\u0015s\u0018100 (which should be relevant in practice for a\nshorter\u0015and/or lower \n). The frequency according to\nEq. (15) is found as != \n(1+Imk\u00152\ns=\u0015)\u0019\n, so that the\ninstability threshold is \fnally found according to Eq. (16)\nas\n@xT(c)\u0019~\u000b\n\u0011\u001b\n\u0015&~\n\"\n1 +\u0012\u00152\nsp\n2~\u000b\u00152\u00132=3#\n; (18)\nwhich is the central result of this Letter.\nNote that Eq. (18) naturally captures also the \u0015!1\nlimit (10) obtained above (thus indicating its general va-\nlidity for extrapolating between both small and large \u0015\nregimes), which we now understand as corresponding to\n~\u000b(\u0015=\u0015s)2\u001d1. In the case of YIG at room temperature,\nwe thus expect Eq. (10) to give a good quantitative esti-\nmate for the threshold bias. The details of the magnetic\npro\fle beyond the instability threshold can in general be\nexpected to be quite complex, as described by the non-\nlinear Eq. (11), especially if one takes into account the\nfeedback of coherent dynamics on the magnon di\u000busion\naccording to Eq. (5). This nonlinear regime is outside\nthe scope of this work.\nDiscussion and outlook .|At a su\u000eciently large\nmagnon \rux in the bulk of the ferromagnet, the trans-\nverse dynamics exhibit also the Doppler-shift instabil-\nity [26], according to the bulk thermomagnonic torque\n/jx@xn[27]. We \fnd the corresponding threshold to be\ngiven byjx\u0018s\n\u0015s, which translates into @xT\u0018s\n\u0015s=&.\nDividing it by the threshold (10), we get @xT=@xT(c)\u0018\n(\u0011=~\u000b)(s\u0015s\u0015=~\u001b). Taking [21] \u001b\u0018(T=Tc)(s2=3l)=~, where\nlis the magnon mean free path, we thus get for this ratio\n\u0018(\u0011=~\u000b)(Tc=T)(s1=3\u0015s\u0015=l). Performing, once again, an\nestimate for YIG at room temperature by taking \u0011=~\u000b\u0018\n10\u00002,Tc=T\u00182,s1=3\u00182/nm,l\u00181\u0016m,\u0015s\u001810 nm,\nand\u0015\u001810\u0016m, we \fnd that @xT=@xT(c)&1, so that\nboth instability scenarios are in fact viable and could po-\ntentially be competing. This could of course be easily\nchecked as the Doppler-shift instability is independent of\nthe heat-\rux direction, while the BEC of magnons dis-\ncussed here is unipolar, corresponding to the heat \rux\ntowards the interface, as sketched in Fig. 1.\nIt needs also be stressed that the ratio \u0011=~\u000b\u001810\u00002\nemployed in this Letter for our estimates corresponds4\nonly to thermal magnons and disregards low-energy\nmagnons that are beyond the Bose-Einstein thermaliza-\ntion description [13, 21]. When \u00160approaches and ul-\ntimately exceeds the magnon gap ~\n, the overpopula-\ntion of magnons pumped at the bottom of the spec-\ntrum could e\u000bectively enhance this factor, approaching\n\u0011=~\u000b!1 in the extreme case (realizing the limit of the\nstrong condensate-cloud coupling studied in Ref. [12]).\nThis innately nonequilibrium regime, which would yield\na lower threshold for magnonic condensation, is, however,\nbeyond our present formalism.\nOnce established, the interfacial condensate of\nmagnons can be readily detected by monitoring the spin\naccumulation (utilizing, for example, the magneto-optic\nKerr e\u000bect) in the adjacent metallic (nonmagnetic) sub-\nstrate or detecting the associated spin pumping by the\ninverse spin Hall e\u000bect (as in the conventional spin See-\nbeck geometry [2]). In the latter case, the theory would\nhave to be complemented with the appropriate treatment\nof spin leakage into and relaxation in the normal metal\n[21]. The condensate can also be used as a starting point\nto study and exploit collective \\conveyor-belt\" heat and\nspin \row [21] tangential to the interface, which would\nre\rect its super\ruid nature.\nThe authors thank Joseph P. Heremans and Roberto\nC. 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Box 15100, FI-00076 Aalto, Finland\n(Dated: October 12, 2018)\nWe present a comprehensive investigation of propagating spin waves in nanometer-thick yttrium\niron garnet (YIG) \flms. We use broadband spin-wave spectroscopy with integrated coplanar waveg-\nuides (CPWs) and microstrip antennas on top of continuous and patterned YIG \flms to characterize\nspin waves with wave vectors up to 10 rad/ \u0016m. All \flms are grown by pulsed laser deposition. From\nspin-wave transmission spectra, parameters such as the Gilbert damping constant, spin-wave dis-\npersion relation, group velocity, relaxation time, and decay length are derived and their dependence\non magnetic bias \feld strength and angle is systematically gauged. For a 40-nm-thick YIG \flm, we\nobtain a damping constant of 3 :5\u000210\u00004and a maximum decay length of 1.2 mm. Our experiments\nreveal a strong variation of spin-wave parameters with magnetic bias \feld and wave vector. Spin-\nwave properties change considerably up to a magnetic bias \feld of about 30 mT and above a \feld\nangle of\u0012H= 20\u000e, where\u0012H= 0\u000ecorresponds to the Damon-Eshbach con\fguration.\nPACS numbers:\nI. INTRODUCTION\nMagnonics aims at the exploitation of spin waves for\ninformation transport, storage, and processing1{7. For\npractical devices, it is essential that spin waves propa-\ngate over long distances in thin \flms. Because of its ul-\ntralow damping constant, ferrimagnetic YIG is a promis-\ning material. Bulk crystals and \u0016m-thick YIG \flms ex-\nhibit a Gilbert damping constant \u000b\u00193\u000210\u00005at GHz\nfrequencies. In recent years, nm-thick YIG \flms with\nultralow damping parameters have also been prepared\nsuccessfully. High-quality YIG \flms have been grown on\nGd3Ga5O12(GGG) single-crystal substrates using liquid\nphase epitaxy8{11, magnetron sputtering12{15, and pulsed\nlaser deposition (PLD)16{25. For thin YIG \flms, damp-\ning constants approaching the value of bulk crystals have\nbeen reported21,22. Meanwhile, YIG-based magnonic\ndevices such as logic gates, transistors, and multiplex-\ners have been demonstrated26{30. Spin-wave transmis-\nsion in nm-thick YIG \flms24,31{36and the excitation of\nshort-wavelength spin waves have been investigated as\nwell37{40. To advance YIG-magnonics further, knowledge\non the transport of spin waves in nm-thick YIG \flms and\nits dependence on wave vector and external magnetic bias\n\feld is essential.\nIn this paper, we present a broadband spin-wave spec-\ntroscopy study of PLD-grown YIG \flms with a thickness\nof 35 nm and 40 nm. Spin-wave transmission spectra are\nrecorded by patterning CPWs and microstrip antennas\non top of continuous and patterned YIG \flms. CPWs are\nused because they generate spin waves with well-de\fned\nwave vectors. This enables extraction of key parame-\nters such as the Gilbert damping constant ( \u000b), the spin-\nwave dispersion relation, group velocity ( \u001dg), relaxation\ntime (\u001c), and decay length ( ld). For a 40 nm YIG \flm,\nwe \fnd\u000b\u00193:5\u000210\u00004and a maximum group velocity\nand decay length of 3.0 km/s and 1.2 mm, respectively.\nWe show that spin-wave properties vary strongly withwave vector up to an in-plane external magnetic bias \feld\n\u00160Hext= 30 mT and below a \feld angle \u0012H= 20\u000e(\u0012H\n= 0 corresponds to the Damon-Eshbach geometry). Be-\nyond these \feld parameters, the dependence of spin-wave\nproperties on wave vector weakens. We demonstrate also\nthat broadband spectroscopy with integrated CPWs and\nmicrostrip antennas provide similar spin-wave parame-\nters.\nThe paper is organized as follows. In Sec. II,\nwe describe the PLD process, broadband spin-wave\nspectroscopy setup, and simulations of the CPW- and\nmicrostrip-antenna excitation spectra. In Sec. III, we\npresent vector network analyzer ferromagnetic resonance\n(VNA-FMR) results and broadband spin-wave transmis-\nsion spectra for CPWs. In Sec. IV, we \ft the ex-\nperimental data and extract parameters of propagating\nspin waves. Spin-wave transmission measurements using\nCPWs and microstrip antennas are compared in Sec. V.\nSection VI summarizes the paper.\nII. EXPERIMENT\nA. PLD of YIG thin \flms\nYIG \flms with a thickness of 35 nm and 40 nm were\ngrown on single-crystal GGG(111) substrates using PLD.\nPrior to loading into the PLD vacuum chamber, the\nsubstrates were ultrasonically cleaned in acetone, iso-\npropanol, and distilled water. The substrates were \frst\ndegassed at 550\u000eC for 15 minutes and then heated to\n800\u000eC at a rate of 5\u000eC per minute in an O 2pressure of\n0.13 mbar. YIG \flms were deposited under these condi-\ntions by ablation from a stoichiometric target using an\nexcimer laser with a pulse repetition rate of 2 Hz and\na \ruence of 1.8 J/cm2. After deposition, the YIG \flms\nwere \frst annealed at 730\u000eC for 10 minutes in 13 mbar\nO2before cooling down to room temperature at a rate ofarXiv:1810.04973v1  [cond-mat.mes-hall]  11 Oct 20182\n-1.0 -0.5 0.0 0.5 1.0-100-50050100\n49 50 51 52 53300 400 500 6000.00.51.0Ms (kA/m)\nMagnetic field (mT)GGG (444)Intensity (a.u.)\n2oYIG (444)\nM/Ms\nTemperature (K)(a) (b)\nFIG. 1: (a) XRD \u0012\u00002\u0012scan of the (444) re\rections from a\nPLD-grown YIG \flm on a GGG(111) substrate. The period\nof Laue oscillations surrounding the (444) peaks corresponds\nto a \flm thickness of 40 nm. (b) Room temperature VSM\nhysteresis loop of the same \flm. The inset shows how the\nYIG saturation magnetization varies with temperature.\n\u00003\u000eC per minute.\nB. Structural and magnetic characterization\nThe crystal structure of our YIG \flms was inspected\nby high-resolution X-ray di\u000braction (XRD) on a Rigaku\nSmartLab system. Figure 1(a) shows a XRD \u0012\u00002\u0012scan\nof a 40-nm-thick YIG \flm on GGG(111). Clear (444) \flm\nand substrate peaks are surrounded by Laue oscillations,\nsignifying epitaxial and smooth \flm growth. We used a\nvibrating sample magnetometer (VSM) in a PPMS Dy-\nnacool system from Quantum Design to characterize the\nmagnetic properties. Figure 1(b) depicts a VSM hystere-\nsis loop of a 40-nm-thick YIG \flm. At room temperature,\nthe coercive \feld of the YIG \flm is only 0.1 mT and the\nsaturation magnetization ( Ms) is 115 kA/m. The evo-\nlution ofMswith temperature is shown in the inset of\nFig. 1(b). From these data, we derive a Curie temper-\nature (TC) of around 500 K. The values of MsandTC\nare similar to those obtained in previous studies on nm-\nthick YIG \flms14,22,23and about 10% smaller compared\nto values of YIG bulk crystals ( Ms= 139 kA/m, TC=\n559 K). Minor o\u000b-stoichiometries in the YIG \flm might\nbe the reason for the small discrepancy41.\nC. Broadband spin-wave spectroscopy\nVNA-FMR and spin-wave transmission measurements\nwere performed using a two-port VNA and a microwave\nprobing station with a quadrupole electromagnet. In\nVNA-FMR experiments, the YIG \flm was placed face-\ndown onto a prepatterned CPW on a GaAs substrate.\nThe signal line and ground lines of this CPW had a\nwidth of 50 \u0016m and 800 \u0016m, respectively, and were sep-\narated by 30 \u0016m. Broadband spin-wave spectroscopyin transmission geometry was conducted by contacting\ntwo integrated CPWs or microstrip antennas on top of\na continuous YIG \flm or YIG waveguide. Most of the\nexperiments were performed with CPWs consisting of 2\n\u0016m-wide signal and ground lines with a separation of 1.6\n\u0016m. For comparison measurements, we used CPWs and\nmicrostrip antennas with 4- \u0016m-wide signal lines. All an-\ntenna structures were fabricated by electron-beam lithog-\nraphy and were composed of 3-nm Ta and 120-nm Au.\nA microwave current provided by the VNA was used to\ngenerate a rf magnetic \feld around one of the CPWs\nor microstrip antennas. We used CST microwave studio\nsoftware to simulate the excitation spectra of the antenna\nstructures (see next section).\nSpin waves that are excited by a rf magnetic \feld pro-\nduce an inductive voltage across a nearby antenna. At\nthe exciting CPW or microstrip antenna, this voltage is\ngiven by42:\nVind/Z\n\u001f(!;k)j\u001a(k)j2dk; (1)\nwhere\u001f(!;k) is the magnetic susceptibility and j\u001a(k)j2\nis the spin-wave excitation spectrum. Propagating spin\nwaves arriving at the receiving CPW or microstrip an-\ntenna produce an inductive voltage:\nVind/Z\n\u001f(!;k)j\u001a(k)j2exp(\u0000i(ks+ \b 0))dk; (2)\nwheresis the propagation distance and \b 0is the initial\nphase of the spin waves. In the experiments, we used the\n\frst and second port of the VNA to measure these induc-\ntive voltages by recording the S12scattering parameter.\nD. Simulations of CPW and microstrip antenna\nexcitation spectra\nWe used CST microwave studio software to simulate\nthe spin-wave excitation spectra of the di\u000berent antenna\nstructures43. This commercial solver of Maxwell's equa-\ntions uses a \fnite integration method to calculate the rf\nmagnetic \feld \u00160hrfand its in-plane ( \u00160hrf\nx,\u00160hrf\ny) and\nout-of-plane ( \u00160hrf\nz) components. Since the excitation\n\feld along the CPW or antenna ( \u00160hrf\nx) is nearly uni-\nform and\u00160hrf\nzis much smaller than \u00160hrf\ny, we Fourier-\ntransformed only the latter component. Figure 2 depicts\nseveral CPW and antenna con\fgurations used in the ex-\nperiments together with their simulated spin-wave exci-\ntation spectra. The large prepatterned CPW on a GaAs\nsubstrate (Fig. 2(a)), which is used for VNA-FMR mea-\nsurements, mainly excites spin waves with k\u00190 rad/\u0016m\n(Fig. 2(d)). The excitation spectrum of the smaller in-\ntegrated CPW with a 2- \u0016m-wide signal line (Fig. 2(b))\nincludes one main spin-wave mode with wave vector k1\n= 0.76 rad/ \u0016m and several high-order modes k2\u0000k7\n(Fig. 2(e)). The 4- \u0016m-wide microstrip antenna (Fig.\n2(c)) mainly excites spin waves with k1ranging from 03\n(a)\nGGS\n02468 1 0 1 20.00.51.0\n-20 -10 0 10 20Amplitude (Normalized)\nWave vector (rad/ m)0Hy (a. u.)\ny (m)S\n02468 1 0 1 20.00.51.0-100 -50 0 50 100Amplitude (Normalized)\nWave vector (rad/ m)GG0Hy (a. u.)\ny (m)S\n02468 1 0 1 20.00.51.0 -10 -5 0 5 10Amplitude (Normalized)\nWave vector (rad/ m)G G0Hy (a. u.)\ny (m)S\n(b) (c)\n(d) (e) (f)\nk1\nk2k3k4k5k6k7k1\nk2k3xyz\nxyz\nx yz\nFIG. 2: (a-c) Schematic illustrations of several measurement con\fgurations used in this study. (a) VNA-FMR measurements\nare performed by placing the YIG/GGG sample face-down onto a CPW. The CPW consists of a 50 \u0016m-wide signal line and\ntwo 800\u0016m-wide ground lines. The gap between the signal and ground lines is 30 \u0016m. (b-c) Spin-wave transmission through\nthe YIG \flm is characterized by patterning two CPWs (b) or two microstrip antennas (c) on top of a YIG \flm. The signal and\nground lines of the CPWs in (b) are 2 \u0016m wide and separated by 1.6 \u0016m gaps. The microstrip antennas, which are marked\nby red arrows in (c), are 4 \u0016m wide. (d-f) Simulated spin-wave excitation spectra of the di\u000berent antenna structures. The\nin-plane rf magnetic \felds ( \u00160hrf\ny) that are produced by passing a microwave current through the CPWs in (a) and (b) or the\nmicrostrip antenna in (c) are shown in the insets.\nto 1.5 rad/\u0016m and some higher order modes at k2\u00192:0\nrad/\u0016m andk3\u00193:8 rad/\u0016m (Fig. 2(f)). The insets of\nFigs. 2(d-f) show the simulated rf magnetic \felds \u00160hrf\ny\nalong they-axis for each antenna structure.\nIII. RESULTS\nA. VNA-FMR\nWe recorded FMR spectra for various in-plane exter-\nnal magnetic bias \felds by measuring the S12scatter-\ning parameter on a 40-nm-thick YIG \flm. As an ex-\nample, the imaginary part of S12recorded with a mag-\nnetic bias \feld \u00160Hext= 80 mT is shown in Fig. 3(a).\nThe spectrum was subtracted a reference measured at\na bias \feld of 200 mT for enhancing signal-to-noise ra-\ntio. The resonance at f= 4:432 GHz is \ftted by a\nLorentzian function. From similar data taken at other\nbias \felds, we extracted the \feld-dependence of FMR\nfrequency and the evolution of resonance linewidth (\u0001 f)\nwith frequency. Figures 3(b) and 3(c) summarize our\nresults. Fitting the data of Fig. 3(b) to the Kittel for-\nmulafres=\r\u00160\n2\u0019p\nHext(Hext+Meff) using\r=2\u0019= 28\nGHz/T, we \fnd Meff= 184 kA/m. The measured\nvalue ofMeffis comparable to those of other PLD-grown\nYIG thin \flms23,24, but it is large compared to Ms(115\nkA/m). Since Meff=Ms-Hani, this means that the\nanisotropy \feld Hani=\u000069 kA/m in our \flm. The\nnegative anisotropy \feld is caused by a lattice mismatch\nbetween the YIG \flm and GGG substrate23. Fitting the\n4.42 4.44 3456681012\n05 0 1 0 0 1 5 00246Im S12\nFrequency (GHz) Frequency (GHz)\nFrequency (GHz)\n0Hext (mT)(a) (b)\n= 3.5 × 10-4f(c)FIG. 3: (a) Imaginary part of the S12scattering parameter\nshowing FMR for an in-plane external magnetic bias \feld of\n80 mT along the CPW. The orange line is a Lorentzian func-\ntion \ft. (b) FMR frequency as a function of external magnetic\nbias \feld. The orange line represents a \ft to the experimen-\ntal data using the Kittel formula. (c) Dependence of FMR\nlinewidth (\u0001 f) on resonance frequency. From a linear \ft to\nthe data, we derive \u000b= 3:5\u000210\u00004.\ndata of Fig. 3(c) using \u0001 f= 2\u000bf+\u001dg\u0001kgives a Gilbert\ndamping constant \u000b= 3:5\u000210\u00004, which is comparable to\nother experiments on PLD-grown \flms17,18,20. In the \ft-\nting formula, \u001dgand \u0001kare the spin-wave group velocity\nand excitation-spectrum width, respectively44.4\n1.8 2.1 2.4 2.7 3.0 10 20 30 40 501234\n0Hext (mT)Frequency (GHz)Im S12k7k5k4 k6k3k2\nFrequency (GHz)k1\n-60 -30 0 30 601.82.12.42.73.03.3\nH (O)Frequency (GHz)(a) (b) (c)\nk1k7\nk1k7\nCPW 1\nCPW 2\n0Hext45 m\nFIG. 4: (a) Spin-wave transmission spectrum (imaginary part of S12) recorded on a 40-nm-thick YIG waveguide with an\nexternal magnetic bias \feld \u00160Hext= 15:5 mT along the CPWs. The inset shows a top-view schematic of the measurement\ngeometry. (b) 2D map of spin-wave transmission spectra measured as a function of magnetic bias \feld strength. (c) Angular\ndependence of spin-wave transmission spectra for a constant bias \feld of 15.5 mT. The \feld angle \u0012H= 0\u000ecorresponds to the\nDamon-Eshbach con\fguration.\nB. Propagating spin waves\nWe measured spin-wave transmission spectra on a 40-\nnm-thick YIG \flm. The measurement geometry con-\nsisted of two CPWs on top of YIG waveguides with 45\u000e\nedges (see the inset of Fig. 4(a)). The CPW parame-\nters were identical to those in Fig. 2(b) and their sig-\nnal lines were separated by 45 \u0016m. During broadband\nspin-wave spectroscopy, spin waves with characteristic\nwave vectors ki(i= 1, 2...) were excited by passing\na rf current through one of the CPWs. After propaga-\ntion through the YIG \flm, the other CPW inductively\ndetected the spin waves. Figure 4(a) shows the imagi-\nnary part of the S12scattering parameter for an external\nmagnetic bias \feld \u00160Hext= 15:5 mT parallel to the\nCPWs (Damon-Eshbach con\fguration). The graph con-\ntains seven envelope-type peaks ( k1\u0000k7) with clear pe-\nriodic oscillations. The peak intensities decrease with\nfrequency because of reductions in the excitation e\u000e-\nciency and spin-wave decay length. The oscillations sig-\nnify spin-wave propagation between the CPWs44. Fig-\nure 4(b) shows a 2D representation of spin-wave trans-\nmission spectra recorded at di\u000berent bias \felds. As the\n\feld strengthens, the frequency gaps between spin-wave\nmodes become smaller. Figure 4(c) depicts the angu-\nlar dependence of S12spectra at a constant magnetic\nbias \feld of 15.5 mT. In this measurement, the in-plane\nmagnetic bias \feld was rotated from -72\u000eto 72\u000e, where\n\u0012H= 0\u000ecorresponds to the Damon-Eshbach con\fgura-\ntion. As the magnetization rotates towards the wave vec-\ntor of propagating spin waves, the frequency and inten-\nsity of thek1\u0000k7modes drop. The frequency evolutions\nof the spin-wave modes in Figs. 4(b) and 4(c) are ex-\nplained by a \rattening of the dispersion relation with\nincreasing magnetic bias \feld strength and angle.\n1 . 71 . 81 . 92 . 02 . 12 . 22 . 3-101Im S12 (Normalized)  Fit\n  Exp.\nFrequency (GHz)k1\nk2FIG. 5: A \ft to the spectrum for \u00160Hext= 15:5 mT and\n\u0012H= 0\u000e(blue squares) using Eq. 3 (orange line).\nIV. DISCUSSION\nA. Fitting of spin-wave transmission spectra\nWe used Eq. 2 to \ft spin-wave transmission spectra.\nIn this equation, \u001f(!;k) is described by a Lorentzian\nfunction, while the excitation spectrum j\u001a(k)j2is ap-\nproximated by a Gaussian function (see Fig. 2(e)). For\nDamon-Eshbach spin waves with kd\u001c1, the wave vec-\ntor is given by k=2\nd(2\u0019f)2\u0000(2\u0019fres)2\n(\r\u00160Ms)2, wheredis the \flm\nthickness. Based on these approximations, we rewrite\nEq. 2 as:\nImS 12/\u0001f\n(f\u0000fres)2+ (\u0001f)2\u0002e\u00004ln2(k\u0000k0)2=\u0001k2\n\u0002sin(ks+ \b);(3)5\n02468 1 0 1 201234\n0369 1 2 1 51.52.02.53.040 mT 15.5 mTFrequency (GHz)\nWave vector (rad/ m)1 mT\n9070604836H = 0Frequency (GHz)\nWave vector (rad/ m)18(a) (b)\nFIG. 6: Spin-wave dispersion relations for di\u000berent external\nmagnetic bias \felds (a) and \feld angles (b). In (a) \u0012H= 0\u000e\nand in (b)\u00160Hext= 15.5 mT. The colored lines represent \fts\nto the disperion relations using Eq. 4.\nwhere \u0001fis theS12envelope width, \u0001 kis the width of\nthe spin-wave excitation spectrum, \b is the initial phase,\nandsis the propagation distance. Figure 5 shows a \ft-\nting result for a spin-wave transmission spectrum with\n\u00160Hext= 15.5 mT and \u0012H= 0\u000e. As input parameters,\nwe usedfres= 1.75 GHz, d= 40 nm,s= 45\u0016m, and\nMeff= 184 kA/m, which are either determined by ge-\nometry or extracted from measurements. \u0001 f, \u0001k,k0\nare \ftting parameters. For the k1peak, we obtained\nthe best \ft for \u0001 f= 0.25 GHz, \u0001 k= 0.6 rad/\u0016m, and\nk1= 0:72 rad/\u0016m. Thek2peak was \ftted with k2= 1:87\nrad/\u0016m. The values of \u0001 k,k1, andk2are in good agree-\nment with the simulated excitation spectrum of the CPW\n(Fig. 2(e)) and \u0001 fmatches the width of the envelope\npeak in the experimental S12spectrum.\nB. Spin-wave dispersion relations\nWe extracted spin-wave dispersion relations for di\u000ber-\nent magnetic bias \felds and \feld angles by \ftting the S12\ntransmission spectra shown in Figs. 4(b) and 4(c). The\nsymbols in Fig. 6 summarize the results. We also cal-\nculated the dispersion relations using the Kalinikos and\nSlavin formula45:\nf=\r\u00160\n2\u0019\u0014\nHext\u0000\nHext+Meff\u0002\n1\u0000Fsin2\u0012H\n+Meff\nHextF(1\u0000F) cos2\u0012H\u0003\u0001\u00151=2\n;(4)\nwithF= 1\u00001\u0000exp(\u0000kd)\nkd. The calculated dispersion re-\nlations for\r=2\u0019= 28 GHz/T, Meff= 184 kA/m, and d\n= 40 nm are shown as lines in Fig. 6.\nThe dispersion curves \ratten with increasing magnetic\nbias \feld. For instance, at \u00160Hext= 1 mT, the frequency\nof propagating spin waves changes from 0.5 GHz to 2.4\nGHz for wave vectors ranging from 0 to 10 rad/ \u0016m. At\n\u00160Hext= 40 mT, the frequency evolution with wave vec-\n0 1 02 03 04 05 00.51.01.52.02.53.0\n02 0 4 0 6 00.30.60.91.21.5g (k1)\ng (k2)\ng (k3)Group velocity g (km/s)\next (mT)\nGroup velocity g (km/s)\nH (o)g (k1)\ng (k2)\ng (k3)(a) (b)FIG. 7: Spin-wave group velocity \u001dgofk1\u0000k3modes as a\nfunction of external magnetic bias \feld (a) and \feld angle (b).\nIn (a)\u0012H= 0\u000eand in (b)\u00160Hext= 15.5 mT.\ntor is reduced to 3 \u00003:7 GHz. This magnetic-\feld de-\npendence of the dispersion relation narrows the spin-wave\ntransmission bands in Fig. 4(b) at large \u00160Hext.\nThe angular dependence of the spin-wave dispersion\ncurves in Fig. 6(b) is explained by in-plane magneti-\nzation rotation from M?k(\u0012H= 0\u000e) towardsMkk\n(\u0012H= 90\u000e). At\u0012H= 0\u000e, dispersive Damon-Eshbach spin\nwaves with positive group velocity propagate between the\nCPWs. The character of excited spin waves changes\ngradually with increasing \u0012Huntil it has fully trans-\nformed into a backward-volume magnetostatic mode at\n\u0012H= 90\u000e. This mode is only weakly dispersive and ex-\nhibits a negative group velocity.\nC. Group velocity\nThe phase relation between signals from the two CPWs\nis given by \b = k\u0002s31,44. Since the phase shift between\ntwo neighboring maxima ( \u000ef) in broadband spin-wave\ntransmission spectra corresponds to 2 \u0019, the group veloc-\nity can be written as:\n\u001dg=@!\n@k=2\u0019\u000ef\n2\u0019=s=\u000ef\u0002s; (5)\nwheres= 45\u0016m in our experiments. Using this equa-\ntion, we extracted the spin-wave group velocity for wave\nvectorsk1\u0000k3from the transmission spectra shown in\nFigs. 4(b) and 4(c). Figure 7 summarizes the variation\nof\u001dgwith external magnetic bias \feld and \feld angle.\nFor weak bias \felds ( \u00160Hext<30 mT), the group ve-\nlocity decreases swiftly, especially if kis small. For in-\nstance,\u001dg(k1) reduces from 3.0 km/s to 1.0 km/s in the\n0\u000030 mT \feld range, while \u001dg(k3) only changes from\n1.2 km/s to 0.8 km/s. At larger external magnetic bias\n\felds,\u001dgdecreases more slowly for all wave vectors. Fig-\nure 7(b) shows how \u001dgvaries as a function of \feld angle\nat\u00160Hext= 15.5 mT. For all wave vectors, the group\nvelocity is largest in the Damon-Eshbach con\fguration\n(\u0012H= 0\u000e). At larger \feld angles, \u001dgdecreases and its6\n0 1 02 03 04 05 0200400600\n02 0 4 0 6 0200250300Relaxation time  (ns)\next (mT) (k1)\n (k2)\nk3)\nRelaxation time (ns)\nH (o)  (k1)\n  (k2)\n  (k3)(a) (b)\nFIG. 8: Spin-wave relaxation time \u001cofk1\u0000k3modes as a\nfunction of external magnetic bias \feld (a) and \feld angle (b).\nIn (a)\u0012H= 0\u000eand in (b)\u00160Hext= 15.5 mT.\n0 1 02 03 04 05 0030060090012001500\n0 1 02 03 04 05 06 00100200300400Decay length ld (m)\n0Hext (mT) ld (k1)\n ld (k2)\n ld (k3)\nDecay length ld (m)\no ld (k1)\n ld (k2)\n ld (k3)(a) (b)\nFIG. 9: Spin-wave decay length ldofk1\u0000k3modes as a\nfunction of external magnetic bias \feld (a) and \feld angle\n(b). In (a) \u0012H= 0\u000eand in (b)\u00160Hext= 15.5 mT.\ndependence on wave vector diminishes. Variations of the\nspin-wave group velocity with wave vector and magnetic-\n\feld strength or angle are explained by a \rattening of the\ndispersion relations, as illustrated by the data in Fig. 6.\nD. Spin-wave relaxation time and decay length\nWe now discuss the relaxation time ( \u001c) and decay\nlength (ld) of spin waves in our YIG \flms. Following\nRef. 46, the relaxation time is estimated by \u001c= 1=2\u0019\u000bf.\nUsing\u000b= 3:5\u000210\u00004and spin-wave transmission data\nfrom Fig. 4, we determined \u001cfor wave vectors k1\u0000k3.\nThe dependence of \u001con external magnetic bias \feld and\n\feld angle is shown in Fig. 8. The maximum spin-wave\nrelaxation time in our 40-nm-thick YIG \flms is approx-\nimately 500 ns. Resembling the spin-wave group veloc-\nity,\u001cis largest for small wave vectors and it decreases\nwith increasing bias \feld (Fig. 8(a)). In contrast to \u001dg,\nthe spin-wave relaxation time is smallest in the Damon-\nEshbach con\fguration ( \u0012H= 0\u000e) and it evolves more\nstrongly with increasing \u0012Hifkis large (Fig. 8(b)). This\nresult is explained by \u001c/1=fand a lowering of thespin-wave frequency if the in-plane bias \feld rotates the\nmagnetization towards k(see Fig. 4(c)).\nThe spin-wave decay length is calculated using ld=\n\u001dg\u0002\u001cand data from Figs. 7 and 8. Figure 9(a) shows\nthe dependence of ldon\u00160Hextfor wave vectors k1\u0000k3.\nThe largest spin-wave decay length in our 40-nm-thick\nYIG \flms is 1.2 mm, which we measured for k1= 0:72\nrad/\u0016m and\u00160Hext= 2 mT. The decay length decreases\nwith magnetic bias \feld to about 100 \u0016m at\u00160Hext=\n50 mT. Figure 9(b) depicts the dependence of ldon the\ndirection of a 15.5 mT bias \feld. The spin-wave decay\nlength decreases substantially with \u0012Hfor smallk, but\nits angular dependence weakens for larger wave vectors.\nThe decay of propagating spin waves between the ex-\nciting and detecting CPW in the broadband spectroscopy\nmeasurement geometry is given by exp( \u0000s=ld)46. Based\non the results of Fig. 9, one would thus expect the in-\ntensity of spin waves to drop with increasing wave vector\nand in-plane bias \feld strength or angle. The spin-wave\ntransmission spectra of Fig. 4 con\frm this behavior.\nE. CPWs versus microstrip antennas\nFinally, we compare broadband spin-wave spec-\ntroscopy measurements on YIG thin \flms using CPWs\nand microstrip antennas. In these experiments, the\nCPWs and antenna structures have 4- \u0016m-wide signal\nlines and they were patterned onto the same 35-nm-thick\nYIG \flm. For comparison, we also recorded transmission\nspectra on 50- \u0016m wide YIG waveguides. The separation\ndistance (s) between the CPWs or microstrip antennas\nwas set to 110 \u0016m or 220\u0016m. Schematics of the di\u000berent\nmeasurement geometries are depicted on the sides of Fig.\n10. Transmission spectra that were obtained for Damon-\nEshbach spin waves in each con\fguration are also shown.\nIn all measurements, we used an in-plane external mag-\nnetic bias \feld of 10 mT. The plots focus on phase os-\ncillations in the \frst-order excitation at k1(higher-order\nexcitations were measured also, but are not shown). The\ndi\u000berently shaped outline of the S12peak for two CPWs\n(left) or two microstrip antennas (right) mimics the pro-\n\fle of their excitation spectra (Fig. 2). As expected from\n\u000ef=\u001dg=s, the period of frequency oscillations ( \u000ef) be-\ncomes smaller if the separation between antennas ( s) is\nenhanced (Figs. 10(c) and 10(f)).\nWe \ftted the spin-wave transmission spectra obtained\nwith CPWs (Figs. 10(a)-(c)) using the same procedure as\ndescribed earlier. Good agreements between experimen-\ntal data (blue squares) and calculations (orange lines)\nwere obtained by inserting Meff= 190\u00064 kA/m, \u0001f=\n0.18 GHz,k= 0.34 rad/ \u0016m, and \u0001k= 0.33 rad/ \u0016m into\nEq. 3. To \ft S12spectra measured by microstrip anten-\nnas, we approximated the wave vector of the excitation\nask=2\nd(2\u0019f)2\u0000(2\u0019fres)2\n(\r\u00160Meff)2H(f\u0000fres), whereHis a Heav-\niside step function47. The best results were achieved for\nMeff= 178\u00062 kA/m, \u0001 f= 0.25 GHz, k= 0 rad/\u0016m\nand \u0001k= 0.65 rad/ \u0016m. From the data comparison in7\n-101\n-101\n1.3 1.4 1.5-101-101\n-101\n1.3 1.4 1.5-101\nFrequency (GHz) Frequency (GHz)110 m\n220 mCPW1\nCPW2\n0Hext\n110 m110 mantenna2\n0Hextantenna1\n220 m(a) (d)\n(b) (e)\n(c) (f)Im S12 (Normalized)\nIm S12(Normalized)110 m\n220 m\nFIG. 10: (a)-(c) Spin-wave transmission spectra measured using CPWs on a continuous YIG \flm (a) and 50- \u0016m-wide YIG\nwaveguides ((b) and (c)). The YIG \flm and waveguides are 35 nm thick and the CPWs are separated by 110 \u0016m ((a) and\n(b)) and 220 \u0016m (c). (d)-(f) Spin-wave transmission spectra measured using microstrip antennas on the same YIG \flm and\nwaveguides. The signal lines of the CPWs and microstrip antennas are 4 \u0016m wide. The orange lines represent \fts to the\nexperimental data using Eq. 3. The measurement geometry for each spectrum is illustrated next to the graphs. In the\nschematics, the green areas depict a continuous YIG \flm or waveguide.\nFig. 10, we conclude that broadband spin-wave spec-\ntroscopy measurements with CPWs and microstrip an-\ntennas yield similar results for Meff. We also note that\ntheS12peak width (\u0001 f) obtained from measurements\non continuous YIG \flms and YIG waveguides are nearly\nidentical (\u0001 f= 0:18 GHz for CPWs, \u0001 f= 0:22 GHz for\nantennas). Thus, patterning of the YIG \flm into waveg-\nuides does not deteriorate the Gilbert damping constant.\nFrom the oscillation periods ( \u000ef) in the transmission\nspectra of Fig. 10, we extracted the properties of prop-\nagating spin waves. By averaging \u000efover the same fre-\nquency range in spectra measured by CPWs and mi-\ncrostrip antennas, we obtained \u001dg= 1:67 km/s and\n\u001dg= 1:53 km/s, respectively. The spin-wave relax-\nation time was determined as \u001c= 225 ns (CPW) and\n\u001c= 237 ns (antenna) and the decay length was extracted\nasld= 375\u0016m (CPW) and ld= 363\u0016m (antenna).\nThese results clearly demonstrate that broadband spin-\nwave spectroscopy measurements on YIG thin \flms us-\ning CPWs or microstrip antennas provide comparable\nresults.\nV. SUMMARY\nIn conclusion, we prepared 35 \u000040 nm thick epitaxial\nYIG \flms with a Gilbert damping constant \u000b= 3:5\u000210\u00004on GGG(111) substrates using PLD. The dependence of\nspin-wave transmission on the strength and angle of an\nin-plane magnetic bias \feld was systematically gauged.\nWe used the measurements to demonstrate strong tun-\ning of the spin-wave group velocity ( \u001dg), relaxation time\n(\u001c), and decay length ( ld) up to a \feld strength of about\n30 mT and above a \feld angle of 20\u000e. Maximum val-\nues of\u001dg= 3:0 km/s,\u001c= 500 ns, and ld= 1:2 mm\nwere extracted for Damon-Eshbach spin waves with k1\n= 0.72 rad/ \u0016m. Moreover, we demonstrated that broad-\nband spin-wave spectroscopy performed with integrated\nCPWs and microstrip antennas yield similar results.\nVI. ACKNOWLEDGEMENTS\nThis work was supported by the European Re-\nsearch Council (Grant Nos. ERC-2012-StG 307502-E-\nCONTROL and ERC-PoC-2018 812841-POWERSPIN).\nS.J.H. acknowledges \fnancial support from the V ais al a\nFoundation. Lithography was performed at the Mi-\ncronova Nanofabrication Centre, supported by Aalto\nUniversity. We also acknowledge the computational re-\nsources provided by the Aalto Science-IT project.\n\u0003Electronic address: huajun.qin@aalto.\f\nyElectronic address: sebastiaan.van.dijken@aalto.\f1A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys.\nD: Appl. Phys. 43, 264002 (2010).8\n2V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. 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Yuan,1R. Lavrijsen,2and R. A. Duine1, 2\n1Institute for Theoretical Physics, Utrecht University,\nPrincetonplein 5, 3584 CC Utrecht, The Netherlands\n2Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: September 29, 2022)\nWe show that interlayer Dzyaloshinskii-Moriya interaction in combination with non-local Gilbert\ndamping gives rise to unidirectional magnetic coupling. That is, the coupling between two magnetic\nlayers | say the left and right layer | is such that dynamics of the left layer leads to dynamics of\nthe right layer, but not vice versa. We discuss the implications of this result for the magnetic sus-\nceptibility of a magnetic bilayer, electrically-actuated spin-current transmission, and unidirectional\nspin-wave packet generation and propagation. Our results may enable a route towards spin-current\nand spin-wave diodes and further pave the way to design spintronic devices via reservoir engineering.\nIntroduction. | Non-reciprocal transmission of elec-\ntrical signals lies at the heart of modern communication\ntechnologies. While semi-conductor diodes, as an exam-\nple of an electronic component that underpins such non-\nreciprocity, have been a mature technology for several\ndecades, new solutions are being actively pursued [1, 2].\nSuch research is spurred on by the emergence of quan-\ntum technologies that need to be read out electrically\nbut should not receive unwanted back-action from their\nelectronic environment.\nComplementary to these developments, spintronics has\nsought to control electronic spin currents and, more\nrecently, spin currents carried by spin waves | i.e.,\nmagnons | in magnetic insulators [3]. Devices that im-\nplement non-reciprocal spin-wave spin currents have been\nproposed [4{7]. Most of these proposals rely on dipolar\ninteractions [8{11] or Dzyaloshinskii-Moriya interactions\n(DMI) [12{16]. Other proposals involve the coupling of\nthe spin waves to additional excitations such that the spin\nwaves are endowed with non-reciprocity. Examples are\nthe coupling of the spin waves to magnetoelastic, optical,\nand microwave excitations [17{22].\nMost of these proposals have in common that they con-\nsider spin-wave dispersions that are asymmetric in wave\nvector. For example, due to the DMI spin waves at one\nparticular frequency have di\u000berent wave numbers and ve-\nlocities for the two di\u000berent directions. There are there-\nfore spin waves travelling in both directions. This may\nbe detrimental for some applications. For example, one\nwould like to shield quantum-magnonic technologies from\nspin-current noise [23], and completely quench the spin-\ncurrent transmission in one of the two directions along a\nwire.\nHere, we propose a set-up that realizes unidirectional\nmagnetic coupling between two magnetic layers or be-\ntween two magnetic moments. The ingredients are DMI\nand dissipative coupling between the two layers or mo-\nments. The dissipative coupling takes the form of a non-\nlocal Gilbert damping and may arise, for example, from\nthe combined action of spin pumping and spin transfer.Then, one magnet emits spin current when it precesses,\nwhich is absorbed by the other. The resulting dissipa-\ntive coupling turns out to, for certain parameters, pre-\ncisely cancel the DMI in one direction. As a result, an\nexcitation of one of the magnets leads to magnetization\ndynamics of the other, but not vice versa. This yields\nspin-wave propagation that is truly uni-directional: for\nspeci\fc direction and magnitude of the external \feld, all\nspin waves travel in one direction only.\nMinimal model. | Let us start with the minimal set-\nup that demonstrates the unidirectional coupling. We\n\frst consider two identical homogeneous magnetic lay-\ners that are coupled only by an interlayer DMI with\nDzyaloshinskii vector Dand by interlayer spin pumping\n(see Fig. 1). The magnetization direction in the layers\nis denoted by mi, wherei2f1;2glabels the two lay-\ners. We also include an external \feld H. The magnetic\nenergy is given by\nE[m1;m2] =D\u0001(m1\u0002m2)\u0000\u00160MsH\u0001(m1+m2);(1)\nwhereMsis the saturation magnetization of both layers\nand\u00160is the vacuum susceptibility. The magnetization\ndynamics of layer 1 is determined by the Landau-Lifshitz-\nGilbert (LLG) equation\n@m1\n@t=\r\nMsm1\u0002\u000eE\n\u000em1+\u000bnlm1\u0002@m2\n@t; (2)\nwhere\ris the gyromagnetic ratio and \u000bnlcharacterizes\nthe strength of the non-local damping that in this set-up\nresults from the combination of spin pumping and spin\ntransfer torques, as described in the introduction. The\nequation of motion for the magnetization dynamics of the\nsecond layer is found by interchanging the labels 1 and\n2 in the above equation. Working out the e\u000bective \feldsarXiv:2209.14179v1  [cond-mat.mes-hall]  28 Sep 20222\nFM\nFM\nm2\nm1xy\nz, H, D\nFIG. 1. Schematic of two magnetic moments coupled by an\ninterlayer DMI and by interlayer spin pumping. The dynam-\nics of m1induces the motion of m2, but not vice versa for\nappropriate parameters.\n\u000eE=\u000emiyields\n@m1\n@t=\r\nMsm1\u0002(m2\u0002D\u0000\u00160MsH) +\u000bnlm1\u0002@m2\n@t;\n(3a)\n@m2\n@t=\r\nMsm2\u0002(D\u0002m1\u0000\u00160MsH) +\u000bnlm2\u0002@m1\n@t;\n(3b)\nwhere the sign di\u000berence in e\u000bective-\feld contribution\nfrom the DMI stems from the asymmetric nature of the\nDMI. We show now that depending on the magnitude and\ndirection of the e\u000bective \feld, this sign di\u000berence leads\nfor one of the layers to cancellation of the torques due\nto interlayer DMI and non-local damping. As the can-\ncellation does not occur for the other layer, and because\nthe DMI and non-local damping are the mechanisms that\ncouple the layers in the model under consideration, this\nleads to uni-directional magnetic coupling.\nTaking the external \feld to be much larger than the\ninterlayer DMI, i.e., \u00160jHj\u001djDj=Ms, and taking \u000bnl\u001c\n1, we may replace @mi=@tby\u0000\r\u00160mi\u0002Hon the right-\nhand side of Eqs. (3) because the external \feld then is\nthe dominant contribution to the precession frequency.\nFor the \feld H=D=\u000bnl\u00160Ms, one then \fnds that\n@m1\n@t=\u0000\r\n\u000bnlMsm1\u0002D; (4a)\n@m2\n@t=2\r\nMsm2\u0002(D\u0002m1)\u0000\r\n\u000bnlMsm2\u0002D:(4b)\nHence, the coupling between the two magnetic layers is\nunidirectional at the \feld H=D=\u000bnl\u00160Ms: the magne-\ntization dynamics of layer 1 leads to dynamics of layer\n2 as evidenced by Eq. (4b), but not vice versa as im-\nplied by Eq. (4a). This one-way coupling is reversed by\nchanging the direction of the \feld to \u0000Hor the sign of\nthe non-local coupling \u000bnl.\nMagnetic susceptibility. | Let us now take into ac-\ncount the Gilbert damping within the layers, exchange,\nand anisotropies and discuss the in\ruence of the unidi-\nrectional coupling on the magnetic susceptibility. Theenergy now reads\nE[m1;m2] =\u0000Jm1\u0001m2+D\u0001(m1\u0002m2)\n\u0000\u00160MsH\u0001(m1+m2)\u0000K\n2\u0000\nm2\n1;z+m2\n2;z\u0001\n;(5)\nwith the constant Kcharacterizing the strength of the\nanisotropy and Jthe exchange. We shall focus on the\nferromagnetic coupling ( J >0) without loss of generality.\nThe LLG equation now becomes\n@m1\n@t=\r\nMsm1\u0002@E\n@m1+\u000bm1\u0002@m1\n@t\n+\u000bnlm1\u0002@m2\n@t; (6)\nwith\u000bthe Gilbert damping constant of each layer, and\nwhere the equation for the second layer is obtained from\nthe above by interchanging the labels 1 and 2. We\ntake the external \feld in the same direction as the\nDzyaloshinskii vector and D=D^z,H=H^z, while\n\u00160MsH;K\u001dD, so that the magnetic layers are aligned\nin the ^z-direction. Linearizing the LLG equation around\nthis direction we write mi= (mi;x;mi;y;1)Tand keep\nterms linear in mi;xandmi;y. Writing\u001ei=mi;x\u0000imi;y,\nwe \fnd, after Fourier transforming to frequency space,\nthat\n\u001f\u00001(!)\u0012\u001e1(!)\n\u001e2(!)\u0013\n= 0: (7)\nTo avoid lengthy formulas, we give explicit results below\nfor the case that J= 0, while plotting the results for\nJ6= 0 in Fig. 2. The susceptibility tensor \u001fij, or magnon\nGreen's function, is given by\n\u001f(!) =1\n((1 +i\u000b)!\u0000!0)2\u0000(\rD=Ms)2\u0000\u000b2\nnl!2)\n\u0002\u0012(1 +i\u000b)!\u0000!0i(\rD=Ms\u0000\u000bnl!)\n\u0000i(\rD=Ms+\u000bnl!) (1 +i\u000b)!\u0000!0\u0013\n;(8)\nwith!0=\r(\u00160H+K=Ms) the ferromagnetic-resonance\n(FMR) frequency of an individual layer. The poles of\nthe susceptibility determine the FMR frequencies of the\ncoupled layers and are, for the typical case that \u000b;\u000b nl\u001c\n1, given by\n!\u0006=!r;\u0006\u0000i\u000b!r;\u0006; (9)\nwith resonance frequency\n!r;\u0006=\r(\u00160H+K=Ms\u0006D=Ms): (10)\nWhen\r\u00160H= (1\u0007\u000bnl)D=(\u000bnlMs)\u0000K=Ms\u0019\nD=(\u000bnlMs)\u0000K=Mswe have for J= 0 that\u001f12(!r;\u0006) = 0\nwhile\u001f21(!r;\u0006)6= 0, signalling the non-reciprocal cou-\npling. That is, the excitation of layer 1 by FMR leads\nto response of magnetic layer 2, while layer 1 does not\nrespond to the excitation of layer 2. For opposite direc-\ntion of \feld the coupling reverses: the excitation of layer3\n|χ21(J=0)|\n|χ21(J=0.5D)|\n|χ21(J=15D)|\n|χ12(J=0)|\n|χ12(J=0.5D)|\n|χ12(J=15D)|\n0.96 0.98 1.00 1.02 1.040100200300400\nω/ωH\nFIG. 2. Magnetic susceptibilities of two magnetic layers as a\nfunction of frequency at di\u000berent exchange couplings. !H\u0011\n\r(\u00160H+K=M s). The resonance frequencies are located at\nthe peak positions. The parameters are D=! H= 0:001;\u000bnl=\n0:001;\u000b= 0:002.\n2 by FMR leads in that case to response of magnetic\nlayer 1, while layer 2 does not respond to the excitation\nof layer 1. As is observed from Fig. 2, for \fnite but\nsmallJ\u001cD, the coupling is not purely unidirectional\nanymore but there is still a large non-reciprocity. For\nJ\u001dD, this non-reciprocity is washed out.\nElectrically-actuated spin-current transmission. | In\npractice, it may be challenging to excite the individual\nlayers independently with magnetic \felds, which would\nbe required to probe the susceptibility that is determined\nabove. The two layers may be more easily probed inde-\npendently by spin-current injection/extraction from ad-\njacent contacts. Therefore, we consider the situation that\nthe two coupled magnetic layers are sandwiched between\nheavy-metal contacts (see Fig. 3(a)). In this set-up, spin\ncurrent may be transmitted between the two contacts\nthrough the magnetic layers.\nFollowing the Green's function formalism developed by\nZheng et al. [24], the spin-current from the left (right)\nlead to its adjacent magnetic layer is determined by the\ntransmission function of the hybrid system T12(T21)\ngiven by\nTij(!) = Trh\n\u0000i(!)G(+)(!)\u0000j(!)G(\u0000)(!)i\n: (11)\nHere,G(+)(!) is the retarded Green's function for\nmagnons in contact with the metallic leads that is de-\ntermined by Dyson's equation\u0002\nG(+)\u0003\u00001(!) =\u001f\u00001(!)\u0000\n\u0006(+)\n1(!)\u0000\u0006(+)\n2(!), where the retarded self energy\n~\u0006(+)\ni(!) accounts for the contact with the metallic lead\ni. These self energies are given by\n~\u0006(+)\n1(!) =\u0000i~\u000b0\n1\u0012\n!0\n0 0\u0013\n; (12)and\n~\u0006(+)\n2(!) =\u0000i~\u000b0\n2\u00120 0\n0!\u0013\n: (13)\nThe rates for spin-current transmission from the heavy\nmetal adjacent to the magnet iinto it, are given by\n\u0000i(!) =\u00002Imh\n\u0006(+)\ni(!)i\n=~. The couplings \u000b0\ni=\n\rRe[g\"#\ni]=4\u0019Msdiare proportional to the real part of the\nspin-mixing conductance per area g\"#\nibetween the heavy\nmetal and the magnetic layer i, and further depend on\nthe thickness diof the magnetic layers. Finally, the ad-\nvanced Green's function is G(\u0000)(!) =\u0002\nG(+)\u0003y.\nIn the analytical results below, we again restrict our-\nselves to the case that J= 0 for brevity, leaving the\ncaseJ6= 0 to plots. Using the above ingredients,\nEq. (11) is evaluated. Taking identical contacts so that\n\u000b0\n1=\u000b0\n2\u0011\u000b0, we \fnd that\nT12=4(\u000b0)2!2(\rD=Ms+\u000bnl!)2\njC(!)j2; (14)\nwhile\nT21=4(\u000b0)2!2(\rD=Ms\u0000\u000bnl!)2\njC(!)j2; (15)\nwith\nC(!) = [!H\u0000(1 +i(\u000b\u0000\u000bnl+\u000b0))!]\u0001\n[!H\u0000(1 +i(\u000b+\u000bnl+\u000b0))!]\u0000(\rD=Ms)2:(16)\nFrom the expression for C(!) it is clear that, since\n\u000b;\u000b nl;\u000b0\u001c1, the transmission predominantly occurs\nfor frequencies equal to the resonance frequencies !r;\u0006\nfrom Eq. (9). Similar to the discussion of the suscepti-\nbilities, we have for \felds \r\u00160H=D=\u000b nl\u0000K=Msthat\nthe transmission T12(!=D=\u000b nl)6= 0, while T21(!=\nD=\u000b nl) = 0. As a result, the spin-current transmis-\nsion is unidirectional at these \felds. For the linear spin-\nconductances Gij, given byGij=R\n~!(\u0000N0(~!))Tij(!),\nwe also have that G126= 0, while G21= 0. Here,\nN(~!) = [e~!=kBT\u00001]\u00001is the Bose-Einstein distri-\nbution function at thermal energy kBT. For the oppo-\nsite direction of external \feld we have G12= 0, while\nG216= 0. Like in the case of the susceptibility discussed\nin the previous section, a \fnite but small exchange cou-\npling makes the spin current transport no longer purely\nunidirectional, while maintaining a large non-reciprocity\n(see Fig. 3(b)).\nSpin-wave propagation. | Besides the unidirectional\ncoupling of two magnetic layers, the above results may\nbe generalized to a magnetic multilayer, or, equivalently,\nan array of coupled magnetic moments that are labeled\nby the index isuch that the magnetization direction of\nthei-th layer is mi. This extension allows us to engi-\nneer unidirectional spin-wave propagation as we shall see4\nm1\nm2\nLead Lead FM FM(a)\n(b)\nT21(J=0)\nT21(J=0.5D)\nT21(J=15D)\nT12(J=0)\nT12(J=0.5D)\nT12(J=15D)\n0.96 0.98 1.00 1.02 1.040.0000.0020.0040.0060.008\nω/ωH\nFIG. 3. (a) Schematic of the system that the two coupled\nmagnetic layers are sandwiched between two heavy-metal con-\ntacts. (b) Transmission of the hybrid system as a function of\nfrequency.\nbelow. We consider the magnetic energy\nE[m] =X\nk[D\u0001(mk\u0002mk+1)\u0000\u00160MsH\u0001mk];(17)\nand \fnd | within the same approximations as for our\ntoy model above | for the magnetization dynamics that\n@mk\n@t=2\r\nMsmk\u0002(D\u0002mk\u00001)\u0000\r\n\u000bnlMsmk\u0002D;(18)\nfor the \feld H=D=\u000bnl\u00160Ms. This shows that for these\n\felds the magnetic excitations travel to the right | cor-\nresponding to increasing index k| only. The direction\nof this one-way propagation is reversed by changing the\nmagnetic \feld to \u0000Hor by changing the sign of the non-\nlocal damping.\nTo study how spin waves propagate in an array of cou-\npled magnetic moments described by the Hamiltonian in\nEq. (17). We start from the ground state mk= (0;0;1)T\nand perturb the left-most spin ( k= 0) to excite spin\nwaves. Since the dynamics of this spin is not in\ruenced\nby the other spins for the \feld H=D=\u000b nl\u00160Ms, its\nsmall-amplitude oscillation can be immediately solved\nas\u001e0(t) =\u001e0(t= 0) exp(\u0000i!0t\u0000\u000b!0t) with\u001ek=\nmk;x\u0000imk;yas used previously. The dynamics of the\nspins to the right of this left-most spin is derived by solv-\ning the LLG equation (18) iteratively, which yields\n\u001ek(t) =\u001e0(t= 0)e\u0000i!0te\u0000\u000b!0t\nk!(\u00002\u000bnl!0t)k;(19)wherek= 0;1;2;:::N\u00001.\nTo guarantee the stability of the magnetization dynam-\nics, the dissipation matrix of the N-spin system should\nbe negative-de\fnite, which imposes a constraint on the\nrelative strength of Gilbert damping and non-local damp-\ning, i.e.,\u000b > 2\u000bnlcos\u0019\nN+1. For an in\fnitely-long chain\nN!1 , we have\u000b>2\u000bnl. Physically, this means that\nthe local dissipation of a spin has to be strong enough to\ndissipate the spin current pumped by its two neighbors.\nFor a spin chain with \fnite number of spins, \u000b= 2j\u000bnljis\nalways su\u000ecient to guarantee the stability of the system.\nTaking this strength of dissipation simpli\fes Eq. (19) to\n\u001ek(t) =\u001e0(t= 0)e\u0000it=(\u000b\u001c)e\u0000t=\u001c\nk!(\u0000t=\u001c)k; (20)\nwhere\u001c\u00001=\u000b!0is the inverse lifetime of the FMR\nmode. This spatial-temporal pro\fle of spins is the same\nas a Poisson distribution with both mean and variance\nequal to\u001b=t=\u001cexcept for a phase modulation, and it\ncan be further approximated as a Gaussian wavepacket\non the time scale t\u001d\u001c, i.e.\n\u001e(x) =\u001e0(t= 0)e\u0000it=(\u000b\u001c)\np\n2\u0019\u001be\u0000(x\u0000\u001b)2\n2\u001b: (21)\nSuch similarity suggests that any local excitation of the\nleft-most spin will generate a Gaussian wavepacket prop-\nagating along the spin chain. The group velocity of the\nmoving wavepacket is v=a=\u001c, whereais the distance\nbetween the two neighboring magnetic moments. The\nwidth of the wavepacket spreads with time as ap\nt=\u001c,\nwhich resembles the behavior of a di\u000busive particle. Af-\nter su\u000eciently long time, the wavepacket will collapse.\nOn the other hand, the excitation is localized and can-\nnot propagate when the right-most spin ( k=N\u00001) is\nexcited, because its left neighbor, being in the ground\nstate, has zero in\ruence on its evolution. These results\ndemonstrate the unidirectional properties of spin-wave\ntransport in our magnetic array.\nDiscussion, conclusion, and outlook. | We have\nshown that the ingredients for unidirectional coupling be-\ntween magnetic layers or moments are that they are cou-\npled only by DMI and non-local Gilbert damping. While\nin practice it may be hard to eliminate other couplings,\nthe DMI and non-local coupling need to be su\u000eciently\nlarger than the other couplings to observe unidirectional\ncoupling.\nThere are several systems that may realize the unidi-\nrectional coupling we propose. A \frst example is that of\ntwo magnetic layers that are coupled by a metallic spacer.\nSuch a spacer would accommodate non-local coupling via\nspin pumping and spin transfer. For a spacer that is\nmuch thinner than the spin relaxation length, we \fnd,\nfollowing Refs. [25{27], that \u000bnl=\r~Re[~g\"#]=4\u0019dMs,\nwith ~g\"#the spin-mixing conductance of the interface\nbetween the magnetic layers and the spacer, dthe thick-\nness of the magnetic layers. For simplicity, we took the5\nmagnetic layers to have equal properties. The two mag-\nnetic layers may be coupled by the recently-discovered\ninterlayer DMI [28, 29], tuning to a point (as a function\nof thickness of the spacer) where the ordinary RKKY ex-\nchange coupling is small. We estimate \u000bnl= 4:5\u000210\u00003\nford= 20 nm, Re[~g\"#] = 4:56\u00021014\n\u00001m\u00002and\nMs= 1:92\u0002105A=m (YIGjPt). The required mag-\nnetic \feld for unidirectional magnetic coupling is then\naround 4.5 T for D= 1 mT. Another possible platform\nfor realizing the unidirectional coupling is the system of\nFe atoms on top of a Pt substrate that was demonstrated\nrecently [30]. Here, the relative strength of the DMI and\nexchange is tuned by the interatomic distance between\nthe Fe atoms. Though not demonstrated in this experi-\nment, the Pt will mediate non-local coupling between the\natoms as well. Hence, this system may demonstrate the\nunidirectional coupling that we proposed.\nThe non-local damping is expected to be generically\npresent in any magnetic material and does not require\nspecial tuning, though it may be hard to determine its\nstrength experimentally. Hence, an attractive implemen-\ntation of the unidirectional coupling would be a magnetic\nmaterial with spins that are coupled only via DMI, with-\nout exchange interactions. While such a material has\nto the best of our knowledge not been discovered yet,\nit is realized transiently in experiments with ultrafast\nlaser pulses [31]. Moreover, it has been predicted that\nhigh-frequency laser \felds may be used to manipulate\nDMI and exchange, even to the point that the former is\nnonzero while the latter is zero [32, 33].\nPossible applications of our results are spin-wave and\nspin-current diodes and magnetic sensors, where a weak\n\feld signal can be ampli\fed and transported through\nthe unidirectional coupling to the remote site to be read\nout without unwanted back-action. Finally, we remark\nthat the unidirectional magnetic coupling that we pro-\npose here may be thought of as reservoir engineering, cf.\nRef. [34]. In our proposal, the reservoir is made up by the\ndegrees of freedom that give rise to the non-local damp-\ning, usually the electrons. We hope that this perspective\nmay pave the way for further reservoir-engineered mag-\nnetic systems\nAcknowledgements. | It is a great pleasure to\nthank Mathias Kl aui and Thomas Kools for discus-\nsions. 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Bauer1, 5, 6\n1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n2Departamento de Física, Universidad de Santiago de Chile, Avda. Ecuador 3493, Santiago, Chile\n3Center for the Development of Nanoscience and Nanotechnology (CEDENNA), 917-0124 Santiago, Chile\n4Departamento de Física, Facultad de Ciencias Físicas y Matemáticas,\nUniversidad de Chile, Casilla 487-3, Santiago, Chile\n5Department of Physics, Beijing Normal University, Beijing 100875, China\n6Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan\n(Dated: June 4, 2019)\nWe address the theory of magnon-phonon interactions and compute the corresponding quasi-\nparticle and transport lifetimes in magnetic insulators with focus on yttrium iron garnet at inter-\nmediate temperatures from anisotropy- and exchange-mediated magnon-phonon interactions, the\nlatter being derived from the volume dependence of the Curie temperature. We find in general weak\neffects of phonon scattering on magnon transport and the Gilbert damping of the macrospin Kittel\nmode. The magnon transport lifetime differs from the quasi-particle lifetime at shorter wavelengths.\nI. INTRODUCTION\nMagnons are the elementary excitations of magnetic\norder, i.e. the quanta of spin waves. They are bosonic\nandcarryspinangularmomentum. Ofparticularinterest\nare the magnon transport properties in yttrium iron gar-\nnet (YIG) due to its very low damping ( \u000b<10\u00004), which\nmakes it one of the best materials to study spin-wave or\nspin caloritronic phenomena [1–6]. For instance, the spin\nSeebeck effect (SSE) in YIG has been intensely studied\nin the past decade [7–13]. Here, a temperature gradi-\nent in the magnetic insulator injects a spin current into\nattached Pt contacts that is converted into a transverse\nvoltage by the inverse spin Hall effect. Most theories ex-\nplain the effect by thermally induced magnons and their\ntransport to and through the interface to Pt [7, 14–19].\nHowever, phonons also play an important role in the SSE\nthrough their interactions with magnons [20–22].\nMagnetoelastic effects in magnetic insulators were ad-\ndressed first by Abrahams and Kittel [23–25], and by\nKaganov and Tsukernik [26]. In the long-wavelength\nregime, the strain-induced magnetic anisotropy is the\nmost important contribution to the magnetoelastic en-\nergy, whereas for shorter wavelengths, the contribution\nfrom the strain-dependence of the exchange interaction\nbecomes significant [27–29]. Rückriegel et al.[28] com-\nputed very small magnon decay rates in thin YIG films\ndue to magnon-phonon interactions with quasi-particle\nlifetimes\u001cqp?480 ns;even at room temperature. How-\never, these authors do not consider the exchange interac-\ntion and the difference between quasi-particle and trans-\nport lifetimes.\nRecently, it has been suggested that magnon spin\ntransport in YIG at room temperature is driven by\nthe magnon chemical potential [3, 30]. Cornelissen et\nal. [3] assume that at room temperature magnon-\nphonon scattering of short-wavelength thermal magnons\nis dominated by the exchange interaction with a scat-\ntering time of \u001cqp\u00181 ps, which is much faster than\nthe anisotropy-mediated magnon-phonon coupling con-sidered in Ref. [28] and efficiently thermalizes magnons\nand phonons to equal temperatures without magnon de-\ncay. Recently, the exchange-mediated magnon-phonon\ninteraction [31] has been taken into account in a Boltz-\nmann approach to the SSE, but this work underestimates\nthe coupling strength by an order of magnitude, as we\nwill argue below.\nIn this paper we present an analytical and numeri-\ncal study of magnon-phonon interactions in bulk ferro-\nmagnetic insulators, where we take both the anisotropy-\nand the exchange-mediated magnon-phonon interactions\ninto account. By using diagrammatic perturbation the-\nory to calculate the magnon self-energy, we arrive at a\nwave-vector dependent expression of the magnon scat-\ntering rate, which is the inverse of the magnon quasi-\nparticle lifetime \u001cqp. The magnetic Grüneisen parameter\n\u0000m=@lnTC=@lnV[32, 33], where TCis the Curie tem-\nperature and Vthe volume of the magnet, gives direct\naccess to the exchange-mediated magnon-phonon inter-\naction parameter. We predict an enhancement in the\nphonon scattering of the Kittel mode at the touching\npoints of the two-magnon energy (of the Kittel mode and\na finite momentum magnon) and the longitudinal and\ntransverse phonon dispersions, for YIG at around 1:3 T\nand4:6 T. We also emphasize the difference in magnon\nlifetimesthatbroadenlightandneutronscatteringexper-\niments, and the transport lifetimes that govern magnon\nheat and spin transport.\nThe paper is organized as follows: in Sec. II we briefly\nreview the theory of acoustic magnons and phonons in\nferro-/ferrimagnets, particularly in YIG. In Sec. III we\nderive the exchange- and anisotropy-mediated magnon-\nphonon interactions for a cubic Heisenberg ferromagnet\nwith nearest neighbor exchange interactions in the long-\nwavelength limit. In Sec. IV we derive the magnon decay\nrate from the imaginary part of the magnon self-energy\nin a diagrammatic approach and in Sec. V we explain\nthe differences between the magnon quasi-particle and\ntransport lifetimes. Our numerical results for YIG are\ndiscussed in Sec. VI. Finally in Sec. VII we summarizearXiv:1906.01042v1  [cond-mat.str-el]  3 Jun 20192\nand discuss the main results of the present work. The va-\nlidity of our long-wavelength approximation is analyzed\nin Appendix A and in Appendix B we explain why sec-\nond order magnetoelastic couplings may be disregarded.\nIn Appendix C we briefly discuss the numerical methods\nused to evaluate the k-space integrals.\nII. MAGNONS AND PHONONS IN\nFERROMAGNETIC INSULATORS\nWithout loss of generality, we focus our treatment on\nyttrium iron garnet (YIG). The magnon band structure\nof YIG has been determined by inelastic neutron scatter-\ning [34–36] and by ab initio calculation of the exchange\nconstants [37]. The complete magnon spectral function\nhasbeencomputedforalltemperaturesbyatomisticspin\nsimulations [38], taking all magnon-magnon interactions\ninto account, but not the magnon-phonon scattering.\nThe pure phonon dispersion is known as well [29, 39]. In\nthe following, we consider the interactions of the acoustic\nmagnons from the lowest magnon band with transverse\nand longitudinal acoustic phonons, which allows a semi-\nanalytic treatment but limits the validity of our results\nto temperatures below 100 K. Since the low-temperature\nvalues of the magnetoelastic constants, sound velocities,\nand magnetic Grüneisen parameter are not available for\nYIG, we use throughout the material parameters under\nambient conditions.\nA. Magnons\nSpinsinteractwitheachotherviadipolarandexchange\ninteractions. We disregard the former since at the energy\nscaleEdip\u00190:02 meV [28] it is only relevant for long-\nwavelength magnons with wave vectors k.6\u0002107m\u00001\nand energies Ek=kB.0:2 K, which are negligible for\nthe thermal magnon transport in the temperature regime\nwe are interested in. The lowest magnon band can then\nbe described by a simple Heisenberg model on a course-\ngrained simple cubic ferromagnet with exchange interac-\ntionJ\nHm=\u0000J\n2X\nhi6=jiSi\u0001Sj\u0000X\nig\u0016BBSz\ni;(2.1)\nwhere the sum is over all nearest neighbors and ~Siis the\nspin operator at lattice site Ri. The lattice constant of\nthe cubic lattice or YIG is a= 12:376\u0017Aand the effective\nspin per unit cell ~S=~Msa3=(g\u0016B)\u001914:2~at room\ntemperature [28] ( S\u001920forT.50 K[40]), where the\ng-factorg\u00192,\u0016Bis the Bohr magneton and Msthe sat-\nuration magnetization. The parameter Jis an adjustable\nparameter that can be fitted to experiments or computed\nfrom first principles. Bis an effective magnetic field that\norients the ground-state magnetization vector to the z\naxis and includes the (for YIG small) magnetocrystallineanisotropyfield. The 1=Sexpansionofthespinoperators\nin terms of Holstein-Primakoff bosons reads [41],\nS+\ni=Sx+iSy\u0019p\n2S[bi+O(1=S)];(2.2)\nS\u0000\ni=Sx\u0000iSy\u0019p\n2Sh\nby\ni+O(1=S)i\n;(2.3)\nSz\ni=S\u0000by\nibi; (2.4)\nwhereby\niandbiare the magnon creation and annihilation\noperators with boson commutation ruleh\nbi;by\nji\n=\u000ei;j.\nThen\nHm!X\nkEkby\nkbk; (2.5)\nwhere the magnon operators by\nkandbkare defined by\nbi=1p\nNX\nkeik\u0001Ribk; (2.6)\nby\ni=1p\nNX\nke\u0000ik\u0001Riby\nk; (2.7)\nandNthe number of unit cells. The dispersion relation\nEk=g\u0016BB+ 4SJX\n\u000b=x;y;zsin2(k\u000ba=2) (2.8)\nbecomes quadratic in the long-wavelength limit ka\u001c1:\nEk=g\u0016BB+Eexk2a2; (2.9)\nwhereEex=SJ. WithEex=kB\u000240 K = 3:45 meV\nthe latter is a good approximation up to k0= 1=a\u0019\n8\u0002108m\u00001[34]. The effective exchange coupling is\nthenJ\u00190:24 meV. The lowest magnon band does not\ndepend significantly on temperature [38], which implies\nthatEex=SJdoes not depend strongly on temper-\nature. The temperature dependence of the saturation\nmagnetization and effective spin Sshould therefore not\naffect the low-energy exchange magnons significantly. By\nusing Eq. (2.9) in the following, our theory is valid for\nk.k0(see Fig. 1) or temperatures T.100 K. In this\nregime the cut-off of an ultraviolet divergence does not\naffect results significantly (see Appendix A). We disre-\ngard magnetostatic interactions that affect the magnon\nspectrum only for very small wave vectors since at low\ntemperatures the phonon scattering is not significant.\nB. Phonons\nWe expand the displacement Xiof the position riof\nunit cellifrom the equilibrium position Ri\nXi=ri\u0000Ri; (2.10)\ninto the phonon eigenmodes Xq\u0015,\nX\u000b\ni=1p\nNX\nq;\u0015e\u000b\nq\u0015Xq\u0015eiq\u0001Ri;(2.11)3\nwhere\u000b2 fx;y;zgandqa wave vector. We define\npolarizations \u00152f1;2;3gfor the elastic continuum [42]\neq1= (cos\u0012qcos\u001eq;cos\u0012qsin\u001eq;\u0000sin\u0012q);(2.12)\neq2=i(\u0000sin\u001eq;cos\u001eq;0); (2.13)\neq3=i(sin\u0012qcos\u001eq;sin\u0012qsin\u001eq;cos\u0012q);(2.14)\nwhere the angles \u0012qand\u001eqare the spherical coordinates\nof\nq=q(sin\u0012qcos\u001eq;sin\u0012qsin\u001eq;cos\u0012q);(2.15)\nwhich is valid for YIG up to 3 THz(12 meV) [29, 39].\nThe phonon Hamiltonian then reads\nHp=X\nq\u0015\u0014P\u0000q\u0015Pq\u0015\n2m+m\n2~2\"2\nq\u0015X\u0000q\u0015Xq\u0015\u0015\n;\n=X\nq\u0015\"q\u0015\u0012\nay\nq\u0015aq\u0015+1\n2\u0013\n; (2.16)\nwherethecanonicalmomenta Pq\u0015obeythecommutation\nrelations [Xq\u0015;Pq0\u00150] =i~\u000eq;\u0000q0\u000e\u0015\u00150and the mass of the\nYIG unit cell m=\u001aa3= 9:8\u000210\u000024kg[27]. The phonon\ndispersions for YIG then read\n\"q\u0015=~c\u0015jqj; (2.17)\nwherec1;2=ct= 3843 m=sis the transverse sound ve-\nlocity andc3=cl= 7209 m=sthe longitudinal velocity\nat room temperature [27]. In terms of phonon creation\nand annihilation operators\nXq\u0015=aq\u0015+ay\n\u0000q\u0015p\n2m\"q\u0015=~2; P q\u0015=1\nirm\"q\u0015\n2\u0010\naq\u0015\u0000ay\n\u0000q\u0015\u0011\n;\n(2.18)\nandh\naq\u0015;ay\nq0\u00150i\n=\u000eq;q0\u000e\u0015;\u00150.\nIn Fig. 1 we plot the longitudinal and transverse\nphonon and the acoustic magnon dispersion relations for\nYIG at zero magnetic field. The magnon-phonon inter-\naction leads to an avoided level crossing at points where\nmagnon and phonon dispersion cross, as discussed in\nRefs. [27] and [28].\nIII. MAGNON-PHONON INTERACTIONS\nWe derive in this section the magnon-phonon interac-\ntions due to the anisotropy and exchange interactions for\na cubic lattice ferromagnet.\nA. Phenomenological magnon-phonon interaction\nIn the long-wavelength/continuum limit ( k.k0) the\nmagnetoelastic energy to lowest order in the deviations\nof magnetization and lattice from equilibrium reads [23–\n26, 28]\n0.0 0.5 1.0 1.5\nk[109m−1]024681012Ek[meV]magnon model\nparabolic approximation\nlongitudinal acoustic phonon\ntransverse acoustic phononFigure 1. Dispersion relations of the acoustic phonons and\nmagnons in YIG at zero magnetic field.\nEme=n\nM2sZ\nd3rX\n\u000b\f[B\u000b\fM\u000b(r)M\f(r)\n+B0\n\u000b\f@M(r)\n@r\u000b\u0001@M(r)\n@r\f\u0015\nX\u000b\f(r);(3.1)\nwheren= 1=a3. The strain tensor X\u000b\fis defined in\nterms of the lattice displacements X\u000b,\nX\u000b\f(r) =1\n2\u0014@X\u000b(r)\n@r\f+@X\f(r)\n@r\u000b\u0015\n;(3.2)\nwith, for a cubic lattice [28],\nB\u000b\f=\u000e\u000b\fBk+ (1\u0000\u000e\u000b\f)B?; (3.3)\nB0\n\u000b\f=\u000e\u000b\fB0\nk+ (1\u0000\u000e\u000b\f)B0\n?: (3.4)\nB\u000b\fis caused by magnetic anisotropies and B0\n\u000b\fby the\nexchange interaction under lattice deformations. For\nYIG at room temperature [27, 33]\nBk=kB\u000247:8 K = 4:12 meV;(3.5)\nB?=kB\u000295:6 K = 8:24 meV;(3.6)\nB0\nk=a2=kB\u00022727 K = 235 meV ;(3.7)\nB0\n?=a2\u00190: (3.8)\nWe discuss the values for B0\nkandB0\n?in Sec. IIIC.\nB. Anisotropy-mediated magnon-phonon\ninteraction\nThe magnetoelastic anisotropy (3.1) is described by\nthe Hamiltonian [28],4\nHan\nmp=X\nq\u0015\u0002\n\u0000q\u0015b\u0000qXq\u0015+ \u0000\u0003\n\u0000q\u0015by\nqXq\u0015\u0003\n+1p\nNX\nq;k;k0\u000ek\u0000k0\u0000q;0X\n\u0015\u0000an\nkk0;\u0015by\nkbk0Xq\u0015\n+1p\nNX\nq;k;k0\u000ek+k0+q;0X\n\u0015\u0000bb\nkk0;\u0015bkbk0Xq\u0015\n+1p\nNX\nq;k;k0\u000ek+k0\u0000q;0X\n\u0015\u0000\u0016b\u0016b\nkk0;\u0015by\nkby\nk0Xq\u0015;(3.9)\nwith interaction vertices\n\u0000q\u0015=B?p\n2Sh\niqzex\nq\u0015+qzey\nq\u0015\n+ (iqx+qy)ez\nq\u0015\u0003\n; (3.10)\n\u0000an\nkk0;\u0015=Uk\u0000k0;\u0015; (3.11)\n\u0000bb\nkk0;\u0015=V\u0000k\u0000k0;\u0015; (3.12)\n\u0000\u0016b\u0016b\nkk0;\u0015=V\u0003\n\u0000k\u0000k0;\u0015; (3.13)\nand\nUq;\u0015=iBk\nSh\nqxex\nq\u0015+qyey\nq\u0015\u00002qzez\nq\u0015i\n;(3.14)\nVq;\u0015=iBk\nSh\nqxex\nq\u0015\u0000qyey\nq\u0015i\n+B?\nSh\nqyex\nq\u0015+qxey\nq\u0015i\n: (3.15)\nThe one magnon-two phonon process is of the same\norder in the total number of magnons and phonons as\nthe two magnon-one phonon processes, but its effect on\nmagnon transport is small, as shown in Appendix B.\nC. Exchange-mediated magnon-phonon interaction\nThe exchange-mediated magnon-phonon interaction is\nobtained under the assumption that the exchange inter-\nactionJijbetween two neighboring spins at lattice sites\nriandrjdepends only on their distance, which leads to\nthe expansion to leading order in the small parameter\n(jri\u0000rjj\u0000a)\nJij=J(jri\u0000rjj)\u0019J+J0\u0001(jri\u0000rjj\u0000a);(3.16)\nwhereais the equilibrium distance and J0=@J=@a.\nWith ri=Ri+XRi;the Heisenberg Hamiltonian (2.1)\nis modulated by\nHex\nmp=\u0000J0X\niX\n\u000b=x;y;z\u0000\nX\u000b\nRi+ae\u000b\u0000X\u000b\nRi\u0001\nSRi\u0001SRi+ae\u000b;\n(3.17)where e\u000bis a unit vectors in the \u000bdirection. Expanding\nthe displacements in terms of the phonon and magnon\nmodes\nHex\nmp=1p\nNX\nq;k;k0\u000ek\u0000k0\u0000q;0X\n\u0015\u0000ex\nkk0;\u0015by\nkbk0Xq\u0015;(3.18)\nwith interaction\n\u0000ex\nkk0;\u0015= 8iJ0SX\n\u000be\u000b\nk\u0000k0;\u0015sin\u0012k\u000ba\n2\u0013\nsin\u0012k0\n\u000ba\n2\u0013\n\u0002sin\u0012(k\u000b\u0000k0\n\u000b)a\n2\u0013\n\u0019iJ0a3SX\n\u000be\u000b\nk\u0000k0;\u0015k\u000bk\u000b0(k\u000b\u0000k0\n\u000b);(3.19)\nwhere the last line is the long-wavelength expansion. The\nmagnon-phonon interaction\n\u0000\u0016bb\nk;k0;\u0015= \u0000ex\nk;k0;\u0015+ \u0000an\nk;k0;\u0015 (3.20)\nconserves the magnon number, while (3.12) and (3.13) do\nnot. Phonon numbers are not conserved in either case.\nThe value of J0for YIG is determined by the magnetic\nGrüneisen parameter [32, 33]\n\u0000m=@lnTC\n@lnV=@lnJ\n@lnV=J0a\n3J;(3.21)\nwhereV=Na3is the volume of the magnet. The only\nassumption here is that the Curie temperature TCscales\nlinearly with the exchange constant J[43]. \u0000mhas been\nmeasured for YIG via the compressibility to be \u0000m=\n\u00003:26[32], and via thermal expansion, \u0000m=\u00003:13[33],\nso we set \u0000m=\u00003:2. For other materials the magnetic\nGrüneisen parameter is also of the order of unity and in\nmany cases \u0000m\u0019\u000010=3[32, 33, 44]. A recent ab initio\nstudy of YIG finds \u0000m=\u00003:1[45].\nComparing the continuum limit of Eq. (3.17) with the\nclassical magnetoelastic energy (3.1)\nB0\nk= 3\u0000mJS2a2=2; (3.22)\nwhereforYIG B0\nk=a2\u0019235 meV . Wedisregard B0\n?since\nit vanishes for nearest neighbor interactions by cubic lat-\ntice symmetry.\nThe coupling strength of the exchange-mediated\nmagnon-phonon interaction can be estimated from the\nexchange energy SJ0a\u0019Eex=SJ[31, 46] following\nAkhiezer et al.[47, 48]. Our estimate of SJ0a= 3\u0000mSJ\nis larger by 3\u0000m, i.e. one order of magnitude. Since the\nscattering rate is proportional to the square of the in-\nteraction strength, our estimate of the scattering rate is\na factor 100larger than previous ones. The assumption\nJ0a\u0019Jis too small to be consistent with the experi-\nmental Grüneisen constant [32, 33]. Ref. [3] educatedly\nguessedJ0a\u0019100J;which we now judge to be too large.5\nFigure2. Feynmandiagramsofinteractionsbetweenmagnons\n(solid lines) and phonons (dashed lines). The arrows indicate\nthe energy-momentum flow. (a) magnon-phonon interconver-\nsion, (b) magnon number-conserving magnon-phonon inter-\naction, (c) and (d) magnon number non-conserving magnon-\nphonon interactions.\nD. Interaction vertices\nThe magnon-phonon interactions in the Hamiltonian\n(3.9) are shown in Fig. 2 as Feynman diagrams. Fig. 2(a)\nillustrates magnon and phonon interconversion, which\nis responsible for the magnon-phonon hybridization and\nlevel splitting at the crossing of magnon and phonon dis-\npersions [27, 28]. The divergence of this diagram at the\nmagnon-phonon crossing points is avoided by either di-\nrect diagonalization of the magnon-phonon Hamiltonian\n[42] or by cutting-off the divergence by a lifetime param-\neter [31]. This process still generates enhanced magnontransport that is observable as magnon polaron anoma-\nlies in the spin Seebeck effect [22] or spin-wave excitation\nthresholds [49, 50], but these are strongly localized in\nphase space and disregarded in the following, where we\nfocus on the magnon scattering rates to leading order in\n1=Sof the scattering processes in Fig. 2(b)-(d).\nIV. MAGNON SCATTERING RATE\nHere we derive the magnon reciprocal quasi-particle\nlifetime\u001c\u00001\nqp=\ras the imaginary part of the wave vector\ndependent self-energy, caused by acoustic phonon scat-\ntering [28],\n\r(k) =\u00002\n~Im\u0006(k;Ek=~+i0+):(4.1)\nThis quantity is in principle observable by inelastic neu-\ntron scattering. The total decay rate\n\r=\rc+\rnc+\rother(4.2)\nis the sum of the magnon number conserving decay rate\n\rcand the magnon number non-conserving decay rate\n\rnc, which are related to the magnon-phonon scattering\ntime\u001cmpand the magnon-phonon dissipation time \u001cmr\nby\n\u001cmp=1\n\rc; \u001cmr=1\n\rnc: (4.3)\n\rotheris caused by magnon-magnon and magnon disorder\nscattering, thereby beyond the scope of this work.\nThe self-energy to leading order in the 1=Sexpansion\nis of second order in the magnon-phonon interaction [28],\n\u00062(k;i!) =1\nNX\nk0\u0015~2\f\f\f\u0000\u0016bb\nk;k0;\u0015\f\f\f2\n2m\"k\u0000k0;\u0015\u0014nB(\"k\u0000k0;\u0015)\u0000nB(Ek0)\ni~!+\"k\u0000k0;\u0015\u0000Ek0+1 +nB(\"k\u0000k0;\u0015) +nB(Ek0)\ni~!\u0000\"k\u0000k0;\u0015\u0000Ek0\u0015\n\u00001\nNX\nk0\u0015~2\f\f\f\u0000bb\nk;k0;\u0015\f\f\f2\n2m\"k\u0000k0;\u0015\u00141 +nB(\"k+k0;\u0015) +nB(Ek0)\ni~!+\"k+k0;\u0015+Ek0+nB(\"k+k0;\u0015)\u0000nB(Ek0)\ni~!\u0000\"k+k0;\u0015+Ek0\u0015\n; (4.4)\nwhere the magnon number conserving magnon-phonon\nscattering vertex \u0000\u0016bb\nk;k0;\u0015= \u0000ex\nk;k0;\u0015+ \u0000an\nk;k0;\u0015and the\nPlanck (Bose) distribution function nB(\") = (e\f\"\u00001)\u00001\nwith inverse temperature \f= 1=(kBT). The Feynman\ndiagrams representing the magnon number conserving\nand non-conserving contributions to the self-energy areshown in Fig. 3.\nWe write the decay rate in terms of four contributions\n\r(k) =\rc\nout(k) +\rnc\nout(k)\u0000\rc\nin(k)\u0000\rnc\nin(k);(4.5)\nwhereoutandindenote the out-scattering and in-\nscattering parts. The contributions to the decay rate\nread [28]6\nk\nqk-q\nk kk\nqq-k\nk k(a) (b)\nFigure 3. Feynman diagrams representing the self-energy\nEq. (4.4) due to (a) magnon number-conserving magnon-\nphonon interactions and (b) magnon number non-conserving\nmagnon-phonon interactions.\n\rc\nout(k) =\u0019~\nmNX\nq;\u0015\f\f\f\u0000\u0016bb\nk;k\u0000q;\u0015\f\f\f2\n\"q\u0015[(1 +nB(Ek\u0000q))nB(\"q\u0015)\u000e(Ek\u0000Ek\u0000q+\"q\u0015)\n+ (1 +nB(Ek\u0000q))(1 +nB(\"q\u0015))\u000e(Ek\u0000Ek\u0000q\u0000\"q\u0015)]; (4.6)\n\rc\nin(k) =\u0019~\nmNX\nq;\u0015\f\f\f\u0000\u0016bb\nk;k\u0000q;\u0015\f\f\f2\n\"q\u0015[nB(Ek\u0000q)(1 +nB(\"q\u0015))\u000e(Ek\u0000Ek\u0000q+\"q\u0015)\n+nB(Ek\u0000q)nB(\"q\u0015)\u000e(Ek\u0000Ek\u0000q\u0000\"q\u0015)]; (4.7)\n\rnc\nout(k) =\u0019~\nmNX\nq;\u0015\f\f\f\u0000bb\nk;q\u0000k;\u0015\f\f\f2\n\"q\u0015[nB(Eq\u0000k)(1 +nB(\"q\u0015))\u000e(Ek+Eq\u0000k\u0000\"q\u0015)]; (4.8)\n\rnc\nin(k) =\u0019~\nmNX\nq;\u0015\f\f\f\u0000bb\nk;q\u0000k;\u0015\f\f\f2\n\"q\u0015[(1 +nB(Eq\u0000k))nB(\"q\u0015)\u000e(Ek+Eq\u0000k\u0000\"q\u0015)]; (4.9)\nwhere the sum is over all momenta qin the Brillouin\nzone. Here the magnon/phonon annihilation rate is pro-\nportional to the Boson number nB, while the creation\nratescaleswith 1+nB. Forexample,intheout-scattering\nrate\rc\nout(k)theincomingmagnonwithmomentum kgets\nscattered into the state k\u0000qand a phonon is either ab-\nsorbedwithprobability \u0018nBoremittedwithprobability\n\u0018(1 +nB). The out- and in-scattering rates are related\nby the detailed balance\n\rc\nin(k)=\rc\nout(k) =\rnc\nin(k)=\rnc\nout(k) =e\u0000\fEk:(4.10)\nFor high temperatures kBT\u001dEk, we may expand the\nBose functions nB(Ek)\u0018kBT=E kand we find \rin\u0018\n\rout\u0018T2and\r=\rout\u0000\rin\u0018T. For low temperatures\nkBT\u001cEk, the out-scattering rate \rout!const:and\nthe in-scattering rate \rin\u0018e\u0000\fEk!0. The scattering\nprocesses (c) and (d) in Fig. 2 conserve energy and linear\nmomentum, but not angular momentum. A loss of an-gular momentum after integration over all wave vectors\ncorresponds to a mechanical torque on the total lattice\nthat contributes to the Einstein-de Haas effect [51].\nV. MAGNON TRANSPORT LIFETIME\nInthissectionwecomparethetransportlifetime \u001ctand\nthe magnon quasi-particle lifetime \u001cqpthat can be very\ndifferent [52–54], but, to the best of our knowledge, has\nnot yet been addressed for magnons. The magnon decay\nrate is proportional to the imaginary part of self energy,\nas shown in Eq. (4.1). On the other hand, the transport\nis governed by transport lifetime \u001ctin the Boltzmann\nequation that agrees with \u001cqponly in the relaxation time\napproximation. The stationary Boltzmann equation for7\nthe magnon distribution can be written as [3, 42]\n@fk(r)\n@r\u0001@Ek\n@(~k)= \u0000in[f]\u0000\u0000out[f];(5.1)\nwherefk(r)is the magnon distribution function. The in\nandoutcontributions to the collision integral are related\nto the previously defined in- and out-scattering rates by\n\u0000in[f] = (1 +fk)\rin[f]; (5.2)\n\u0000out[f] =fk\rout[f]; (5.3)\nwhere the equilibrium magnon distribution nB(Ek)is re-\nplaced by the non-equilibrium distribution function fk.\nThe factor (1 +fk)corresponds to the creation of a\nmagnon with momentum kin the in-scattering process\nand the factor fkto the annihilation in the out-scattering\nprocess. The phonons are assumed to remain at thermal\nequilibrium, so we disregard the phonon drift contribu-\ntion that is expected in the presence of a phononic heat\ncurrent.\nMagnon transport is governed by three linear response\nfunctions, i.e. spin and heat conductivity and spin See-\nbeck coefficient [42]. These can be obtained from the ex-\npansion of the distribution function in terms of temper-\nature and chemical potential gradients and correspond\nto two-particle Green functions with vertex corrections,\nthat reflect the non-equilibrium in-scattering processes,\ncaptured by a transport lifetime \u001ctthat can be different\nfrom the quasi-particle (dephasing) lifetime \u001cqpdefined\nby the self-energy. We define the transport life time of\na magnon with momentum kin terms of the collision\nintegral\n\u0000out[f]\u0000\u0000in[f] =1\n\u001ck;t[f](fk(r)\u0000f0;k);(5.4)\nwithf0;k=nB(Ek)and we assume a thermalized quasi-\nequilibrium distribution function\nfk(r) =nB\u0012Ek\u0000\u0016(r)\nkBT(r)\u0013\n; (5.5)\nwhere\u0016is the magnon chemical potential. We linearize\nthe function fkin terms of small deviations \u000efkfrom\nequilibrium f0;k,\n\u000efk=fk\u0000f0;k: (5.6)\nleading to [3]\n\u000efk=\u001ck;t[f]@f0;k\n@Ek@Ek\n@(~k)\u0001\u0012\nr\u0016+Ek\u0000\u0016\nTrT\u0013\n;(5.7)\nwhere the gradients of chemical potential r\u0016and tem-\nperature rTdrive the magnon current. In the relax-\nation time approximation we disregard the dependence\nof\u001ck;t[f]on\u000efand recover the quasi-particle lifetime\n\u001ck;t!\u001ck;qp.Tofirstorderinthephononoperatorsandsecondorder\nin the magnon operators the collision integral for magnon\nnumber non-conserving processes,\n\u0000nc\nout[f]\u0000\u0000nc\nin[f]\n=\u0019~\nmNX\nq\u0015j\u0000bb\nk;q\u0000k;\u0015j2\n\"q\u0015\u000e(Ek+Eq\u0000k\u0000\"q\u0015)\n\u0002[(1 +nq\u0015)fkfq\u0000k\u0000nq\u0015(1 +fq\u0000k)(1 +fk)];\n(5.8)\nwhere the interaction vertex \u0000bb\nk;k0;\u0015is given by Eq. (3.12)\nandnq\u0015=nB(\"q\u0015). By using the expansion (5.6) in the\ncollision integral that vanishes at equilibrium,\n\u0000out[f0]\u0000\u0000in[f0] = 0; (5.9)\nwe arrive at\n1\n\u001cnc\nk;t=\u0019~\nmNX\nq\u0015j\u0000bb\nk;q\u0000k;\u0015j2\n\"q\u0015\u000e(Ek+Eq\u0000k\u0000\"q\u0015)\n\u0002\u0014\nnB(Ek\u0000q)\u0000nq\u0015+\u000efq\u0000k\n\u000efk(nB(Ek)\u0000nq\u0015)\u0015\n:\n(5.10)\nFor the magnon number conserving process the deriva-\ntion is similar and we find\n1\n\u001cc\nk;t=\u0019~\nmNX\nq\u0015j\u0000\u0016bb\nk;k\u0000q;\u0015j2\n\"q\u0015\"\n\u000e(Ek\u0000Ek\u0000q+\"q\u0015)\n\u0002\u0012\nnq\u0015\u0000nB(Ek\u0000q)\u0000\u000efk\u0000q\n\u000efk(nB(Ek) +nq\u0015+ 1)\u0013\n+\u000e(Ek\u0000Ek\u0000q\u0000\"q\u0015)\n\u0002\u0012\n1 +nB(Ek\u0000q) +nq\u0015+\u000efk\u0000q\n\u000efk(nB(Ek)\u0000nq\u0015)\u0013#\n;\n(5.11)\nwith interaction vertex \u0000\u0016bb\nk;k0;\u0015given by Eq. (3.20). Due\nto the\u000efk\u0000q=\u000efkterm this is an integral equation. It\ncan be solved iteratively to generate a geometric series\nreferred to as vertex correction in diagrammatic theo-\nries. By simply disregarding the in-scattering with terms\n\u000efk\u0000q=\u000efkthetransportlifetimereducestothethequasi-\nparticle lifetime of the self-energy. We leave the general\nsolution of this integral equation for future work, but\nargue in Sec. VID that the vertex corrections are not\nimportant in our regime of interest.\nVI. NUMERICAL RESULTS\nA. Magnon decay rate\nIn the following we present and analyze our results\nfor the magnon decay rates in YIG. We first consider8\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nk[109m−1]01020304050γc(k) [106s−1]\n(100)\n(001)\n(110)\n(111)\n(011)0.0 0.5 1.0−0.10.00.10.2\nFigure 4. Magnon decay rate in YIG due to magnon-phonon\ninteractions for magnons propagating along various directions\natT= 50 KandB= 0. We denote the propagation direction\nby(lmn), i.e.lex+mey+nez. The inset shows the relative\ndeviation\u000e\rc=\rcfrom the (100) direction.\n0.00 0.01 0.02 0.03 0.04 0.05\nkx[109m−1]0.000.050.100.150.200.250.300.350.40γc(k) [103s−1]γc, total\nγc, anisotropy\nγc, exchange\nFigure 5. Comparison of the contributions from exchange-\nmediated and anisotropy-mediated magnon-phonon interac-\ntions to the magnon number conserving scattering rate \rcat\nT= 50 KandB= 0.\nthe case of vanishing effective magnetic field ( B= 0)\nand discuss the magnetic field dependence in Sec. VIC.\nSince our model is only valid in the long-wavelength ( k<\n8×108m\u00001) and low-temperature ( T.100 K) regime,\nwe focus first on T= 50 Kand discuss the temperature\ndependence in Sec. VIB.\nInFig.4weshowthemagnonnumberconservingdecay\nrate\rc(k), which is on the displayed scale dominated by\nthe exchange-mediated magnon-phonon interaction and\nis isotropic for long-wavelength magnons.\nIn Fig. 5 we compare the contribution from the\nexchange-mediated magnon-phonon interaction ( \rc\u0018\nk4) and from the anisotropy-mediated magnon-phonon\ninteraction ( \rc\u0018k2). We observe a cross-over at\nk\u00194\u0002107m\u00001: for much smaller wave numbers, the\nexchange contribution can be disregarded and for larger\nwave numbers the exchange contribution becomes domi-\nnant.\nThe magnon number non-conserving decay rate \rncin\nFig. 6 is much smaller than the magnon-conserving one.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nk[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](100)\n(001)\n(110)\n(111)\n(011)Figure 6. Magnon decay rate in YIG due to magnon num-\nber non-conserving magnon-phonon interactions for magnons\npropagating along various directions at T= 50 KandB= 0.\nThis is consistent with the low magnetization damping\nof YIG, i.e. the magnetization is long-lived. We observe\ndivergent peaks at the crossing points (shown in Fig. 1)\nwith the exception of the (001) direction. These diver-\ngences occur when magnons and phonons are degenerate\natk= 0:48\u0002109m\u00001(1:2 meV) andk= 0:9\u0002109m\u00001\n(4:3 meV), respectively, at which the Boltzmann formal-\nism does not hold; a treatment in the magnon-polaron\nbasis [42] or a broadening parameter [31] would get rid\nof the singular behavior. The divergences are also sup-\npressed by arbitrarily small effective magnetic fields (see\nSec. VIC). There are no peaks along the (001) direc-\ntion because in the (001) direction the vertex function\nVq;\u0015(see Eq. (3.15)) vanishes for q= (0;0;kz). For\nk >~cl=(D(p\n8\u00002)) = 1:085\u0002109m\u00001the decay rate\n\rncvanishes because the decay process does not conserve\nenergy (\u000e(Ek+Eq\u0000k\u0000\"q\u0015) = 0).\nB. Temperature dependence\nAbove we focused on T= 50 Kand explained that we\nexpect a linear temperature dependence of the magnon\ndecay rates at high, but not low temperatures. Fig. 7\nshowsourresultsforthetemperaturedependenceat kx=\n108m\u00001. Deviations from the linear dependence at low\ntemperatures occurs when quantum effects set in, i.e. the\nRayleigh-Jeans distribution does not hold anymore,\n1\ne\"=(kBT)\u000016\u0019kBT\n\": (6.1)\nC. Magnetic field dependence\nThe numerical results presented above are for a mono-\ndomain magnet in the limit of small applied magnetic\nfields. A finite magnetic field Balong the magnetization\ndirectioninducesanenergygap g\u0016BBinthemagnondis-\npersion, which shifts the positions of the magnon-phonon9\n0 2 4 6 8 10\nT[K]0.00.10.20.30.40.50.60.7γ[103s−1]γc\nγnc\nFigure7. Temperaturedependenceofthemagnondecayrates\n\rncand\rcatB= 0,kx= 108m\u00001andky=kz= 0, i.e.\nalong (100).\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nkx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1]B= 0 T\nB= 0.1 T\nB= 0.5 T\nB= 1 T\nB= 2 T\nFigure 8. Magnetic field dependence of the magnon number\nnon-conserving magnon decay rate in YIG at T= 50 Kwith\nmagnon momentum along (100).\ncrossingpointsto longerwavelengths. Themagneticfield\nsuppresses the (unphysical) sharp peaks at the crossing\npoints (see Fig. 8) that are caused by the divergence of\nthePlanckdistributionfunctionforavanishingspinwave\ngap.\nIn the magnon number conserving magnon-phonon\ninteractions, the magnetic field dependence cancels in\nthe delta function and enters only in the Bose func-\ntion vianB(magnetic freeze-out). Fig. 9 shows that\nthe magnetic field mainly affects magnons with energies\n.2g\u0016BB= 0:23(B=T) meV.\nAs shown in Fig. 10 the magnon decay by phonons\ndoes not vanish for the k= 0Kittel mode, but only\nin the presence of a spin wave gap E0=g\u0016BB. Both\nmagnon conserving and non-conserving scattering pro-\ncesses contribute. The divergent peaks at B\u00191:3 Tand\nB\u00194:6 Tin\rncare caused by energy and momentum\nconservation in the two-magnon-one-phonon scattering\nprocess,\n\u000e(Ek=0+Eq\u0000\"q\u0015) =\u000e(2g\u0016BB+Eexq2a2\u0000~c\u0015q);(6.2)\nwhen the gradient of the argument of the delta function\n0.0 0.1 0.2 0.3 0.4 0.5\nkx[109m−1]0246810δγc/γcB= 1 T\nB= 10 TFigure 9. Relative deviation \u000e\rc=\rcfrom theB= 0result of\nthe magnon number conserving magnon decay rate in YIG at\nT= 50 Kwith magnon momentum along (100).\nvanishes,\nrq(Ek=0+Eq\u0000\"q\u0015) = 0; (6.3)\ni.e., the two-magnon energy Ek=0+Eqtouches either the\ntransverse or longitudinal phonon dispersion \"q\u0015. The\ntotal energy of the two magnons is equivalent to the en-\nergy of a single magnon with momentum qbut in a field\n2B, resulting in the divergence at fields that are half of\nthose for the magnon-polaron observed in the spin See-\nbeck effect [31, 42]. The two-magnon touching condition\ncan be satisfied in all directions of the phonon momen-\ntumq, which therefore contributes to the magnon decay\nrate when integrating over the phonon momentum q. For\nk6= 0this two-magnon touching condition can only be\nfulfilled for phonons along a particular direction and the\ndivergence is suppressed.\nThe magnon decay rate is related to the Gilbert damp-\ning\u000bkas~\rk= 2\u000bkEk[55]. We find that phonons\ncontribute only weakly to the Gilbert damping, \u000bnc\n0=\n~\rnc\n0=(2E0)\u001810\u00008atT= 50 K, which is much smaller\nthan the total Gilbert damping \u000b\u001810\u00005in YIG, but\nthe peaks at 1:3 Tand4:6 Tmight be observable. The\nphonon contribution to the Gilbert damping scales lin-\nearly with temperature, so is twice as large at 100 K. At\nlow temperatures ( T.100 K) Gilbert damping in YIG\nhas been found to be caused by two-level systems [56]\nand impurity scattering [40], while for higher tempera-\ntures magnon-phonon [57] and magnon-magnon scatter-\ning involving optical magnons [34] have been proposed to\nexplain the observed damping. Enhanced damping as a\nfunction of magnetic field at higher temperatures might\nreveal other van Hove singularities in the joint magnon-\nphonon density of states.\nD. Magnon transport lifetime\nWe do not attempt a full solution of the integral equa-\ntions (5.10) and (5.11) for the transport lifetime. How-\never, we can still estimate its effect by the observation10\n0 1 2 3 4 5 6\nMagnetic field [T]01020304050γ[103s−1]γc(k= 0)\nγnc(k= 0)\nFigure 10. Magnetic field dependence of the magnon decay\nrates in YIG at k= 0andT= 50 K.\nthat the ansatz \u001c\u00001\nk;t\u0018kncan be an approximate solu-\ntion of the Boltzmann equation with in-scattering.\nOur results for the magnon number conserving interac-\ntion are shown in Fig. 11 (for rT= 0and finite r\u0016jjex),\nwhere\rt=\u001c\u00001\nt. We consider the cases n= 0;2;4,\nwheren= 0or\u001ck;t= const:would be the solution for\na short-range scattering potential. For very long wave-\nlengths (k.4\u0002107m\u00001) the inverse quasi-particle life-\ntime\u001c\u00001\nk;qp\u0018k2and for shorter wavelengths \u001c\u00001\nk;qp\u0018k4.\nn= 2is a self-consistent solution only for very small\nk.4\u0002107m\u00001, while\u001c\u00001\nk;qp\u0018k4is a good ansatz up\ntok.0:3\u0002109m\u00001. We see that the transport life-\ntime approximately equals the quasi-particle lifetime in\nthe regime of the validity of the n= 4power law.\nFor the magnon number non-conserving processes in\nFig. 12 the quasi-particle lifetime behaves as \u001c\u00001\nk;qp\u0018k2.\nThe ansatz n= 2turns out to be self-consistent and we\nsee deviations of the transport lifetime from the quasi-\nparticlelifetimefor k&5\u0002107m\u00001. Theplotonlyshows\nour results for k<1\u0002108m\u00001because our assumption\nof an isotropic lifetime is not valid for higher momenta\nin this case.\nWe conclude that for YIG in the long-wavelength\nregime the magnon transport lifetime (due to magnon-\nphonon interactions) should be approximately the same\nas the quasi-particle lifetime, but deviations at shorter\nwavelengths require more attention.\nVII. SUMMARY AND CONCLUSION\nWe calculated the decay rate of magnons in YIG\ninduced by magnon-phonon interactions in the long-\nwavelength regime ( k.1\u0002109m\u00001). Our model\ntakes only the acoustic magnon and phonon branches\ninto account and is therefore valid at low to intermedi-\nate temperatures ( T.100 K). The exchange-mediated\nmagnon-phonon interaction has been recently identified\nas a crucial contribution to the overall magnon-phonon\ninteraction in YIG at high temperatures [3, 29, 45]. We\nemphasize that its coupling strength can be derived from\n0.0 0.1 0.2 0.3 0.4 0.5\nkx[109m−1]0.00.51.01.52.0γc\nt(k) [106s−1]quasi-particle\n1/τ∼k0\n1/τ∼k2\n1/τ∼k4Figure 11. Inverse of the magnon transport lifetime in YIG\n(with magnon momentum along (100)) due to magnon num-\nber conserving magnon-phonon interactions at T= 50 Kand\nB= 0for magnons along the (100) direction.\n0.00 0.02 0.04 0.06 0.08 0.10\nkx[109m−1]0.00.20.40.60.81.01.21.41.6γnc\nt(k) [103s−1]quasi-particle\n1/τ∼k0\n1/τ∼k2\n1/τ∼k4\nFigure 12. Inverse of the magnon transport lifetime in YIG\n(with magnon momentum along (100)) due to magnon num-\nber non-conserving interactions at T= 50 KandB= 0.\nexperimental values of the magnetic Grüneisen parame-\nter\u0000m=@lnTC=@lnV[32, 33]. In previous works this\ninteraction has been either disregarded [28], underesti-\nmated [29, 46], or overestimated [3].\nIn the ultra-long-wavelength regime the wave vector\ndependent magnon decay rate \r(k)is determined by the\nanisotropy-mediated magnon-phonon interaction with\n\r(k)\u0018k2, while for shorter wavelengths k&4\u0002107m\u00001\nthe exchange-mediated magnon-phonon interaction be-\ncomes dominant, which scales as \r(k)\u0018k4. The magnon\nnumber non-conserving processes are caused by spin-\norbit interaction, i.e., the anisotropy-mediated magnon-\nphonon interaction, and are correspondingly weak.\nIn a finite magnetic field the average phonon scatter-\ningcontribution, fromthemechanismunderstudy, tothe\nGilbert damping of the k= 0macrospin Kittel mode is\nabout three orders of magnitude smaller than the best\nvalues for the Gilbert damping \u000b\u001810\u00005. However, we\npredict peaks at 1:3 Tand4:6 T, that may be experi-\nmentally observable in high-quality samples.\nThe magnon transport lifetime, which is given by the\nbalance between in- and out-scattering in the Boltz-11\nmann equation, is in the long-wavelength regime approx-\nimately the same as the quasi-particle lifetime. However,\nthe magnon quasi-particle and transport lifetime differ\nmore significantly at shorter wavelengths. A theory for\nmagnon transport at room temperature should therefore\ninclude the “vertex corrections”.\nA full theory of magnon transport at high temperature\nrequires a method that takes the full dispersion relations\nof acoustic and optical phonons and magnons into ac-\ncount. This would also require a full microscopic descrip-\ntion of the magnon-phonon interaction, since the magne-\ntoelastic energy used here only holds in the continuum\nlimit.\nACKNOWLEDGMENTS\nN. V-S thanks F. Mendez for useful discussions. This\nwork is part of the research program of the Stichting voor\nFundamenteel Onderzoek der Materie (FOM), which is\nfinancially supported by the Nederlandse Organisatie\nvoor Wetenschappelijk Onderzoek (NWO) as well as a\nGrant-in-Aid for Scientific Research on Innovative Area,\n”Nano Spin Conversion Science” (Grant No. 26103006),\nCONICYT-PCHA/Doctorado Nacional/2014-21140141,\nFondecyt Postdoctorado No. 3190264, and Fundamen-\ntal Research Funds for the Central Universities.\nAppendix A: Long-wavelength approximation\nThe theory is designed for magnons with momen-\ntumk < 0:8\u0002109m\u00001and phonons with momen-\ntumq < 2:5\u0002109m\u00001(corresponding to phonon en-\nergies/frequencies \u001412 meV/3 THz), but relies on high-\nmomentum cut-off parameters kcbecause of the assump-\ntion of quadratic/linear dispersion of magnon/phonons.\nWe see in Fig. 13 that the scattering rates only weakly\ndepend onkc.\nThe dependence of the scattering rate on the phonon\nmomentum cut-off qcis shown in Fig. 14. qc= 3:15\u0002\n109m\u00001corresponds to an integration over the whole\nBrillouin zone, approximated by a sphere. From these\nconsiderations we estimate that the long-wavelength ap-\nproximation is reliable for k.8\u0002108m\u00001. Opti-\ncal phonons (magnons) that are thermally excited for\nT?100 K (300 K) are not considered here.\nAppendix B: Second order magnetoelastic coupling\nThe magnetoelastic energy is usually expanded only to\nfirst order in the displacement fields. Second order terms\ncan become important e.g. when the first order terms\nvanish. Thisisthecaseforone-magnontwo-phononscat-tering processes. The first order term\nX\nq\u0015\u0002\n\u0000q\u0015b\u0000qXq\u0015+ \u0000\u0003\n\u0000q\u0015by\nqXq\u0015\u0003\n(B1)\nonlycontributeswhenphononandmagnonmomentaand\nenergies cross, giving rise to magnon polaron modes [42].\nIn other areas of reciprocal space the second order term\nshould therefore be considered. Eastman [58, 59] derived\nthe second-order magnetoelastic energy and determined\nthecorrespondingcouplingconstantsforYIG.Inmomen-\ntum space, the relevant contribution to the Hamiltonian\nis of the form\nH2p1m=1p\nNX\nk;q1;\u00151;q2;\u00152\u0000\n\u000eq1+q2+k;0\u0000b\nq1\u00151;q2\u00152Xq1\u00151Xq2\u00152bk\n+\u000eq1+q2\u0000k;0\u0000\u0016b\nq1\u00151;q2\u00152Xq1\u00151Xq2\u00152by\nk\u0011\n;(B2)\nwhere the interaction vertices are symmetrized,\n\u0000b\nq1\u00151;q2\u00152=1\n2\u0010\n~\u0000b\nq1\u00151;q2\u00152+~\u0000b\nq2\u00152;q1\u00151\u0011\n;(B3)\nand obey\n\u0000b\nq1\u00151;q2\u00152=\u0010\n\u0000\u0016b\n\u0000q1\u00151;\u0000q2\u00152\u0011\u0003\n: (B4)\nThe non-symmetrized vertex function is\n~\u0000b\nq1\u00151;q2\u00152=1\na2p\n2S[B144(iI1\u0000I1;x$y)\n+B155(iI2\u0000I2;x$y)\n+B456(iI3\u0000I3;x$y)]; (B5)\nwith\nI1=a2ex\nq1\u00151qx\n1h\ney\nq2\u00152qz\n2+ez\nq2\u00152qy\n2i\n;(B6)\nI2=a2h\ney\nq1\u00151qy\n1+ez\nq1\u00151qz\n1i\n\u0002h\ney\nq2\u00152qz\n2+ez\nq2\u00152qy\n2i\n; (B7)\nI3=a2\u0002\nex\nq1\u00151qz\n1+ez\nq1\u00151qx\n1\u0003\n\u0002h\nex\nq2\u00152qy\n2+ey\nq2\u00152qx\n2i\n; (B8)\nandx$ydenotes an exchange of xandy. The relevant\ncoupling constants in YIG are [58, 59]\nB144=\u00006\u000648 meV; (B9)\nB155=\u000044\u00066 meV; (B10)\nB456=\u000032\u00068 meV: (B11)\nThe magnon self-energy (see Fig. 15) reads\n\u00062p1m(k;i!) =\u00002\nNX\nq1;\u00151;q2;\u001521\n\fX\n\n\u000eq1+q2+k;0\n\u0002\f\f\u0000b\nq1\u00151;q2\u00152\f\f2F\u00151(q1;\n)F\u00152(q2;\u0000\n\u0000!):\n(B12)12\n0.0 0.2 0.4 0.6 0.8 1.0\nkx[109m−1]0102030405060γc(k) [106s−1](a)\nkc= 3.15×109m−1\nkc= 0.8×109m−1\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nkx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](b)\nkc= 3.15×109m−1\nkc= 0.8×109m−1\nFigure 13. Dependence the magnon decay rate along (100) on the high magnon momentum cut-off kcfor the (a) magnon\nnumber conserving ( \rc) and (b) non-conserving ( \rnc) contributions at T= 50 KandB= 0.\n0.0 0.2 0.4 0.6 0.8 1.0\nkx[109m−1]0102030405060γc(k) [106s−1](a)\nqc= 3.15×109m−1\nqc= 2.5×109m−1\nqc= 2×109m−1\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nkx[109m−1]0.000.050.100.150.200.250.30γnc(k) [106s−1](b)\nqc= 3.15×109m−1\nqc= 2×109m−1\nFigure 14. Dependence the magnon decay rate along (100) on the high phonon momentum cut-off qcfor the (a) magnon number\nconserving ( \rc) and (b) non-conserving ( \rnc) contributions at T= 50 KandB= 0.\nwith phonon propagator\nF\u0015(q;\n) =~2\nm1\n~2\n2+\"2\nq\u0015: (B13)\nand leads to a magnon decay rate\n\rnc\n2p(k) =\u00002\n~Im\u0006 2p1m(k;i!!Ek=~+i0+)\n=\u0019~3\nm2NX\nq1;\u00151;q2;\u00152\u000eq1+q2+k;01\n\"1\"2\f\f\u0000b\nq1\u00151;q2\u00152\f\f2\n\u0002f2\u000e(Ek+\"1\u0000\"2) [n1\u0000n2]\n+\u000e(Ek\u0000\"1\u0000\"2) [1 +n1+n2]g; (B14)\nwhere\nn1=nB(\"q1\u00151); n2=nB(\"q2\u00152); (B15)\n\"1=\"q1\u00151; \"2=\"q2\u00152: (B16)\nThe first term in curly brackets on the right-hand-side\nof Eq. (B14) describes annihilation and creation of a\nphonon as a sum of out-scattering minus in-scattering\ncontributions,\nn1(1 +n2)\u0000(1 +n1)n2=n1\u0000n2;(B17)while the second term can be understood in terms of\nout-scattering by the creation of two phonons and the\nin-scattering by annihilation of two phonons,\n(1 +n1)(1 +n2)\u0000n1n2= 1 +n1+n2:(B18)\nFor this one-magnon-two-phonon process the quasi-\nparticle and the transport lifetimes are the same,\n\u001ct=\u001cqp; (B19)\nsince this process involves only a single magnon that is\neither annihilated or created. The collision integral is\nthen independent of the magnon distribution of other\nmagnons and the transport lifetime reduces to the quasi-\nparticle lifetime.\nThe two-phonon contribution to the magnon scatter-\ning rate in YIG at T= 50 Kand along (100) direction\nas shown in Fig. 16 is more than two orders of magni-\ntude smaller than that from one-phonon processes and\ntherefore disregarded in the main text. The numerical\nresults depend strongly on the phonon momentum cutoff\nqc, even in the long-wavelength regime, which implies\nthat the magnons in this process dominantly interact13\nk k\nq'q\nFigure 15. Feynman diagram representing the self-energy\nEq. (B12) due to one-magnon-two-phonon processes.\n0.0 0.2 0.4 0.6 0.8 1.0 1.2\nkx[109m−1]0.00.10.20.30.40.5γnc\n2p(k) [103s−1]qc= 3.15×109m−1\nqc= 2.5×109m−1\nqc= 2×109m−1\nFigure 16. Two-phonon contribution to the magnon number\nnon-conserving magnon scattering rate with magnon momen-\ntum along (100) for different values of the phonon momentum\ncutoffqcatT= 50 KandB= 0.\nwith short-wavelength, thermally excited phonons. In-\ndeed, the second order magnetoelastic interaction (B5) is\nquadratic in the phonon momenta, which favors scatter-\ningwithshort-wavelengthphonons. Ourlong-wavelength\napproximation therefore becomes questionable and the\nresults may be not accurate at T= 50 K, but this should\nnot change the main conclusion that we can disregard\nthese diagrams.\nOur finding that the two-phonon contributions are\nso small can be understood in terms of the dimension-\nful prefactors of the decay rates (Eqs. (4.8-4.9) and\n(B14)): The one-phonon decay rate is proportional to\n~=(ma2)\u00197\u0002106s\u00001, while the two-phonon decay\nrate is proportional to ~3=(m2a4\")\u001933 s\u00001, where\n\"\u00191 meVis a typical phonon energy. The coupling con-\nstants for the magnon number non-conserving processes\nareBk;?\u00185 meVwhile the strongest two phonon cou-\npling which enhances the two-phonon process by about\na factor 100, but does not nearly compensate the pref-\nactor. The two phonon process is therefore three orders\nof magnitudes smaller than the contribution of the one\nphonon process. The physical reason appears to be the\nlarge mass density of YIG, i.e. the heavy yttrium atoms.\nAppendix C: Numerical integration\nThe magnon decay rate is given be the weighted den-\nsity of statesI=Z\nBZd3qf(q)\u000e(\"(q)); (C1)\nthat contain the Dirac delta function \u000e(\")that can be\neliminated to yield\nI=X\nqiZ\nAid2qf(q)\njr\"(q)j; (C2)\nwhere the qiare the zeros of \"(q)andAithe surfaces\ninside the Brillouin zone with \"(q) =\"(qi). The calcu-\nlation these integrals is a standard numerical problem in\ncondensed matter physics.\nFor aspherical Brillouin zone of radius qcand spherical\ncoordinates (q;\u0012;\u001e ),\nI=Z\u0019\n0d\u0012Z2\u0019\n0d\u001eZqc\n0dqq2sin(\u0012)f(q;\u0012;\u001e )\u000e(\"(q;\u0012;\u001e )):\n(C3)\nWhen\"(qi;\u0012;\u001e) = 0\n\u000e(\"(q;\u0012;\u001e )) =X\nqi(\u0012;\u001e)\u000e(q\u0000qi(\u0012;\u001e))\nj\"0(qi(\u0012;\u001e);\u0012;\u001e)j;(C4)\nwhere\"0=@\"=@qand\nI=Z\u0019\n0d\u0012Z2\u0019\n0d\u001eX\nqi(\u0012;\u001e)<qcq2\ni(\u0012;\u001e) sin(\u0012)\n\u0002f(qi(\u0012;\u001e);\u0012;\u001e)\nj\"0(qi(\u0012;\u001e);\u0012;\u001e)j;(C5)\nwhich is particularly useful when the zeros of \"(q;\u0012;\u001e )\ncan be calculated analytically for linear and quadratic\ndispersion relations.\nWe can also evaluate the integral Ifully numerically\nby broadening the delta function [60] e.g. replacing it by\na Gaussian [60],\n\u000e(\")!1p\u0019\u001bexp\u0012\n\u0000\"2\n\u001b2\u0013\n; (C6)\nwhere\u001bis the broadening parameter. 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Softw. 17, 437 (1991)."    },    {        "title": "1303.2895v1.Thermally_excited_spin_waves_in_a_nano_structure__thermal_gradient_vs__constant_temperature.pdf",        "content": "1\nThermally excited spin waves in a nano-structure: thermal gradient\nvs. constant temperature\nSimone Borlenghi1, Mattero Franchin2, Hans Fangohr2, Lars Bergqvist1,4, and Anna Delin1,3,4,\n1Department of Materials and Nanophysics, School of Information and Communication Technology,\nElectrum 229, Royal Institute of Technology, SE-16440 Kista, Sweden\n2Department of Engineering and the Environment, University of Southampton, SO17 1BJ Southampton, United Kingdom\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden\n4SeRC (Swedish e-Science Research Center), KTH, SE-10044 Stockholm, Sweden.\nUsing micromagnetic simulations, we have investigated spin dynamics in a nanostructure in the presence of thermal fluctuations.\nIn particular, we have studied the effects of a uniform temperature and of a uniform thermal gradient. In both cases, the stochastic\nfield leads to an increase of the precession angle of the magnetization, and to a mild decreas of the linewidth of the resonance peaks.\nOur results indicate that the Gilbert damping parameter plays the role of control parameter for the amplification of spin waves.\nI. I NTRODUCTION\nRecent experiments have shown that a temperature gradient\nacross a magnetic material (conductor or insulator) generates\na pure spin current. This phenomenon, known as the Spin-\nSeebeck Effect (SSE) [1], [2] has opened a new research field\nin which temperature, magnetism and electronic transport are\nconsidered simultaneusly [3].\nIn insulating ferromagnets, the spin current cannot be car-\nried by electrons and therefore a spin wave spin current [2]\nassociated with the magnetization dynamics in the sample is\nat the core of SSE. This effect is not limited to ferromagnets:\nin a recent experiment Padron-Hernandez et al. [4] suggest\nthat Spin Waves (SW), excited by a radio frequency generator\nin an yttrium iron garnet sample, are amplified in presence\nof a uniform thermal gradient through the sample. In their\npaper, they suggest that heat flow acts as a thermal torque that\nopposes the damping, in a similar way as spin torque does in\nspin transfer nano-oscillators [5].\nIn conducting ferromagnets, pure spin currents are generated\nby itinerant electrons with opposite spins that flow in opposite\ndirections as well as by magnons. Experiments and theoretical\nanalysis performed so far have mainly focused on the effect\nof thermal gradient at the interface between different materials\nthrough the Inverse Spin-Hall Effect [6], [7], or on domain\nwall motion [8], [9]. More recently, Machado et. al. [10]\ninvestigated through micromagnetic simulations the role of\nthermal gradient on the vortex core magnetization dynamics\nin a Permalloy disk.\nSo far, a micromagnetic study of the effect of heat flow on\nSW amplification in a ferromagnet, and a comparison with\nthe uniform temperature case, is missing. The possibility to\npropagate SW using a thermal gradient suggests many possible\napplications [11], and an understanding of the direct effect of\nheat flow on the magnetization dynamics is highly desirable.\nIn this paper we present detailed micromagnetic simulations\nof the SW spectrum of a nanoscructure in the presence\nCorresponding author: S. Borlenghi (email: simonebg@kth.se.)of a uniform and non uniform temperature distribution. For\npractical reasons we have chosen to use the parameters valid\nfor Permalloy (Py). However, our results should not been\ninterpreted as quantitatively specific for Py, but rather as a\nqualitative description of the effect of heat flow in a nanos-\ntructure. While the studies performed so far have focused on\nthe local properties of Spin-Seebeck devices (i.e. spin current\npropagation), we have here chosen to investigate the effect of\nthermal fluctuations on SW.\nII. F ORMULATION OF THE PROBLEM\nThe dynamics of the magnetization in a ferromagnet is\ndescribed, at the length scale of the exchange length, by the\nclassical LLG equation of motion [12], [13]\n@M\n@t=\u0000j\r0jM\u0002He\u000b+\u000b\nMsM\u0002@M\n@t; (1)\nwhere\r0=\u00002:21\u0002105m/(As) is the gyromagnetic ratio,\n\u000bis the dimensionless Gilbert damping parameter and Ms\nis the saturation magnetization of the sample. The first term\nat the right-hand side of Eq.(1) is the adiabatic torque, which\naccounts for the precession of the magnetization Maround the\neffective field He\u000b. The effective field itself is the functional\nderivative of the Gibbs free energy of the system with respect\nto the magnetization [14]. The second term on the right-hand\nside, proportional to \u000b, accounts for energy dissipation. In\ngeneral, He\u000bcontains a Zeeman term, which describes the\ninteraction of the precessing magnetization with the applied\nfield, as well as exchange, shape anisotropy and demagnetizing\nfield. [14].\nOur numerical simulations have been performed with Nmag\n[15], a micromagnetic package based on finite-element dis-\ncretization of the sample, which is represented by a network\nof sites (mesh). In this package, the magnetization dynamics at\neach sitekof the mesh is described by the LLG equation, and\nthe interactions with neighbouring sites are taken into account\nin the computation of the effective field.arXiv:1303.2895v1  [cond-mat.mes-hall]  12 Mar 20132\nFigure 1. (Color online) a)Magnetization vector precessing around the field\nHext, aligned with the zaxis. The transverse component \u0016\u0018\u0019mx+imy\nis the SW complex amplitude, which describes the magnetization precession\nat the Larmor frequency in the x-yplane. b)Cartoon of the Py nanopillar\nstudied in this paper, showing the mesh used in the computations. A uniform\nthermal gradient @xTis applied in the xdirection, and a uniform magnetic\nfieldHextis applied in the zdirection.\nTemperature is introduced by adding to the effective field\nHk\ne\u000bat sitek, a stochastic field Hk\nth, which is assumed to\nbe a Gaussian random process with zero mean and amplitudeD\nHk\nth;iHl\nth;jE\n= 2Dk\u000eij\u000ekl\u000e(t\u0000t0). Herei;j=x;y;z stand\nfor the cartesian components of the field, while k;lrefers to\nthe sites on the mesh. The fluctuation amplitude Dkis given\nby [16]\nDk=2\u000bkBTk\nMs\r0\u00160Vk\u0001t; (2)\nwherekBis the Boltzmann constant, \u00160is the vacuum\nmagnetic permeability, Tkis the temperature at site k,Vkis the\nvolume containing the magnetic moment at site kand\u0001tis\nthe integration time step. We have taken Vkas the average\nvolume per site, given by the total volume of the sample\ndivided by the number of sites of the mesh. In agreement\nwith the results of Ref. [17], we have neglected the weak\ntemperature dependence of \u000b. Notice that Eq. (2) is valid\nin a spin dynamics atomistic description, where each site\ncorresponds to a single precessing spin. In a micromagnetic\nframework, where each site corresponds to a large number of\nprecessing spins inside a finite volume, Dkhas to be multiplied\nby a scaling factor, which for Py is equal to 10 (see Ref. [18]).\nIII. N UMERICAL SIMULATIONS\nThe sample used in our simulations, shown in Fig. (1b)\nis a cuboid with dimensions of 100\u000230\u000230nm. The\nmesh contains 4100 nodes, giving a lattice distance smaller\nthan 3 nm. This is of the order of the Py exchange length.\nThe micromagnetic parameters are those of Py: the exchange\nstiffness of is J= 1:3\u000210\u000011J/m, while for the saturation\nmagnetization we have taken 0:86\u0002106A/m. In most of our\nsimulations, the sample is saturated by an external field Hext\nof 10 kOe applied in the zdirection, which corresponds to\nthe precession axis of the magnetization. The computations\nperformed at different fields will be clearly indicated. The\ntemperature Tis uniform along the yandzdirections, whilea uniform thermal gradient @xTis applied in the xdirection.\nThis configuration, with the field orthogonal to the thermal\ngradient, is the one commonly used in SSE experiments [1],\n[2], [4].\nThe quantity of interest in our simulations is the normalized\nmagnetization averaged over the volume Vof the sample:\nhm(t)i=1\nVM sR\nVM(r;t)dV. In particular, the complex SW\namplitude\n\u0018(t) =hMxi+ihMyip\nMs(Ms+Mz); (3)\ndescribes the transverse magnetization precessing in the x-y\nplane [see Fig. (1a)]. for small oscillations considered here,\nMz\u0019Ms, so that\u0018\u0019hmxi+ihmyi. The LLG equation,\nwritten in terms of this variable, reads [5]\nd\u0018\ndt=i!\u0018+ \u0000\u000b\ne\u000b\u0018: (4)\nFor a single spin, the solution \u0018(t)\u0019exp [(i!\u0000\u0000\u000b\ne\u000b)t]is an\noscillating signal with frequency !0=\rHe\u000b. In an extended\nconfined ferromagnet, the effective field is spatially dependent,\nso that the solution consists of a discrete set of SW modes with\nfrequencies !`, whose spatial profile depends on the geometry\nof the system [14], [19]. In the cuboid geometry, they consist\nof sine and cosine.\nThe time decay of the signal is controlled by the damping\nrate\u0000\u000b\ne\u000b, which in general is a function of \u000band of the reso-\nnant frequency [5]. For small precession angles and damping\nparameter considered here, each SW mode has an effective\ndamping \u0000\u000b\ne\u000b\u0019\u000b!`.\nThe SW power spectrum, given by the absolute value of the\nFourier transform of the SW amplitude, consists of Lorentzians\ncentered at the resonance frequencies, whose linewidths (full\nwidth at half height) correspond to \u0000\u000b\ne\u000b[5], [14]. When the\nsystem absorbs energy, the heights of the resonance peaks in-\ncrease, while their linewidths decrease [14]. This corresponds\nto and increase of the precession angle.\nIn our simulations, we started from an initial condition\nwhere the magnetization is uniformly tilted 8\u000ein thexdirec-\ntion with respect to the precession axis z. We then computed\nthe time evolution for 10 ns with a time step of 1 ps.\nFig. (2a) shows the typical output mx(t)andmy(t)of our\nsimulations at zero temperature, computed for \u000b=0.01, while\nFig. (2b) shows the corresponding power spectrum, which is\ndominated by the two modes f`,`= 1;2with frequencies 31.2\nand 40.1 GHz correspondingly. These low energy modes are\nthe only ones visible in the linear regime, since their damping\nis relatively weak. Fig. (2d) shows their spatial profile. These\nmodes have different linewidths, and consequently different\neffective dampings \u0000\u000b;`\ne\u000b.\nWe performed the computations have in a wide range of\nthermal gradients, spanning between 0 and 102K/nm. Fig.\n2c) shows the magnetization as a function of temperature, in\ngood agreement with Ref. [18]. T he gradients at which SW\namplification is observed are comprised between 10\u00001and10\nK/nm, which correspond to temperatures between 3 and 300\nK in the hottest part of the system, well below the Curie point.3\nFigure 2. (Color online) a)Time evolution of the transverse components of\nthe magnetization mxandmy, computed for \u000b= 0:01.b)Power spectrum\nof the system, with the two dominating modes and the corresponding effective\ndamping rate. c)Magnetization as a function of temperature. d)Spatial profile\nof the precessing component of the magnetization mx, which gives the profile\nof the standing SW modes in the system, the mode f1corresponds to the SW\nalong thezdirection, while the degenerate mode f2to the SW along the x\nandydirections.\nThe low-temperature side is set at 0 K, so that the system is\nstudied in the simplest possible condition.\nThe heights of the SW peaks fluctuate considerably from\nsample to sample, due to different realizations of the thermal\nfield. Below follows the results pertaining to simulations\nperformed on a single sample. An analysis of the signal\naveraged over many samples is performed in the subsequent\nsection.\nFig. (3) shows the power spectrum as a function of fre-\nquency and thermal gradient, for \u000b= 0:01. Both the color\ncode and the gradient are in logarithmic scale. The two modes\nf1andf2are clearly visible. Their frequency is independent\nof@xT, while their amplitude grows up to a factor 3, and\nreaches its maximum around 10K/nm. For larger gradients,\nthe temperature in the sample approaches the Curie point, so\nthat the magnetization drops and the spectrum is dominated\nby noise. Other thermally excited SW modes are visible at\nfrequencies around 70 GHz. Because of their higher damping,\ntheir height is almost one order of magnitude smaller than f1\nandf2, so that they give a negligible contribution to the SW\npower.\nIn our system, the damping parameter \u000bplays a key role,\nsince it is responsible both for the dissipation and the strength\nof the thermal field. To obtain better insight regarding the\neffect of the thermal gradient, we computed the linewidth (full\nwidth at half maximum) of the modes f`, as a function of @xT,\nfor different values of \u000b[Fig. (4) a) and b)]. Interestingly, we\nfind that the linewidth of the two modes f`, shows a quadratic\nFigure 3. (Color online) Power spectrum (in color code) as a function of\nfrequency and thermal gradient @xT. Two modes f1(31.2 GHz) and f2\n(40.1 GHz) are visible. The frequency of both modes is independent of @xT,\nwhile their amplitude grows as a function of it, and reaches its maximum\naround 10K/nm.\ndependence on the damping parameter \u000band on the thermal\ngradient.\n\u0000\u000b`\ne\u000b(@xT) = \u0000`\n+(\u000b)\u0000b`(\u000b)@xT\u0000(b`(\u000b)@xT)2;(5)\nwhere \u0000`\n+(\u000b)is the positive damping rate (i.e. the linewidth\nat zero thermal gradient) of mode `. From our numerical\nsimulations, we can extract \u0000`\n+(\u000b), which fits the functions\n\u00001\n+(\u000b) = 166\u000b=(1 + 7:5\u0002103\u000b2\u00005:5\u0002106\u000b3);(6)\n\u00002\n+(\u000b) = 180\u000b=(1 + 4:8\u0002103\u000b2\u00002:9\u0002105\u000b3);\nand of the coefficients b`(\u000b)\nb1(\u000b) = 1:7\u000210\u00009\u000b+ 4:5\u000210\u00005\u000b2; (7)\nb2(\u000b) = 3:53\u000210\u00008\u000b+ 7:69\u000210\u00005\u000b2:\nFor low values of \u000b(up to 3\u000210\u00003) the quadratic correction\nis negligible.\nLet us now turn to a discussion of the amplification of\nSW signal given by the temperature gradient. Fig. (4c) shows\nthe gainA\u000b(i.e.the ratio between the SW power at a given\n@xTand the SW power at @xT= 0) as a function of @xT,\ncalculated for different values of \u000b. Between 0 and 10 K/nm,\nthe signal grows as a function of @xT. At gradients larger than\n10 K/nm, corresponding to an average temperature of 1500 K\n(the Curie point of Py), the magnetization drops abruptly and\nthe signal is destroyed. Fig. (4d) shows the maximum gain\nAmax(computed at @xT= 10 K/nm) as a function of \u000b, which\nfits the functionAmax(\u000b) = 1 +c\u000b+ (c\u000b)2, withc= 129:9.\nFig. (4e) shows the gain as a function of the thermal gradient,\nfor\u000b= 10\u00002and an applied field between 1 and 9 KOe.\nThe amplification starts at a critical gradient gc. This critical\nthreshold, which is different for each mode, is plotted as a4\nFigure 4. (Color online) a)andb)Linewidths of the modes f1andf2(in\nGHz ) vs@xT(in Log scale), computed for different values of \u000b. Their\nlinewidths decrease quadratically with the thermal gradient (see text). c)Gain\nA\u000bvs@xT, computed for different values \u000b.A\u000bgrows exponentially with\nthe gradient. d)Maximum gainAmax (at@xT= 10 K/nm) vs\u000b. The\nmaximum gain increases quadratically with \u000b(see text). e)Gain Vs thermal\ngradient (in Log scale), computed for \u000b= 0:01and different values of the\napplied field. f)Critical gradient gc(in Log scale) vs \u000b, for both modes.\nfunction of \u000bin Fig. (4f). Remarkably, an increase in \u000bof\na factor six corresponds to a decrease in gcof two orders of\nmagnitude.\nThus, both the critical threshold and the SW amplification\nare dramatically affected by the Gilbert damping parameter \u000b,\nbut they do not depend on the applied field. A simple analysis\ncan give a qualitative understanding of this phenomenon. The\nSW power spectrum for the SW amplitude of a single spin\nobeying Eq.(4), in presence of thermal fluctuation, reads [20]\np(!) =D\n(!2\u0000!2\n0)2+ \u0000+(\u000b)2; (8)\nwhich is a Lorentzian centered around the resonance frequency\n!0\u0019\r0Hext, with damping rate \u0000+\u0019!0\u000b. The strength of\nthe fluctuations D/\u000bTis given in Eq. (2). Thus, the height\nof the resonant peaks is proportional to \u000b. On the contrary,\nThe applied field Hextcontrols the resonance frequency and\nthe damping rate, but not the height of the peaks, so that it\ndoes not influence the amplification. However, this qualitative\nconsideration is strictly valid only for a single spin in the linear\nregime, where the random field acts as an additive noise. In the\ncase of many interacting spins, and in a non-linear regime, the\nnoise acts in a multiplicative way [21]–[23], and the damping\nrate itself is not anymore linear function !0\u000b. Thus, the\nbehaviour of the resonant peaks is expected to depend strongly\non the geometry of the system, and on the intensity of the\nthermal gradient. The transition between linear and nonlinear\nregimes, and between additive and multiplicative noise, has\nnot yet been investigated in within the SSE.\nFigure 5. (Color online) Computations of gain and noise/signal ratio for\n\u000b= 0:003 (black dots) and \u000b= 0:01(red squares), and an applied field\nof 10 KOe. Panels a)andb)show respectively the gain and the noise/signal\nratio for a system with uniform thermal gradient (in Log scale). Panels c)\nandd)show respectively the gain and the noise/signal ratio for a system with\nuniform temperature.\nIV. C OMPARISON BETWEEN THERMAL GRADIENT AND\nCONSTANT TEMPERATURE\nThe results shown in the previous section suggest that a\nthermal gradient is an effective means to amplify SW in\na ferromagnet. However, a uniform temperature distribution\nmight lead to a similar effect. In both the isothermal and the\nuniform gradient case, only a part of the thermal excitation\ncontributes to SW amplification, the rest being wasted into\nthermal noise.\nIn this section we analyse these issues, comparing the effects\nof thermal gradient with the ones of uniform temperature.\nFor this purpose, we have averaged the SW spectrum over\n24 samples with different realization of the random thermal\nfield, and we have analyzed the average amplification and the\nnoise/signal ratio for a system with thermal gradient and for\nan isothermal one. The computations were performed for an\napplied field of 10 KOe, and for two values of the Gilbert\ndamping parameter, \u000b= 3\u000210\u00003and\u000b= 10\u00001.\nThe gradient in the non-isothermal system spans between\n0 and 10 K/nm, while the temperature in the isothermal one\nspans between 0 and 1500 K. The panels a) and c) in Fig. (5)\nshow the average gain of the SW signal respectively for the\nnon-isothermal and for the isothermal system, with the error\nbars indicating the variance of the average signal. In fact,\nthe amplification is similar in both systems, where SW are\namplified up to a factor 2 (for \u000b= 10\u00002), with large error\nbars exceeding 2.4.\nApart from a slight overestimation of the amplification\neffect, the study performed on a single sample agree with\nthe sample-averaged results. In the non-isothermal system,\nthe maximum amplification is between at 10 K/nm, which\ncorresponds to an average temperature between of 150 K in\nthe system, and a maximum temperature on the hotter side\nof 300 K. In the isothermal system, a similar amplification\nis reached at a temperature of 300 K. The panels b) and d)5\nshow the noise/signal ratios for the two systems, whose values\nare similar in the amplification region. Remarkably, the Gilbert\ndamping parameter \u000bhas a negligible influence on noise/signal\nratio.\nV. C ONCLUSIONS\nWe have performed a numerical study, which analyses\nthe effect of thermal excitations on SW amplification in a\nnanostructure. Our simulations suggest that the system should\nbehave in a similar way with a uniform temperature and\na uniform gradient. The Gilbert damping parameter affects\ndramatically the SW amplification: a system with higher\ndamping dissipates energy at a higher rate, but is also more\neffective in absorbing thermal energy.\nIn this study we have not taken into account the spin\ntransfer effect, since our objective consists in studying the\neffect of thermal gradient on the LLG equation in the simplest\npossible configuration, where the only source that affects the\nmagnetization dynamics is the stochastic thermal field. A\nthorough simulation of the spin-Seebeck effect, which takes\ninto account the effect of electronic transport, will be the\nsubject of a future paper.\nConcerning possible experimental study of SW excitations\nthrough a temperature gradient, the paper of Naletov et al. [19]\nanalyses the SW modes excited by various means (rf field and\nrf current) in a perpendicularly magnetized nanopillar, using\na magnetic resonance force microscope. This is an effective\nmeans to investigate samples buried under several electrodes,\nand is sensitive to all the SW modes excited in the system.\nWe believe that an experimental investigation performed with\na setup similar to the one of Ref. [19] could elucidate the effect\nof thermal gradient on SW modes with different symmetries.\nVI. A CKNOWLEDGMENT\nWe gratefully acknowledge the Carl Tryggers foundation\nand the Göran Gustafssons foundation for financial support.\nThis work was financed through the EU project NexTec, VR\n(the Swedish Research Council), and SSF (Swedish Founda-\ntion for Strategic Research). The computations were performed\non resources provided by the Swedish National Infrastructure\nfor Computing (SNIC) at the National Supercomputer Centre\nin LinkÃ˝ uping (NSC). We wish to thank M. Rasander and J.\nChico for reviewing the manuscript.\nREFERENCES\n[1] K. Uchida et al. Observation of the spin seebeck effect. Nature , 455:778–\n781, 2008.\n[2] K. Uchida et al. Spin seebeck insulator. Nat. Mater. , 9:894–897, 2010.\n[3] J. Sinova. Spin seebeck effect: Thinks globally but acts locally. Nat.\nMater. , 9:880–881, 2010.\n[4] E. Padrón-Hernández et al. Amplification of spin waves by thermal\nspin-transfer torque. Phys. Rev. Lett. , 107:197203, 2011.\n[5] A. Slavin and V . Tiberkevich. Nonlinear auto-oscillator theory of\nmicrowave generation by spin-polarized current. IEEE Transactions on\nMagnetics , 45(4):1875 –1918, 2009.\n[6] H. Adachi et al. arXiv:1010.2325v2 , 2010.\n[7] Xiao et al. Theory of magnon-driven spin seebeck effect. Phys. Rev. B ,\n81:214418, 2010.\n[8] Hals et al. Thermopower and thermally induced domain wall motion\nin (ga, mn)as. Solid State Communications , 150(11â ˘A¸ S12):461 – 465,\n2010. Spin Caloritronics.[9] D. Hinzke and U. Nowak. Domain wall motion by the magnonic spin\nseebeck effect. Phys. Rev. Lett. , 107:027205, 2011.\n[10] Tiago S. Machado, Tatiana G. Rappoport, and Luiz C. Sampaio. V ortex\ncore magnetization dynamics induced by thermal excitation. Applied\nPhysics Letters , 100(11):112404, 2012.\n[11] G. E. W. Bauer. arXiv:1107.4395v1 , 2011.\n[12] L. D. Landau and E. M. Lifshitz. To the theory of the dispersion of the\nferromagnetic-body permeability. In Collected papers . Ed. Pergamon,\n1965.\n[13] T.L. Gilbert. A phenomenological theory of damping in ferromagnetic\nmaterials. IEEE, Transaction on Magnetics , 40(6):3443 – 3449, 2004.\n[14] A. G. Gurevich and G. A. Melkov. Magnetization Oscillation and\nWaves . CRC Press, 1996.\n[15] T. Fischbacher et al. A systematic approach to multiphysics extensions\nof finite-element-based micromagnetic simulations: Nmag. Magnetics,\nIEEE Transactions on , 43(6):2896 –2898, 2007.\n[16] E. Martinez et al. Thermal effects in domain wall motion: Micromag-\nnetic simulations and analytical model. Phys. Rev. B , 75:174409, 2007.\n[17] P. Lubitz, M. Rubinstein, J. J. Krebs, and S.-F. Cheng. Frequency and\ntemperature dependence of ferromagnetic linewidth in exchange biased\npermalloy. Journal of Applied Physics , 89(11):6901–6903, 2001.\n[18] G. Grinstein and R. H. Koch. Coarse graining in micromagnetics. Phys.\nRev. Lett. , 90:207201, 2003.\n[19] V . V . Naletov et al. Identification and selection rules of the spin-\nwave eigenmodes in a normally magnetized nanopillar. Phys. Rev. B ,\n84:224423, 2011.\n[20] N. Saito M. Toda, R. Kubo. Statistical physics . Springer-Verlag, 1983.\n[21] José Luis Garcia-Palacios and Francisco J. Lázaro. Langevin-dynamics\nstudy of the dynamical properties of small magnetic particles. Phys.\nRev. B , 58:14937–14958, Dec 1998.\n[22] Werner Scholz, Thomas Schrefl, and Josef Fidler. Micromagnetic\nsimulation of thermally activated switching in fine particles. Journal\nof Magnetism and Magnetic Materials , 233(3):296 – 304, 2001.\n[23] O. Chubykalo, R. Smirnov-Rueda, J.M. Gonzalez, M.A. Wongsam, R.W.\nChantrell, and U. Nowak. Brownian dynamics approach to interacting\nmagnetic moments. Journal of Magnetism and Magnetic Materials ,\n266(1â ˘A¸ S2):28 – 35, 2003. Proceedings of the 4th International\nConference on Fine Particle Magnetism (ICFPM)."    },    {        "title": "1708.02008v2.Chiral_damping__chiral_gyromagnetism_and_current_induced_torques_in_textured_one_dimensional_Rashba_ferromagnets.pdf",        "content": "arXiv:1708.02008v2  [cond-mat.mes-hall]  31 Aug 2017Chiral damping, chiral gyromagnetism and current-induced torques in textured\none-dimensional Rashba ferromagnets\nFrank Freimuth,∗Stefan Bl¨ ugel, and Yuriy Mokrousov\nPeter Gr¨ unberg Institut and Institute for Advanced Simula tion,\nForschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, German y\n(Dated: May 17, 2018)\nWe investigate Gilbert damping, spectroscopic gyromagnet ic ratio and current-induced torques\nin the one-dimensional Rashba model with an additional nonc ollinear magnetic exchange field. We\nfind that the Gilbert damping differs between left-handed and right-handed N´ eel-type magnetic\ndomain walls due to the combination of spatial inversion asy mmetry and spin-orbit interaction\n(SOI), consistent with recent experimental observations o f chiral damping. Additionally, we find\nthat also the spectroscopic gfactor differs between left-handed and right-handed N´ eel- type domain\nwalls, which we call chiral gyromagnetism. We also investig ate the gyromagnetic ratio in the Rashba\nmodel with collinear magnetization, where we find that scatt ering corrections to the gfactor vanish\nfor zero SOI, become important for finite spin-orbit couplin g, and tend to stabilize the gyromagnetic\nratio close to its nonrelativistic value.\nI. INTRODUCTION\nIn magnetic bilayer systems with structural inversion\nasymmetry the energies of left-handed and right-handed\nN´ eel-type domain walls differ due to the Dzyaloshinskii-\nMoriya interaction (DMI) [1–4]. DMI is a chiral interac-\ntion, i.e., it distinguishes between left-handed and right-\nhanded spin-spirals. Not only the energy is sensitive to\nthe chirality of spin-spirals. Recently, it has been re-\nported that the orbital magnetic moments differ as well\nbetween left-handed and right-handed cycloidal spin spi-\nrals in magnetic bilayers [5, 6]. Moreover, the experi-\nmental observation of asymmetry in the velocity of do-\nmain walls driven by magnetic fields suggests that also\nthe Gilbert damping is sensitive to chirality [7, 8].\nIn this work we show that additionally the spectro-\nscopic gyromagnetic ratio γis sensitive to the chirality\nof spin-spirals. The spectroscopic gyromagnetic ratio γ\ncan be defined by the equation\ndm\ndt=γT, (1)\nwhereTis the torque that acts on the magnetic moment\nmand dm/dtis the resulting rate of change. γenters\nthe Landau-Lifshitz-Gilbert equation (LLG):\ndˆM\ndt=γˆM×Heff+αGˆM×dˆM\ndt,(2)\nwhereˆMis a normalized vector that points in the direc-\ntionofthemagnetizationandthetensor αGdescribesthe\nGilbert damping. The chiralityofthe gyromagneticratio\nprovides another mechanism for asymmetries in domain-\nwall motion between left-handed and right-handed do-\nmain walls.\nNot only the damping and the gyromagnetic ratio\nexhibit chiral corrections in inversion asymmetric sys-\ntems but also the current-induced torques. Amongthese torques that act on domain-walls are the adia-\nbatic and nonadiabatic spin-transfer torques [9–12] and\nthe spin-orbit torque [13–16]. Based on phenomenologi-\ncal grounds additional types of torques have been sug-\ngested [17]. Since this large number of contributions\nare difficult to disentangle experimentally, current-driven\ndomain-wall motion in inversion asymmetric systems is\nnot yet fully understood.\nThe two-dimensionalRashbamodel with an additional\nexchange splitting has been used to study spintronics\neffects associated with the interfaces in magnetic bi-\nlayer systems [18–22]. Recently, interest in the role of\nDMI in one-dimensional magnetic chains has been trig-\ngered [23, 24]. For example, the magnetic moments in\nbi-atomic Fe chains on the Ir surface order in a 120◦\nspin-spiral state due to DMI [25]. Apart from DMI, also\nother chiral effects, such as chiral damping and chiral\ngyromagnetism, are expected to be important in one-\ndimensional magnetic chains on heavy metal substrates.\nThe one-dimensional Rashba model [26, 27] with an ad-\nditional exchange splitting can be used to simulate spin-\norbit driven effects in one-dimensional magnetic wires on\nsubstrates [28–30]. While the generalized Bloch theo-\nrem[31]usuallycannotbeusedtotreatspin-spiralswhen\nSOI is included in the calculation, the one-dimensional\nRashba model has the advantage that it can be solved\nwith the help of the generalized Bloch theorem, or with a\ngauge-field approach [32], when the spin-spiral is of N´ eel-\ntype. WhenthegeneralizedBlochtheoremcannotbeem-\nployed one needs to resort to a supercell approach [33],\nuse open boundary conditions [34, 35], or apply pertur-\nbation theory [6, 9, 36–39] in order to study spintronics\neffects in noncollinear magnets with SOI. In the case of\nthe one-dimensional Rashba model the DMI and the ex-\nchangeparameterswerecalculatedbothdirectlybasedon\nagauge-fieldapproachandfromperturbationtheory[38].\nThe results from the two approaches were found to be in\nperfect agreement. Thus, the one-dimensional Rashba2\nmodel provides also an excellent opportunity to verify\nexpressions obtained from perturbation theory by com-\nparisonto the resultsfromthe generalizedBlochtheorem\nor from the gauge-field approach.\nIn this work we study chiral gyromagnetism and chi-\nral damping in the one-dimensional Rashba model with\nan additional noncollinear magnetic exchange field. The\none-dimensional Rashba model is very well suited to\nstudy these SOI-driven chiral spintronics effects, because\nit can be solved in a very transparent way without the\nneed for a supercell approach, open boundary conditions\nor perturbation theory. We describe scattering effects by\nthe Gaussian scalar disorder model. To investigate the\nrole of disorder for the gyromagnetic ratio in general, we\nstudyγalso in the two-dimensional Rashba model with\ncollinear magnetization. Additionally, we compute the\ncurrent-induced torques in the one-dimensional Rashba\nmodel.\nThis paper is structured as follows: In section IIA we\nintroduce the one-dimensional Rashba model. In sec-\ntion IIB we discuss the formalism for the calculation\nof the Gilbert damping and of the gyromagnetic ratio.\nIn section IIC we present the formalism used to calcu-\nlate the current-induced torques. In sections IIIA, IIIB,\nand IIIC we discuss the gyromagnetic ratio, the Gilbert\ndamping, and the current-induced torques in the one-\ndimensionalRashbamodel, respectively. Thispaperends\nwith a summary in section IV.\nII. FORMALISM\nA. One-dimensional Rashba model\nThe two-dimensional Rashba model is given by the\nHamiltonian [19]\nH=−/planckover2pi12\n2me∂2\n∂x2−/planckover2pi12\n2me∂2\n∂y2+\n+iαRσy∂\n∂x−iαRσx∂\n∂y+∆V\n2σ·ˆM(r),(3)\nwhere the first line describes the kinetic energy, the first\ntwotermsin thesecondline describethe RashbaSOI and\nthe last term in the second line describes the exchange\nsplitting. ˆM(r) is the magnetization direction, which\nmay depend on the position r= (x,y), andσis the\nvector of Pauli spin matrices. By removing the terms\nwith the y-derivatives from Eq. (3), i.e., −/planckover2pi12\n2me∂2\n∂y2and\n−iαRσx∂\n∂y, one obtains a one-dimensional variant of the\nRashba model with the Hamiltonian [38]\nH=−/planckover2pi12\n2me∂2\n∂x2+iαRσy∂\n∂x+∆V\n2σ·ˆM(x).(4)\nEq. (4) is invariant under the simultaneous rotation\nofσand of the magnetization ˆMaround the yaxis.Therefore, if ˆM(x) describes a flat cycloidal spin-spiral\npropagating into the xdirection, as given by\nˆM(x) =\nsin(qx)\n0\ncos(qx)\n, (5)\nwe can use the unitary transformation\nU(x) =/parenleftBigg\ncos(qx\n2)−sin(qx\n2)\nsin(qx\n2) cos(qx\n2)/parenrightBigg\n(6)\nin order to transform Eq. (4) into a position-independent\neffective Hamiltonian [38]:\nH=1\n2m/parenleftbig\npx+eAeff\nx/parenrightbig2−m(αR)2\n2/planckover2pi12+∆V\n2σz,(7)\nwherepx=−i/planckover2pi1∂/∂xis thexcomponent of the momen-\ntum operator and\nAeff\nx=−m\ne/planckover2pi1/parenleftbigg\nαR+/planckover2pi12\n2mq/parenrightbigg\nσy (8)\nis thex-component of the effective magnetic vector po-\ntential. Eq. (8) shows that the noncollinearity described\nbyqacts like an effective SOI in the special case of the\none-dimensional Rashba model. This suggests to intro-\nduce the concept of effective SOI strength\nαR\neff=αR+/planckover2pi12\n2mq. (9)\nBased on this concept of the effective SOI strength\none can obtain the q-dependence of the one-dimensional\nRashba model from its αR-dependence at q= 0. That a\nnoncollinear magnetic texture provides a nonrelativistic\neffective SOI has been found also in the context of the\nintrinsic contribution to the nonadiabatic torque in the\nabsence of relativistic SOI, which can be interpreted as\na spin-orbit torque arising from this effective SOI [40].\nWhile the Hamiltonian in Eq. (4) depends on position\nxthrough the position-dependence of the magnetization\nˆM(x) in Eq. (5), the effective Hamiltonian in Eq. (7) is\nnot dependent on xand therefore easy to diagonalize.\nB. Gilbert damping and gyromagnetic ratio\nIn collinear magnets damping and gyromagnetic ratio\ncan be extracted from the tensor [16]\nΛij=−1\nVlim\nω→0ImGR\nTi,Tj(/planckover2pi1ω)\n/planckover2pi1ω, (10)\nwhereVis the volume of the unit cell and\nGR\nTi,Tj(/planckover2pi1ω) =−i∞/integraldisplay\n0dteiωt/angbracketleft[Ti(t),Tj(0)]−/angbracketright(11)3\nis the retarded torque-torque correlation function. Tiis\nthei-th component of the torque operator [16]. The dc-\nlimitω→0 in Eq. (10) is only justified when the fre-\nquency of the magnetization dynamics, e.g., the ferro-\nmagnetic resonance frequency, is smaller than the relax-\nationrateoftheelectronicstates. In thin magneticlayers\nand monoatomicchains on substratesthis is typically the\ncase due to the strong interfacial disorder. However, in\nvery pure crystalline samples at low temperatures the\nrelaxation rate may be smaller than the ferromagnetic\nresonance frequency and one needs to assume ω >0 in\nEq. (10) [41, 42]. The tensor Λdepends on the mag-\nnetization direction ˆMand we decompose it into the\ntensorS, which is even under magnetization reversal\n(S(ˆM) =S(−ˆM)), and the tensor A, which is odd un-\nder magnetization reversal ( A(ˆM) =−A(−ˆM)), such\nthatΛ=S+A, where\nSij(ˆM) =1\n2/bracketleftBig\nΛij(ˆM)+Λij(−ˆM)/bracketrightBig\n(12)\nand\nAij(ˆM) =1\n2/bracketleftBig\nΛij(ˆM)−Λij(−ˆM)/bracketrightBig\n.(13)\nOne can show that Sis symmetric, i.e., Sij(ˆM) =\nSji(ˆM), while Ais antisymmetric, i.e., Aij(ˆM) =\n−Aji(ˆM).\nThe Gilbert damping may be extracted from the sym-\nmetric component Sas follows [16]:\nαG\nij=|γ|Sij\nMµ0, (14)\nwhereMis the magnetization. The gyromagnetic ratio\nγis obtained from Λ according to the equation [16]\n1\nγ=1\n2µ0M/summationdisplay\nijkǫijkΛijˆMk=1\n2µ0M/summationdisplay\nijkǫijkAijˆMk.\n(15)\nIt is convenient to discuss the gyromagnetic ratio in\nterms of the dimensionless g-factor, which is related to\nγthrough γ=gµ0µB//planckover2pi1. Consequently, the g-factor is\ngiven by\n1\ng=µB\n2/planckover2pi1M/summationdisplay\nijkǫijkΛijˆMk=µB\n2/planckover2pi1M/summationdisplay\nijkǫijkAijˆMk.(16)\nDue to the presence of the Levi-Civita tensor ǫijkin\nEq. (15) and in Eq. (16) the gyromagnetic ratio and the\ng-factoraredetermined solelyby the antisymmetriccom-\nponentAofΛ.\nVarious different conventions are used in the literature\nconcerning the sign of the g-factor [43]. Here, we define\nthe sign of the g-factor such that γ >0 forg >0 and\nγ <0 forg <0. According to Eq. (1) the rate of change\nofthemagneticmomentisthereforeparalleltothetorqueforpositive gandantiparalleltothetorquefornegative g.\nWhile we are interested in this work in the spectroscopic\ng-factor, and hence in the relation between the rate of\nchange of the magnetic moment and the torque, Ref. [43]\ndiscusses the relation between the magnetic moment m\nandtheangularmomentum Lthatgeneratesit, i.e., m=\nγstaticL. Since differentiation with respect to time and\nuse ofT= dL/dtleads to Eq. (1) our definition of the\nsigns ofgandγagrees essentially with the one suggested\nin Ref. [43], which proposes to use a positive gwhen the\nmagnetic moment is parallel to the angular momentum\ngeneratingitandanegative gwhenthemagneticmoment\nis antiparallel to the angular momentum generating it.\nCombining Eq. (14) and Eq. (15) we can express the\nGilbert damping in terms of AandSas follows:\nαG\nxx=Sxx\n|Axy|. (17)\nIntheindependentparticleapproximationEq.(10)can\nbe written as Λij= ΛI(a)\nij+ΛI(b)\nij+ΛII\nij, where\nΛI(a)\nij=1\nh/integraldisplayddk\n(2π)dTr/angbracketleftbig\nTiGR\nk(EF)TjGA\nk(EF)/angbracketrightbig\nΛI(b)\nij=−1\nh/integraldisplayddk\n(2π)dReTr/angbracketleftbig\nTiGR\nk(EF)TjGR\nk(EF)/angbracketrightbig\nΛII\nij=1\nh/integraldisplayddk\n(2π)d/integraldisplayEF\n−∞dEReTr/angbracketleftbigg\nTiGR\nk(E)TjdGR\nk(E)\ndE\n− TidGR\nk(E)\ndETjGR\nk(E)/angbracketrightbigg\n.(18)\nHere,dis the dimension ( d= 1 ord= 2 ord= 3),GR\nk(E)\nis the retarded Green’s function and GA\nk(E) = [GR\nk(E)]†.\nEFis the Fermi energy. ΛI(b)\nijis symmetric under the\ninterchange of the indices iandjwhile ΛII\nijis antisym-\nmetric. The term ΛI(a)\nijcontains both symmetric and\nantisymmetric components. Since the Gilbert damping\ntensor is symmetric, both ΛI(b)\nijand ΛI(a)\nijcontribute to\nit. Since the gyromagnetic tensor is antisymmetric, both\nΛII\nijand ΛI(a)\nijcontribute to it.\nIn order to account for disorder we use the Gaus-\nsian scalardisordermodel, wherethe scatteringpotential\nV(r) satisfies /angbracketleftV(r)/angbracketright= 0 and /angbracketleftV(r)V(r′)/angbracketright=Uδ(r−r′).\nThis model is frequently used to calculate transport\nproperties in disordered multiband model systems [44],\nbut it has also been combined with ab-initio electronic\nstructure calculations to study the anomalous Hall ef-\nfect [45, 46] and the anomalous Nernst effect [47] in tran-\nsition metals and their alloys.\nIn the clean limit, i.e., in the limit U→0, the an-\ntisymmetric contribution to Eq. (18) can be written as4\nAij=Aint\nij+Ascatt\nij, where the intrinsic part is given by\nAint\nij=/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn,m[fkn−fkm]ImTi\nknmTj\nkmn\n(Ekn−Ekm)2\n= 2/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn/summationdisplay\nll′fknIm/bracketleftbigg∂/angbracketleftukn|\n∂ˆMl∂|ukn/angbracketright\n∂ˆMl′/bracketrightbigg\n×\n×/summationdisplay\nmm′ǫilmǫjl′m′ˆMmˆMm′.\n(19)\nThe second line in Eq. (19) expresses Aint\nijin terms of\nthe Berry curvature in magnetization space [48]. The\nscattering contribution is given by\nAscatt\nij=/planckover2pi1/summationdisplay\nnm/integraldisplayddk\n(2π)dδ(EF−Ekn)Im/braceleftBigg\n−/bracketleftbigg\nMi\nknmγkmn\nγknnTj\nknn−Mj\nknmγkmn\nγknnTi\nknn/bracketrightbigg\n+/bracketleftBig\nMi\nkmn˜Tj\nknm−Mj\nkmn˜Ti\nknm/bracketrightBig\n−/bracketleftbigg\nMi\nknmγkmn\nγknn˜Tj\nknn−Mj\nknmγkmn\nγknn˜Ti\nknn/bracketrightbigg\n+/bracketleftBigg\n˜Ti\nknnγknm\nγknn˜Tj\nkmn\nEkn−Ekm−˜Tj\nknnγknm\nγknn˜Ti\nkmn\nEkn−Ekm/bracketrightBigg\n+1\n2/bracketleftbigg\n˜Ti\nknm1\nEkn−Ekm˜Tj\nkmn−˜Tj\nknm1\nEkn−Ekm˜Ti\nkmn/bracketrightbigg\n+/bracketleftBig\nTj\nknnγknm\nγknn1\nEkn−Ekm˜Ti\nkmn\n−Ti\nknnγknm\nγknn1\nEkn−Ekm˜Tj\nkmn/bracketrightBig/bracerightBigg\n.\n(20)\nHere,Ti\nknm=/angbracketleftukn|Ti|ukm/angbracketrightare the matrix elements of\nthe torque operator. ˜Ti\nknmdenotes the vertex corrections\nof the torque, which solve the equation\n˜Ti\nknm=/summationdisplay\np/integraldisplaydnk′\n(2π)n−1δ(EF−Ek′p)\n2γk′pp×\n×/angbracketleftukn|uk′p/angbracketright/bracketleftBig\n˜Ti\nk′pp+Ti\nk′pp/bracketrightBig\n/angbracketleftuk′p|ukm/angbracketright.(21)\nThe matrix γknmis given by\nγknm=−π/summationdisplay\np/integraldisplayddk′\n(2π)dδ(EF−Ek′p)/angbracketleftukn|uk′p/angbracketright/angbracketleftuk′p|ukm/angbracketright\n(22)\nand the Berry connection in magnetization space is de-\nfined as\niMj\nknm=iTj\nknm\nEkm−Ekn. (23)\nThe scattering contribution Eq. (20) formally resembles\nthe side-jump contribution to the AHE [44] as obtainedfrom the scalar disorder model: It can be obtained by\nreplacing the velocity operators in Ref. [44] by torque\noperators. We find thatin collinearmagnetswithoutSOI\nthis scattering contribution vanishes. The gyromagnetic\nratio is then given purely by the intrinsic contribution\nEq. (19). This is an interesting difference to the AHE:\nWithout SOI all contributions to the AHE are zero in\ncollinear magnets, while both the intrinsic and the side-\njump contributions are generally nonzero in the presence\nof SOI.\nIn the absence of SOI Eq. (19) can be expressed in\nterms of the magnetization [48]:\nAint\nij=−/planckover2pi1\n2µB/summationdisplay\nkǫijkMk. (24)\nInserting Eq. (24) into Eq. (16) yields g=−2, i.e., the\nexpected nonrelativistic value of the g-factor.\nTheg-factor in the presence of SOI is usually assumed\nto be given by [49]\ng=−2Mspin+Morb\nMspin=−2M\nMspin,(25)\nwhereMorbis the orbital magnetization, Mspinis the\nspin magnetization and M=Morb+Mspinis the total\nmagnetization. The g-factor obtained from Eq. (25) is\nusually in good agreementwith experimental results [50].\nWhen SOI is absent, the orbital magnetization is zero,\nMorb= 0, and consequently Eq. (25) yields g=−2 in\nthat case. Eq. (16) can be rewritten as\n1\ng=Mspin\nMµB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk=Mspin\nM1\ng1,(26)\nwith\n1\ng1=µB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk. (27)\nFrom the comparison of Eq. (26) with Eq. (25) it follows\nthat Eq. (25) holds exactly if g1=−2 is satisfied. How-\never, Eq. (27) usually yields g1=−2 only in collinear\nmagnets when SOI is absent, otherwise g1/negationslash=−2. In the\none-dimensionalRashbamodel the orbitalmagnetization\nis zero,Morb= 0, and consequently\n1\ng=µB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk. (28)\nThe symmetric contribution can be written as Sij=\nSint\nij+SRR−vert\nij+SRA−vert\nij, where\nSint\nij=1\nh/integraldisplayddk\n(2π)dTr/braceleftbig\nTiGR\nk(EF)Tj/bracketleftbig\nGA\nk(EF)−GR\nk(EF)/bracketrightbig/bracerightbig\n(29)5\nand\nSRR−vert\nij=−1\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TRR\niGR\nk(EF)TjGR\nk(EF)/bracerightBig\n(30)\nand\nSAR−vert\nij=1\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TAR\niGR\nk(EF)TjGA\nk(EF)/bracerightBig\n,\n(31)\nwhereGR\nk(EF) =/planckover2pi1[EF−Hk−ΣR\nk(EF)]−1is the retarded\nGreen’s function, GA\nk(EF) =/bracketleftbig\nGR\nk(EF)/bracketrightbig†is the advanced\nGreen’s function and\nΣR(EF) =U\n/planckover2pi1/integraldisplayddk\n(2π)dGR\nk(EF) (32)\nis the retarded self-energy. The vertex corrections are\ndetermined by the equations\n˜TAR=T+U\n/planckover2pi12/integraldisplayddk\n(2π)dGA\nk(EF)˜TAR\nkGR\nk(EF) (33)\nand\n˜TRR=T+U\n/planckover2pi12/integraldisplayddk\n(2π)dGR\nk(EF)˜TRR\nkGR\nk(EF).(34)\nIn contrast to the antisymmetric tensor A, which be-\ncomes independent of the scattering strength Ufor suf-\nficiently small U, i.e., in the clean limit, the symmetric\ntensorSdepends strongly on Uin metallic systems in\nthe clean limit. Sint\nijandSscatt\nijdepend therefore on U\nthrough the self-energy and through the vertex correc-\ntions.\nIn the case of the one-dimensional Rashba model, the\nequations Eq. (19) and Eq. (20) for the antisymmet-\nric tensor Aand the equations Eq. (29), Eq. (30) and\nEq. (31) for the symmetric tensor Scan be used both\nfor the collinear magnetic state as well as for the spin-\nspiral of Eq. (5). To obtain the g-factor for the collinear\nmagnetic state, we plug the eigenstates and eigenvalues\nof Eq. (4) (with ˆM=ˆez) into Eq. (19) and into Eq. (20).\nIn the case of the spin-spiral of Eq. (5) we use instead the\neigenstates and eigenvalues of Eq. (7). Similarly, to ob-\ntain the Gilbert damping in the collinear magnetic state,\nwe evaluate Eq. (29), Eq. (30) and Eq. (31) based on\nthe Hamiltonian in Eq. (4) and for the spin-spiral we use\ninstead the effective Hamiltonian in Eq. (7).\nC. Current-induced torques\nThe current-induced torque on the magnetization can\nbe expressed in terms of the torkance tensor tijas [15]\nTi=/summationdisplay\njtijEj, (35)whereEjis thej-th component of the applied elec-\ntric field and Tiis thei-th component of the torque\nper volume [51]. tijis the sum of three terms, tij=\ntI(a)\nij+tI(b)\nij+tII\nij, where [15]\ntI(a)\nij=e\nh/integraldisplayddk\n(2π)dTr/angbracketleftbig\nTiGR\nk(EF)vjGA\nk(EF)/angbracketrightbig\ntI(b)\nij=−e\nh/integraldisplayddk\n(2π)dReTr/angbracketleftbig\nTiGR\nk(EF)vjGR\nk(EF)/angbracketrightbig\ntII\nij=e\nh/integraldisplayddk\n(2π)d/integraldisplayEF\n−∞dEReTr/angbracketleftbigg\nTiGR\nk(E)vjdGR\nk(E)\ndE\n− TidGR\nk(E)\ndEvjGR\nk(E)/angbracketrightbigg\n.(36)\nWe decompose the torkance into two parts that are,\nrespectively, even and odd with respect to magnetiza-\ntion reversal, i.e., te\nij(ˆM) = [tij(ˆM) +tij(−ˆM)]/2 and\nto\nij(ˆM) = [tij(ˆM)−tij(−ˆM)]/2.\nIn the clean limit, i.e., for U→0, the even torkance\ncan be written as te\nij=te,int\nij+te,scatt\nij, where [15]\nte,int\nij= 2e/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn/negationslash=mfknImTi\nknmvj\nkmn\n(Ekn−Ekm)2(37)\nis the intrinsic contribution and\nte,scatt\nij=e/planckover2pi1/summationdisplay\nnm/integraldisplayddk\n(2π)dδ(EF−Ekn)Im/braceleftBigg\n/bracketleftBig\n−Mi\nknmγkmn\nγknnvj\nknn+Aj\nknmγkmn\nγknnTi\nknn/bracketrightBig\n+/bracketleftBig\nMi\nkmn˜vj\nknm−Aj\nkmn˜Ti\nknm/bracketrightBig\n−/bracketleftBig\nMi\nknmγkmn\nγknn˜vj\nknn−Aj\nknmγkmn\nγknn˜Ti\nknn/bracketrightBig\n+/bracketleftBig\n˜vj\nkmnγknm\nγknn˜Ti\nnn\nEkn−Ekm−˜Ti\nkmnγknm\nγknn˜vj\nknn\nEkn−Ekm/bracketrightBig\n+1\n2/bracketleftBig\n˜vj\nknm1\nEkn−Ekm˜Ti\nkmn−˜Ti\nknm1\nEkn−Ekm˜vj\nkmn/bracketrightBig\n+/bracketleftBig\nvj\nknnγknm\nγknn1\nEkn−Ekm˜Ti\nkmn\n−Ti\nknnγknm\nγknn1\nEkn−Ekm˜vj\nkmn/bracketrightBig/bracerightBigg\n.\n(38)\nis the scattering contribution. Here,\niAj\nknm=ivj\nknm\nEkm−Ekn=i\n/planckover2pi1/angbracketleftukn|∂\n∂kj|ukm/angbracketright(39)\nis the Berry connection in kspace and the vertex correc-\ntions of the velocity operator solve the equation\n˜vi\nknm=/summationdisplay\np/integraldisplaydnk′\n(2π)n−1δ(EF−Ek′p)\n2γk′pp×\n×/angbracketleftukn|uk′p/angbracketright/bracketleftbig\n˜vi\nk′pp+vi\nk′pp/bracketrightbig\n/angbracketleftuk′p|ukm/angbracketright.(40)6\nThe odd contribution can be written as to\nij=to,int\nij+\ntRR−vert\nij+tAR−vert\nij, where\nto,int\nij=e\nh/integraldisplayddk\n(2π)dTr/braceleftbig\nTiGR\nk(EF)vj/bracketleftbig\nGA\nk(EF)−GR\nk(EF)/bracketrightbig/bracerightbig\n(41)\nand\ntRR−vert\nij=−e\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TRR\niGR\nk(EF)vjGR\nk(EF)/bracerightBig\n(42)\nand\ntAR−vert\nij=e\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TAR\niGR\nk(EF)vjGA\nk(EF)/bracerightBig\n.(43)\nThe vertex corrections ˜TAR\niand˜TRR\niof the torque op-\nerator are given in Eq. (33) and in Eq. (34), respectively.\nWhile the even torkance, Eq. (37) and Eq. (38), be-\ncomes independent of the scattering strength Uin the\nclean limit, i.e., for U→0, the odd torkance to\nijdepends\nstrongly on Uin metallic systems in the clean limit [15].\nIn the case of the one-dimensional Rashba model, the\nequations Eq. (37) and Eq. (38) for the even torkance\nte\nijand the equations Eq. (41), Eq. (42) and Eq. (43) for\nthe odd torkance to\nijcan be used both for the collinear\nmagnetic state as well as for the spin-spiral of Eq. (5).\nTo obtain the even torkance for the collinear magnetic\nstate, we plug the eigenstates and eigenvalues of Eq. (4)\n(withˆM=ˆez) into Eq. (37) and into Eq. (38). In the\ncase of the spin-spiral of Eq. (5) we use instead the eigen-\nstates and eigenvalues of Eq. (7). Similarly, to obtain the\nodd torkance in the collinear magnetic state, we evaluate\nEq. (41), Eq. (42) and Eq. (43) based on the Hamilto-\nnian in Eq. (4) and for the spin-spiral we use instead the\neffective Hamiltonian in Eq. (7).\nIII. RESULTS\nA. Gyromagnetic ratio\nWe first discuss the g-factor in the collinear case, i.e.,\nwhenˆM(r) =ˆez. Inthis casetheenergybandsaregiven\nby\nE=/planckover2pi12k2\nx\n2m±/radicalbigg\n1\n4(∆V)2+(αRkx)2.(44)\nWhen ∆ V/negationslash= 0 orαR/negationslash= 0 the energy Ecan become\nnegative. The band structure of the one-dimensional\nRashba model is shown in Fig. 1 for the model param-\netersαR=2eV˚A and ∆ V= 0.5eV. For this choice of\nparameters the energy minima are not located at kx= 0\nbut instead at\nkmin\nx=±/radicalBig\n(αR)4m2−1\n4/planckover2pi14(∆V)2\n/planckover2pi12αR,(45)-0.4 -0.2 0 0.2 0.4\nk-Point kx [Å-1]00.511.5Band energy [eV]\nFIG. 1: Band structure of theone-dimensional Rashbamodel.\nand the corresponding minimum of the energy is given\nby\nEmin=−m(αR)4+1\n4/planckover2pi14\nm(∆V)2\n2/planckover2pi12(αR)2. (46)\nThe inverse g-factor is shown as a function of the SOI\nstrength αRin Fig. 2 for the exchange splitting ∆ V=\n1eV and Fermi energy EF= 1.36eV. At αR= 0 the\nscattering contribution is zero, i.e., the g-factor is de-\ntermined completely by the intrinsic Berry curvature ex-\npression, Eq. (24). Thus, at αR= 0 it assumes the value\n1/g=−0.5, which is the expected nonrelativistic value\n(see the discussion below Eq. (24)). With increasing SOI\nstrength αRthe intrinsic contribution to 1 /gis more and\nmore suppressed. However, the scattering contribution\ncompensates this decrease such that the total 1 /gis close\nto its nonrelativistic value of −0.5. The neglect of the\nscattering corrections at large values of αRwould lead in\nthis case to a strong underestimation of the magnitude\nof 1/g, i.e., a strong overestimation of the magnitude of\ng.\nHowever, at smaller values of the Fermi energy, the\ngfactor can deviate substantially from its nonrelativis-\ntic value of −2. To show this we plot in Fig. 3 the in-\nverseg-factor as a function of the Fermi energy when\nthe exchange splitting and the SOI strength are set to\n∆V= 1eV and αR=2eV˚A, respectively. As discussed in\nEq. (44) the minimal Fermi energyis negativ in this case.\nThe intrinsic contribution to 1 /gdeclines with increas-\ning Fermi energy. At large values of the Fermi energy\nthis decline is compensated by the increase of the vertex\ncorrections and the total value of 1 /gis close to −0.5.\nPrevious theoretical works on the g-factor have not\nconsidered the scattering contribution [52]. It is there-\nfore important to find out whether the compensation\nof the decrease of the intrinsic contribution by the in-7\n00.511.52\nSOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/gscattering\nintrinsic\ntotal\nFIG. 2: Inverse g-factor vs. SOI strength αRin the one-\ndimensional Rashba model.\n0 1 2 3 4 5 6\nFermi energy [eV]-0.6-0.4-0.201/gscattering\nintrinsic\ntotal\nFIG. 3: Inverse g-factor vs. Fermi energy in the one-\ndimensional Rashba model.\ncrease of the extrinsic contribution as discussed in Fig. 2\nand Fig. 3 is peculiar to the one-dimensional Rashba\nmodel or whether it can be found in more general cases.\nFor this reason we evaluate g1for the two-dimensional\nRashba model. In Fig. 4 we show the inverse g1-factor\nin the two-dimensional Rashba model as a function of\nSOI strength αRfor the exchange splitting ∆ V= 1eV\nand the Fermi energy EF= 1.36eV. Indeed for αR<\n0.5eV˚A the scattering corrections tend to stabilize g1at\nits non-relativistic value. However, in contrast to the\none-dimensional case (Fig. 2), where gdoes not deviate\nmuch from its nonrelativistic value up to αR= 2eV˚A,\ng1starts to be affected by SOI at smaller values of αR\nin the two-dimensional case. According to Eq. (26) the\nfullgfactor is given by g=g1(1+Morb/Mspin). There-\nfore, when the scattering corrections stabilize g1at its00.511.52\nSOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/g1\nscattering\nintrinsic\ntotal\nFIG. 4: Inverse g1-factor vs. SOI strength αRin the two-\ndimensional Rashba model.\nnonrelativistic value the Eq. (25) is satisfied. In the two-\ndimensional Rashba model Morb= 0 when both bands\nare occupied. For the Fermi energy EF= 1.36eV both\nbands are occupied and therefore g=g1for the range of\nparameters used in Fig. 4.\nThe inverse g1of the two-dimensional Rashba model\nis shown in Fig. 5 as a function of Fermi energy for the\nparameters ∆ V= 1eV and αR= 2eV˚A. The scattering\ncorrection is as large as the intrinsic contribution when\nEF>1eV. While the scattering correction is therefore\nimportant, it is not sufficiently large to bring g1close to\nits nonrelativistic value in the energy range shown in the\nfigure, which is a major difference to the one-dimensional\ncase illustrated in Fig. 3. According to Eq. (26) the g\nfactor is related to g1byg=g1M/Mspin. Therefore, we\nshow in Fig. 6 the ratio M/Mspinas a function of Fermi\nenergy. AthighFermienergy(whenbothbandsareoccu-\npied) the orbital magnetization is zeroand M/Mspin= 1.\nAt low Fermi energy the sign of the orbital magnetiza-\ntionis oppositeto the signofthe spin magnetizationsuch\nthat the magnitude of Mis smaller than the magnitude\nofMspinresulting in the ratio M/Mspin<1.\nNext, we discuss the g-factor of the one-dimensional\nRashba model in the noncollinear case. In Fig. 7 we\nplot the inverse g-factor and its decomposition into the\nintrinsic and scattering contributions as a function of\nthe spin-spiral wave vector q, where exchange splitting,\nSOI strength and Fermi energy are set to ∆ V= 1eV,\nαR= 2eV˚A andEF= 1.36eV, respectively. Since\nthe curves are not symmetric around q= 0, the g-\nfactor at wave number qdiffers from the one at −q, i.e.,\nthegyromagnetism in the Rashba model is chiral . At\nq=−2meαR//planckover2pi12theg-factorassumesthevalueof g=−2\nand the scattering corrections are zero. Moreover, the\ncurves are symmetric around q=−2meαR//planckover2pi12. These8\n0 2 4 6\nFermi energy [eV]-0.5-0.4-0.3-0.2-0.101/g1\nscattering\nintrinsic\ntotal\nFIG. 5: Inverse g1-factor 1 /g1vs. Fermi energy in the two-\ndimensional Rashba model.\n-2 0 2 4 6\nFermi energy [eV]00.511.52M/Mspin\nFIG. 6: Ratio of total magnetization and spin magnetization ,\nM/Mspin, vs. Fermi energy in the two-dimensional Rashba\nmodel.\nobservationscan be explained by the concept of the effec-\ntive SOI introduced in Eq. (9): At q=−2meαR//planckover2pi12the\neffective SOI is zero and consequently the noncollinear\nmagnet behaves like a collinear magnet without SOI at\nthis value of q. As we have discussed above in Fig. 2, the\ng-factor of collinear magnets is g=−2 when SOI is ab-\nsent, which explains why it is also g=−2 in noncollinear\nmagnets with q=−2meαR//planckover2pi12. If only the intrinsic con-\ntribution is considered and the scattering corrections are\nneglected, 1 /gvaries much stronger around the point of\nzero effective SOI q=−2meαR//planckover2pi12, i.e., the scattering\ncorrections stabilize gat its nonrelativistic value close to\nthe point of zero effective SOI.-2 -1 0 1\nWave vector q [Å-1]-0.8-0.6-0.4-0.201/g\nscattering\nintrinsic\ntotal\nFIG. 7: Inverse g-factor 1 /gvs. wave number qin the one-\ndimensional Rashba model.\n0 1 2 3 4\nScattering strength U [(eV)2Å]-0.4-0.200.20.4Gilbert Damping αG\nxx\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 8: Gilbert damping αG\nxxvs. scattering strength Uin the\none-dimensional Rashba model without SOI. In this case the\nvertex corrections and the intrinsic contribution sum up to\nzero.\nB. Damping\nWe first discuss the Gilbert damping in the collinear\ncase, i.e., we set ˆM(r) =ˆezin Eq. (4). The xxcom-\nponent of the Gilbert damping is shown in Fig. 8 as\na function of scattering strength Ufor the following\nmodel parameters: exchange splitting ∆ V=1eV, Fermi\nenergyEF= 2.72eV and SOI strength αR= 0. All\nthree contributions are individually non-zero, but the\ncontribution from the RR-vertex correction (Eq. (30)) is\nmuchsmallerthanthe onefromthe AR-vertexcorrection\n(Eq. (31)) and much smaller than the intrinsic contribu-\ntion (Eq. (29)). However, in this case the total damping\nis zero, because a non-zero damping in periodic crystals\nwith collinear magnetization is only possible when SOI\nis present [53].9\n1 2 3 4\nScattering strength U [(eV)2Å]050100150200250300Gilbert Damping αG\nxx\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 9: Gilbert damping αG\nxxvs. scattering strength Uin the\none-dimensional Rashba model with SOI.\nIn Fig. 9 we show the xxcomponent of the Gilbert\ndamping αG\nxxas a function of scattering strength Ufor\nthe model parameters ∆ V= 1eV, EF= 2.72eV and\nαR= 2eV˚A. ThedominantcontributionistheAR-vertex\ncorrection. The damping as obtained based on Eq. (10)\ndiverges like 1 /Uin the limit U→0, i.e., proportional\nto the relaxation time τ[53]. However, once the relax-\nation time τis larger than the inverse frequency of the\nmagnetization dynamics the dc-limit ω→0 in Eq. (10)\nis not appropriate and ω >0 needs to be used. It has\nbeenshownthattheGilbertdampingisnotinfinite inthe\nballistic limit τ→ ∞whenω >0 [41, 42]. In the one-\ndimensional Rashba model the effective magnetic field\nexerted by SOI on the electron spins points in ydirec-\ntion. Since a magnetic field along ydirection cannot lead\ntoatorquein ydirectionthe yycomponentoftheGilbert\ndamping αG\nyyis zero (not shown in the Figure).\nNext, we discuss the Gilbert damping in the non-\ncollinear case. In Fig. 10 we plot the xxcomponent\nof the Gilbert damping as a function of spin spiral\nwave number qfor the model parameters ∆ V= 1eV,\nEF= 1.36eV,αR= 2eV˚A, and the scattering strength\nU= 0.98(eV)2˚A. The curves are symmetric around\nq=−2meαR//planckover2pi12, because the damping is determined by\nthe effective SOI defined in Eq. (9). At q=−2meαR//planckover2pi12\nthe effective SOI is zero and therefore the total damp-\ning is zero as well. The damping at wave number qdif-\nfers from the one at wave number −q, i.e.,the damp-\ning is chiral in the Rashba model . Around the point\nq=−2meαR//planckover2pi12the damping is described by aquadratic\nparabola at first. In the regions -2 ˚A−1< q <-1.2˚A−1\nand 0.2˚A−1< q <1˚A−1this trend is interrupted by a W-\nshape behaviour. In the quadratic parabola region the\nlowest energy band crosses the Fermi energy twice. As\nshown in Fig. 1 the lowest band has a local maximum at-2-1.5-1-0.500.51\nWave vector q [Å-1]05101520Gilbert damping αxxG RR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 10: Gilbert damping αG\nxxvs. spin spiral wave number q\nin the one-dimensional Rashba model.\nq= 0. In the W-shape region this local maximum shifts\nupwards, approaches the Fermi level and finally passes it\nsuch that the lowest energy band crosses the Fermi level\nfour times. This transition in the band structure leads to\noscillations in the density of states, which results in the\nW-shape behaviour of the Gilbert damping.\nSince the damping is determined by the effective SOI,\nwe can use Fig. 10 to draw conclusions about the damp-\ning in the noncollinear case with αR= 0: We only need\nto shift all curves in Fig. 10 to the right such that they\nare symmetric around q= 0 and shift the Fermi energy.\nThus, for αR= 0 the Gilbert damping does not vanish\nifq/negationslash= 0. Since for αR= 0 angular momentum transfer\nfrom the electronic system to the lattice is not possible,\nthe damping is purely nonlocal in this case, i.e., angular\nmomentum is interchanged between electrons at differ-\nent positions. This means that for a volume in which\nthe magnetization of the spin-spiral in Eq. (5) performs\nexactly one revolution between one end of the volume\nand the other end the total angular momentum change\nassociated with the damping is zero, because the angu-\nlar momentum is simply redistributed within this volume\nand there is no net change of the angular momentum.\nA substantial contribution of nonlocal damping has also\nbeen predicted for domain walls in permalloy [35].\nIn Fig. 11 we plot the yycomponent of the Gilbert\ndamping as a function of spin spiral wave number qfor\nthe model parameters ∆ V= 1eV,EF= 1.36eV,αR=\n2eV˚A, and the scattering strength U= 0.98(eV)2˚A. The\ntotaldampingiszerointhiscase. Thiscanbeunderstood\nfrom the symmetry properties of the one-dimensional\nRashba Hamiltonian, Eq. (4): Since this Hamiltonian is\ninvariant when both σandˆMare rotated around the\nyaxis, the damping coefficient αG\nyydoes not depend on\nthe position within the cycloidal spin spiral of Eq. (5).10\n-3 -2 -1 0 1 2\nWave vector q [Å-1]-0.4-0.200.20.4Gilbert Damping αG\nyyRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 11: Gilbert damping αG\nyyvs. spin spiral wave number q\nin the one-dimensional Rashba model.\nTherefore, nonlocal damping is not possible in this case\nandαG\nyyhas to be zero when αR= 0. It remains to be\nshown that αG\nyy= 0 also for αR/negationslash= 0. However, this fol-\nlows directly from the observation that the damping is\ndetermined by the effective SOI, Eq. (9), meaning that\nany case with q/negationslash= 0 and αR/negationslash= 0 can always be mapped\nonto a case with q/negationslash= 0 and αR= 0. As an alternative\nargumentation we can also invoke the finding discussed\nabovethat αG\nyy= 0 in the collinearcase. Since the damp-\ning is determined by the effective SOI, it follows that\nαG\nyy= 0 also in the noncollinear case.\nC. Current-induced torques\nWe first discuss the yxcomponent of the torkance. In\nFig. 12 we show the torkance tyxas a function of the\nFermi energy EFfor the model parameters ∆ V= 1eV\nandαR= 2eV˚A when the magnetization is collinear and\npoints in zdirection. We specify the torkance in units of\nthe positive elementary charge e, which is a convenient\nchoice for the one-dimensional Rashba model. When\nthe torkance is multiplied with the electric field, we ob-\ntain the torque per length (see Eq. (35) and Ref. [51]).\nSince the effective magnetic field from SOI points in\nydirection, it cannot give rise to a torque in ydirec-\ntion and consequently the total tyxis zero. Interest-\ningly, the intrinsic and scattering contributions are indi-\nvidually nonzero. The intrinsic contribution is nonzero,\nbecause the electric field accelerates the electrons such\nthat/planckover2pi1˙kx=−eEx. Therefore, the effective magnetic\nfieldBSOI\ny=αRkx/µBchanges as well, i.e., ˙BSOI\ny=\nαR˙kx/µB=−αRExe/(/planckover2pi1µB). Consequently, the electron\nspin is no longer aligned with the total effective magnetic\nfield (the effective magnetic field resulting from both SOI-1 0 1 2 3 4 5 6\nFermi energy [eV]-0.2-0.100.10.2Torkance tyx [e]scattering\nintrinsic\ntotal\nFIG. 12: Torkance tyxvs. Fermi energy EFin the one-\ndimensional Rashba model.\nand from the exchange splitting ∆ V), when an electric\nfield is applied. While the total effective magnetic field\nlies in the yzplane, the electron spin acquires an xcom-\nponent, because it precesses around the total effective\nmagnetic field, with which it is not aligned due to the\napplied electric field [54]. The xcomponent of the spin\ndensity results in a torque in ydirection, which is the\nreason why the intrinsic contribution to tyxis nonzero.\nThe scattering contribution to tyxcancels the intrinsic\ncontribution such that the total tyxis zero and angular\nmomentum conservation is satisfied.\nUsing the concept of effective SOI, Eq. (9), we con-\nclude that tyxis also zero for the noncollinear spin-spiral\ndescribed by Eq. (5). Thus, both the ycomponent of the\nspin-orbit torque and the nonadiabatic torque are zero\nfor the one-dimensional Rashba model.\nTo show that tyx= 0 is a peculiarity of the one-\ndimensional Rashba model, we plot in Fig. 13 the\ntorkance tyxin the two-dimensional Rashba model. The\nintrinsic and scattering contributions depend linearly on\nαRfor small values of αR, but the slopes are opposite\nsuch that the total tyxis zero for sufficiently small αR.\nHowever, for largervalues of αRthe intrinsic and scatter-\ning contributions do not cancel each other and therefore\nthe total tyxbecomes nonzero, in contrast to the one-\ndimensional Rashba model, where tyx= 0 even for large\nαR. Several previous works determined the part of tyx\nthat is proportionalto αRin the two-dimensionalRashba\nmodel and found it to be zero [21, 22] for scalar disor-\nder, which is consistent with our finding that the linear\nslopes of the intrinsic and scattering contributions to tyx\nare opposite for small αR.\nNext, we discuss the xxcomponent of the torkance\nin the collinear case ( ˆM=ˆez). In Fig. 14 we plot\nthe torkance txxvs. scattering strength Uin the one-11\n00.511.52\nSOI strength αR [eVÅ]-0.00500.0050.01Torkance tyx [e/Å]\nscattering\nintrinsic\ntotal\nFIG. 13: Nonadiabatic torkance tyxvs. SOI parameter αRin\nthe two-dimensional Rashba model.\ndimensional Rashba model for the parameters ∆ V=\n1eV,EF= 2.72eV and αR= 2eV˚A. The dominant con-\ntribution is the AR-type vertex correction (see Eq. (43)).\ntxxdiverges like 1 /Uin the limit U→0 as expected for\nthe odd torque in metallic systems [15].\nIn Fig. 15 and Fig. 16 we plot txxas a function of\nspin-spiral wave number qfor the model parameters\n∆V= 1eV,EF= 2.72eV and U= 0.18(eV)2˚A. In Fig. 15\nthe case with αR= 2eV˚A is shown, while Fig. 16 illus-\ntrates the case with αR= 0. In the case αR= 0 the\ntorkance txxdescribes the spin-transfer torque (STT). In\nthe case αR/negationslash= 0 the torkance txxis the sum of contribu-\ntions from STT and spin-orbit torque (SOT). The curves\nwithαR= 0 andαR/negationslash= 0 are essentially related by a shift\nof ∆q=−2meαR//planckover2pi12, which can be understood based on\nthe concept of the effective SOI, Eq. (9). Thus, in the\nspecial case of the one-dimensional Rashba model STT\nand SOT are strongly related.\nIV. SUMMARY\nWe study chiral damping, chiral gyromagnetism and\ncurrent-induced torques in the one-dimensional Rashba\nmodel with an additional N´ eel-type noncollinear mag-\nnetic exchange field. In order to describe scattering ef-\nfects we use a Gaussian scalar disorder model. Scat-\ntering contributions are generally important in the one-\ndimensional Rashba model with the exception of the gy-\nromagnetic ratio in the collinear case with zero SOI,\nwhere the scattering correctionsvanish in the clean limit.\nIn the one-dimensional Rashba model SOI and non-\ncollinearity can be combined into an effective SOI. Us-\ning the concept of effective SOI, results for the mag-\nnetically collinear one-dimensional Rashba model can be\nused to predict the behaviour in the noncollinear case.1 2 3 4\nScattering strength U [(eV)2Å]-6-4-20Torkance txx [e]\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 14: Torkance txxvs. scattering strength Uin the one-\ndimensional Rashba model.\n-2 -1 0 1\nWave vector q [Å-1]-4-2024Torkance txx [e]RR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 15: Torkance txxvs. wave vector qin the one-\ndimensional Rashba model with SOI.\nIn the noncollinear Rashba model the Gilbert damp-\ning is nonlocal and does not vanish for zero SOI. The\nscattering corrections tend to stabilize the gyromagnetic\nratio in the one-dimensional Rashba model at its non-\nrelativistic value. 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Kar\nimec, Kapeldreef 75, 3001 Heverlee, Belgium\n(Dated: June 16, 2021)\nUsing broadband ferromagnetic resonance, we measure the damping parameter of [Co(5 Å)/Pt(3 Å)] \u00026mul-\ntilayers whose growth was optimized to maximize the perpendicular anisotropy. Structural characterizations in-\ndicate abrupt interfaces essentially free of intermixing despite the miscible character of Co and Pt. Gilbert damp-\ning parameters as low as 0.021 can be obtained despite a magneto-crystalline anisotropy as large as 106J/m3.\nThe inhomogeneous broadening accounts for part of the ferromagnetic resonance linewidth, indicating some\nstructural disorder leading to a equivalent 20 mT of inhomogenity of the effective field. The unexpectedly rel-\natively low damping factor indicates that the presence of the Pt heavy metal within the multilayer may not be\ndetrimental to the damping provided that intermixing is avoided at the Co/Pt interfaces.\nI. INTRODUCTION\nThanks to their large perpendicular magnetic anisotropy,\ntheir confortable magneto-optical signals and their easy\ngrowth by physical vapor deposition1, the [Co/Pt] multilay-\ners are one of the most popular system in spintronics. Early\nin spintronics history this model system was used to study\nthe physics of domain wall propagation2, for the develop-\nment of advanced patterning techniques3and for the assess-\nment of micromagnetic theories4. More recently they have\nbeen extensively used as high quality fixed layers in per-\npendicularly magnetized tunnel junctions, in particular in the\nmost advanced prototypes of spin-transfer-torque magnetic\nrandom access memories memories5. Despite the widespread\nuse of Co/Pt multilayers, their high frequency properties,\nand in particular their Gilbert damping parameter remains\nlargely debated with experimental values that can differ by\norders of magnitude from60.02 to 20 times larger7and the-\noretical calculations from circa 0.035 in Co 50Pt50alloys8\nto slightly smaller or substantially larger values in multilay-\ners made of chemically pure layers9. Direct measurements\nby conventional ferromagnetic resonance (FMR) are scarce\nas the high anisotropy of the material pushes the FMR fre-\nquencies far above610 GHz and results in a correlatively\nlow permeability that challenges the sensitivity of commer-\ncial FMR instruments10. As a result most of the measure-\nments of the damping of Co/Pt systems were made by all-\noptical techniques11,12in small intervals of applied fields. Un-\nfortunately this technique requires the static magnetization to\nbe tilted away from the out-of-plane axis and this tilt ren-\nders difficult the estimation of the contribution of the ma-\nterial disorder to the observed FMR linewidth using the es-\ntablished protocols13; this is problematic since in Co-Pt sys-\ntems there contributions of inhomogeneity line broadening\nand two-magnon scattering by the structural disorder (rough-\nness, interdiffusion, granularity,...) are often large14,15.\nIt is noticeable that past reports on the damping of Co/Pt\nsystems concluded that it ought to be remeasured in sam-\nples with atomically flat interfaces11. Besides, this measure-\nment should be done in out-of-plane applied field since thiseases the separation of the Gilbert damping contribution to\nthe linewidth from the contribution of structural disorder16.\nIn this paper, we measure the damping parameter of [Co(5\nÅ)/Pt(3 Å)]\u00026multilayers whose growth was optimized to\nmaximize perpendicular anisotropy anisotropy. The sputter-\ndeposition is performed at an extremely low17Argon pressure\nin remote plasma conditions which enables very abrupt inter-\nfaces that are essentially free of intermixing. We show that in\ncontrast to common thinking, the Gilbert damping parameter\nof Co/Pt multilayers can be low; its effective value is 0.021\nbut it still likely16includes contributions from spin-pumping\nthat our protocol can unfortunately not suppress.\nII. EXPERIMENTAL\nOur objective is to report the high frequency properties of\nCo/Pt multilayers that were optimized for high anisotropy.\nThe multilayer is grown by sputter-deposition on a Ru (50 Å)\nbuffer and capped with a Ru(70 Å)/Ta(70 Å)/Ru(100 Å)/Ta(10\nÅ, cap) sequence (bottom to top order). The Ru buffer was\nchosen because it does not mix with Co-based multilayers\neven under tough annealing conditions18. The stacks were\ndeposited by physical vapor deposition in a Canon-Anelva\nEC7800 300 mm system on oxidized silicon substrates at\nroom temperature. The Argon plasma pressure is kept at 0.02\nPa, i.e. substantially lower than the usual conditions of 0.1-0.5\nPa used in typical deposition machines17. As this multilayer\nis meant to be the reference layer of bottom-pinned magnetic\ntunnel junctions, in some samples (fig. 1) the non-magnetic\ncap is replaced the following sequence: Ta cap / Fe 60Co20B20\n/ MgO / Fe 60Co20B20/ Ta / [Co(5 Å)/Pt(3 Å)] \u00024/ Ru sim-\nilar to as in ref. 19 and 20 to form a bottom-pinned mag-\nnetic tunnel junction with properties designed for spin-torque\napplications21. All samples were annealed at 300\u000eC for 30\nminutes in an out-of-plane field of 1 T.arXiv:1807.04977v1  [cond-mat.mtrl-sci]  13 Jul 20182\nRu[Co5Å/Pt3Å]×6 RuMgOFeCoBFeCoBTaRuTa[Co5Å/Pt3Å]×4Co5Å(a)(b)(c)\nFIG. 1. (Color online). Structure and anisotropy of a Co-Pt multi-\nlayer. (a) Transmission Electron Micrograph of a magnetic tunnel\njunction that embodies our Co/Pt as hard multilayer at the bottom of\nthe reference synthetic antiferromagnet, similar to that of ref. 21. (b)\nEasy axis and (c) hard axis hysteresis loops of the hard multilayer\nwhen covered with Ru(70 Å)/Ta(70 Å)/Ru(100 Å)/Ta(10 Å, cap)\nIII. STRUCTURE\nX-ray reflectivity scans (not shown) indicate Bragg reflex-\nions at 2\u0012= 11 , 22.2 and 33.6 deg., consistent with the mul-\ntilayer periodicity of 8 Å. Consistently, the Pt to Co inter-\nmixing is sufficiently low that well formed 3Å Pt spacers can\nbe seen the Transmission Electron Micrograph after anneal-\ning [Fig. 1(a)]. Almost no roughness is observed throughout\nthe Co/Pt multilayer. We emphasize that this quality of inter-\nfaces is almost equivalent to that obtained in Molecular Beam\nEpitaxy conditions22. Indeed Co and Pt are strongly miscible\nsuch that hyperthermal (high energy) deposition techniques\nlike sputter deposition do not easily yield this low degree of\nintermixing, except when the deposition is conducted under\nsufficiently low plasma pressure in remote plasma conditions,\ni.e. when the substrate-to-target distance is large to avoid di-\nrect plasma exposure to the film being deposited.\nIV . ANISOTROPY\nThe magnetic material properties were measured by vibrat-\ning sample magnetometry (VSM) and Vector Network Ana-\nlyzer ferromagnetic resonance23in both easy (z) and hard axis\n(x) configurations. For VNA-FMR the sample is mechanically\npressed on the surface of a 50 microns wide coplanar waveg-\nuide terminated by an open circuit; data analysis is conducted\nfollowing the methods described in ref. 24. The VSM signal\nindicated a magnetization Ms= 8:5\u0002105kA/m if assuming\na magnetic thickness of 48 Å, i.e. assuming that the [Co(5\nÅ)/Pt(3 Å)]\u00026multilayer can be described as a single mate-\nrial. The loops indicate a perpendicular anisotropy with full\nremanence. The reversal starts at 46.8 mT and completes be-\n/s45/s50 /s45/s49 /s48 /s49 /s50/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48\n/s72/s101/s102/s102\n/s107/s49/s43/s72\n/s107/s50\n/s105/s110/s45/s112/s108/s97/s110/s101/s32\n/s102/s105/s101/s108/s100/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s48/s72/s32/s40/s84/s41/s72/s101/s102/s102\n/s107/s49/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32\n/s102/s105/s101/s108/s100FIG. 2. (Color online). FMR frequencies versus in-plane (cross sym-\nbols) or out-of-plane (square symbols) applied field. The bold lines\nare fits using Eq. 1 and 2, yielding \u00160(Hk1\u0000Ms) = 1:320\u00060:005T\nand\u00160Hk2= 0:120\u00060:015T.\nfore 48 mT with a tail-free square hysteresis loop. Careful\nattempts to demagnetize the sample using an acperpendicular\nfield failed to produce a multidomain state at remanence. This\nindicates that the lowest nucleation field in the whole sample\nis larger that the domain wall propagation field everywhere in\nthe film. This low propagation field indicates qualitatively that\nthe effective anisotropy field is very uniform. The hard axis\nloop indicates an in-plane saturation field of \u00191:3\u00060:1T in\nline with the expectations for such composition3. The round-\ning of the hard axis loop near saturation and its slight hys-\nteretic remanence [Fig. 1(c)] impedes a more precise deduc-\ntion of the anisotropy fields from the sole hard axis loop.\nWe shall instead use the ferromagnetic resonance data\nbecause magnetization eigenfrequencies constitute absolute\nmeasurements of the effective fields acting on the mag-\nnetization. Fig. 2 gathers the measured FMR frequen-\ncies measured for in-plane and out-of-plane applied fields\nfrom -2.5 to 2.5 T. To analyze the microwave susceptibil-\nity data, we assume an energy density that reads E=\n1\n2\u00160Hk1MSsin2\u0012+1\n4\u00160Hk2MSsin4\u0012with\u0012the (suppos-\nedly uniform) angle between the magnetization and the sam-\nple normal. Our convention is that the first and second order\nmagneto-crystalline anisotropy fields Hk1= 2K1=(\u00160MS)\nandHk2= 4K2=(\u00160MS)are positive when they favor per-\npendicular magnetization, i.e. \u0012= 0.\nIn that framework, the ferromagnetic resonance frequencies\nin out-of-plane and in-plane applied fields saturating the mag-\nnetization as:\n!perp=\r0(Hz+Hk1\u0000Ms) (1)\nand\n!in-plane =\r0p\nHx(Hx\u0000Hk1\u0000Hk2+Ms); (2)\nwhere\r0=j\rj\u00160is the gyromagnetic ratio. (For in-plane\nfieldsHxlower thanHx;sat=Hk1\u0000Hk2\u0000Msthe magne-\ntization is tilted. A straightforward energy minimization was3\nused to yield magnetization tilt \u0012that was subsequently in-\njected to a Smit and Beljers equation to yield the FMR fre-\nquency). The best fit to the experimental data is obtained\nfor\u00160(Hk1\u0000Ms) = 1:320\u00060:005 T (corresponding to\nK1= 106J/m3) and\u00160Hk2= 0:120\u00060:015T. Note that\nthe second order anisotropy is small but non negligible such\nthat the effective anisotropy fields deduced from easy and axis\naxis measurements would differ by circa 10% if Hk2was dis-\nregarded.\nV . GILBERT DAMPING\nA. Models\nWe now turn to the analysis of the FMR linewidth (Fig.\n3). As common in FMR, the linewidth comprises an intrin-\nsic Gilbert damping part and an extrinsic additional contribu-\ntion linked to the lateral non uniformity of the local effective\nfieldsHk1\u0000Ms. This can be gathered in a characteristic\nfield \u0001H0measuring the disorder relevant for FMR. In out-\nof-plane field FMR experiments, the proportionality between\neffective fields and resonance frequencies (Eq. 1) allows to\nwrite simply \u0001H0=1\n\r0\u0001!j!!0, and for the perpendicular\nmagnetization we follow the usual convention16and write:\n1\n\r0\u0001!perp= 2\u000b(Hz+Hk1\u0000Ms) + \u0001H0 (3)\nor equivalently \u0001!perp= 2\u000b!perp+\r0\u0001H0.\nFor in-plane magnetization, the intrinsic linewidth above\nthe in-plane saturation field is\n1\n\r0\u0001!Gilbert\nin-plane =\u000b(2Hx\u0000Hk1\u0000Hk2+Ms) (4)\nThe resonance frequency (Eq. 2) is non linear with the ef-\nfective fields such that the non uniformity \u0001H0of the local\neffective fields translates in a linewidth broadening through\nthe term\n1\n\r0\u0001!disorder\nin-plane =d!in-plane\nd(Ms\u0000Hk1)\u0001H0 (5)\nwhere the derivative term ispHx\n2pHx\u0000Hk1\u0000Hk2+Ms. In case of\nfinite disorder, this factor diverges at the spatially-averaged\nin-plane saturation field Hx;sat.\nB. Results\nFor each applied field, the real and imaginary parts of the\ntransverse permeability \u0016(f)were fitted with the one expected\nfor the uniform precession mode25with three free param-\neters: the FMR frequency !FMR=(2\u0019), the FMR linewidth\n\u0001!=(2\u0019))and a scaling (sensitivity) factor common to both\nreal and imaginary parts of \u0016(f)as illustrated in Fig. 3b.When plotting the symmetric lorentzian-shaped imagi-\nnary part of the transverse permeability versus the asymet-\nric lorentzian-shaped real part of the permeability for fre-\nquencies ranging from dcto infinity, a circle of diameter\nMs=[2\u000b(Hz+Hk1\u0000Ms)]should be obtained for a spatially\nuniform sample18. The finite disorder \u0001H0distorts the exper-\nimental imaginary part of the permeability towards a larger\nand more gaussian shape. It can also damp and smoothen the\npositive and negative peaks of the real part of the permeabil-\nity; when the applied field is such that the inhomogeneous\nbroadening is larger than the intrinsic Gilbert linewidth, this\nresults in a visible ellipticity of the polar plot of \u0016(f). In our\nexperimental polar plot of \u0016(f)(Fig. 3a) the deviations from\nperfect circularity are hardly visible which indicates that the\ninhomogeneous broadening is not the dominant contribution\nto the sample FMR linewidth in out-of-plane field conditions.\nTo confirm this point we have plotted in Fig. 3c the de-\npendence of FMR linewidth with FMR frequency for out-of-\nplane applied fields. A linear fit yields \u000b= 0:021\u00060:002and\n\u0001H0\u001940mT. A substantial part of the measured linewidth\nthus still comes from the contribution of the lateral inhomo-\ngeneity of the effective anisotropy field within the film. As a\nresult, low field measurements of the FMR linewidth would\nbe insufficient to disentangle the Gilbert contribution and the\nstructural disorder contributions to the total FMR linewidth.\nThe in-plane applied field FMR linewidth can in principle\nbe used to confirm this estimate of the damping factor. Un-\nfortunately we experience a weak signal to noise ratio in in-\nplane field FMR experiments such that only a crude estimation\nof the linewidth was possible.Within the error bar, it is inde-\npendent from the applied field from 1.7 to 2.5 T (not shown)\nwhich indicates that the disorder still substantially contributes\nto the linewidth even at our maximum achievable field. At\n2.5 T the linewidth was1\n2\u0019\u0001!in-plane\u00193:0\u00060:3GHz. This\nis consistent width the expectations of that would predict 2.2\nGHz of intrinsic contribution (Eq. 4) and 0.4 GHz of intrinsic\ncontribution (Eq. 5).\nVI. DISCUSSION\nWe conclude that the damping of Co/Pt multilayers can be\nof the order of 0.02 even for multilayers with anisotropies\namong the strongest reported (see ref. 26 for a survey of\nthe anisotropy of Co/Pt multilayers). Note that \u000b\u00190:021is\nstill a higher bound, as we are unable to measure and subtract\nthe spin-pumping contribution. Measuring the spin-pumping\ncontribution would require to vary the cap and buffer layer\nthicknesses without affecting the multilayer structure which\nis difficult to achieve. Still, we can conclude that the damp-\ning of Co/Pt multilayers lies in the same range as other high\nanisotropy multilayers like Co/Ni (ref. 18 and 27) and Co/Pd\n(ref. 16) systems.\nThis conclusion is in stark contrast with the common\nthinking7that Co/Pt systems alway s have a large damping.\nThis widespread opinion is based on the standard models\nof magneto-crystalline anisotropy28and damping29that pre-\ndicts that they both scale with the square of the spin-orbit4\n/s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s52/s52 /s52/s54 /s52/s56 /s53/s48 /s53/s50 /s53/s52 /s53/s54/s45/s49/s48/s48/s49/s48/s45/s49/s48 /s48 /s49/s48\n/s45/s49/s48/s48/s72/s97/s108/s102/s32/s70/s77/s82/s32/s108/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s71/s72/s122/s41\n/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s48/s46/s54/s32/s71/s72/s122/s32/s43/s32/s48/s46/s48/s50/s49 /s102/s80/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121/s32/s40/s114/s101/s97/s108/s32/s112/s97/s114/s116/s41/s80/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121/s32\n/s40/s105/s109/s97/s103/s105/s110/s97/s114/s121/s32/s112/s97/s114/s116/s41\n/s73/s109/s97/s103/s105/s110/s97/s114/s121/s32/s48/s46/s48/s52\n/s32/s50/s102/s32 /s102\n/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s32/s80/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121/s82/s101/s97/s108\n/s40/s99/s41/s40/s98/s41/s77/s97/s99/s114/s111/s115/s112/s105/s110/s32/s102/s105/s116/s32/s32\n/s40/s97/s41\nFIG. 3. (Color online). Gilbert damping of the Co/Pt multilayer. (a)\nImaginary part versus real part of the permeability for a field of 0.45\nT applied perpendicularly to the plane. The bold lines are theoretical\nmacrospin permeability curves with linewidth parameters (i.e. effec-\ntive damping) of 0.04. (b) Same data but versus frequency. (c): FMR\nhalf linewidth versus FMR frequency. The bold line is a guide to the\neye with a slope \u000b= 0:021and zero frequency intercept of 0.6 GHz.\ncoupling\u0018, which is particularly large in the Pt atoms. We\nemphasize that this expectation of large damping is not sys-\ntematically verified: in studies that make a thorough anal-\nysis of the effects of structural disorder, no correlation wasfound between anisotropy and damping in comparable mate-\nrial systems11,16. Rather, a large correlation was found be-\ntweenHk1and\u0001H0, indicating that when the anisotropy is\nstrong, any local inhomogeneity thereof has a large impact\non the FMR linewidth. Owing to the difficulty of achiev-\ning well-defined Co/Pt interfaces, we believe that past con-\nclusions on the large damping of Co/Pt systems were based\non systems likely to present some intermixing at the inter-\nface; indeed the presence of impurities with large spin-orbit\ncoupling considerably degrades (increases) the damping of a\nmagnetic material30and synchonously degrades (decreases)\nthe magneto-crystalline anisotropy31.\nVII. CONCLUSION\nIn summary, we have studied high anisotropy [Co(5 Å)/Pt(3\nÅ)]\u00026multilayers grown by low pressure remote plasma\nsputter deposition. The deposition conditions were tuned\nto achieve abrupt interfaces with little intermixing. Broad-\nband ferromagnetic resonance was used to measure the first\nand second order uniaxial anisotropy fields. With the mag-\nnetization measured by vibrating sample magnetometry, this\nyields an anisotropy energy of 1MJ/m3. The inhomogeneous\nbroadening accounts for part of the ferromagnetic resonance\nlinewidth, indicating some structural disorder leading to a\nequivalent 40 mT (or equivalently 600 MHz) of inhomogenity\nof the effective field in out-of-plane applied fields. This FMR-\nrelevant inhomogeneity is comparable to the coercivity of 47\nmT. Despite the large anisotropy a Gilbert damping parameter\nas low as 0.021\u00060.002 is obtained. This unexpectedly rela-\ntively low damping factor indicates that the presence of the Pt\nheavy metal within the multilayer can in some condition not\nbe detrimental to the damping. 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Couet, J. Swerts, and A. Furnemont, Applied\nPhysics Letters 108, 172409 (2016).21T. Devolder, J.-V . Kim, F. Garcia-Sanchez, J. Swerts, W. Kim,\nS. Couet, G. Kar, and A. Furnemont, Physical Review B 93,\n024420 (2016).\n22D. Weller, L. Folks, M. Best, E. E. Fullerton, B. D. Terris, G. J.\nKusinski, K. M. Krishnan, and G. Thomas, Journal of Applied\nPhysics 89, 7525 (2001).\n23C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and\nP. P. Freitas, Journal of Applied Physics 101, 074505 (2007).\n24C. Bilzer, T. Devolder, P. Crozat, and C. Chappert, IEEE Trans-\nactions on Magnetics 44, 3265 (2008).\n25T. Devolder, Physical Review B 96, 104413 (2017).\n26V . W. Guo, B. Lu, X. Wu, G. Ju, B. Valcu, and D. Weller, Journal\nof Applied Physics 99, 08E918 (2006).\n27J.-M. L. Beaujour, W. Chen, K. Krycka, C.-C. Kao, J. Z. Sun, and\nA. D. Kent, The European Physical Journal B 59, 475 (2007).\n28P. Bruno, Physical Review B 39, 865 (1989).\n29V . Kambersky, Physical Review B 76(2007), 10.1103/Phys-\nRevB.76.134416.\n30J. O. Rantscher, R. D. McMichael, A. Castillo, A. J. Shapiro, W. F.\nEgelhoff, B. B. Maranville, D. Pulugurtha, A. P. Chen, and L. M.\nConnors, Journal of Applied Physics 101, 033911 (2007).\n31T. Devolder, Physical Review B 62, 5794 (2000)."    },    {        "title": "1704.03326v1.CoFeAlB_alloy_with_low_damping_and_low_magnetization_for_spin_transfer_torque_switching.pdf",        "content": "arXiv:1704.03326v1  [cond-mat.mtrl-sci]  11 Apr 2017CoFeAlB alloy with low damping and low magnetization for spi n transfer torque\nswitching\nA. Conca,1,∗T. Nakano,2T. Meyer,1Y. Ando,2and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2Department of Applied Physics, Tohoku University, Japan\n(Dated: June 29, 2021)\nWe investigate the effect of Al doping on the magnetic propert ies of the alloy CoFeB. Comparative\nmeasurements of the saturation magnetization, the Gilbert damping parameter αand the exchange\nconstantasafunctionoftheannealingtemperature forCoFe B andCoFeAlBthinfilmsare presented.\nOur results reveal a strong reduction of the magnetization f or CoFeAlB in comparison to CoFeB.\nIf the prepared CoFeAlB films are amorphous, the damping para meterαis unaffected by the Al\ndoping in comparison to the CoFeB alloy. In contrast, in the c ase of a crystalline CoFeAlB film, α\nis found to be reduced. Furthermore, the x-ray characteriza tion and the evolution of the exchange\nconstant with the annealing temperature indicate a similar crystallization process in both alloys.\nThe data proves the suitability of CoFeAlB for spin torque sw itching properties where a reduction\nof the switching current in comparison with CoFeB is expecte d.\nThe alloy CoFeB is widely used in magnetic tunnel-\ning junctions in combination with MgO barriers due to\nthe large magnetoresistance effect originating in the spin\nfiltering effect [1–4]. For the application in magnetic ran-\ndom accessmemories, the switching ofthe magnetization\nof the free layer via spin transfer torque (STT) with spin\npolarised currents is a key technology. However, the re-\nquired currents for the switching process are still large\nand hinder the applicability of this technique. The criti-\ncal switching current density for an in-plane magnetized\nsystem is given by [5]\nJc0=2eαMStf(HK+Hext+2πMS)\n/planckover2pi1η(1)\nwhereeis the electron charge, αis the Gilbert damping\nparameter, MSis the saturation magnetization, tfis the\nthickness of the free layer, Hextis the external field, HK\nis the effective anisotropy field and ηis the spin transfer\nefficiency. Fromtheexpressionitisclearthat, concerning\nmaterial parameters, Jc0is ruled by the product αM2\nS.\nFor out-of-plane oriented layers, the term 2 πMSvanishes\nand then JC0is proportional to αMS[6]. Even in the\ncase of using pure spin currents created by the Spin Hall\neffect, the required currents are proportional to factors\nof the form αnMSwithn= 1,1/2 [7]. A proper strat-\negy to reduce the critical switching currents is then de-\nfined by reducing the saturation magnetization. This can\nbe achieved by the development of new materials or the\nmodification of known materials with promising prop-\nerties. Since the compatibility with a MgO tunneling\nbarrier and the spin filtering effect must be guaranteed\ntogether with industrial applicability, the second option\nis clearly an advantage by reducing MSin the CoFeB al-\nloy. In this case, a critical point is that this reduction\nmust not be associated with an increase of the damping\nparameter α.In the last years, several reports on doped CoFeB al-\nloys have proven the potential of this approach. The\nintroduction of Cr results in a strong reduction of MS\n[8–10], however, it is sometimes also causing an increase\nof the damping parameter [8]. The reduction of MSby\ndoping CoFeB with Ni is smaller compared to a doping\nwith Cr but it additionally leads to a reduction of α[8].\nFIG. 1. (Color online) θ/2θ-scans for 40 nm thick films of\nCo40Fe40B20(top) and Co 36Fe36Al18B10(bottom) showing\nthe evolution of crystallization with the annealing temper a-\nture.2\nFIG. 2. (Color online) Evolution of the saturation magnetiz a-\ntion for CoFeB and CoFeAlB with the annealing temperature\nTann.\nIn constrast, the reduction of magnetization with V is\ncomparable to Cr [9] but to our knowledge no values for\nαhave been published. In the case of doping of CoFeB\nby Cr or by V, a reduction of the switching current has\nbeen shown [8, 9].\nIn this Letter, we report on results on Al doped CoFeB\nalloy thin films characterized by ferromagnetic resonance\nspectroscopy. The dependence of MS, the Gilbert damp-\ning parameter αand the exchange constant on the an-\nnealing temperature is discussed together with the crys-\ntalline structure of the films and the suitability for STT\nswitching devices.\nThe samples are grown on Si/SiO 2substrates us-\ning DC (for metals) and RF (for MgO) sput-\ntering techniques. The layer stack of the sam-\nples is Si/SiO 2/Ta(5)/MgO(2)/FM(40)/MgO(2)/Ta(5)\nwhere FM = Co 40Fe40B20(CoFeB) or Co 36Fe36Al18B10\n(CoFeAlB). Here, the values in brackets denote the layer\nthicknesses in nm. In particular, the FM/MgO interface\nis chosen since it is widely used for STT devices based\non MTJs. This interface is also required to promote the\ncorrect crystallizationof the CoFeB layerupon annealing\nsince the MgO layer acts as a template for a CoFe bcc\n(100)-oriented structure [1–3] with consequent B migra-\ntion.\nThe dynamic properties and material parameters were\nstudied by measuring the ferromagnetic resonance using\na strip-line vector network analyzer (VNA-FMR). For\nthis, the samples were placed face down and the S 12\ntransmission parameter was recorded. A more detailed\ndescription of the FMR measurement and analysis pro-\ncedure is shown in previous work [11, 12]. Brillouin light\nspectroscopy (BLS) was additionally used for the mea-\nsurement of the exchange constant. The crystalline bulk\nproperties of the films were studied by X-ray diffractom-\netry (XRD) using the Cu-K αline.\nFigure 1 shows the θ/2θ-scans for CoFeB (top) andFIG. 3. (Color online) Linewidth at a fixed frequency of 18\nGHz (a) and Gilbert dampingparameter αdependenceon the\nannealing temperature T ann(b). The αvalue for Tann= 500◦\nis only a rough estimation since the large linewidth value do es\nnot allow for a proper estimation. The inset shows the linear\ndependence of the linewidth on the frequency exemplarily fo r\nCoFeAlB annealed at 350◦C and 400◦C. The red lines are a\nlinear fit.\nCoFeAlB (bottom) samples annealed at different tem-\nperatures T ann. The appearance of the CoFe diffractions\npeaks, as shown by the arrowsin Fig. 1 indicate the start\nof crystallization at high annealing temperatures of more\nthan 400◦C. In the case of lower annealing temperatures\nor the as-deposited samples, the FM layer is in an amor-\nphous state. The first appearance of the (200) diffraction\npeak occurs at the same point for both alloys showing a\nverysimilarthermalevolution. Thissimplifiesasubstitu-\ntion ofCoFeB by the Al alloyin tunneling junctions since\nthe same annealing recipes can be applied. This is criti-\ncal since the used values must be also optimized for the\nquality of the tunneling barrier itself or the perpendicu-\nlar anisotropy induced by the FM/MgO interface. The\n(110) CoFe peak is also present for both material compo-\nsitions owing to a partial texturing of the film. However,\nthe larger intensity of the (200) peak is not compatible\nwith a random crystallite orientation but with a domi-\nnant (100) oriented film [13, 14]. This is needed since the\nspin filtering effect responsible for the large magnetore-\nsistance effect in MgO-based junctions requires a (100)3\nFIG.4. (Color online)Dependenceoftheproduct αM2\nSonthe\nannealing temperature T annfor CoFeB and CoFeAlB. This\nquantityisrulingtheswitchingcurrentinin-planemagnet ized\nSTT devices as shown in Eq. 1.\norientation.\nThe dependence of the FMR frequency on the external\nmagnetic field is described by Kittel’s formula [15]. The\nvalue ofMeffextracted from the Kittel fit is related with\nthe saturation magnetization of the sample and the in-\nterfacial properties by Meff=MS−2K⊥\nS/µ0MSdwhere\nK⊥\nSis the interface perpendicular anisotropy constant.\nFor the thickness used in this work (40 nm) and physi-\ncally reasonable K⊥\nSvalues, the influence of the interface\nis negligible and therefore Meff≈MS. For details about\nthe estimation of Meffthe reader is referred to [12].\nFigure 2 shows the obtained values for MSfor all sam-\nples. AstrongreductionforCoFeAlBin comparisonwith\nstandard CoFeB is observed and the relative difference is\nmaintained for all T ann. The evolution with annealing is\nvery similar for both alloys. Significantly, the increase in\nMSstartsforvaluesofT annlowerthan expectedfromthe\nappearance of the characteristic CoFe diffraction peaks\nin the XRD data (see Fig. 1). This shows that the mea-\nsurement of MSis the more sensitive method to probe\nthe change of the crystalline structure.\nFor CoFeB a saturationvalue around MS≈1500kA/m\nis reached at T ann= 450◦C. This is compatible with val-\nues reported for CoFe (1350-1700 kA/m) [16, 17] and\nCoFeB (1350-1500 kA/m) [17, 18]. On the contrary, for\nCoFeAlB the introduction of Al reduces the magnetiza-\ntion of the samples and the annealing does not recover\nto CoFe-like values.\nFigure 3(a) shows the dependence of the magnetic\nfield linewidth on T annmeasured at a fixed frequency\nof 18 GHz. From the linear dependence of this linewidth\non the FMR frequency, the Gilbert damping parameter\nis extracted (as exemplarily shown for the CoFeAlB al-\nloy in the inset in Fig. 3(b)) and the results are shown\nin Fig. 3(b). For T annvalues up to 350◦C, where theFIG. 5. (Color online) Dependence of the exchange con-\nstantAexon the annealing temperature T annfor CoFeB and\nCoFeAlB. The top panels show typical BLS spectra for ma-\nterials (see text).\namorphousphaseisstill dominating, almostnodifference\nbetween both alloys is observed. With increasing tem-\nperature the damping increases for both alloys but the\nevolution is different. For CoFeAlB the increase starts\nalmost abruptly at T ann= 400◦C, reaches a maximum\naroundα= 0.02 and then decreases again to α= 0.012\nfor Tann= 500◦C. In contrast, the increase for CoFeB\nis more smoothly with T annand increases stadily with\nhigher T ann. In fact, due to the large linewidths reached\nfor Tann= 500◦C, the value of αcannot be properly\nestimated and only a lower limit of 0.03-0.04 can be\ngiven. This situation is represented by the dashed line\nin Fig. 3(b). It is important to note here that when the\ncrystallization process is fulfilled (i.e. for T ann= 500◦C)\nαis much lower for the Al doped alloy. This is rele-\nvant for the application in tunneling junctions where a\nfull crystallization is required for the presence of the spin\nfiltering effect originating large magnetoresistance values\nin combination with MgO barriers [4].\nFor further comparison of both alloys, the quantity\nαM2\nShasbeen calculatedand plotted in Fig. 4. As shown\nin Eq. 1, this value is ruling the critical switching current\nin in-plane magnetized systems. We observe for the al-\nloys showing a mostly amorphous phase (T ann<400◦C)\na slight improvement for CoFeAlB in comparison with\nCoFeB due to the lower MS. However, for fully crys-\ntalline films (T ann= 500◦C), the CoFeAlB shows a much\nsmaller value for αM2\nS. Since a full crystalline phase is\nneeded for any application of this alloy in MTJ-based de-\nvices, this denotes a major advantage of this compound\ncompared to standard.\nThe exchange constant Aexis a critical parameter that\nis strongly influenced by the introduction of Al. Its esti-\nmationinrequiredformodelingthespintorqueswitching\nbehaviorofthe alloys. The accessto the constantisgiven4\nby the dependence of the frequency of the perpendicular\nstanding spin-wave (PSSW) modes on the external static\nmagnetic field [19]. As shown in previous works [12, 20],\nitispossibletoobservethePSSWmodesinmetallicfilms\nwith a standard VNA-FMR setup. However, the signal\nis strongly reduced compared to the FMR peak. For the\nsamples presented in this paper, the PSSW peak could\nnot be observed for T ann>400◦C since the increased\ndamping leads to a broadening and lowering of the peak\nwhich prevents the estimation of Aex. For this reason,\nBLS spectroscopy is used for the measurement of the fre-\nquency position of the PSSW modes. This technique has\nalargersensitivityforthePSSWmodesthanVNA-FMR.\nFigure 5(c) shows the evolution of Aexupon annealing\nfor both alloys. For the films dominated by the amor-\nphous phase the value is much lower for CoFeAlB which\nis also compatible with the lower magnetization. How-\never, asthe crystallizationevolves,theexchangeconstant\nincreases stronger than for CoFeB and the same value is\nobtained for the fully crystallized films. This fact points\nto a similar role of Al and B during the crystallization\nprocess: when the CoFe crystallitesform, the light atoms\nare expelled forming a Al-B-rich matrix embedding the\nmagnetic crystallites. This explains also the similar evo-\nlution observed in the XRD data shown in Fig. 1. The\nlower maximal magnetization obtained for the CoFeAlB\ncan be explained by the reduced CoFe content but also\na certain number of residual Al and B atoms in the crys-\ntallites, which may differ for both alloys.\nTheAexvalues for as-deposited CoFeB films are very\nsimilar to previous reports [12, 20, 21]. Concerning the\nvalues for the crystallized samples, since the properties\nare strongly dependent on the B content and of the ra-\ntio between Co and Fe as well as on the exact annealing\nconditions, a comparison with literature has to be made\ncarefully. Nevertheless, the maximal value and the evo-\nlution with T annfor CoFeB is similar to the one reported\nby some of the authors [12]. Also results for alloys with\nthe same B content arecompatiblewith ourdata [22, 23].\nCoFeB films with reduced B content show larger values\n[17], the same is true for CoFe alloys with values between\n3.84-2.61 ×1011J/m depending on the exact stoichiome-\ntry [16, 17]. This may again be a hint that a rest of Al\nor B is present in the CoFe crystallites.\nIn summary, the presented experimental results show\nthat CoFeAlB is a good candidate as alternative to\nCoFeB for spin torque switching devices due to the re-\nduction of the factor αM2\nSwhich dominates the critical\nswitching current. This reduction was found to originate\nfrom a strong reduction of the saturation magnetization\nandadecreaseddampingparameter αforfullycrystalline\nCoFeAlB films. Furthermore, the results reveal a larger\nthermal stability of the damping properties in CoFeAlB\ncompared to CoFeB. The absolute values of MSand the\nexchange constant Aexfor crystalline films point to a for-\nmation of CoFe crystallines with a non-vanishing contentof the lights atoms embedded in a B or Al matrix.\nFinancial support by M-era.Net through the\nHEUMEM project, the DFG in the framework of\nthe research unit TRR 173 Spin+X and by the JSPS\nCore-to-Core Program is gratefully acknowledged.\n∗conca@physik.uni-kl.de\n[1] S.YuasaandD.D.Djayaprawira, J.Phys.D:Appl.Phys.\n40, R337-R354 (2007).\n[2] S. Yuasa,Y. Suzuki, T. Katayama, and K. Ando,\nAppl. Phys. Lett. 87, 242503 (2005).\n[3] Y. S. Choi, K. Tsunekawa, Y. Nagamine, and\nD. Djayaprawira, J. Appl. Phys. 101, 013907 (2007).\n[4] X.-G. Zhang and W. H. Butler, J. Phys.: Condens. Mat-\nter15R1603, (2003).\n[5] Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen,\nL.-C. Wang, and Y. Huai, J. Phys. D: Appl. Phys. 19,\n165209 (2007).\n[6] K. L. Wang, J. G. Alzate, and P.K. Amiri, J. Phys. D:\nAppl. Phys. 46, 074003 (2013).\n[7] T. Taniguchi, S. Mitani, and M. Hayashi, Phys. Rev. B\n92, 024428 (2015).\n[8] K. Oguz, M. Ozdemir, O. Dur, and J. M. D. Coey,\nJ. Appl. Phys. 111, 113904 (2012).\n[9] H. Kubota, A. Fukushima, K. Yakushiji, S. Yakata,\nS. Yuasa, K. Ando, M. Ogane, Y.Ando, andT. Miyazaki,\nJ. Appl. Phys. 105, 07D117 (2009).\n[10] Y. Cui, M. Ding, S. J. Poon, T. P. Adl, S. Keshavarz,\nT. Mewes, S. A. Wolf, and J. Lu, J. Appl. Phys. 114,\n153902 (2013).\n[11] A. Conca, S. Keller, L. Mihalceanu, T. Kehagias,\nG. P. Dimitrakopulos, B. Hillebrands, and E. Th. Pa-\npaioannou, Phys. Rev. B 93, 134405 (2016).\n[12] A. Conca, E. Th. Papaioannou, S. Klingler, J. Greser,\nT. Sebastian, B. Leven, J. L¨ osch, and B. Hillebrands,\nAppl. Phys. Lett. 104, 182407 (2014).\n[13] G. Concas, F. Congiu, G. Ennas, G. Piccaluga, and\nG. Spano. J. of Non-Crystalline Solids 330, 234 (2003).\n[14] C. Y. You, T. Ohkubo, Y. K. Takahashi, and K. Hono,\nJ. Appl. Phys. 104, 033517 (2008).\n[15] C. Kittel, Phys. Rev. 73, 155 (1948).\n[16] X. Liu, R. Sooryakumar, C. J. Gutierrez, and\nG. A. Prinz, J. Appl. Phys. 75, 7021 (1994).\n[17] C. Bilzer, T. Devolder, J.-V. Kim, G. Counil, C. Chap-\npert, S. Cardoso, and P. P. Freitas, J. Appl. Phys. 100,\n053903 (2006).\n[18] X. Liu, W. Zhang, M. J. Carter, and G. Xiao, J. Appl.\nPhys.110, 033910 (2011).\n[19] S. O. Demokritov, B. Hillebrands, Spin Dynamics in\nConfined Magnetic Structures I , Springer, Berlin, (2002).\n[20] A. Conca, J. Greser, T. Sebastian, S. Klingler, B. Obry,\nB. Leven, and B. Hillebrands, J. Appl. Phys. 113, 213909\n(2013).\n[21] J. Cho, J.Jung, K.-E.Kim, S.-I.Kim, S.-Y.Park, andM.-\nH. Jung, C.-Y. You, J. of Magn and Magn. Mat. 339, 36\n(2013).\n[22] A. Helmer, S. Cornelissen, T. Devolder, J.-V. Kim,\nW. van Roy, L. Lagae, and C. Chappert, Phys. Rev. B\n81, 094416 (2010).\n[23] H.Sato, M.Yamanouchi, K.Miura, S.Ikeda, R.Koizumi,5\nF. Matsukura, and H. Ohno, IEEE Magn. Lett., 3,\n3000204 (2012)."    },    {        "title": "1504.00199v1.Multiscale_modeling_of_ultrafast_element_specific_magnetization_dynamics_of_ferromagnetic_alloys.pdf",        "content": "Multiscale modeling of ultrafast element-specific magnetization dynamics of\nferromagnetic alloys\nD. Hinzke1,∗U. Atxitia1,2, K. Carva3,4, P. Nieves5, O. Chubykalo-Fesenko5, P. M. Oppeneer4, and U. Nowak1\n1Fachbereich Physik, Universit¨ at Konstanz, D-78457 Konstanz, Germany\n2Zukunftskolleg at Universit¨ at Konstanz, D-78457 Konstanz, Germany\n3Faculty of Mathematics and Physics, DCMP, Charles University,\nKe Karlovu 5, CZ-12116 Prague 2, Czech Republic\n4Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden and\n5Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain\n(Dated: March 31, 2015)\nA hierarchical multiscale approach to model the magnetization dynamics of ferromagnetic ran-\ndom alloys is presented. First-principles calculations of the Heisenberg exchange integrals are linked\nto atomistic spin models based upon the stochastic Landau-Lifshitz-Gilbert (LLG) equation to\ncalculate temperature-dependent parameters (e.g., effective exchange interactions, damping param-\neters). These parameters are subsequently used in the Landau-Lifshitz-Bloch (LLB) model for\nmulti-sublattice magnets to calculate numerically and analytically the ultrafast demagnetization\ntimes. The developed multiscale method is applied here to FeNi (permalloy) as well as to copper-\ndoped FeNi alloys. We find that after an ultrafast heat pulse the Ni sublattice demagnetizes faster\nthan the Fe sublattice for the here-studied FeNi-based alloys.\nI. INTRODUCTION\nExcitation of magnetic materials by powerful femtosec-\nond laser pulses leads to magnetization dynamics on the\ntimescale of exchange interactions. For elemental fer-\nromagnets the emerging dynamics can be probed us-\ning conventional magneto-optical methods1,2. For mag-\nnets composed of several distinct elements, such as fer-\nrimagnetic or ferromagnetic alloys, the individual spin\ndynamics of the different elements can be probed em-\nploying ultrafast excitation in combination with the\nfemtosecond-resolved x-ray magnetic circular dichroism\n(XMCD) technique3,4. An astonishing example of such\nelement-specific ultrafast magnetization dynamics was\nfirst measured on ferrimagnetic GdFeCo alloys5. There,\nit was observed that the rare-earth Gd sublattice demag-\nnetizes in around 1.5 ps whereas the transition metal\nFeCo sublattice has a much shorter demagnetization time\nof 300 fs. Similar element-specific spin dynamics was\nalso observed in CoGd and CoTb alloys6,7. The element-\nselective technique allowed moreover to observe for the\nfirst time the element-specific dynamics of the so-called\n“all-optical switching” (AOS)8in GdFeCo alloys, find-\ning that it unexpectedly proceeds through a transient-\nferromagnetic-like state (TFLS) where the FeCo sublat-\ntice magnetization points in the same direction as that\nof the Gd sublattice before complete reversal5,9. Recent\ntheoretical works supported the distinct demagnetiza-\ntion times observed experimentally10–12and their cru-\ncial role on the TFLS. AOS has been also demonstrated\nfor other rare-earth transition-metal ferrimagnetic alloys\nas TbFe13, TbCo14, TbFeCo15, DyCo16, HoFeCo16, syn-\nthetic ferrimagnets16–18and very recently in the hard-\nmagnetic ferromagnet FePt19.\nAlthough the full theoretical explanation of the\nthermally driven AOS process is still a topic of\ndebate9,12,20–23, the distinct demagnetization rates ofeach of the constituting elements has been suggested as\nthe main driving mechanism for the AOS observed on\nantiferromagnetically coupled alloy9,10,12. These findings\nhave highlighted the question how ultrafast demagneti-\nzation would proceed in ferromagnetically coupled two-\nsublattice materials such as permalloy (Py). Unlike rare-\nearth transition-metal alloys which consists of two intrin-\nsically different metals, Py is composed of Fe (20 %) and\nNi (80 %) which have a rather similar magnetic nature,\ndue to a partially filled 3 dshell. Thus, it is a priori not\nclear if their spin dynamics should be the same or differ-\nent.\nRecent measurements have addressed this question.\nUsing extreme ultraviolet pulses from high-harmonic\ngeneration sources Mathias et al.24probed element-\nspecifically the ultrafast demagnetization in Py and ob-\ntained the same demagnetization rates for each element,\nFe and Ni, but with a 10 to 70 fs delay between them.\nFrom a theoretical viewpoint an important question\nis which materials parameter are defining for the ultra-\nfast demagnetization. Thus far, different criteria have\nbeen suggested25,26. For single-element ferromagnets,\nKazantseva et al.25estimated, based on phenomenolog-\nical arguments, that the timescale for the demagnetiza-\ntion processes is limited by τdemag≈µ/(2λγkBTpulse).\nHere,τdemag depends not only on the elemental atomic\nmagnetic moment, µ, but also on the electron tempera-\nture,Tpulse, and on the damping constant λ. Assuming\nthat the damping constants λand gyromagnetic ratios γ\nare equal for Fe and Ni the demagnetization time would\ntherefore only vary due to the different magnetic mo-\nments of the constituting elements. In that case, the\ndemagnetization time of Fe is larger than the one for Ni\n(sinceµFe>µNi, see Table I below).\nA similar criterion (as in Ref. 25 for single-element fer-\nromagnets) has been suggested by Koopmans et al.26on\nthe basis of the ratio between the magnetic moment andarXiv:1504.00199v1  [cond-mat.mtrl-sci]  1 Apr 20152\nthe Curie temperature, µ/TC. Since for ferromagnetic al-\nloys each element has the same Curie temperature, this\ncriterion would lead to the same conclusions as Kazant-\nseva et al. ; the different atomic magnetic moments of Fe\nand Ni are responsible for the different demagnetization\ntimes. Furthermore, Atxitia et al.10have theoretically\nestimated the demagnetization times in GdFeCo alloys\nproposing that the demagnetization times scale with the\nratio of the magnetic moment to the exchange energy of\neach element and a similar relation is expected for ferro-\nmagnetic alloys. The demagnetization times of Fe and Ni\nin Py were also theoretically investigated by Schellekens\nand Koopmans in Ref. 11 where a modified microscopic\nthree temperature model (M3TM)26was used. Thereby,\nthey obtained a perfect agreement with experimental re-\nsults of Mathias et al. ,24but only assuming an at least\n4 times larger damping constant for Fe. However, this\nwork does not provide a simple general criterion, valid\nfor other ferromagnetic alloys.\nWe have developed a hierarchical multiscale approach\n(cf. Ref. 27) to investigate the element-specific spin dy-\nnamics of ferromagnetic alloys and to obtain a deeper\ninsight into the underlying mechanisms. First, we con-\nstruct and parametrize a model spin Hamiltonian for\nFeNi alloys on the basis of first-principles calculations\n[Sec. II A]. This model spin Hamiltonian in combina-\ntion with extensive numerical atomistic spin dynamics\nsimulations based on the stochastic LLG equation are\nused to calculate the equilibrium properties [Sec. II B]\nas well as the demagnetization process after the appli-\ncation of a step heat pulse. The second step of the pre-\nsented multiscale model links the atomistic spin model to\nthe macroscopic two-sublattices Landau-Lifshitz-Bloch\n(LLB) equation of motion recently derived by Atxitia\net al.28[Sec. III]. The analytical LLB approach allows\nfor efficient simulations, and most importantly, provides\ninsight in the element-specific demagnetization times of\nFeNi alloys.\nII. FROM FIRST PRINCIPLES TO ATOMISTIC\nSPIN MODEL\nA. Building the spin Hamiltonian\nTo start with, we construct an atomistic, classical spin\nHamiltonianHon the basis of first-principles calcula-\ntions. In particular, we consider three relevant alloys:\nFe50Ni50, Fe20Ni80(Py) and Py 60Cu40. The first two al-\nloys will allow us to assess the influence of the Fe and Ni\ncomposition, while the last two alloys will permit us to\nstudy the effect of the inclusion of non-magnetic impuri-\nties on the demagnetization times. This was motivated\nby the work of Mathias et al.24who studied the influence\nof Cu doping on the Fe and Ni demagnetization times in\nan Py 60Cu40alloy.\nTo obtain the spin Hamiltonian we have employed spin-\ndensity functional theory calculations to map the behav-ior of the magnetic material onto an effective Heisenberg\nHamiltonian, which can be achieved in various ways29,30.\nHere we use the two-step approach suggested by Licht-\nenstein et al.31. The first step represents the calculation\nof the self-consistent electronic structure for a collinear\nspin structure at zero temperature. In the second step,\nexchange parameters of an effective classical Heisenberg\nHamiltonian are determined using the one-electron Green\nfunctions. This method has been rather successful in ex-\nplaining magnetic thermodynamic properties of a broad\nclass of magnetic materials32–34.\nThe self-consistent electronic structure was calcu-\nlated using the tight-binding linear muffin-tin orbital\n(TB-LMTO) approach32within the local spin-density\napproximation35to the density functional theory.\nImportantly, the materials we investigate here are al-\nloys. Hence, it is assumed that atoms are distributed\nrandomly on the host fcc lattice. The effect of disor-\nder was described by the coherent-potential approxima-\ntion (CPA)36. The same radii for constituent atoms were\nused in the TB-LMTO-CPA calculations. We have used\naround a million k-points in the full Brillouin zone to\nresolve accurately energy dispersions close to the Fermi\nlevel.\nThe calculations of the Heisenberg exchange constants\nJijin ferromagnets can be performed with a reason-\nable numerical effort by employing the magnetic force\ntheorem29,31. It allows to express the infinitesimal\nchanges of the total energy using changes in one-particle\neigenvalues due to non-self-consistent changes of the ef-\nfective one-electron potential accompanying the infinites-\nimal rotations of spin quantization axes, i.e., without any\nadditional self-consistent calculations besides that for the\ncollinear ground state. The resulting pair exchange in-\nteractions are given by\nJij=1\nπIm/integraldisplayEF\n−∞dE/integraldisplay\nΩidr/integraldisplay\nΩjdr/primeBex(r)G↑\n+Bex(r/prime)G↓\n−,(1)\nwithG↑\n+=G↑(r,r/prime,E+) andG↓\n−=G↓(r/prime,r,E−).EF\ndenotes the Fermi level and Ω ithei-th atomic cell,\nσ=↑,↓is the spin index, E+= limα→0E+ iα,Gσ\nare spin-dependent one-electron retarded Green func-\ntions, andBexis the magnetic field from the exchange-\ncorrelation potential. The validity of this approxima-\ntion has been examined more quantitatively in several\nstudies.37–39Theab initio calculated distance-dependent\nexchange constants for the Fe 20Ni80alloy, i.e., the ex-\nchange within the Fe sublattice (Fe-Fe), the Ni sublat-\ntice (Ni-Ni) as well as between the Fe and Ni sublattices\n(Fe-Ni), are shown in Fig. 1. The calculated magnetic\nmoments for all three alloys considered here are given in\nTable I.\nIn our hierarchical multiscale approach, these com-\nputed material parameters (the exchange constant ma-\ntrix as well as the magnetic moments) are now used as\nmaterial parameters for our numerical simulations based\non an atomistic Heisenberg spin Hamiltonian. We con-\nsider thereto classical spins S/epsilon1\ni=µ/epsilon1\ni/µ/epsilon1\niwith/epsilon1randomly3\nTABLE I. Ab initio calculated magnetic moments µ/epsilon1and experimental lattice constants ∆ used in the atomistic Langevin spin\ndynamics simulations. Effective exchange parameters calculated from ab initio calculations, J/epsilon1,δ\n0=/summationtext\njJ/epsilon1δ\n0j, where the sum is\nhere over all neighbors j. Curie temperatures as calculated from the atomistic simulations, TLLG\nC, and the experimental value,\nTexp\nC.\nalloyµFeµNi∆JNi−Ni\n0JFe−Fe\n0JFe−Ni\n0TLLG\nCTexp\nC\n[µB] [µB] [nm] [J ×10−21] [J×10−21] [J×10−21] [K] [K]\nPy 2.637 0.628 0.3550406.2419 32.3162 26.3654 650 85024\nNi50Fe502.470 0.730 0.3588416.6265 25.3789 25.0656 850\nPy60Cu402.645 0.429 0.3550 2 .6623 56.2789 22.6442 340 40624\ncutoffFe-NiNi NiFe-Fe\nrij[nm]Jij[meV]\n2.5 2 1.5 1 0.514\n8\n4\n2\n0\n-2\n-4\nFIG. 1. (Color online) Ab initio calculated exchange constants\nJijfor the Fe 20Ni80alloy of the distance rijbetween atoms\niandj. Results are given for the three different possible\nsublattice interactions ( JFe−Fe,JNi−Ni, andJFe−Ni). Note\nour hyperbolic scaling. In our atomistic spin simulations the\nexchange constants are taken into account up to a distance rij\n(cutoff) where they are finally small enough to be neglected.\nrepresenting iron ( µ/epsilon1\ni=µFe\ni) or nickel magnetic moments\n(µ/epsilon1\ni=µNi\ni) on the fcc sublattice. For the Cu-doped\nPy60Cu40alloy the calculated magnetic moments on Cu\nvanish, i.e. µCu\ni= 0.\nThe spin Hamiltonian for unit vectors, S/epsilon1\ni, representing\nthe normalized magnetic moments of the i-th atom on\neither the Fe or Ni sublattice reads\nH=−/summationdisplay\nij/parenleftBigJij\n2S/epsilon1\ni·Sδ\nj (2)\n−µ0µ/epsilon1\niµδ\ni\n8π3(S/epsilon1\ni·eij)(eij·Sδ\nj)−S/epsilon1\ni·Sδ\nj\nr3\nij/parenrightBig\n.\nThe first sum represents the exchange energy of mag-\nnetic moments, either on Ni or on Fe sites, distributed\nrandomly with the required concentrations. The ex-\nchange interaction matrices Jij(corresponding to JNi−Ni,\nJFe−Ni, orJNi−Ni) are those from the ab initio calcula-tions (as shown for Py in Fig. 1). These have been taken\ninto account up to a distance of six unit cells (cutoff also\nshown in Fig. 1) until they are finally small enough to\nbe neglected. The second sum describes the magnetic\ndipole-dipole coupling.\nNote, that the exchange interaction given by the ma-\ntricesJijis incorporated in our atomistic spin dynamics\nsimulations via the Fast Fourier transformation method\n(see Ref. 42 for more details). As a side effect, we are able\nto calculate the dipolar interaction without any addi-\ntional computational effort so that we take them into ac-\ncount although they will not influence our results much.\nSince we are interested in thermal properties we\nuse Langevin dynamics, i.e. numerical solutions of the\nstochastic LLG equation of motion\n(1 + (λ/epsilon1\ni)2)µ/epsilon1\ni\nγ/epsilon1\ni˙S/epsilon1\ni=−S/epsilon1\ni×[Hi+λ/epsilon1\ni(S/epsilon1\ni×Hi)],(3)\nwith the gyromagnetic ratio γ/epsilon1\ni, and a dimensionless\nGilbert damping constant λ/epsilon1\nithat describes the coupling\nto the heat-bath and corresponding either to Fe or to Ni.\nThermal fluctuations are included as an additional noise\ntermζiin the internal fields Hi=−∂H\n∂S/epsilon1\ni+ζi(t) with\n/angbracketleftζi(t)/angbracketright= 0,/angbracketleftζiη(0)ζjθ(t)/angbracketright=2kBTλ/epsilon1\niµ/epsilon1\ni\nγ/epsilon1\niδijδηθδ(t),(4)\nwherei,jdenotes lattice sites occupied either by Fe or\nNi andη,θare Cartesian components. All algorithms we\nuse are described in detail in Ref. 43.\nB. Equilibrium properties: element-specific\nmagnetization\nFirst, we investigate the element-specific zero-field\nequilibrium magnetizations for Fe and Ni sublattices.\nThose magnetizations are calculated as the spatial and\ntime average of the sum of local magnetic moments,\nm/epsilon1=/angbracketleftS/epsilon1/angbracketrightwith/epsilon1representing either Fe or Ni. For\nour numerical studies, we assume identical damping con-\nstants (λ=λ/epsilon1\ni) as well as gyromagnetic ratios ( γ=γ/epsilon1\ni\n= 1.76·1011(Ts)−1) for both, Fe or Ni. We perform4\n00.51mǫ\nPy60Cu40\nFe50Ni50Py\n00.51mǫPy60Cu40\nFe50Ni50Py\n00.51\n0 200 400 600 800mǫ\nT[K]Py60Cu40\nFe50Ni50PyFe, MFA\nNi, MFA\nFe, Atomistic\nNi, Atomistic\nFIG. 2. (Color online) Element-specific zero-field equilibrium\nmagnetizations m/epsilon1of either Fe or Ni as a function of tem-\nperature calculated by a rescaled mean-field approximation\n(MFA) (lines) and by the atomistic spin dynamics simulation\n(open symbols). In the MFA the exchange parameters are\nrenormalized by equalizing the Curie temperatures TCcom-\nputed with atomistic simulations with those obtained from\nthe rescaled MFA. System size 128 ×128×128, damping\nparameterλ= 1.0.\nour Langevin spin dynamics simulations for two differ-\nent FeNi alloys, namely Fe 50Ni50and Py, as well as for\npermalloy diluted with copper, Py 60Cu40. All material\nparameters used in our simulations are given in Table I.\nThe temperature dependence of the normalized\nelement-specific magnetizations m/epsilon1are shown in Fig.\n2. The calculated values of the Curie temperatures are\ngiven in Table I together with known experimental val-\nues. Both, the numerical and experimental values, are\nin good agreement. The element-specific magnetizations\nas well as the total magnetization (not shown in Fig. 2)\nof the alloys share the same Curie temperature while in\nthe temperature range below the Curie temperature their\ntemperature dependence is different for the two sublat-\ntices; the normalized magnetization of Ni is lower than\nthat of Fe.\nThe element-specific magnetizations calculated within\nthe framework of a rescaled mean-field approximation\n(MFA) are shown as well. This approach will be dis-\ncussed in detail in Sec. III below where these curves serve\nas material parameters for the simulations based on the\nLLB equation of motion also introduced in the next sec-\ntion.III. FROM ATOMISTIC SPIN MODEL TO\nMACROSCOPIC MODEL\nA. Two-sublattices Landau-Lifshitz-Bloch equation\nWithin the hierarchical multiscale approach, the\nmacroscopic (micromagnetic) equation of motion valid at\nelevated temperatures is the LLB equation27. Initially,\nthe macroscopic LLB equation of motion was derived by\nGaranin for single-species ferromagnets only. Garanin\nfirst calculated the Fokker-Planck equation for a single\nspin coupled to a heat-bath, thereafter a non-equilibrium\ndistribution function for the thermal averaged spin polar-\nization was assumed to drive the non-equilibrium dynam-\nics. Second, the exchange interactions between atomic\nspins were introduced using the mean field approximation\n(MFA) with respect to the spin-spin interactions. This\nlast step reduces to the replacement of the ferromagnetic\nspin Hamiltonian Hwith the MFA Hamiltonian HMFAin\nthe single (macro)spin solution.\nThe LLB formalism was recently broadened to de-\nscribe the distinct dynamics of two-sublattices mag-\nnets, both antiferromagnetically or ferromagnetically\ncoupled28. The derivation of such equations follows sim-\nilar steps as for the ferromagnetic LLB version but con-\nsidering sublattice specific spin-spin exchange interac-\ntions and MFA exchange fields, /angbracketleftH/epsilon1\nMFA/angbracketrightconf. For the ex-\nchange field the random lattice model is used by gener-\nating the random average with respect to disorder con-\nfigurations/angbracketleft.../angbracketrightconf. The corresponding set of coupled\nLLB equations for each sublattice reduced magnetiza-\ntionm/epsilon1=/angbracketleftS/epsilon1/angbracketright=M/epsilon1/M/epsilon1\ns, whereM/epsilon1\nsis the saturation\nmagnetization at 0 K, has the form\n˙m/epsilon1=γ/epsilon1[m/epsilon1×/angbracketleftH/epsilon1\nMFA/angbracketrightconf]−Γ/epsilon1\n⊥[m/epsilon1×[m/epsilon1×m/epsilon1\n0]]\n(m/epsilon1)2\n−Γ/epsilon1\n/bardbl/parenleftbigg\n1−m/epsilon1m/epsilon1\n0\n(m/epsilon1)2/parenrightbigg\nm/epsilon1. (5)\nHere, m/epsilon1\n0=L(ξ/epsilon1\n0)ξ/epsilon1\n0\nξ/epsilon1\n0is the transient (dynamical) magne-\ntization to which the non-equilibrium magnetization m/epsilon1\ntends to relax, and where ξ/epsilon1\n0≡µ/epsilon1\nkBT/angbracketleftH/epsilon1\nMFA/angbracketrightconfis the ther-\nmal reduced field, ξ/epsilon1\n0≡|ξ/epsilon1\n0|, andL(ξ) = coth (ξ)−1/ξis\nthe Langevin function and L/prime(ξ) = dL(ξ)/dξ. The par-\nallel (Γ/epsilon1\n/bardbl) and perpendicular (Γ/epsilon1\n⊥) relaxation rates in Eq.\n(5) are given by\nΓ/epsilon1\n/bardbl= Λ/epsilon1\nN1\nξ/epsilon1\n0L(ξ/epsilon1\n0)\nL/prime(ξ/epsilon1\n0)and Γ/epsilon1\n⊥=Λ/epsilon1\nN\n2/parenleftbiggξ/epsilon1\n0\nL(ξ/epsilon1\n0)−1/parenrightbigg\n.(6)\nΛ/epsilon1\nN= 2kBTγ/epsilon1λ/epsilon1/µ/epsilon1is the characteristic diffusion relax-\nation rate. The damping parameters λ/epsilon1have the same\norigin as those used in the atomistic simulations.\nThe first and the second terms on the right-hand side\nof Eq. (5) describe the transverse motion of the mag-\nnetization. These dynamics are much slower than the\nlongitudinal magnetization dynamics given by the third\nterm in this equation. Therefore, in the following we will5\nFeNi\nFIG. 3. (Color online) Schematics of the magnetic unit cell\nused in the mean-field approximation for the FeNi alloys. The\nunit cell shown by the box contains two spins, one Fe and one\nNi. The only interaction among spins located at the same unit\ncellris defined by JNi−Fe\n0 . The self-interactions are neglected,\nJNi−Ni\n0 (r,r) =JFe−Fe\n0 (r,r) = 0. The rest of the interactions\nare among spins located in neighboring unit cells randr/prime.\nneglect the transverse components (in Eq. (5)) and keep\nonly the longitudinal one,\n˙m/epsilon1=−Γ/epsilon1\n/bardbl(m/epsilon1−m/epsilon1\n0). (7)\nIn spite of the fact that the form of Eq. (7) is similar\nto the well known Bloch equation, the quantity m0=\nm/epsilon1\n0/parenleftbig\nm/epsilon1,mδ/parenrightbig\n(withδthe 2-nd type of element) is not\nthe equilibrium magnetization but changes dynamically\nthrough the dependence of the effective field /angbracketleftH/epsilon1\nMFA/angbracketrightconf\non both sublattice magnetizations. Moreover, the rate\nparameter Γ/epsilon1\n/bardbl= Γ/epsilon1\n/bardbl/parenleftbig\nm/epsilon1\n0,mδ\n0/parenrightbig\ncontains highly non-linear\nterms inm/epsilon1\n0andmδ\n0.\nTherefore, the analytical solution of Eq. (7) and thus\na deeper physical interpretation of the relaxation rates is\ndifficult without any further approximations. However,\nEq. (7) can be easily solved numerically with the aim to\ndirectly compare the solutions to those of the atomistic\nspin simulations. This is discussed in more detail in the\nnext subsections.\nB. From atomistic spin model to\nLandau-Lifshitz-Bloch equation\nNext, to solve Eq. (5) or Eq. (7), one needs to calculate\n/angbracketleftH/epsilon1\nMFA/angbracketrightconffor the here-considered FeNi alloys. An ade-\nquate definition of such a field will allow us to directly\ncompare the magnetization dynamics from our atomistic\nspin simulation with the LLB macroscopic approach.\nHowever, a quantitative comparison between both a\nstandard MFA and atomistic spin model calculations of\nthe equilibrium properties is usually not possible. This is\ndue to the fact that the Curie temperature gained with\nthe MFA approach is overestimated due to the inher-\nent poor approximation of the spin-spin correlations. Al-\nthough, rescaling the exchange parameters conveniently\nin such a way that the Curie temperature calculated\nwith the MFA approach agrees with atomistic simula-\ntions leads to a good agreement of both methods. Hence,we first present the standard MFA for disordered two-\nsublattices magnets, thereafter, we will deal with the\nrescaling of the exchange parameters.\nThe MFA Hamiltonian of the full spin Hamiltonian for\nFeNi alloys (see Eq. (2) introduced in Sec. II) can be\nwritten as\nHMFA=H00−µFe/summationdisplay\niHFe\nMFA·SFe\ni−µNi/summationdisplay\niHNi\nMFA·SNi\ni,(8)\nwhere the dipolar interaction is neglected. The mean\nfield acting on each site ican be separated in two contri-\nbutions; a) the contribution from neighbors of the same\ntypej/epsilon1and b) those of the other type jδ,\nµ/epsilon1/angbracketleftH/epsilon1\nMFA/angbracketrightconf=/summationdisplay\n/epsilon1j/epsilon1J/epsilon1\nj/epsilon1/angbracketleftSj/epsilon1/angbracketright+/summationdisplay\n/epsilon1jδJ/epsilon1\njδ/angbracketleftSjδ/angbracketright,(9)\nwhere sums run over the nearest neighbours. When the\nhomogenous magnetization approximation is applied (i.e.\n/angbracketleftSjFe/angbracketright=mFeand/angbracketleftSjNi/angbracketright=mNifor all sites) one can de-\nfineJ/epsilon1/epsilon1\n0=/summationtext\n/epsilon1j/epsilon1J/epsilon1\nj/epsilon1andJ/epsilon1δ\n0=/summationtext\n/epsilon1jδJ/epsilon1\njδ. A sketch of the\nexchange interaction within the present MFA model is\npresented in Fig. 3. The impurity model is mapped to a\nregular spin lattice where the unit cell (orange box) con-\ntains the two spin species, Fe and Ni, and the exchange\ninteractions among them are weighted in terms of the\nconcentration of each species.\nThe equilibrium magnetization of each sublattice m/epsilon1\ne\ncan be obtained via the self-consistent solution of the\nCurie-Weiss equations m/epsilon1\ne=L(µ/epsilon1\nkBT/angbracketleftH/epsilon1\nMFA/angbracketrightconf).\nFig. 2 shows good agreement of the calculated m/epsilon1\ne(T)\nusing the MFA and the atomistic spin model for the\nthree system studied in the present work. The exchange\ninteractions are rescaled as J/epsilon1δ\n0,MFA/similarequal(1.65/2)J/epsilon1δ\n0, for\nFe50Ni50and Py. For Py 60Cu40it is in agreement with\nJ/epsilon1δ\n0,MFA= (1.78/2)J/epsilon1δ\n0. Here, the atomistic calculations is\nnot as accurate for intermediate temperatures as for the\nother two alloys. This could be because of the increased\ncomplexity introduced by the inclusion of Cu impurities\nwhich cannot be fully described by the MFA.\nC. De- and remagnetization due to a heat pulse\nIn the following, we study the reaction of the element-\nspecific magnetization to a temperature step in Py as well\nas in Py diluted with Cu. In the first part of the temper-\nature step the system is heated up to T= 0.8TCand in\nthe second part it is cooled down to Tpulse = 0.5TC. The\nheat pulse roughly mimics the effect of heating due to a\nshort laser pulse. The first part of the temperature step\ntriggers the demagnetization while the second one trig-\ngers the remagnetization process. We perform atomistic\nas well as LLB simulation of the de- and remagnetization\nof the two sublattices after the application of a step heat\npulse of 500-fs duration.\nThe reaction of the Fe and Ni sublattice magnetiza-\ntions is shown in Fig. 4. While the temperature step6\nFIG. 4. (Color online) Calculated z-component of the nor-\nmalized element-specific magnetization m/epsilon1\nzvs. time for Py\n(top panel) and Py 60Cu40(bottom panel). In both cases the\nquenching of the element-specific magnetizations for Fe and\nNi due to a temperature step of Tpulse = 0.8TCare shown,\ncomputed with atomistic Langevin spin dynamics (open sym-\nbols) as well as LLB simulations (lines). System size 64 ×64\n×64, damping parameter λ= 0.02.\nis switched on, the two sublattices relax to the corre-\nsponding equilibrium value of the sublattice magnetiza-\ntionsm/epsilon1(Tpulse). Note, that these equilibrium values\nare different for the two sublattices in agreement with\nthe temperature-dependent equilibrium element-specific\nmagnetizations shown in Fig. 2.\nBecause of that, the different demagnetization time\nscales are not well distinguishable in Fig. 4. Thus,\nwe use the normalized magnetization, m/epsilon1\nnorm = (m/epsilon1−\nm/epsilon1\nmin)/(m/epsilon1\n(t=0)−m/epsilon1\nmin) of the sublattices, rather than\nm/epsilon1to directly compare the demagnetization times. The\ndemagnetization time after excitation with a tempera-\nture pulse isfaster for Ni than for Fe (Fig. 5 (top panel))\nfor the first 200 fs, while one can see that for times larger\nthan 200 fs both elements demagnetize at the same rate\n(Fig. 5 (bottom panel)). Experiments on Py suggest that\nthe time shift between distinct and similar demagnetiza-\ntion rates in Py is of around 10–70 femtoseconds24.\nD. Understanding relaxation times within the\nLandau-Lifshitz-Bloch formalism\nThe relaxation rates of the Fe and Ni sublattices\ncan be understood by discussing the linearized form of\nEq. (7). Here, the expansion of Γ/epsilon1\n/bardblandm/epsilon1\n0around\n0.010.11∆m/epsilon1z(t)/∆m/epsilon1z(0)τFe/τNi=1.8\n0.40.60.810.1 0.2 0.3 0.4 0.5m/epsilon1ztime [ps]τFe/τNi=1.8τNi=τFeNiFeFIG. 5. (Color online) Top panel: Normalized magnetiza-\ntion dynamics of Fe and Ni sublattices after the application\nof a heat pulse T= 0.8TCas computed with the atomistic\nspin model. The ratio between the Fe and Ni demagneti-\nzation times is 1.8. The intersection of the linear fit to the\nabscissa gives the relaxation time for each sublattice. Bot-\ntom panel: plot of the unnormalized magnetization dynamics\nwhich shows that after the first 0.2 picosecond the element-\nspecific demagnetization proceeds at the same rate.\nτ−τ+\nT/TCτ[ps]\n1 0.75 0.5 0.25 02.0 1.5 1.0 0.5 0.0T/TCτ+/τ−\n1 0.75 0.5 0.25010\n86420\nFIG. 6. (Color online) Relaxation times of the dynamical\nsystem obtained by the LLB equation as a function of tem-\nperature. Inset: The ratio between the relaxation times.\ntheir equilibrium values m/epsilon1\neis considered28and leads\nto∂(∆m)/∂t=A/bardbl∆mwith ∆ m= (∆m/epsilon1,∆mδ) and\nm/epsilon1(δ)=m/epsilon1(δ)\ne+ ∆m/epsilon1(δ). Furthermore, the characteristic\nmatrixA/bardbldrives the dynamics of this linearized equation\nand has the form\nA/bardbl=/parenleftBigg\n−γ/epsilon1α/epsilon1\n/bardbl/Λ/epsilon1/epsilon1γ/epsilon1α/epsilon1\n/bardblJ/epsilon1δ\n0/µ/epsilon1\nγδαδ\n/bardblJδ/epsilon1\n0/µδ−γδαδ\n/bardbl/Λδδ/parenrightBigg\n, (10)\nwith\nΛ/epsilon1δ=J/epsilon1δ\n0\nµ/epsilon1m/epsilon1\ne\nmδeand Λ/epsilon1/epsilon1=/tildewideχ/epsilon1\n/bardbl\n1 +J/epsilon1δ\n0\nµ/epsilon1/tildewideχδ\n/bardbl, (11)7\nwhere/tildewideχ/epsilon1\n/bardblare the longitudinal susceptibilities which can\nbe evaluated in the MFA approximation as\n/tildewideχ/epsilon1\n/bardbl=J/epsilon1δ\n0µδLδL/epsilon1+µ/epsilon1L/epsilon1(kBT−Jδ\n0Lδ)\n(kBT−Jδ\n0Lδ)(kBT−J/epsilon1\n0L/epsilon1)−J/epsilon1δ\n0Jδ/epsilon1\n0LδL/epsilon1,(12)\nwithL/epsilon1=L/prime(ξ/epsilon1\ne) andLδ=L/prime(ξδ\ne). We note that the\nlongitudinal susceptibility in Eq. (12) depends on the ex-\nchange parameter (Curie temperature) and the atomic\nmagnetic moments of both sublattices.\nNext, the longitudinal damping parameter in Eq. (10)\nis defined as α/epsilon1= (2kBTλ/epsilon1m/epsilon1\ne)/µ/epsilon1H/epsilon1\ne,ex, whereH/epsilon1\ne,exis\nthe average exchange field for the sublattice /epsilon1at equi-\nlibrium, defined by the MFA expression (9). The longi-\ntudinal fluctuations are defined by the exchange energy,\naccording to the expression above. However, the longitu-\ndinal relaxation time is not simply inversely proportional\nto the damping parameter. Instead the relaxation pa-\nrameters in Eq. (10) do also depend on the longitudinal\nsusceptibilities which give the main contribution to their\ntemperature dependence.\nIt is important to note that the matrix elements in\nEq. (10) are temperature as well as (sublattice) material\nparameter dependent. The general solution of the char-\nacteristic equation, |A/bardbl−Γ±I|= 0, gives two different\neigenvalues, Γ±= 1/τ±, corresponding to the eigenvec-\ntorsv±. Here,Iis the unit matrix. The computed tem-\nperature dependence of the relaxation times τ±is pre-\nsented in Fig. 6. More interestingly, we observe that the\nratio between relaxation times τ+/τ−[inset Fig. 6] is al-\nmost constant for temperature below 0 .5TCand it has a\nvalue of 1.8 which compares well with atomistic simula-\ntions [Fig. 5]. At elevated temperatures, one relaxation\ntimeτ+will dominate the magnetization dynamics of\nboth sublattices.\nIn Fig. 7(a) we present the temperature dependence of\nthe longitudinal damping parameters and in Fig. 7(b)\nthe temperature dependence of the parameters Γ/epsilon1δ=\nα/epsilon1\n/bardbl/Λ/epsilon1δ. These parameters define the element-specific\nlongitudinal dynamics. In Figs. 7 (c) and (d) the temper-\nature dependent α/epsilon1\n/bardbl/αδ\n/bardbland Λ/epsilon1/epsilon1/Λδδare shown. It can\nbe seen that at least in the range of low temperatures the\nmagnetization dynamics is mainly defined by Γ/epsilon1/epsilon1/greatermuchΓ/epsilon1δ.\nThe general solution of the linearized LLB system for\nthe two sublattices can be written as\n∆mFe(t) =AFeexp (−t/τ+) +BFeexp (−t/τ−)\n∆mNi(t) =ANiexp (−t/τ+) +BNiexp (−t/τ−),(13)\nwhere the coefficients AFe(Ni)andBFe(Ni)will depend\nof the eigenvectors v±and the initial magnetic state,\n∆mFe(0) and ∆mNi(0). For instance\nAFe= ∆mFe(0)/bracketleftBig\n1−∆mNi(0)\n∆mFe(0)x+/bracketrightBig\nx−\nx−−x+, (14)\nwherex+=vFe\n+/vNi\n+andx−=vFe\n−/vNi\n−, is the ratio be-\ntween he eigenvector components. The other coefficients\n00.20.40.6α/bardbl/λ(a) (b)\n(c) (d)\n11.522.53\n0 0.2 0.4 0.6 0.8 1αNi/bardbl/αFe/bardbl\nT/TC(a) (b)\n(c) (d)0.010.1110\nΓ [ps−1] (a) (b)\n(c) (d)\n0 0.2 0.4 0.6 0.8 100.511.522.53\nT/TC(a) (b)\n(c) (d)Fe\nNi\nΓFe−Ni\nΓNi−Fe\nΓFe−Fe\nΓNi−Ni\nΛNi−Ni/ΛFe−Fe\n/tildewideχNi\n/bardbl//tildewideχFe\n/bardblFIG. 7. (Color online) (a) Temperature dependence of the in-\ndividual longitudinal damping parameters for Fe and Ni. (b)\nMatrix elements of the dynamical system defining the magne-\ntization dynamics. (c) Ratio between the individual damping\nparameters. (d) Ratio between the “effective” susceptibilities\nΛ/epsilon1/epsilon1and the actual susceptibilities /tildewideχ/epsilon1\n/bardbl.\nare calculated similarly. This complexity prohibits a gen-\neral analysis of the results. Thus, although the general\nsolution is clearly a bi-exponential decay, one can wonder\nwhen the one exponential decay approximation will give\na good estimate for the individual relaxation dynamics.\nTwo interesting scenarios exist: First, the relaxation\ntimesτ+andτ−could have very different time scales\nand thus one can separate the solution on short and long\ntime scales, defined by τ−andτ+, respectively. This\nis an interesting scenario for ultrafast magnetization dy-\nnamics where only the fast time scale will be relevant.\nFig. 6 shows the ratio τ+/τ−and we can observe that\nthe scenario τ+/τ−/greatermuch1 only happens for temperatures\napproaching TC. As we have seen in the atomistic simula-\ntions, after an initial distinct quenching of each sublattice\nmagnetization, both sublattice demagnetize at the same\nrate but slower than the initial rates (see Fig. 5).\nThe second scenario occurs when AFe≈∆mFe(0) and\nBNi≈∆mNi(0), even if τ+andτ−are of the same or-\nder. This happens, for example, either when the coupling\nbetween sublattices is very weak, or at relatively low tem-\nperatures, see Fig. 6. In this case the system can be con-\nsidered as two uncoupled ferromagnets (although with\nrenormalized parameters), meaning that the matrix in\nEq. (10) defining the dynamics is almost diagonal. Thus,\nwe can approximately associate each eigenvalue of Eq.\n(10) to each sublattice, τ−=τNiandτ+=τFe. The in-\nset in Fig. 6 shows the ratio τ+/τ−for the whole range of\ntemperatures. At low-to-intermediate temperatures we\nfind thatτ+/τ−≈1.8. This is in good agreement with8\nTABLE II. Theoretical results: ab initio calculated ratio be-\ntween the mean exchange interaction at T= 0 K, the ratio\nbetween atomic magnetic moments and the quotient of these\nratios. Results of simulations: atomistic spin model calcu-\nlated ratio between κexponents and relaxation times. The\nratio between the magnetic atomic moments and the expo-\nnentsκis predicted in the main text to give the ratio between\nrelaxation times.\ntheoretical simulations\nalloy/tildewideJFe\n0\n/tildewideJNi\n0µFe\nµNiµFe\nµNi/tildewideJNi\n0\n/tildewideJFe\n0κFe\nκNiτFe\nτNiµFe\nµNiκFe\nκNi\nFe50Ni50 1.592 3.38 2.12 1.492 2.10 2.25\nPy 2.685 4.198 1.563 2.3 1.8 1.8\nPy60Cu404.412 6.17 1.398 2.95 2.1 2.05\natomistic simulations, see Fig. 5(a), and it clearly shows\nthat the relaxation times ratio is not defined by the ratio\nbetween atomic magnetic moments, µFe/µNi≈4.\nIn the case that the longitudinal relaxation rates are\ndefined by the diagonal elements of the matrix (10) and\nTis not close to TCthe longitudinal relaxation time can\nbe estimated as\nτ/epsilon1/similarequal1\n2γ/epsilon1λ/epsilon1m/epsilon1eH/epsilon1e,ex. (15)\nThus the ratio between the relaxation rates of Ni and Fe\n(for the same gyromagnetic ratio value, the same cou-\npling parameter and not too close to TC) is defined by\nτNi\nτFe=/parenleftbiggλFe\nλNiµNi\nµFe/parenrightbigg/tildewideJFe\n0mFe\ne\n/tildewideJNi\n0mNie. (16)\nWe recall that /tildewideJ/epsilon1\n0m/epsilon1\ne=J/epsilon1\n0m/epsilon1\ne+J/epsilon1δ\n0mδ\neis the average ex-\nchange energy for the sublattice /epsilon1at equilibrium. Thus,\nthe interpretation of the ratio of the relaxation times is\nstraightforward. The low temperature value of the ra-\ntio/tildewideJFe\n0//tildewideJNi\n0is presented in Table II for the three alloys\nstudied here. The second column presents the ratio be-\ntween atomic magnetic moments, and the third column\nthe estimated ratio between relaxation times under the\nassumption of equal damping parameter at each sublat-\ntice.\nThe estimated ratios for relaxation times are in rather\ngood agreement with the atomistic simulations (fifth col-\numn) for Fe 50Ni50and Py, however for Py 60Cu40the es-\ntimation is not that good. We have to remember that the\nMFA re-scaling of the exchange parameters did not give\na completely satisfactory result for the shape of m(T) in\nthis alloy (see Fig. 2(a)). Thus, since the re-scaled ex-\nchange parameter does not work completely well at the\nlow-to-intermediate temperature interval, we further in-\nvestigate this case (Py 60Cu40) by relating the obtained\nrelation in Eq. (16) for the ratio τNi/τFeto the slopes of\nthe curvesm(T).This can be easily done by using the linear decrease of\nmagnetization at low temperature, m(T)≈1−κT/T C,\nwhereκ=WkB/J0for classical spin models, here W\nis the Watson integral44. Thus, the ratio between the\nslopes ofm(T) for each sublattice is directly related to\nthe ratio between the exchange values, /tildewideJδ\n0, as follows,\nκFe/κNi=/tildewideJNi\n0//tildewideJFe\n0. It is worth noting that the equilib-\nrium magnetization as a function of temperature can be\nfitted to the power law m(T) = (1−T/T C)κwhich in\nturn gives the low temperature limit m(T) = 1−κT/T C.\nAnd more importantly, it gives a link of the dynamics to\nthe equilibrium thermodynamic properties through the\nratio\nτNi\nτFe=λFe\nλNiµNi\nµFeκNi\nκFe. (17)\nNext, we fit the numerically evaluated m(T) curves to\nthe power law mFe(Ni)(T) = (1−T/T C)κFe(Ni)forT <\n0.5TC. This allows us to directly estimate the ratio be-\ntween the relaxation times for the three alloys, see Table\nII. We can see that the relation in Eq. (17) agrees well\nfor the three alloys even for Py 60Cu40.\nFor a more general case, for instance at elevated tem-\nperatures, where the one-exponential solution is not a\ngood approximation, we have to solve numerically for the\ncoefficients of each exponential decay A/epsilon1andB/epsilon1. Apart\nfrom the exchange interactions and temperature depen-\ndence,A/epsilon1andB/epsilon1also depend on the initial conditions\nδm/epsilon1(0) =m/epsilon1(0)−m/epsilon1\ne.\nE. Effect on distinct local damping parameters on\nthe magnetization dynamics\nThe intrinsic (atomistic) damping parameters λ/epsilon1are\nnot be necessarily the same for both sublattices. To in-\nvestigate the effect of different damping parameters we\nconsider that the magnetic system is initially at equi-\nlibrium at room temperature T= 300 K. Then a heat\npulseTpulse is applied for 1 ps. We define τFe(Ni)the\ntime at which the normalized magnetization, mnorm(t) =\n(m(t)−mmin)/(m(t=0)−mmin) is 1/e. The results for a\nbroad parameter space of λFe/λNiand heat pulse tem-\nperatureTpulse (scaled toTC) are shown in Fig.\nreffig:PhaseDiagramRelaxTimesPy. The line where\nτNi/τFe= 1 lies at low pulse temperature (linear limit\nin the LLB) λFe/λNi= 1.563. The critical ratio/parenleftBig\nλFe\nλNi/parenrightBig\ncris close to the one which could be predicted from Eq.\n(17) assuming τNi/τFe= 1:\n/parenleftbiggλFe\nλNi/parenrightbigg\ncr=µFe\nµNiκFe\nκNi. (18)\nEstimations of this critical ratio at low temperatures can\nbe found in Table II. The ratio is around 2 for all the\nalloys.9\n0.50.7511.25\n12345Tpulse/TC\nλFe/λNi\n00.511.522.53τNi\nτFe10.5\n1.522.5\nFIG. 8. (Color online) Ratio between the relaxation times τ\nof the Fe and Ni sublattices in Py after the application of a\nheat pulse of temperature Tpulse for a range of values of the\nratio of intrinsic damping parameters, λFe/λFe. Black lines\nrepresentλFe/λNivalues where the ratio between relaxation\ntimesτNiandτFeis constant with the value given by the label.\nThe results presented in Fig. 8 show a variety of possi-\nble situations that can be encountered in experiments on\nalloys with two magnetic sublattices. They show that in\nthe case of equal coupling to the heat-bath, the Ni sub-\nlattice demagnetizes faster than the Fe sublattics in all\ntemperature ranges. The situation may be changed if Fe\nis as least twice stronger coupled to the heat-bath than\nNi. This conclusion is not inconsistent with the dispro-\nportional couplings that were assumed in Ref. 11. Thus,\nFe can demagnetize faster than Ni (as reported in Ref.\n24) only if Fe is stronger coupled to the heat-bath.\nIV. DISCUSSION AND CONCLUSION\nElement-specific magnetization dynamics in multi-\nsublattice magnets has attracted a lot of attention\nlately24,45,46. The case of GdFeCo ferrimagnetic al-\nloys is paradigmatic since this was the first material\nwhere the so-called ultrafast all-optical switching (AOS)\nof the magnetization has been observed8. The element-\ndependent magnetization dynamics in GdFeCo alloys has\nmeanwhile been thoroughly studied9,10,12,20–23. From a\nfundamental view point, however, it is also important to\nunderstand the element-specific magnetization dynamics\nin multi-element ferromagnetic alloys. This is challenging\nfrom a modeling perspective and, moreover, contradict-\ning results have been observed in NiFe alloys24,47.\nTo treat such alloys we have developed here a hierar-\nchical multiscale approach for disordered multisublattice\nferromagnets. The electronic structure ab initio calcula-\ntions of the exchange integrals between atomic spins in\nFeNi alloys serves as as an accurate foundation to definea classical Heisenberg spin Hamiltonian which in turn has\nbeen used to calculate the element-specific magnetization\ndynamics of atomic spins through computer simulations\nbased on the stochastic LLG equation. Our simulations\npredict consistently a faster demagnetization of the Ni\nas compared to the Fe. These findings are however in\ncontrast to the dynamics measured by Mathias et al.24\nFrom a modeling perspective, we have linked informa-\ntion obtained from computer simulations of the atom-\nistic Heisenberg Hamiltonian to large scale continuum\ntheory on the basis of the recently derived finite temper-\nature LLB model for two sublattice magnets28. The LLB\nmodel is rather general, it can be applied not only to fer-\nromagnetic alloys, as we have done in the present work,\nbut also to ferrimagnetic alloys.10Thanks to analytical\nexpressions coming from the LLB model we have been\nable to interpret the distinct element-specific dynamics\nin FeNi alloys in terms of the strength of the exchange\ninteraction acting on each sublattice. Assuming equal\ndamping parameters for Fe and Ni, the difference is not\nonly coming from the different atomic moments. Analyt-\nical expressions derived for the ratio between demagneti-\nzation times in Fe and in Ni compare very well to numer-\nical results from computer simulations of the atomistic\nspin model. To investigate the effect of different intrin-\nsic damping parameters we have restrained ourselves to\nuse the LLB approach which is computationally less ex-\npensive than the atomistic spin dynamic simulations on\na large system of atomic spins. Our investigation thus\nprepares a route to an easier characterization, prediction\nand hence, control of the thermal magnetic properties\nof disordered multi-sublattice magnets, something which\nwill be valuable for technological purposes.\nAs for the applicability of our multiscale approach to\nferrimagnetic materials, one would obviously need accu-\nrately calculated exchange integrals as a starting point.\nComputing these for rare-earth transition metals alloys\nmight not straightforward, as the rare-earth ions con-\ntain mostly localized f-electrons with a sizable orbital\ncontribution to the atomic moment. However it is ex-\npected that for ferrimagnetic alloys, or multilayers with\nantiparallel alignment, composed of transition metals\nthis task will be easier. Initial theoretical comparisons\nof the element-specific demagnetization in GdFeCo were\ndone recently by Atxitia el al.10who obtained a good\nagreement with experimental observations. 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Eschenlohr, PhD thesis, Helmholtz Zentrum Berlin\n(2012)."    },    {        "title": "1802.02415v1.Breaking_the_current_density_threshold_in_spin_orbit_torque_magnetic_random_access_memory.pdf",        "content": "arXiv:1802.02415v1  [cond-mat.mes-hall]  7 Feb 2018Breaking the current density threshold in spin-orbit-torq ue magnetic random access\nmemory\nYin Zhang,1,2H. Y. Yuan,3,∗X. S. Wang,4,1and 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\n3Department of Physics, Southern University of Science and T echnology of China, Shenzhen 518055, China\n4School of Microelectronics and Solid-State Electronics,\nUniversity of Electronic Science and Technology of China, C hengdu, Sichuan 610054, China\n(Dated: March 25, 2022)\nSpin-orbit-torque magnetic random access memory (SOT-MRA M) is a promising technology\nfor the next generation of data storage devices. The main bot tleneck of this technology is the\nhigh reversal current density threshold. This outstanding problem of SOT-MRAM is now solved\nby using a current density of constant magnitude and varying flow direction that reduces the\nreversal current density threshold by a factor of more than t he Gilbert damping coefficient. The\nEuler-Lagrange equation for the fastest magnetization rev ersal path and the optimal current\npulse are derived for an arbitrary magnetic cell. The theore tical limit of minimal reversal current\ndensity and current density for a GHz switching rate of the ne w reversal strategy for CoFeB/Ta\nSOT-MRAMs are respectively of the order of 105A/cm2and 106A/cm2far below 107A/cm2and\n108A/cm2in the conventional strategy. Furthermore, no external mag netic field is needed for a\ndeterministic reversal in the new strategy.\nSubject Areas: Magnetism, Nanophysics, Spintronics\nI. INTRODUCTION\nFast and efficient magnetization reversal is of not only\nfundamentally interesting, but also technologically im-\nportant for high density data storage and massive in-\nformation processing. Magnetization reversal can be in-\nduced by magnetic field [1–3], electric current through\ndirect [4–9] and/or indirect [10–22] spin angular mo-\nmentum transfer from polarized itinerant electrons to\nmagnetization, microwaves [23], laser light [24], and\neven electric fields [25]. While the magnetic field in-\nduced magnetization reversal is a matured technology,\nit suffers from scalability and field localization problems\n[8, 26] for nanoscale devices. Spin transfer torque mag-\nnetic random-access memory is an attractive technol-\nogy in spintronics [26] although Joule heating, device\ndurability and reliability are challenging issues [11, 26].\nIn an spin-orbit-torque magnetic random access mem-\nory (SOT-MRAM) whose central component is a heavy-\nmetal/ferromagnet bilayer, an electric current in the\nheavy-metal layer generates a pure spin current through\nthe spin-Hall effect [10, 11] that flows perpendicularly\ninto the magnetic layer. The spin current, in turn, pro-\nduces spin-orbit torques (SOT) through spin angular\nmomentum transfer [4, 5] and/or Rashba effect [16–22].\nSOT-MRAM is a promising technology because writing\ncharge current does not pass through the memory cells\nso that the cells do not suffer from the Joule heating\nand associated device damaging. In principle, such de-\nvices are infinitely durable due to negligible heating from\n∗[Corresponding author:]yuanhy@sustc.edu.cn\n†[Corresponding author:]phxwan@ust.hkspin current [11]. However, the reversal current density\nthreshold (above 107A/cm2[14, 15] for realistic materi-\nals) in the present SOT-MRAM architecture is too high.\nTo have a reasonable switching rate (order of GHz), the\ncurrent density should be much larger than 108A/cm2\n[14, 15] that is too high for devices. In order to lower the\nminimalreversalcurrentdensityaswellastoswitchmag-\nnetization states at GHz rate at a tolerable current den-\nsity in SOT-MRAM, it is interesting to find new reversal\nschemes (strategies) that can achieveabove goals. In this\npaper, we show that a proper current density pulse of\ntime-dependent flow direction and constant magnitude,\nmuch lower than the conventional threshold, can switch\na SOT-MRAM at GHz rate. Such a time-dependent cur-\nrent pulse can be realized by using two perpendicular\ncurrentspassingthrough the heavy-metallayer. The the-\noretical limit of minimal reversal current density of the\nnew reversal strategy for realistic materials can be of the\norder of 105A/cm2, far below 107A/cm2in the con-\nventional strategy that uses a direct current (DC), both\nbased on macrospin approximation. The validity of the\nmacrospin model is also verified by micromagnetic simu-\nlations.\nII. MACROSPIN MODEL AND RESULTS\nA. Model\nOur new reversal strategy for an SOT-MRAM, whose\ncentral component is a ferromagnetic/heavy-metal bi-\nlayer lying in the xy-plane with initial spin along the\n+z-direction as shown in Fig. 1, uses a current den-\nsityJ=JcosΦˆx+JsinΦˆygenerated from two time-2\nm\nFM \nHM JxJx\nJyJy\nFIG. 1. Schematic illustration of new reversal scheme for\nSOT-MRAMs. Two perpendicular currents flowin the heavy-\nmetal layer of a ferromagnet/heavy-metal bilayer to genera te\na current whose direction can vary in the xy-plane.\ndependent electric currents flowing along the x- and the\ny-directions, where Φ is a time-dependent angle between\nJand the x-axis and Jis a constant total current den-\nsity. The magnetic energy density is ε=−Kcos2θwith\nKbeing the anisotropy coefficient and θbeing the polar\nangle of the magnetization. In the absence of an electric\ncurrent, the system has two stable states m= +ˆzand\nm=−ˆzwheremis the unit direction of magnetization\nM=Mmof magnitude M. The electric current gen-\nerates a transverse spin current perpendicularly flowing\ninto the ferromagnetic layer via the spin-Hall effect [10],\nandthenproducesaneffectiveSOTonthemagnetization[4, 5, 16], i.e.\n/vector τ=−am×(m׈s)+βam׈s, (1)\nwhere the first term on the right-hand-side is the\nSlonczewski-liketorquewhile thesecondtermis thefield-\nlike torque. The spin-polarization direction is ˆ s=ˆJ׈z\n(for other type of spin-Hall effect, see Note [27]) with ˆJ\nbeing the unit vector of current density. a=¯h\n2edθSHJ\nmeasures SOT where ¯ h,e, anddare respectively the\nPlank constant, the electron charge, and the sample\nthickness. θSHis the spin Hall angle which measures\nthe conversion efficiency between the spin current and\ncharge current. βmeasures the field-like torque and can\nbe an arbitrary real number since this torque may also\nbe directly generated from the Rashba effect [16].\nThemagnetizationdynamicsunderanin-planecurrent\ndensityJis governed by the generalized dimensionless\nLandau-Lifshitz-Gilbert (LLG) equation,\n∂m\n∂t=−m×heff+αm×∂m\n∂t+/vector τ, (2)\nwhereαis the Gilbert damping constant that is typically\nmuch smaller than unity. The effective field is heff=\n−∇mεfrom energy density ε. Time, magnetic field and\nenergy density are respectively in units of ( γM)−1,M\nandµ0M2, where γandµ0are respectively the gyro-\nmagnetic ratio and vacuum magnetic permeability. In\nthis unit system, a=¯h\n2edµ0M2θSHJbecomes dimension-\nless.\nThe magnetization mcan be conveniently described\nby a polar angle θand an azimuthal angle φin thexyz-\ncoordinate. In terms of θandφ, the generalized LLG\nequation becomes\n(1+α2)˙θ=−αKsin2θ+a(1−αβ)cosθsin(Φ−φ)+a(α+β)cos(Φ−φ)≡F1, (3a)\n(1+α2)˙φsinθ=Ksin2θ−a(1−αβ)cos(Φ−φ)+a(α+β)cosθsin(Φ−φ)≡F2. (3b)\nB. Derivation of the Euler-Lagrange equation\nThe goal is to reverse the initial state θ= 0 to the\ntarget state θ=πby SOT. There are an infinite number\nof paths that connect the initial state θ= 0 with the\ntarget state θ=π, and each of these paths can be used\nas a magnetization reversal route. For a given reversal\nroute, there are an infinite number of current pulses that\ncan reverse the magnetization. The theoretical limit of\nminimal current density Jcis defined as the smallest val-\nues of minimal reversal current densities of all possible\nreversal routes. Then it comes two interesting and im-\nportant questions: 1) What is Jcabove which there is at\nleast one reversal route that the current density can re-\nverse the magnetization along it? 2) For a given J > Jc,what are the optimal reversal route and the optimal cur-\nrent pulse Φ( t) that can reverse the magnetization at the\nhighest speed?\nDividing Eq. (3b) by Eq. (3a), one can obtain the\nfollowing constraint,\nG≡∂φ\n∂θsinθF1−F2= 0. (4)\nThe magnetization reversal time Tis\nT=/integraldisplayπ\n0dθ\n˙θ=/integraldisplayπ\n01+α2\nF1dθ. (5)\nThe optimization problem here is to find the optimal\nreversal route φ(θ) and the optimal current pulse Φ( t)3\nsuch that Tis minimum under constraint (4). Using the\nLagrange multiplier method, the optimal reversal route\nand the optimal current pulse satisfy the Euler-Lagrange\nequations [28, 29],\n∂F\n∂φ=d\ndθ(∂F\n∂(∂φ/∂θ)),∂F\n∂Φ=d\ndθ(∂F\n∂(∂Φ/∂θ)),(6)\nwhereF= (1 +α2)/F1+λGandλis the Lagrange\nmultipliers which can be determined self-consistently by\nEq. (6) and constrain (4). Given a current density of\nconstant magnitude J, Eq. (6) may or may not have a\nsolution of φ(θ) that continuously passing through θ=\n0 andθ=π. If such a solution exists, then φ(θ) is\nthe optimal path for the fastest magnetization reversal\nand the corresponding solution of Φ( t) is the optimal\ncurrent pulse. The theoretical limit of minimal reversal\ncurrent density is then the smallest current density Jc\nbelow which the optimal reversal path does not exist.\nC. The optimal current pulse and theoretical limit\nof minimal reversal current density\nFrom Eqs. (3a), (3b) and (4) as well as F= (1 +\nα2)/F1+λG, theEuler-Lagrangeequationof (6)becomes\nλd\ndθ(F1sinθ) = 0, (7a)\n1+α2\nF2\n1∂F1\n∂φ−λ∂G\n∂φ=−1+α2\nF2\n1∂F1\n∂Φ+λ∂G\n∂Φ= 0.(7b)\nFrom Eq. (7a), one has λ/ne}ationslash= 0 orλ= 0. Ifλ/ne}ationslash= 0,F1must\nbeF1=C/sinθ(C/ne}ationslash= 0) so that (1+ α2)˙θ=C/sinθ→\n∞asθ→0 orπ. This solution is not physical, and\nshouldbe discarded. Therefore, the only allowedsolution\nmust be λ= 0, and one has ∂F1/∂Φ = 0 according to\nEq. (7b). Interestingly, this is exactly the condition of\nmaximal ˙θ=F1/(1+α2) as Φ varies. Φ satisfies tan(Φ −\nφ) =1−αβ\nα+βcosθ, or\nΦ = tan−1(1−αβ\nα+βcosθ)+φ+π(β <−α) (8a)\nΦ = tan−1(1−αβ\nα+βcosθ)+φ (β >−α).(8b)\nSubstituting Eq. (8) into the LLG equation (3), θ(t)\nandφ(t) are determined by the following equations,\n˙θ=1\n1+α2[aP(θ)−αKsin2θ], (9a)\n˙φ=1\n1+α2[2Kcosθ−a(α+β)(1−αβ)sinθ\nP(θ)],(9b)\nwhereP(θ) =/radicalbig\n(α+β)2+(1−αβ)2cos2θ. To reverse\nmagnetization from θ= 0 toθ=π,amust satisfy a >\nαKsin(2θ)/P(θ) according to Eq. (9a) so that ˙θis no\nnegative for all θ. Obviously, ˙θ= 0 atθ=π/2 whenFIG. 2. The log α-dependence of Jcfor various βare plotted\nas the solid curves for model parameters of M= 3.7×105\nA/m,K= 5.0×103J/m3,θSH= 0.084 and d= 0.6 nm. As\na comparison, Jdc\ncis also plotted as the dashed lines.\nβ=−α. The magnetization reversal is not possible in\nthis case, and β=−αisasingularpoint. The theoretical\nlimit of minimal reversal current density Jcforβ/ne}ationslash=−α\nis\nJc=2αeKd\nθSH¯hQ, (10)\nwhereQ≡max{sin2θ/P(θ)}forθ∈[0,π].\nIn comparison with the current density threshold [13,\n14, 18] (Jdc\nc) in the conventional strategy for β= 0,\nJdc\nc=2eKd\nθSH¯h(1−H√\n2K), (11)\nthe minimal reversal current density is reduced by more\nthan a factor of α. HereH(≃22 Oe in experiments)\nis a small external magnetic needed for a deterministic\nreversal in conventional strategy. Using CoFeB/Ta pa-\nrameters of M= 3.7×105A/m,K= 5.0×103J/m3,\nθSH= 0.084 and d= 0.6 nm [11, 14, 15], Fig. 2 shows\nlogα-dependenceof Jc(solidlines)and Jdc\nc(dashedlines)\nforβ= 0 (black), 0 .3 (red) and −0.3 (blue), respectively.\nBothJdc\ncandJcdepend on β. The lower the damping of\na magnetic material is, the smaller our minimum switch-\ning current density will be. For a magnetic material of\nα= 10−5, the theoretical limit of minimal reversal cur-\nrent density can be five order of magnitude smaller than\nthe value in the conventional strategy.\nFor a given J > Jc, the shortest reversal time is given\nby Eqs. (5) and (9a):\nT=/integraldisplayπ\n01+α2\naP(θ)−αKsin2θdθ. (12)\nThe optimal reversal path is given by φ(θ) =/integraltextθ\n0˙φ\n˙θdθ′\nwhere˙θand˙φare given by Eqs. (9a) and (9b). Eq.\n(9a) gives t(θ) =/integraltextθ\n0(1+α2)/(aP(θ)−αKsin2θ)dθ′and\nthenθ(t) is just θ(t) =t−1(θ). Thus, Φ( θ,φ),φ(θ)\nandθ(t) giveφ(t) =φ(θ(t)) and Φ( t) = Φ(θ(t),φ(t)).4\n(d) (e) mz\nmxmymz\nmy\nmx(a) (b) (c) \n(f) \nmymz\nmx\nFIG. 3. Model parameters of M= 3.7×105A/m,K= 5.0×103J/m3,θSH= 0.084,α= 0.008 and d= 0.6 nm are used to\nmimic CoFeB/Ta bilayer, and β= 0.3 for (a), (c), (d) and (f) while β= 0.1 for (b) and (e). The theoretical limit of minimum\nreversal current density is Jc= 1.56×105A/cm2forβ= 0.1 andJc= 1.28×105A/cm2forβ= 0.3. Optimal current pulses\n((a)-(c)) and fastest reversal routes ((d)-(f)) are for J= 1.92×106A/cm2((a) and (d)), and for J= 9.0×106A/cm2((b),\n(c)), (e) and (f).\nUsing the same parameters as those for Fig. 2 with\nα= 0.008 and various β, Fig. 3 shows the optimal\ncurrent pulses ((a)-(c)) and the corresponding fastest\nmagnetization reversal routes ((d)-(f)) for β= 0.3 and\nJ= 1.92×106A/cm2≈15Jc((a) and (d)), for β= 0.1\nandJ= 9.0×106A/cm2≈58Jc((b) and (e)), and for\nβ= 0.3 andJ= 9.0×106A/cm2≈70Jc((c) and (f)). It\nisknownthatTahaslesseffecton α[11]. Theminimalre-\nversal current density Jcunder the optimal current pulse\nis 1.56×105A/cm2forβ= 0.1 and 1.28×105A/cm2\nforβ= 0.3 which is far below Jdc\nc= 9.6×106A/cm2for\nthe same material parameters [15]. The multiple oscilla-\ntions ofmxandmyreveal that the reversal is a spinning\nprocess and optimal reversal path winds around the two\nstable states many times. Correspondingly, the driving\ncurrent makes also many turns as shown by the multiple\noscillations of JxandJy. The number of spinning turns\ndependsonhowfar JisfromJc. Thecloser JtoJcis, the\nnumber of turns is larger. The number of turns is about\n5 in Figs. 3(a) and 3(d) for J≈15Jcand one turn for\nJ >50Jcas shown in Figs. 3(b), 3(c), 3(e) and 3(f), so\nthat the reversal is almost ballistic. The reversaltime for\nβ= 0.3 andJ= 1.92×106A/cm2is about 10 nanosec-\nonds, for β= 0.1 andJ= 9.0×106A/cm2is about 3.3\nnanoseconds, and for β= 0.3 andJ= 9.0×106A/cm2is\nabout 2.1 nanoseconds. Figure 4 is the reversaltime Tas\na function of current density Junder the optimal current\npulse for the same parameters as those for Fig. 2. TheFIG. 4. Magnetization reversal time Tunder the optimal\ncurrent pulses as a function of Jfor various αandβ.\nreversaltime quickly decreases to nanoseconds as current\ndensity increases. In a real experiment, there are many\nuncertainties so that the current pulse may be different\nfrom the optimal one. To check whether our strategy is\nrobust again small fluctuations, we let the current pulse\nin Fig. 3(c) deviate from its exact value. Numerical\nsimulations show that the magnetization reversal is not\nsignificantly influenced at least when the deviation be-\ntween the real current and optimal current is less than\nfive percents.5\nIII. VERIFICATION OF MACROSPIN MODEL\nBY MICROMAGNETIC SIMULATION\nIn our analysis, the memory cell is treated as a\nmacrospin. A nature question is how good the macrospin\nmodel is for a realistic memory device. To answer this\nquestion, we carried out micromagnetic simulations by\nusing Newton-Raphson algorithm [30] for two memory\ncells of 150 nm ×150nm×0.6 nm (Figs. 5(a), (b), (d) and\n(e)) and 250 nm ×250 nm×0.6 nm (Figs. 5(c) and (f)).\nTo model the possible edge pinning effect due to mag-\nnetic dipole-dipole interaction, we consider square-shape\ndevices instead of cylinder shape device whose edge pin-\nning is negligible. To make a quantitative comparison,\nthe material parameters are the same as those used in\nFig. 3. In our simulations, the unit cell size is 2 nm ×2\nnm×0.6 nm. For a fair comparison, the optimal current\npulses shown in Figs. 3(a) and (c) of respective current\ndensityJ= 1.92×106A/cm2andJ= 9.0×106A/cm2\nwere applied to the memory cell of 150 nm ×150 nm×0.6\nnm. The symbols in Figs. 5(a) and (b) are the time evo-\nlution of averaged magnetization mx,myandmzwhile\nthesolidlinesarethetheoreticalpredictionsofmacrospin\nmodel shown in Figs. 3(d) and (f). The perfect agree-\nments prove the validity of the macrospin approximation\nfor our device of such a size. To further verify that the\nmemory device can be treated as a macrospin, Figs. 5(d)\nand (e) are the spin configurations in the middle of the\nreversal at t= 5.5 ns for Fig. 5(a) and at t= 1.2 ns for\nFig. 5(b). Thefactthatallspinsalignalmostinthesame\ndirection verifies the validity of the macro spin model. In\nreal experiments, non-uniformity of current density is in-\nevitable. To demonstrate the macrospin model is still\nvalid, we let current density linearly varies from 9 .5×106\nA/cm2on the leftmost column of cells to 8 .5×106A/cm2\nonthe rightmostcolumn ofcells. As expected, thereis no\nnoticeable difference with the data shown in Figs. 5(b)\nand (e).\nFor the large memory device of 250 nm ×250 nm×0.6\nnm, the optimal current pulse shown in Fig. 3(c) of cur-\nrent density J= 9.0×106A/cm2was considered. The\ntime evolution of averaged magnetization mx,myand\nmzare plotted in Fig. 5(c), with the symbols for simula-\ntions andsolid linesfor the macrospinmodel. They agree\nvery well although there is a small deviation for device\nof such a large size. Figure 5(f) is the spin configurations\nin the middle of the reversal at t= 1.2 ns for Fig. 5(c).\nThe marcospin model is not too bad although all spins\nare not perfectly aligned in this case.\nIn summary, for a normal SOT-MRAM device of size\nless than 300 nm [11, 15], macrospin model describes\nmagnetization reversal well. However, for a larger sam-\nple size and lower current density ( J <106A/cm2for\nthe same material parameters as those used in Fig. 3),\nonly the spins in sample center can be reversed while the\nspins near sample edges are pinned.(a) (b) (c) \n(d) (e) (f) t = 5.5 ns t = 1.2 ns t = 1.2 ns \nFIG. 5. (a)-(c) Time evolution of the average magnetization :\ncycles for micromagnetic simulations and solid lines are th e-\noretical predictions from macrospin model. (a) and (b) are\nfor the memory cell of 150 nm ×150 nm×0.6 nm and optimal\ncurrent pulse of current density of J= 1.92×106A/cm2and\nJ= 9.0×106A/cm2, respectively. (c) is for the memory\ncell of 250 nm ×250 nm×0.6 nm and optimal current pulse of\ncurrent density of J= 9.0×106A/cm2. (d)-(f) Spin configu-\nrations respectively corresponding to (a)-(c) in the middl e of\nmagnetization reversal at t= 5.5 ns and 1.2 ns. The cell size\nin micromagnetic simulation is 2 nm ×2 nm×0.6 nm.\nIV. DISCUSSION\nObviously, the strategy present here can easily be gen-\neralized to the existing spin-transfer torque MRAM. The\nmathematics involved are very similar, and one expects\na substantial current density reduction is possible there\nif a proper optimal current pulse is used. Of course, how\nto generate such a current pulse should be much more\nchallenge than that for SOT-MRAM where two perpen-\ndicular currents can be used. In the conventional strat-\negy that uses a DC-current, a static magnetic field along\ncurrent flow is required for a deterministic magnetiza-\ntion reversal [13, 14, 18]. Although several field-free de-\nsigns have been proposed [19, 20], an antiferromagnet is\nneeded to create an exchange bias which plays the role\nof an applied magnetic field. As we have shown, such a\nrequirement or complication is not needed in our strat-\negy. Ourstrategydoesnothaveanotherproblemexisting\nin the conventional strategy in which the magnetization\ncan only be placed near θ=π/2 [13, 14, 18] so that\nthe system falls into the target state by itself through\nthe damping. Therefore, one would like to use materials\nwith largerdamping in the conventionalstrategyin order\nto speed up this falling process. In contrast, our strategy\npreferslowdampingmaterials,andreversalisalmostbal-\nlistic when current density is large enough ( >50Jcin the\ncurrent case). To reverse the magnetization from θ=π\ntoθ= 0, one only needs to reverse the current direction\nof the optimal current pulse. One should notice that\nthe Euler-Lagrange equation allows us to easily obtain\nthe optimal reversal current pulse and theoretical limit6\nof the minimal reversal current density for an arbitrary\nmagnetic cell such as in-plane magnetized layer [11] and\nbiaxial anisotropy.\nV. CONCLUSION\nIn conclusion, weinvestigatedthe magnetizationrever-\nsal of SOT-MRAMs, and propose a new reversal strat-\negy whose minimal reversal current density is far be-\nlow the existing current density threshold. 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The rest procedures\nare similar to what was done in the main text.\n[28] X. R. Wang, P. Yan, J. Lu and C. He, Euler equation\nof the optimal trajectory for the fastest magnetization re-\nversal of nanomagnetic structures , Europhys. Lett. 84,\n27008 (2008).\n[29] G. Arfken, Mathematical Methods for Physicists, 3rd ed.\n(Orlando, FL: Academic Press, 1985).\n[30] M. d’Aquino, C. Serpico, G. Miano, I. D. Mayergoyz and\nG. Bertotti, J. Appl. Phys. 97, 10E319 (2005)."    },    {        "title": "1510.03571v2.Nonlocal_torque_operators_in_ab_initio_theory_of_the_Gilbert_damping_in_random_ferromagnetic_alloys.pdf",        "content": "arXiv:1510.03571v2  [cond-mat.mtrl-sci]  19 Nov 2015Nonlocal torque operators in ab initio theory of the Gilbert damping in random\nferromagnetic alloys\nI. Turek∗\nInstitute of Physics of Materials, Academy of Sciences of th e Czech Republic, ˇZiˇ zkova 22, CZ-616 62 Brno, Czech Republic\nJ. Kudrnovsk´ y†and V. Drchal‡\nInstitute of Physics, Academy of Sciences of the Czech Repub lic,\nNa Slovance 2, CZ-182 21 Praha 8, Czech Republic\n(Dated: July 5, 2018)\nWe present an ab initio theory of theGilbert dampingin substitutionally disorder ed ferromagnetic\nalloys. The theory rests on introduced nonlocal torques whi ch replace traditional local torque\noperators in the well-known torque-correlation formula an d which can be formulated within the\natomic-sphereapproximation. Theformalism is sketchedin asimpletight-bindingmodel andworked\nout in detail in the relativistic tight-binding linear muffin -tin orbital (TB-LMTO) method and\nthe coherent potential approximation (CPA). The resulting nonlocal torques are represented by\nnonrandom, non-site-diagonal and spin-independent matri ces, which simplifies the configuration\naveraging. The CPA-vertex corrections play a crucial role f or the internal consistency of the theory\nand for its exact equivalence to other first-principles appr oaches based on the random local torques.\nThis equivalence is also illustrated by the calculated Gilb ert damping parameters for binary NiFe\nand FeCo random alloys, for pure iron with a model atomic-lev el disorder, and for stoichiometric\nFePt alloys with a varying degree of L1 0atomic long-range order.\nPACS numbers: 72.10.Bg, 72.25.Rb, 75.78.-n\nI. INTRODUCTION\nThe dynamics of magnetization of bulk ferromagnets,\nutrathin magnetic films and magnetic nanoparticles rep-\nresents an important property of these systems, espe-\ncially in the context of high speed magnetic devices for\ndata storage. While a complete picture of magnetization\ndynamics including, e.g., excitation ofmagnonsand their\ninteraction with other degrees of freedom, is still a chal-\nlenge for the modern theory of magnetism, remarkable\nprogresshas been achieved during the last years concern-\ning the dynamics of the total magnetic moment, which\ncan be probed experimentally by means of the ferromag-\nnetic resonance1or by the time-resolved magneto-optical\nKerr effect.2Time evolution of the macroscopic magne-\ntization vector Mcan be described by the well-known\nLandau-Lifshitz-Gilbert (LLG) equation3,4\ndM\ndt=Beff×M+M\nM×/parenleftbigg\nα·dM\ndt/parenrightbigg\n,(1)\nwhereBeffdenotes an effective magnetic field (with the\ngyromagnetic ratio absorbed) acting on the magnetiza-\ntion,M=|M|, and the quantity α={αµν}denotes\na symmetric 3 ×3 tensor of the dimensionless Gilbert\ndamping parameters ( µ,ν=x,y,z). The first term in\nEq. (1) defines a precession of the magnetization vector\naround the direction of the effective magnetic field and\nthe second term describes a damping of the dynamics.\nThe LLG equation in itinerant ferromagnets is appropri-\nate for magnetization precessions very slow as compared\nto precessions of the single-electron spin due to the ex-\nchange splitting and to frequencies of interatomic elec-\ntron hoppings.A large number of theoretical approaches to the\nGilbert damping has been workedout during the last two\ndecades; here we mention only schemes within the one-\nelectron theory of itinerant magnets,5–20where the most\nimportant effects of electron-electron interaction are cap-\ntured by means of a local spin-dependent exchange-\ncorrelation (XC) potential. These techniques can be\nnaturally combined with existing first-principles tech-\nniques based on the density-functional theory, which\nleads to parameter-freecalculations of the Gilbert damp-\ning tensor of pure ferromagnetic metals, their ordered\nand disordered alloys, diluted magnetic semiconductors,\netc. One part of these approaches is based on a static\nlimit of the frequency-dependent spin-spin correlation\nfunction of a ferromagnet.5–8,15,16Other routes to the\nGilbert damping employ relaxations of occupation num-\nbers of individual Bloch electron states during quasi-\nstatic nonequilibrium processes or transition rates be-\ntween different states induced by the spin-orbit (SO)\ninteraction.9–12,14,20The dissipation of magnetic energy\naccompanying the slow magnetization dynamics, evalu-\nated within a scattering theory or the Kubo linear re-\nsponse formalism, leads also to explicit expressions for\nthe Gilbert damping tensor.13,17–19Most of these formu-\nlations yield relations equivalent to the so-called torque-\ncorrelation formula\nαµν=−α0Tr{Tµ(G+−G−)Tν(G+−G−)},(2)\ninwhich thetorqueoperators Tµareeither duetothe XC\nor SO terms of the one-electron Hamiltonian. In Eq. (2),\nwhich has a form of the Kubo-Greenwood formula and is\nvalid for zero temperature of electrons, the quantity α0is\nrelated to the system magnetization (and to fundamental2\nconstants and units used, see Section IIB), the trace is\ntaken over the whole Hilbert space of valence electrons,\nandthesymbols G±=G(EF±i0)denotetheone-particle\nretarded and advanced propagators (Green’s functions)\nat the Fermi energy EF.\nImplementation of the above-mentioned theories in\nfirst-principles computational schemes proved opposite\ntrends of the intraband and interband contributions to\nthe Gilbert damping parameter as functions of a phe-\nnomenological quasiparticle lifetime broadening.7,11,12\nThese qualitative studies have recently been put on a\nmore solid basis by considering a particular mechanism\nof the lifetime broadening, namely, a frozen temperature-\ninduced structural disorder, which represents a realistic\nmodel for a treatment of temperature dependence of the\nGilbert damping.21,22This approach explained quanti-\ntatively the low-temperature conductivity-like and high-\ntemperature resistivity-like trends of the damping pa-\nrameters of iron, cobalt and nickel. Further improve-\nmentsofthemodel, includingstatictemperature-induced\nrandomorientationsoflocalmagneticmoments, haveap-\npeared recently.23\nTheab initio studies have also been successful in re-\nproduction and interpretation of values and concentra-\ntion trends of the Gilbert damping in random ferromag-\nnetic alloys, such as the NiFe alloy with the face-centered\ncubic (fcc) structure (Permalloy)17,22and Fe-based al-\nloys with the body-centered cubic (bcc) structure (FeCo,\nFeV,FeSi).19,22,24Otherstudiesaddressedalsotheeffects\nof doping the Permalloy and bcc iron by 5 dtransition-\nmetal elements19,20,22and of the degree of atomic long-\nrange order in equiconcentration FeNi and FePt alloys\nwith the L1 0-type structures.20Recently, an application\nto halfmetallic Co-based Heusler alloys has appeared as\nwell.25The obtained results revealed correlations of the\ndamping parameter with the density of states at the\nFermienergyandwiththesizeofmagneticmoments.22,24\nIn a one-particle mean-field-like description of a ferro-\nmagnet, the total spin is not conserved due to the XC\nfield and the SO interaction. The currently employed\nformsofthetorqueoperators Tµinthe torque-correlation\nformula (2) reflect these two sources; both the XC- and\nthe SO-induced torques are local and their equivalence\nfor the theory of Gilbert damping has been discussed\nby several authors.15,16,26In the case of random alloys,\nthis equivalence rests on a proper inclusion of vertex cor-\nrections in the configuration averaging of the damping\nparameters αµνas two-particle quantities.\nThe purpose of the present paper is to introduce an-\nother torque operator that can be used in the torque-\ncorrelationformula(2) andto discussits properties. This\noperatoris due to intersiteelectronhopping andit is con-\nsequently nonlocal; in contrast to the local XC- and SO-\ninduced torques which are random in random crystalline\nalloys, the nonlocal torque is nonrandom, i.e., indepen-\ndent on the particular configuration of a random alloy,\nwhich simplifies the configuration averaging of Eq. (2).\nWe show that a similar nonlocal effective torque appearsin the fully relativistic linear muffin-tin orbital (LMTO)\nmethod in the atomic-sphere approximation (ASA) used\nrecently for calculations of the conductivity tensor in\nspin-polarized random alloys.27,28Here we discuss theo-\nretical aspects of the averaging in the coherent-potential\napproximation (CPA)29,30and illustrate the developed\nab initio scheme byapplicationsto selected binaryalloys.\nWe also compare the obtained results with those of the\nLMTO-supercell technique17and with other CPA-based\ntechniques, the fully relativisticKorringa-Kohn-Rostoker\n(KKR) method19,22and the LMTO method with a sim-\nplified treatment of the SO interaction.20\nThe paper is organized as follows. The theoretical for-\nmalism is contained in Section II, with a general discus-\nsion of various torque operators and results of a simple\ntight-binding model presented in Section IIA. The fol-\nlowingSection IIB describes the derivation of the LMTO\ntorque-correlation formula with nonlocal torques; tech-\nnical details are left to Appendix A concerning linear-\nresponse calculations with varying basis sets and to Ap-\npendix B regarding the LMTO method for systems with\na tilted magnetization direction. Selected formal proper-\nties of the developed theory are discussed in Section IIC.\nApplications of the developed formalism can be found\nin Section III. Details of numerical implementation are\nlisted in Section IIIA followed by illustrating examples\nforsystemsofthreedifferent kinds: binarysolidsolutions\nof 3dtransition metals in Section IIIB, pure iron with a\nsimple model of random potential fluctuations in Section\nIIIC, and stoichiometric FePt alloys with a partial long-\nrange order in Section IIID. The main conclusions are\nsummarized in Section IV.\nII. THEORETICAL FORMALISM\nA. Torque-correlation formula with alternative\ntorque operators\nThe torque operators Tµentering the torque-\ncorrelation formula (2) are closely related to compo-\nnents of the time derivative of electron spin. For spin-\npolarized systems described by means of an effective\nSchr¨ odinger-Paulione-electronHamiltonian H, actingon\ntwo-componentwavefunctions, thecompletetimederiva-\ntive of the spin operator is given by the commutation re-\nlationtµ=−i[σµ/2,H], where ¯ h= 1 is assumed and σµ\n(µ=x,y,z) denote the Pauli spin matrices. Let us write\nthe Hamiltonian as H=Hp+Hxc, whereHpincludes all\nspin-independent terms and the SO interaction (Hamil-\ntonian of a paramagnetic system) while Hxc=Bxc(r)·σ\ndenotes the XC term due to an effective magnetic field\nBxc(r). The complete time derivative (spin torque) can\nthen be written as tµ=tso\nµ+txc\nµ, where\ntso\nµ=−i[σµ/2,Hp], txc\nµ=−i[σµ/2,Hxc].(3)\nAs discussed, e.g., in Ref. 15, the use of the complete\ntorquetµinthetorque-correlationformula(2)leadsiden-3\ntically to zero; the correct Gilbert damping coefficients\nαµνfollow from Eq. (2) by using either the SO-induced\ntorquetso\nµ, or the XC-induced torque txc\nµ. Note that only\ntransverse components (with respect to the easy axis of\nthe ferromagnet)of the vectors tsoandtxcare needed for\nthe relevant part of the Gilbert damping tensor (2).\nThe equivalence of both torque operators (3) for the\nGilbert damping can be extended. Let us consider a sim-\nple system described by a model tight-binding Hamilto-\nnianH, written now as H=Hloc+Hnl, where the first\ntermHlocis a lattice sum of local atomic-like terms and\nthe nonlocal second term Hnlincludes all intersite hop-\nping matrix elements. Let us assume that all effects of\nthe SO interaction and XC fields are contained in the\nlocal term Hloc, so that the hopping elements are spin-\nindependent and [ σµ,Hnl] = 0. (Note that this assump-\ntion, often used in model studies, is satisfied only ap-\nproximatively in real ferromagnets with different widths\nof the majority and minority spin bands.) Let us write\nexplicitly Hloc=/summationtext\nR(Hp\nR+Hxc\nR), whereRlabelsthe lat-\ntice sites and where Hp\nRcomprises the spin-independent\npart and the SO interaction of the Rth atomic poten-\ntial while Hxc\nRis due to the local XC field of the Rth\natom. The operators Hp\nRandHxc\nRact only in the sub-\nspace of the Rth site; the subspaces of different sites\nare orthogonal to each other. The total spin operator\ncan be written as σµ/2 = (1/2)/summationtext\nRσRµ, where the local\noperator σRµis the projection of σµon theRth sub-\nspace. Let us assume that each term Hp\nRis spherically\nsymmetric and that Hxc\nR=Bxc\nR·σR, where the effec-\ntive field Bxc\nRof theRth atom has a constant size and\ndirection. Let us introduce local orbital-momentum op-\neratorsLRµand their counterparts including the spin,\nJRµ=LRµ+ (σRµ/2), which are generators of local\ninfinitesimal rotations with respect to the Rth lattice\nsite, and let us define the corresponding lattice sums\nLµ=/summationtext\nRLRµandJµ=/summationtext\nRJRµ=Lµ+(σµ/2). Then\nthe local terms Hp\nRandHxc\nRsatisfy, respectively, commu-\ntation rules [ JRµ,Hp\nR] = 0 and [ LRµ,Hxc\nR] = 0. By using\nthe above assumptions and definitions, the XC-induced\nspin torque (3) due to the XC term Hxc=/summationtext\nRHxc\nRcan\nbe reformulated as\ntxc\nµ=−i/summationdisplay\nR[σRµ/2,Hxc\nR] =−i/summationdisplay\nR[JRµ,Hxc\nR] (4)\n=−i/summationdisplay\nR[JRµ,Hp\nR+Hxc\nR] =−i[Jµ,Hloc]≡tloc\nµ.\nThe last commutator defines a local torque operator tloc\nµ\ndue to the local part of the Hamiltonian Hlocand the op-\neratorJµ,incontrasttothespinoperator σµ/2inEq.(3).\nLet us define the complementary nonlocal torque tnl\nµdue\nto the nonlocal part of the Hamiltonian Hnl, namely,\ntnl\nµ=−i[Jµ,Hnl] =−i[Lµ,Hnl], (5)\nand let us employ the fact that the complete time deriva-\ntive of the operator Jµ, i.e., the torque ˜tµ=−i[Jµ,H] =\ntloc\nµ+tnl\nµ, leads identically to zero when used in Eq. (2).This fact implies that the Gilbert damping parame-\nters can be also obtained from the torque-correlation\nformula with the nonlocal torques tnl\nµ. These torques\nare equivalent to the original spin-dependent local XC-\nor SO-induced torques; however, the derived nonlocal\ntorques are spin-independent, so that commutation rules\n[tnl\nµ,σν] = 0 are satisfied.\nInordertoseetheeffect ofdifferent formsofthe torque\noperators, Eqs. (3) and (5), we have studied a tight-\nbinding model of p-orbitals on a simple cubic lattice with\nthe ground-state magnetization along zaxis. The local\n(atomic-like) terms of the Hamiltonian are specified by\nthe XC term bσRzand the SO term ξLR·σR, which\nare added to a random spin-independent p-level at en-\nergyǫ0+DR, whereǫ0denotes the nonrandom center of\nthep-band while the random parts DRsatisfy configu-\nration averages /an}bracketle{tDR/an}bracketri}ht= 0 and /an}bracketle{tDR′DR/an}bracketri}ht=γδR′Rwith\nthe disorder strength γ. The spin-independent nonlocal\n(hopping) part of the Hamiltonian has been confined to\nnonrandom nearest-neighbor hoppings parametrized by\ntwoquantities, W1(ppσhopping) and W′\n1(ppπhopping),\nsee, e.g., page 36 of Ref. 31. The particular values have\nbeen set to b= 0.3,ξ= 0.2,EF−ǫ0= 0.1,γ= 0.05,\nW1= 0.3 andW′\n1=−0.1 (the hoppings were chosen\nsuch that the band edges for ǫ0=b=ξ=γ= 0 are±1).\nTheconfigurationaverageofthe propagators /an}bracketle{tG±/an}bracketri}ht=¯G±\nand of the torque correlation (2) was performed in the\nself-consistentBornapproximation(SCBA)includingthe\nvertex corrections. Since all three torques, Eqs. (3) and\n(5), are nonrandom operators in our model, the only rel-\nevant component of the Gilbert damping tensor, namely\nαxx=αyy=α, could be unambiguously decomposed in\nthe coherent part αcohand the incoherent part αvcdue\nto the vertex corrections.\nThe results are summarized in Fig. 1 which displays\nthe torque correlation α/α0as a function of the SO cou-\nplingξ(Fig. 1a) and the XC field b(Fig. 1b). The total\nvalueα=αcoh+αvcis identical for all three forms of\nthe torque operator, in contrast to the coherent parts\nαcohwhich exhibit markedly different values and trends\nas compared to each other and to the total α. This re-\nsult is in line with conclusions drawn by the authors of\nRef. 15, 16, and 26 proving the importance of the ver-\ntex corrections for obtaining the same Gilbert damping\nparameters from the SO- and XC-induced torques. The\nonly exception seems to be the case of the SO splitting\nmuch weaker than the exchange splitting, where the ver-\ntex corrections for the SO-induced torque can be safely\nneglected, see Fig. 1a. This situation, encountered in\n3dtransition metals and their alloys, has been treated\nwith the SO-induced torque on an ab initio level with ne-\nglectedvertexcorrectionsinRef. 11and12. Onthe other\nhand, the use of the XC-induced torque calls for a proper\nevaluation of the vertex corrections; their neglect leads\ntoquantitativelyandphysicallyincorrectresultsasdocu-\nmented by recent first-principles studies.19,22The vertex\ncorrectionsareindispensablealsoforthe nonlocaltorque,\nin particular for correct vanishing of the total torque cor-4\n 0 2 4\n 0  0.1  0.2torque  correlation\nspin-orbit  coupling(a)\ntotcoh-nl\ncoh-xc\ncoh-so\n 0 2 4\n 0  0.1  0.2  0.3torque  correlation\nexchange  field(b)\ntotcoh-nl\ncoh-xc\ncoh-so\nFIG. 1. (Color online) The torque correlation α/α0, Eq. (2),\nin a tight-binding p-orbital model treated in the SCBA as\na function of the spin-orbit coupling ξ(a) and of the ex-\nchange field b(b). The full diamonds display the total torque\ncorrelation (tot) and the open symbols denote the coherent\ncontributions αcoh/α0calculated with the SO-induced torque\n(coh-so), the XC-induced torque (coh-xc), Eq. (3), and the\nnonlocal torque (coh-nl), Eq. (5).\nrelation both in the nonrelativistic limit ( ξ→0, Fig. 1a)\nand in the nonmagnetic limit ( b→0, Fig. 1b).\nFinally, let us discuss briefly the general equivalence of\nthe SO- and XC-induced spin torques, Eq. (3), in the\nfully relativistic four-component Dirac formalism.32,33\nThe Kohn-Sham-Dirac Hamiltonian can be written as\nH=Hp+Hxc, whereHp=cα·p+mc2β+V(r) and\nHxc=Bxc(r)·βΣ, wherecis the speed of light, mde-\nnotes the electron mass, p={pµ}refers to the momen-\ntum operator, V(r) is the spin-independent part of the\neffective potential and the α={αµ},βandΣ={Σµ}\narethe well-known4 ×4matricesofthe Diractheory.34,35Then the XC-induced torque is txc=Bxc(r)×βΣ,\nwhich is currently used in the KKR theory of the Gilbert\ndamping.19,22The SO-induced torque is tso=p×cα,\ni.e., it is given directly by the relativistic momentum ( p)\nand velocity ( cα) operators. One can see that the torque\ntsoislocalbutindependent oftheparticularsystemstud-\nied. A comparison of both alternatives, concerning the\ntotal damping parameters as well as their coherent and\nincoherent parts, would be desirable; however, this task\nis beyond the scope of the present study.\nB. Effective torques in the LMTO method\nIn ourab initio approach to the Gilbert damping, we\nemploy the torque-correlation formula (2) with torques\nderived from the XC field.15,19,22The torque operators\nare constructed by considering infinitesimal deviations of\nthe direction of the XC field of the ferromagnet from its\nequilibrium orientation, taken asa reference state. These\ndeviations result from rotations by small angles around\naxesperpendiculartothe equilibrium direction ofthe XC\nfield; componentsofthetorqueoperatorarethengivenas\nderivatives of the one-particle Hamiltonian with respect\nto the rotation angles.36\nFor practical evaluation of Eq. (2) in an ab initio tech-\nnique (such as the LMTO method), one has to consider\na matrix representation of all operators in a suitable or-\nthonormal basis. The most efficient techniques of the\nelectronic structure theory require typically basis vectors\ntailored to the system studied; in the present context,\nthis leads naturally to basis sets depending on the angu-\nlar variables needed to define the torque operators. Eval-\nuation of the torque correlation using angle-dependent\nbases is discussed in Appendix A, where we prove that\nEq. (2) can be calculated solely from the matrix ele-\nments of the Hamiltonian and their angular derivatives,\nsee Eq. (A7), whereas the angular dependence of the ba-\nsis vectorsdoes not contribute directly to the final result.\nThe relativistic LMTO-ASA Hamiltonian matrix for\nthe reference system in the orthogonal LMTO represen-\ntation is given by37–39\nH=C+(√\n∆)+S(1−γS)−1√\n∆, (6)\nwhere the C,√\n∆ andγdenote site-diagonal matrices\nof the standard LMTO potential parameters and Sis\nthe matrix of canonical structure constants. The change\nof the Hamiltonian matrix Hdue to a uniform rotation\nof the XC field is treated in Appendix B; it is sum-\nmarized for finite rotations in Eq. (B7) and for angu-\nlar derivatives of Hin Eqs. (B8) and (B9). The resol-\nventG(z) = (z−H)−1of the LMTO Hamiltonian (6)\nfor complex energies zcan be expressed using the auxil-\niary resolvent g(z) = [P(z)−S]−1, which represents an\nLMTO-counterpart of the scattering-path operator ma-\ntrix of the KKR method.32,33The symbol P(z) denotes\nthe site-diagonal matrix of potential functions; their an-\nalytic dependence on zand on the potential parameters5\ncan be found elsewhere.27,37The relation between both\nresolvents leads to the formula28\nG+−G−=F(g+−g−)F+, (7)\nwhere the same abbreviation F= (√\n∆)−1(1−γS) as in\nEq. (B8) was used and g±=g(EF±i0) .\nThe torque-correlation formula (2) in the LMTO-ASA\nmethod follows directly from relations (A7), (B8), (B9)\nand (7). The components of the Gilbert damping tensor\n{αµν}in the LLG equation (1) can be obtained from a\nbasic tensor {˜αµν}given by\n˜αµν=−α0Tr{τµ(g+−g−)τν(g+−g−)},(8)\nwhere the quantities\nτµ=−i[Jµ,S] =−i[Lµ,S] (9)\ndefine components of an effective torque in the LMTO-\nASA method. The site-diagonal matrices JµandLµ\n(µ=x,y,z) are Cartesian components of the total and\norbital angular momentum operator, respectively, see\ntext aroundEqs.(B8) and (B9). The tracein (8) extends\nover all orbitals of the crystalline solid and the prefactor\ncan be written as α0= (2πMspin)−1, where Mspinde-\nnotes the spin magnetic moment of the whole crystal in\nunits of the Bohr magneton µB.15,19,22\nLet us discuss properties of the effective torque (9).\nIts form is obviously identical to the nonlocal torque (5).\nThe matrix τµis non-site-diagonal, but—for a random\nsubstitutional alloy on a nonrandom lattice—it is non-\nrandom (independent on the alloy configuration). More-\nover, it is given by a commutator of the site-diagonal\nnonrandom matrix Jµ(orLµ) and the LMTO structure-\nconstantmatrix S. Thesepropertiespointtoacloseanal-\nogy between the effective torque and the effective veloci-\nties in the LMTO conductivity tensor based on a concept\nof intersite electron hopping.27,28,40Let us mention that\nexisting ab initio approaches employ random torques,\neither the XC-induced torque in the KKR method19,22\nor the SO-induced torque in the LMTO method.20An-\nother interesting property of the effective torque τµ(9)\nis its spin-independence which follows from the spin-\nindependence of the matrices LµandS.\nThe explicit relation between the symmetric tensors\n{αµν}and{˜αµν}canbeeasilyformulatedfortheground-\nstate magnetization along zaxis; then it is given simply\nbyαxx= ˜αyy,αyy= ˜αxx, andαxy=−˜αxy. These\nrelations reflect the fact that an infinitesimal deviation\ntowards xaxis results from an infinitesimal rotation of\nthe magnetization vector around yaxis and vice versa.\nNote that the other components of the Gilbert damp-\ning tensor ( αµzforµ=x,y,z) are not relevant for the\ndynamics of small deviations of magnetization direction\ndescribed by the LLG equation (1). For the ground-\nstate magnetization pointing along a general unit vector\nm= (mx,my,mz), one has to employ the Levi-Civita\nsymbolǫµνλin order to get the Gilbert damping tensorαas\nαµν=/summationdisplay\nµ′ν′ηµµ′ηνν′˜αµ′ν′, (10)\nwhereηµν=/summationtext\nλǫµνλmλ. The resultingtensor(10) satis-\nfies the condition α·m= 0 appropriate for the dynamics\nof small transverse deviations of magnetization.\nThe application to random alloys requires configura-\ntion averaging of ˜ αµν(8). Since the effective torques τµ\nare nonrandom, one can write a unique decomposition\nof the average into the coherent and incoherent parts,\n˜αµν= ˜αcoh\nµν+ ˜αvc\nµν, where the coherent part is expressed\nby means of the averaged auxiliary resolvents ¯ g±=/an}bracketle{tg±/an}bracketri}ht\nas\n˜αcoh\nµν=−α0Tr{τµ(¯g+−¯g−)τν(¯g+−¯g−)}(11)\nand the incoherent part (vertex corrections) is given as a\nsum of four terms, namely,\n˜αvc\nµν=−α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)Tr/an}bracketle{tτµgpτνgq/an}bracketri}htvc.(12)\nIn this work, the configuration averaging has been done\nin the CPA. Details concerning the averaged resolvents\ncan be found, e.g., in Ref. 39 and the construction of the\nvertex corrections for transport properties was described\nin Appendix to Ref. 30.\nC. Properties of the LMTO torque-correlation\nformula\nThe damping tensor (8) has been formulated in the\ncanonical LMTO representation. In the numerical im-\nplementation, the well-known transformation to a tight-\nbinding(TB)LMTOrepresentation41,42isadvantageous.\nThe TB-LMTO representation is specified by a diag-\nonal matrix βof spin-independent screening constants\n(βR′ℓ′m′s′,Rℓms=δR′Rδℓ′ℓδm′mδs′sβRℓin a nonrelativis-\ntic basis) and the transformation of all quantities be-\ntween both LMTO representations has been discussed in\nthe literature for pure crystals42as well as for random\nalloys.28,39,43The same techniques can be used in the\npresent case together with an obvious commutation rule\n[Jµ,β] = [Lµ,β] = 0. Consequently, the conclusions\ndrawn are the same as for the conductivity tensor:28the\ntotal damping tensor (8) as well as its coherent (11) and\nincoherent (12) parts in the CPA are invariant with re-\nspect to the choice of the LMTO representation.\nIt should be mentioned that the central result, namely\nthe relations(8) and (9), is not limited to the LMTO the-\nory, but it can be translatedinto the KKRtheory aswell,\nsimilarly to the conductivity tensor in the formalism of\nintersite hopping.40The LMTO structure-constant ma-\ntrixSandtheauxiliaryGreen’sfunction g(z)willbethen\nreplacedrespectivelybytheKKRstructure-constantma-\ntrix and by the scattering-path operator.32,33Note, how-6\never, that the total ( Jµ) and orbital ( Lµ) angular mo-\nmentum operatorsin the effective torques (9) will be rep-\nresented by the same matrices as in the LMTO theory.\nLet us mention for completeness that the present\nLMTO-ASA theory allows one to introduce effective lo-\ncal (but random) torques as well. This is based on the\nfact that only the Fermi-level propagators g±defined by\nthe structure constant matrix Sand by the potential\nfunctions at the Fermi energy, P=P(EF), enter the\nzero-temperature expression for the damping tensor ˜ αµν\n(8). Since the equation of motion ( P−S)g±= 1 implies\nimmediately S(g+−g−) =P(g+−g−) and, similarly,\n(g+−g−)S= (g+−g−)P, one can obviously replace the\nnonlocal torques τµ(9) in the torque-correlation formula\n(8) by their local counterparts\nτxc\nµ= i[P,Jµ], τso\nµ= i[P,Lµ].(13)\nThese effective torques are represented by random, site-\ndiagonal matrices; the τxc\nµandτso\nµcorrespond, respec-\ntively, to the XC-induced torque used in the KKR\nmethod22and to the SO-induced torque used in the\nLMTO method with a simplified treatment of the SO-\ninteraction.20In the case of random alloys treated in the\nCPA, the randomness of the local torques (13) calls for\nthe approach developed by Butler44for the averaging of\nthe torque-correlationcoefficient (8). One can provethat\nthe resulting damping parameters ˜ αµνobtained in the\nCPA with the local and nonlocal torques are fully equiv-\nalent to each other; this equivalence rests heavily on a\nproper inclusion of the vertex corrections45and it leads\nto further important consequences. First, the Gilbert\ndamping tensor vanishes exactly for zero SO interaction,\nwhich follows from the use of the SO-induced torque τso\nµ\nand from the obvious commutation rule [ P,Lµ] = 0 valid\nfor the spherically symmetric potential functions (in the\nabsence of SO interaction). This result is in agreement\nwith thenumericalstudyofthe toymodel inSectionIIA,\nsee Fig. 1a for ξ= 0. On an ab initio level, this prop-\nerty has been obtained numerically both in the KKR\nmethod22and in the LMTO method.26Second, the XC-\nandSO-inducedlocaltorques(13)withintheCPAareex-\nactly equivalent as well, as has been indicated in a recent\nnumerical study for a random bcc Fe 50Co50alloy.26In\nsummary, the nonlocaltorques(9) andboth localtorques\n(13) can be used as equivalent alternatives in the torque-\ncorrelation formula (8) provided that the vertex correc-\ntions are included consistently with the CPA-averaging\nof the single-particle propagators.\nIII. ILLUSTRATING EXAMPLES\nA. Implementation and numerical details\nThe numerical implementation of the described the-\nory and the calculations have been done with similar\ntools as in our recent studies of ground-state46and 3 6 9\n 0  10  20  30103 α\nε  (µRy)fcc Ni80Fe20bcc Fe80Co20\n(x 10)\nFIG. 2. The Gilbert damping parameters αof random fcc\nNi80Fe20(full circles) and bcc Fe 80Co20(open squares) alloys\nas functions of the imaginary part of energy ε. The values of\nαfor the Fe 80Co20alloy are magnified by a factor of 10.\ntransport27,28,47properties. The ground-state magne-\ntization was taken along zaxis and the selfconsistent\nXC potentials were obtained in the local spin-density ap-\nproximation (LSDA) with parametrization according to\nRef. 48. The valence basis comprised s-,p-, andd-type\norbitalsand the energyargumentsforthe propagators¯ g±\nand the CPA-vertex corrections were obtained by adding\na tiny imaginary part ±εto the real Fermi energy. We\nhave found that the dependence of the Gilbert damping\nparameter on εis quite smooth and that the value of\nε= 10−6Ry is sufficient for the studied systems, see\nFig. 2 for an illustration. Similar smooth dependences\nhavebeenobtainedalsoforotherinvestigatedalloys,such\nas Permalloy doped by 5 delements, Heusler alloys, and\nstoichiometric FePt alloys with a partial atomic long-\nrange order. In all studied cases, the number Nofk\nvectors needed for reliable averaging over the Brillouin\nzone (BZ) was properly checked; as a rule, N∼108in\nthe full BZ was sufficient for most systems, but for di-\nluted alloys (a few percent of impurities), N∼109had\nto be taken.\nB. Binary fcc and bcc solid solutions\nThe developed theory has been applied to random bi-\nnary alloys of 3 dtransition elements Fe, Co, and Ni,\nnamely, to the fcc NiFe and bcc FeCo alloys. The most\nimportant results, including a comparison to other exist-\ningab initio techniques, are summarized in Fig. 3. One\ncan see a good agreement of the calculated concentration\ntrends of the Gilbert damping parameter α=αxx=αyy\nwith the results of an LMTO-supercell approach17and\nof the KKR-CPA method.22The decrease of αwith in-7\n 0 4 8 12\n 0  0.2  0.4  0.6103 α\nFe  concentrationfcc  NiFe(a)\nthis work\nLMTO-SC\n 0 2 4 6\n 0  0.2  0.4  0.6103 α\nCo  concentrationbcc  FeCo(b)\nthis work\nKKR-CPA\nFIG. 3. (Color online) The calculated concentration depen-\ndences of the Gilbert damping parameter αfor random fcc\nNiFe (a) and bcc FeCo (b) alloys. The results of this work\nare marked by the full diamonds, whereas the open circles\ndepict the results of other approaches: the LMTO supercell\n(LMTO-SC) technique17and the KKR-CPA method.22\ncreasingFecontentintheconcentratedNiFealloyscanbe\nrelatedto the increasingalloymagnetization17andto the\ndecreasing strength of the SO-interaction,20whereas the\nbehaviorinthedilutelimitcanbeexplainedbyintraband\nscattering due to Fe impurities.11,12,14In the case of the\nFeCo system, the minimum of αaround 20% Co, which\nis also observed in room-temperature experiments,49,50\nis related primarily to a similar concentration trend of\nthe density of states at the Fermi energy,22though the\nmaximum of the magnetization at roughly the same alloy\ncomposition51might partly contribute as well.\nA more detailed comparison of all ab initio results is\npresented in Table I for the fcc Ni 80Fe20random alloy\n(Permalloy). The differences in the values of αfrom the\ndifferenttechniquescanbe ascribedtovarioustheoretical\nfeatures and numericaldetails employed, such asthe sim-TABLE I. Comparison of the Gilbert damping parameter α\nfor the fcc Ni 80Fe20random alloy (Permalloy) calculated by\nthe present approach and by other techniques using the CPA\nor supercells (SC). The last column displays the coherent pa rt\nαcohof the total damping parameter according to Eq. (11).\nThe experimental value corresponds to room temperature.\nMethod α αcoh\nThis work, ε= 10−5Ry 4 .9×10−31.76\nThis work, ε= 10−6Ry 3 .9×10−31.76\nKKR-CPAa4.2×10−3\nLMTO-CPAb3.5×10−3\nLMTO-SCc4.6×10−3\nExperimentd8×10−3\naReference 22.\nbReference 20.\ncReference 17.\ndReference 49.\nplified treatment of the SO-interaction in Ref. 20 instead\nof the fully relativistic description, or the use of super-\ncells in Ref. 17 instead of the CPA. Taking into account\nthat calculated residual resistivities for this alloy span a\nwide interval between 2 µΩcm, see Ref. 27 and 52, and\n3.5µΩcm, see Ref. 17, one can consider the scatter of the\ncalculated values of αin Table I as little important. The\ntheoretical values of αare smaller systematically than\nthe measured values, typically by a factor of two. This\ndiscrepancy might be partly due to the effects of finite\ntemperatures as well as due to additional structural de-\nfects of real samples.\nA closer look at the theoretical results reveals that the\ntotal damping parameters αareappreciablysmallerthan\nthemagnitudesoftheircoherentandvertexparts,seeTa-\nble I for the case of Permalloy. This is in agreement with\ntheresultsofthemodelstudyinSectionIIA;similarcon-\nclusions about the importance of the vertex corrections\nhave been done with the XC-induced torques in other\nCPA-based studies.19,22,26The present results prove that\nthis unpleasant feature of the nonlocal torques does not\nrepresent a serious obstacle in obtaining reliable values\nof the Gilbert damping parameter in random alloys. We\nnote that the vertex corrections can be negligible in ap-\nproaches employing the SO-induced torques, at least for\nsystems with the SO splittings much weaker than the XC\nsplittings,12such as the binary ferromagnetic alloys of 3 d\ntransition metals,26see also Section IIA.\nC. Pure iron with a model disorder\nAs it has been mentioned in Section I, the Gilbert\ndamping of pure ferromagnetic metals exhibits non-\ntrivial temperature dependences, which have been re-\nproduced by means of ab initio techniques with vari-\nous levels of sophistication.11,12,21,23In this study, we\nhave simulated the effect of finite temperatures by intro-8\n 0 3 6 9\n0123402040103 α\nρ  (µΩcm)\n103 δ2  (Ry2)bcc  Fe\nFIG. 4. (Color online) The calculated Gilbert damping pa-\nrameter α(full squares) and the residual resistivity ρ(open\ncircles) of pure bcc iron as functions of δ2, where δis the\nstrength of a model atomic-level disorder.\nducing static fluctuations of the one-particle potential.\nThe adopted model of atomic-level disorderassumes that\nrandom spin-independent shifts ±δ, constant inside each\natomic sphere and occurring with probabilities 50% of\nbothsigns,areaddedtothenonrandomselfconsistentpo-\ntential obtained at zero temperature. The Fermi energy\niskeptfrozen,equaltoitsselfconsistentzero-temperature\nvalue. This model can be easily treated in the CPA; the\nresulting Gilbert damping parameter αof pure bcc Fe as\na function of the potential shift δis plotted in Fig. 4.\nThecalculateddependence α(δ) isnonmonotonic, with\na minimum at δ≈30 mRy. This trend is in a qualita-\ntive agreement with trends reported previously by other\nauthors, who employed phenomenological models of the\nelectron lifetime11,12as well as models for phonons and\nmagnons.21,23The origin of the nonmonotonic depen-\ndenceα(δ) has been identified on the basis of the band\nstructure of the ferromagnetic system as an interplay be-\ntween the intraband contributions to α, dominating for\nsmall values of δ, and the interband contributions, domi-\nnating for large values of δ.7,11,12Since the present CPA-\nbased approach does not use any bands, we cannot per-\nform a similar analysis.\nThe obtained minimum value of the Gilbert damping,\nαmin≈10−3(Fig. 4), agrees reasonably well with the\nvalues obtained by the authors of Ref. 11, 12, 21, and\n23. This agreement indicates that the atomic-level dis-\norder employed here is equivalent to a phenomenological\nlifetime broadening. For a rough quantitative estimation\nof the temperature effect, one can employ the calculated\nresistivity ρof the model, which increases essentially lin-\nearly with δ2, see Fig. 4. Since the metallic resistivity\ndue to phonons increases linearly with the temperature\nT(for temperatures not much smaller than the Debye\ntemperature), one can assume a proportionality between 10 20 30 40\n0 0.5 1152535103 α\nDOS(EF)  (states/Ry)\nLRO  parameter  SL10  FePt\nFIG. 5. (Color online) The calculated Gilbert damping pa-\nrameter α(full squares) and the total DOS (per formula unit)\nat the Fermi energy (open circles) of stoichiometric L1 0FePt\nalloys as functions of the LRO parameter S.\nδ2andT. The resistivity of bcc iron at the Curie tem-\nperature TC= 1044 K due to lattice vibrations can be\nestimated around 35 µΩcm,23,53which sets an approx-\nimate temperature scale to the data plotted in Fig. 4.\nHowever, a more accurate description of the temperature\ndependence of the Gilbert damping parameter cannot be\nobtained, mainly due to the neglected true atomic dis-\nplacements and the noncollinearity of magnetic moments\n(magnons).23\nD. FePt alloys with a partial long-range order\nSince important ferromagnetic materials include or-\ndered alloys, we address here the Gilbert damping in sto-\nichiometric FePt alloys with L1 0atomic long-range order\n(LRO). Their transport properties47and the damping\nparameter20have recently been studied by means of the\nTB-LMTO method in dependence on a varying degree\nof the LRO. These fcc-based systems contain two sublat-\nticeswith respectiveoccupationsFe 1−yPtyandPt 1−yFey\nwherey(0≤y≤0.5) denotes the concentration of anti-\nsite atoms. The LRO parameter S(0≤S≤1) is then\ndefined as S= 1−2y, so that S= 0 corresponds to the\nrandom fcc alloy and S= 1 corresponds to the perfectly\nordered L1 0structure.\nThe resulting Gilbert damping parameter is displayed\nin Fig. 5 as a function of S. The obtained trend with a\nbroadmaximumat S= 0andaminimumaround S= 0.9\nagrees very well with the previous result.20The values of\nαin Fig. 5 are about 10% higher than those in Ref. 20,\nwhich can be ascribed to the fully relativistic treatment\nin the present study in contrast to a simplified treatment\nof the SO interaction in Ref. 20. The Gilbert damping9\nin the FePt alloys is an order of magnitude stronger than\nin the alloys of 3 delements (Section IIIB) owing to the\nstronger SO interaction of Pt atoms. The origin of the\nslow decrease of αwith increasing S(for 0≤S≤0.9)\ncan be explained by the decreasing total density of states\n(DOS) at the Fermi energy, see Fig. 5, which represents\nan analogy to a similar correlation observed, e.g., for bcc\nFeCo alloys.22\nAll calculated values of αshown in Fig. 5, correspond-\ning to 0 ≤S≤0.985, are appreciably smaller than\nthe measured one which amounts to α≈0.06 reported\nfor a thin L1 0FePt epitaxial film.54The high measured\nvalue of αmight be thus explained by the present cal-\nculations by assuming a very small concentration of an-\ntisites in the prepared films, which does not seem too\nrealistic. Another potential source of the discrepancy\nlies in the thin-film geometry used in the experiment.\nMoreover, the divergence of αin the limit of S→1\n(Fig. 5) illustrates a general shortcoming of approaches\nbased on the torque-correlation formula (2), since the\nzero-temperature Gilbert damping parameter of a pure\nferromagnet should remain finite. A correct treatment\nof this case, including the dilute limit of random alloys\n(Fig. 3), must take into account the full interacting sus-\nceptibility in the presence of SO interaction.15,55Pilotab\ninitiostudies in this direction have recently appeared for\nnonrandom systems;56,57however, their extension to dis-\nordered systems goes far beyond the scope of this work.\nIV. CONCLUSIONS\nWe have introduced nonlocal torques as an alterna-\ntive to the usual local torque operators entering the\ntorque-correlation formula for the Gilbert damping ten-\nsor. Within the relativistic TB-LMTO-ASA method,\nthis idea leads to effective nonlocal torques as non-site-\ndiagonal and spin-independent matrices. For substitu-\ntionally disordered alloys, the nonlocal torques are non-\nrandom, which allows one to develop an internally con-\nsistent theory in the CPA. The CPA-vertex corrections\nproved indispensable for an exact equivalence of the non-\nlocal nonrandom torques with their local random coun-\nterparts. The concept of the nonlocal torques is not lim-\nited to the LMTO method and its formulation both in\na semiempirical TB theory and in the KKR theory is\nstraightforward.\nThe numerical implementation and the results for bi-\nnary solid solutions show that the total Gilbert damping\nparameters from the nonlocal torques are much smaller\nthan magnitudes of the coherent parts and of the ver-\ntex corrections. Nevertheless, the total damping param-\neters for the studied NiFe, FeCo and FePt alloys compare\nquantitatively very well with results of other ab initio\ntechniques,17,20,22which indicates a fair numerical sta-\nbility of the developed theory.\nThe performed numerical study of the Gilbert damp-\ning in pure bcc iron as a function of an atomic-level dis-order yields a nonmonotonic dependence in a qualitative\nagreementwith the trends consisting of the conductivity-\nlike and resistivity-like regions, obtained from a phe-\nnomenological quasiparticle lifetime broadening7,11,12or\nfrom the temperature-induced frozen phonons21,22and\nmagnons.23Future studies should clarify the applicabil-\nity of the introduced nonlocal torques to a full quanti-\ntative description of the finite-temperature behavior as\nwell as to other torque-related phenomena, such as the\nspin-orbit torques due to applied electric fields.58,59\nACKNOWLEDGMENTS\nThe authors acknowledge financial support by the\nCzech Science Foundation (Grant No. 15-13436S).\nAppendix A: Torque correlation formula in a matrix\nrepresentation\nIn this Appendix, evaluation of the Kubo-Greenwood\nexpression for the torque-correlation formula (2) is dis-\ncussed in the case of the XC-induced torque operators\nusing matrix representations of all operators in an or-\nthonormal basis that varies due to the varying direc-\ntion of the XC field. All operators are denoted by a\nhat, in order to be distinguished from matrices repre-\nsenting these operators in the chosen basis. Let us con-\nsider a one-particle Hamiltonian ˆH=ˆH(θ1,θ2) depend-\ning on two real variables θj,j= 1,2, and let us denote\nˆT(j)(θ1,θ2) =∂ˆH(θ1,θ2)/∂θj. In our case, the variables\nθjplay the role of rotation angles and the operators ˆT(j)\nare the corresponding torques. Let us denote the resol-\nvents of ˆH(θ1,θ2) at the Fermi energy as ˆG±(θ1,θ2) and\nlet us consider a special linear response coefficient (argu-\nmentsθ1andθ2are omitted here and below for brevity)\nc= Tr{ˆT(1)(ˆG+−ˆG−)ˆT(2)(ˆG+−ˆG−)} (A1)\n= Tr{(∂ˆH/∂θ1)(ˆG+−ˆG−)(∂ˆH/∂θ2)(ˆG+−ˆG−)}.\nThis torque-correlation coefficient equals the Gilbert\ndamping parameter (2) with the prefactor ( −α0) sup-\npressed. For its evaluation, we introduce an orthonormal\nbasis|χm(θ1,θ2)/an}bracketri}htand represent all operators in this ba-\nsis. This leads to matrices H(θ1,θ2) ={Hmn(θ1,θ2)},\nG±(θ1,θ2) ={(G±)mn(θ1,θ2)}andT(j)(θ1,θ2) =\n{T(j)\nmn(θ1,θ2)}, where\nHmn=/an}bracketle{tχm|ˆH|χn/an}bracketri}ht,(G±)mn=/an}bracketle{tχm|ˆG±|χn/an}bracketri}ht,\nT(j)\nmn=/an}bracketle{tχm|ˆT(j)|χn/an}bracketri}ht=/an}bracketle{tχm|∂ˆH/∂θj|χn/an}bracketri}ht,(A2)\nand, consequently, to the response coefficient (A1) ex-\npressed by using the matrices (A2) as\nc= Tr{T(1)(G+−G−)T(2)(G+−G−)}.(A3)\nHowever, in evaluation of the last expression, atten-\ntion has to be paid to the difference between the ma-\ntrixT(j)(θ1,θ2) and the partial derivative of the matrix10\nH(θ1,θ2) with respect to θj. This difference follows from\nthe identity ˆH=/summationtext\nmn|χm/an}bracketri}htHmn/an}bracketle{tχn|, which yields\nT(j)\nmn=∂Hmn/∂θj+/summationdisplay\nk/an}bracketle{tχm|∂χk/∂θj/an}bracketri}htHkn\n+/summationdisplay\nkHmk/an}bracketle{t∂χk/∂θj|χn/an}bracketri}ht, (A4)\nwhere we employed the orthogonality relations\n/an}bracketle{tχm(θ1,θ2)|χn(θ1,θ2)/an}bracketri}ht=δmn. Their partial derivatives\nyield\n/an}bracketle{tχm|∂χn/∂θj/an}bracketri}ht=−/an}bracketle{t∂χm/∂θj|χn/an}bracketri}ht ≡Q(j)\nmn,(A5)\nwhere we introduced elements of matrices Q(j)={Q(j)\nmn}\nforj= 1,2. Note that the matrices Q(j)(θ1,θ2) reflect\nexplicitlythe dependenceofthebasisvectors |χm(θ1,θ2)/an}bracketri}ht\nonθ1andθ2. The relation (A4) between the matrices\nT(j)and∂H/∂θ jcan be now rewritten compactly as\nT(j)=∂H/∂θ j+[Q(j),H]. (A6)\nSince the last term has a form of a commutator with the\nHamiltonianmatrix H, theuseofEq.(A6)intheformula\n(A3) leads to the final matrix expression for the torque\ncorrelation,\nc= Tr{(∂H/∂θ 1)(G+−G−)(∂H/∂θ 2)(G+−G−)}.(A7)\nThe equivalence of Eqs. (A3) and (A7) rests on the rules\n[Q(j),H] = [EF−H,Q(j)] and (EF−H)(G+−G−) =\n(G+−G−)(EF−H) = 0 and on the cyclic invariance of\nthe trace. It is also required that the matrices Q(j)are\ncompatible with periodic boundary conditions used in\ncalculations of extended systems, which is obviously the\ncase for angular variables θjrelated to the global changes\n(uniform rotations) of the magnetization direction.\nThe obtained result means that the original response\ncoefficient (A1) involving the torques as angular deriva-\ntives of the Hamiltonian can be expressed solely by us-\ning matrix elements of the Hamiltonian in an angle-\ndependent basis; theangulardependence ofthebasisvec-\ntors does not enter explicitly the final torque-correlation\nformula (A7).\nAppendix B: LMTO Hamiltonian of a ferromagnet\nwith a tilted magnetic field\nHere we sketch a derivation of the fully relativis-\ntic LMTO Hamiltonian matrix for a ferromagnet with\nthe XC-field direction tilted from a reference direction\nalong an easy axis. The derivation rests on the form of\nthe Kohn-Sham-Dirac Hamiltonian in the LMTO-ASA\nmethod.37–39The symbols with superscript 0 refer to the\nreferencesystem,thesymbolswithoutthissuperscriptre-\nfer to the system with the tilted XC field. The operators\n(Hamiltonians, rotation operators) are denoted by sym-\nbols with a hat. The spin-dependent parts of the ASApotentials due to the XC fields are rigidly rotated while\nthe spin-independent parts are unchanged, in full anal-\nogy to the approach employed in the relativistic KKR\nmethod.19,22\nThe ASA-Hamiltonians of both systems are given by\nlattice sums ˆH0=/summationtext\nRˆH0\nRandˆH=/summationtext\nRˆHR, where\nthe individual site-contributions are coupled mutually by\nˆHR=ˆURˆH0\nRˆU+\nR, whereˆURdenotesthe unitaryoperator\nof a rotation (in the orbital and spin space) around the\nRth lattice site which brings the local XC field from its\nreference direction into the tilted one. Let |φ0\nRΛ/an}bracketri}htand\n|˙φ0\nRΛ/an}bracketri}htdenote, respectively, the phi and phi-dot orbitals\nof the reference Hamiltonian ˆH0\nR, then\n|φRΛ/an}bracketri}ht=ˆUR|φ0\nRΛ/an}bracketri}ht,|˙φRΛ/an}bracketri}ht=ˆUR|˙φ0\nRΛ/an}bracketri}ht(B1)\ndefine the phi and phi-dot orbitals of the Hamiltonian\nˆHR. The orbital index Λ labels all linearly indepen-\ndentsolutions(regularattheorigin)ofthespin-polarized\nrelativistic single-site problem; the detailed structure of\nΛ can be found elsewhere.37–39Let us introduce further\nthe well-known empty-space solutions |K∞,0\nRN/an}bracketri}ht(extending\nover the whole real space), |Kint,0\nRN/an}bracketri}ht(extending over the\ninterstitial region), and |K0\nRN/an}bracketri}htand|J0\nRN/an}bracketri}ht(both trun-\ncated outside the Rth sphere), needed for the definition\nof the LMTOs of the reference system.41,42,60Their in-\ndexN, which defines the spin-spherical harmonics of the\nlarge component of each solution, can be taken either in\nthe nonrelativistic ( ℓms) form or in its relativistic ( κµ)\ncounterpart. We define further\n|ZRN/an}bracketri}ht=ˆUR|Z0\nRN/an}bracketri}htforZ=K∞, K, J. (B2)\nIsotropyofthe emptyspaceguaranteesrelations(for Z=\nK∞,K,J)\n|ZRN/an}bracketri}ht=/summationdisplay\nN′|Z0\nRN′/an}bracketri}htUN′N,\n|Z0\nRN/an}bracketri}ht=/summationdisplay\nN′|ZRN′/an}bracketri}htU+\nN′N, (B3)\nwhereU={UN′N}denotes a unitary matrix represent-\ning the rotation in the space of spin-spherical harmonics\nand where U+\nN′N≡(U+)N′N= (UNN′)∗= (U−1)N′N;\nthe matrix Uis the same for all lattice sites Rsince we\nconsider only uniform rotations of the XC-field direction\ninside the ferromagnet. The expansion theorem for the\nenvelope orbital |K∞,0\nRN/an}bracketri}htis\n|K∞,0\nRN/an}bracketri}ht=|Kint,0\nRN/an}bracketri}ht+|K0\nRN/an}bracketri}ht\n−/summationdisplay\nR′N′|J0\nR′N′/an}bracketri}htS0\nR′N′,RN,(B4)\nwhereS0\nR′N′,RNdenote elements of the canonical\nstructure-constant matrix (with vanishing on-site ele-\nments,S0\nRN′,RN= 0) of the reference system. The use\nof relations (B3) in the expansion (B4) together with an\nabbreviation\n|Kint\nRN/an}bracketri}ht=/summationdisplay\nN′|Kint,0\nRN′/an}bracketri}htUN′N (B5)11\nyields the expansion of the envelope orbital |K∞\nRN/an}bracketri}htas\n|K∞\nRN/an}bracketri}ht=|Kint\nRN/an}bracketri}ht+|KRN/an}bracketri}ht\n−/summationdisplay\nR′N′|JR′N′/an}bracketri}ht(U+S0U)R′N′,RN,(B6)\nwhereUandU+denote site-diagonal matrices with el-\nementsUR′N′,RN=δR′RUN′Nand (U+)R′N′,RN=\nδR′RU+\nN′N. Note the same form of expansions (B4) and\n(B6), with the orbitals |Z0\nRN/an}bracketri}htreplaced by the rotated or-\nbitals|ZRN/an}bracketri}ht(Z=K∞,K,J), with the interstitial parts\n|Kint,0\nRN/an}bracketri}htreplacedbytheirlinearcombinations |Kint\nRN/an}bracketri}ht, and\nwith the structure-constant matrix S0replaced by the\nproduct U+S0U.\nThe non-orthogonal LMTO |χ0\nRN/an}bracketri}htfor the reference\nsystem is obtained from the expansion (B4), in which all\norbitals|K0\nRN/an}bracketri}htand|J0\nRN/an}bracketri}htare replaced by linear com-\nbinations of |φ0\nRΛ/an}bracketri}htand|˙φ0\nRΛ/an}bracketri}ht. A similar replacement of\nthe orbitals |KRN/an}bracketri}htand|JRN/an}bracketri}htby linear combinations of\n|φRΛ/an}bracketri}htand|˙φRΛ/an}bracketri}htin the expansion (B6) yields the non-\northogonal LMTO |χRN/an}bracketri}htfor the system with the tilted\nXC field. The coefficients in these linear combinations—\nobtained from conditions of continuous matching at the\nsphere boundaries and leading directly to the LMTO po-\ntentialparameters—areidenticalforboth systems, asfol-\nlows from the rotationrelations (B1) and (B2). For these\nreasons, the only essential difference between both sys-\ntems in the construction of the non-orthogonal and or-\nthogonal LMTOs (and of the accompanying Hamiltonian\nand overlap matrices in the ASA) is due to the difference\nbetween the matrices S0andU+S0U.\nAs a consequence, the LMTO Hamiltonian matrix in\nthe orthogonalLMTO representationfor the system with\na tilted magnetizationis easilyobtained fromthat forthe\nreference system, Eq. (6), and it is given by\nH=C+(√\n∆)+U+SU(1−γU+SU)−1√\n∆,(B7)\nwhere the C,√\n∆ andγare site-diagonal matrices of\nthe potential parameters of the reference system and\nwhere we suppressed the superscript 0 at the structure-\nconstant matrix Sof the reference system. Note that\nthe dependence of Hon the XC-field direction is con-\ntained only in the similarity transformation U+SUof\nthe original structure-constant matrix Sgenerated by\nthe rotation matrix U. For the rotation by an angle\nθaround an axis along a unit vector n, the rotation\nmatrix is given by U(θ) = exp( −in·Jθ), where the\nsite-diagonal matrices J≡(Jx,Jy,Jz) with matrix\nelements Jµ\nR′N′,RN=δR′RJµ\nN′N(µ=x,y,z) reduce\nto usual matrices of the total (orbital plus spin) angu-\nlar momentum operator. The limit of small θyields\nU(θ)≈1−in·Jθ, which leads to the θ-derivative of\nthe Hamiltonian matrix (B7) at θ= 0:\n∂H/∂θ= i(F+)−1[n·J,S]F−1, (B8)\nwhere we abbreviated F= (√\n∆)−1(1−γS) andF+=\n(1−Sγ)[(√\n∆)+]−1. Since the structure-constant matrixSis spin-independent, the total angular momentum op-\neratorJin (B8) canbe replacedbyits orbitalmomentum\ncounterpart L≡(Lx,Ly,Lz), so that\n∂H/∂θ= i(F+)−1[n·L,S]F−1.(B9)\nThe relations (B8) and (B9) are used to derive the\nLMTO-ASA torque-correlation formula (8).\nAppendix C: Equivalence of the Gilbert damping in\nthe CPA with local and nonlocal torques\n(Supplemental Material)\n1. Introductory remarks\nThe problem of equivalence of the Gilbert damping\ntensor expressed with the local (loc) and nonlocal (nl)\ntorques can be reduced to the problem of equivalence of\nthese two expressions:\nαloc=α0Tr/an}bracketle{t(g+−g−)[P,K](g+−g−)[P,K]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)Tr/an}bracketle{tgp[P,K]gq[P,K]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)βloc\npq, (C1)\nand\nαnl=α0Tr/an}bracketle{t(g+−g−)[K,S](g+−g−)[K,S]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)Tr/an}bracketle{tgp[K,S]gq[K,S]/an}bracketri}ht\n=α0/summationdisplay\np=±/summationdisplay\nq=±sgn(pq)βnl\npq. (C2)\nThe symbols Tr and /an}bracketle{t.../an}bracketri}htand the quantities α0,g±,P\nandShavethe samemeaning as in the main text and the\nquantity Ksubstitutes any of the operators (matrices)\nJµorLµ. Note that owing to the symmetric nature of\ntheoriginaldampingtensors,theanalysiscanbeconfined\nto scalar quantities αlocandαnldepending on a general\nsite-diagonal nonrandom operator K. The choice of K=\nKµin (C1) and (C2) produces the diagonal elements of\nboth tensors, whereas the choice of K=Kµ±Kνfor\nµ/ne}ationslash=νleads to all off-diagonal elements. The quantities\nβloc\npqandβnl\npqare expressions of the form\nβloc= Tr/an}bracketle{tg1(P1K−KP2)g2(P2K−KP1)/an}bracketri}ht,\nβnl= Tr/an}bracketle{tg1[K,S]g2[K,S]/an}bracketri}ht, (C3)\nwhere the g1andg2replace the gpandgq, respectively.\nFor an internal consistency of these and following expres-\nsions, we have also introduced P1=P2=P.\nThis supplement contains a proof of the equivalence\nofβlocandβnland, consequently, of αlocandαnl. The\nCPA-average in βnlwith a nonlocal nonrandom torque\nhas been done using the theory by Velick´ y29as worked\nout in detail within the present LMTO formalism by\nCarva et al.30whereas the averaging in βlocinvolving\na local but random torque has been treated using the\napproach by Butler.4412\n2. Auxiliary quantities and relations\nSincethenecessaryformulasoftheCPAinmultiorbital\ntechniques30,44are little transparent, partly owing to the\ncomplicated indices of two-particle quantities, we employ\nhere a formalism with the lattice-site index Rkept but\nwith all orbital indices suppressed.\nThe Hilbert spaceis a sum ofmutually orthogonalsub-\nspaces of individual lattice sites R; the corresponding\nprojectors will be denoted by Π R. A number of rele-\nvant operators are site-diagonal, i.e., they can be written\nasX=/summationtext\nRXR, where the site contributions are given\nbyXR= ΠRX=XΠR= ΠRXΠR. Such operators\nare, e.g., the random potential functions, Pj=/summationtext\nRPj\nR,\nand the nonrandom coherent potential functions Pj=/summationtext\nRPj\nR, wherej= 1,2. The operator Kin (C3) is site-\ndiagonal as well, but its site contributions KRwill not\nbe used explicitly in the following.\nAmong the number ofCPA-relationsfor single-particle\nproperties, we will use the equation of motion for the\naverage auxiliary Green’s functions ¯ gj(j= 1,2),\n¯gj(Pj−S) = (Pj−S)¯gj= 1, (C4)\nas well as the definition of random single-site t-matrices\ntj\nR(j= 1,2) with respect to the effective CPA-medium,\ngiven by\ntj\nR= (Pj\nR−Pj\nR)[1+ ¯gj(Pj\nR−Pj\nR)]−1.(C5)\nThe operators tj\nRare site-diagonal, being non-zero only\nin the subspace of site R. The last definition leads to\nidentities\n(1−t1\nR¯g1)P1\nR=P1\nR+t1\nR(1−¯g1P1\nR),\nP2\nR(1−¯g2t2\nR) =P2\nR+(1−P2\nR¯g2)t2\nR,(C6)\nwhich will be employed below together with the CPA-\nselfconsistency conditions /an}bracketle{ttj\nR/an}bracketri}ht= 0 (j= 1,2).\nFor the purpose of evaluation of the two-particle aver-\nages in (C3), we introduce several nonrandom operators:\nf12= ¯g1K−K¯g2, ζ12= ¯g1[K,S]¯g2,(C7)\nand a site-diagonal operator γ12=/summationtext\nRγ12\nR, where\nγ12=P1K−KP2, γ12\nR=P1\nRK−KP2\nR.(C8)\nBy interchanging the superscripts 1 ↔2 in (C7) and\n(C8), one can also get quantities f21,ζ21,γ21andγ21\nR;\nthis will be implicitly understood in the relations below\nas well. The three operators f12,ζ12andγ12satisfy a\nrelation\nf12+ζ12+¯g1γ12¯g2= 0, (C9)\nwhich can be easily proved from their definitions (C7)\nand (C8) and from the equation of motion (C4). An-\nother quantityto be used in the followingis a nonrandomsite-diagonal operator ϑ12related to the local torque and\ndefined by\nϑ12\nR=/an}bracketle{t(1−t1\nR¯g1)(P1\nRK−KP2\nR)(1−¯g2t2\nR)/an}bracketri}ht,\nϑ12=/summationdisplay\nRϑ12\nR. (C10)\nIts site contributions can be rewritten explicitly as\nϑ12\nR=γ12\nR+/an}bracketle{tt1\nR(f12+ ¯g1γ12\nR¯g2)t2\nR/an}bracketri}ht.(C11)\nThe last relation follows from the definition (C10), from\ntheidentities(C6)andfromtheCPA-selfconsistencycon-\nditions. Moreover,the site contributions ϑ12\nRandγ12\nRsat-\nisfy a sum rule\nγ12\nR=/summationdisplay\nR′′/an}bracketle{tt1\nR¯g1γ12\nR′¯g2t2\nR/an}bracketri}ht+/an}bracketle{tt1\nRζ12t2\nR/an}bracketri}ht+ϑ12\nR,(C12)\nwhere the prime at the sum excludes the term with R′=\nR. This sum rule can be proved by using the definitions\nofζ12(C7) and γ12\nR(C8) and by employing the previous\nrelation for ϑ12\nR(C11) and the equation of motion (C4).\nThe treatment of two-particle quantities requires the\nuse of a direct product a⊗bof two operators aandb.\nThis is equivalent to the concept of a superoperator, i.e.,\na linear mapping defined on the vector space of all linear\noperators. In this supplement, superoperators are de-\nnoted by an overhat, e.g., ˆ m. In the present formalism,\nthe direct product of two operators aandbcan be iden-\ntified with a superoperator ˆ m=a⊗b, which induces a\nmapping\nx/ma√sto→ˆmx= (a⊗b)x=axb, (C13)\nwherexdenotes an arbitrary usual operator. This defi-\nnition leads, e.g., to a superoperator multiplication rule\n(a⊗b)(c⊗d) = (ac)⊗(db). (C14)\nIn the CPA, the most important superoperators are\nˆw12=/summationdisplay\nR/an}bracketle{tt1\nR⊗t2\nR/an}bracketri}ht (C15)\nand\nˆχ12=/summationdisplay\nRR′′\nΠR¯g1ΠR′⊗ΠR′¯g2ΠR(C16)\nwhere the prime at the double sum excludes the terms\nwithR=R′. The quantity ˆ w12represents the irre-\nducible CPA-vertex and the quantity ˆ χ12corresponds to\narestrictedtwo-particlepropagatorwithexcludedon-site\nterms. By using these superoperators, the previous sum\nrule (C12) can be rewritten compactly as\n(ˆ1−ˆw12ˆχ12)γ12= ˆw12ζ12+ϑ12,(C17)\nwhereˆ1 = 1⊗1 denotes the unit superoperator.13\nLet us introduce finally a symbol {x;y}, wherexand\nyare arbitrary operators, which is defined by\n{x;y}= Tr(xy). (C18)\nThissymbolissymmetric, {x;y}={y;x}, linearinboth\narguments and it satisfies the rule\n{(a⊗b)x;y}={x;(b⊗a)y},(C19)\nwhich follows from the cyclic invariance of the trace. An\nobvious consequence of this rule are relations\n{ˆw12x;y}={x; ˆw21y},\n{ˆχ12x;y}={x; ˆχ21y}, (C20)\nwhere ˆw21and ˆχ21are defined by (C15) and (C16) with\nthe superscript interchange 1 ↔2.\n3. Expression with the nonlocal torque\nThe configuration averaging in βnl(C3), which con-\ntains the nonrandom operator [ K,S], leads to two terms\nβnl=βnl,coh+βnl,vc, (C21)\nwhere the coherent part is given by\nβnl,coh= Tr{¯g1[K,S]¯g2[K,S]}(C22)\nand the vertex corrections can be compactly written as30\nβnl,vc={(ˆ1−ˆw12ˆχ12)−1ˆw12ζ12;ζ21},(C23)\nwith all symbols and quantities defined in the previous\nsection. The coherent part can be written as a sum of\nfour terms,\nβnl,coh=βnl,coh\nA+βnl,coh\nB+βnl,coh\nC+βnl,coh\nD,\nβnl,coh\nA= Tr{S¯g1KS¯g2K},\nβnl,coh\nB= Tr{¯g1SK¯g2SK},\nβnl,coh\nC=−Tr{¯g1KS¯g2SK},\nβnl,coh\nD=−Tr{S¯g1SK¯g2K}, (C24)\nwhich can be further modified using the equation of mo-\ntion (C4) and its consequences, e.g., S¯gj=Pj¯gj−1. For\nthe first term βnl,coh\nA, one obtains:\nβnl,coh\nA= Tr{P1¯g1KP2¯g2K}+Tr{KK}\n−Tr{KP2¯g2K}−Tr{P1¯g1KK}.(C25)\nThe last three terms do not contribute to the sum over\nfour pairs of indices ( p,q), where p,q∈ {+,−}, in\nEq. (C2). For this reason, they can be omitted for the\npresent purpose, which yields expressions\n˜βnl,coh\nA= Tr{P1¯g1KP2¯g2K},\n˜βnl,coh\nB= Tr{¯g1P1K¯g2P2K}, (C26)where the second relation is obtained in the same way\nfrom the original term βnl,coh\nB. A similar approach can\nbe applied to the third term βnl,coh\nC, which yields\nβnl,coh\nC=−Tr{¯g1KP2¯g2P2K}\n+Tr{¯g1KP2K}+Tr{¯g1KSK}.(C27)\nThe last term does not contribute to the sum over four\npairs (p,q) in Eq. (C2), which leads to expressions\n˜βnl,coh\nC= Tr{¯g1KP2K}−Tr{¯g1KP2¯g2P2K},\n˜βnl,coh\nD= Tr{P1K¯g2K}−Tr{P1¯g1P1K¯g2K},(C28)\nwhere the second relation is obtained in the same way\nfrom the original term βnl,coh\nD. The sum of all four con-\ntributions in (C26) and (C28) yields\n˜βnl,coh=˜βnl,coh\nA+˜βnl,coh\nB+˜βnl,coh\nC+˜βnl,coh\nD\n= Tr{¯g1KP2K}+Tr{P1K¯g2K}\n+Tr{¯g1γ12¯g2γ21}, (C29)\nwhere weused the operators γ12andγ21defined by (C8).\nThe total quantity βnl(C21) is thus equivalent to\n˜βnl=˜βnl,coh+βnl,vc\n= Tr{¯g1KP2K}+Tr{P1K¯g2K}\n+Tr{¯g1γ12¯g2γ21}+βnl,vc,(C30)\nwhere the tildes mark omission of terms irrelevant for the\nsummation over ( p,q) in Eq. (C2).\n4. Expression with the local torque\nThe configuration averagingin βloc(C3), involving the\nrandom local torque, leads to a sum of two terms:44\nβloc=βloc,0+βloc,1, (C31)\nwhere the term βloc,0is given by a simple lattice sum\nβloc,0=/summationdisplay\nRβloc,0\nR,\nβloc,0\nR= Tr/angbracketleftbig\n¯g1(1−t1\nR¯g1)(P1\nRK−KP2\nR)\nׯg2(1−t2\nR¯g2)(P2\nRK−KP1\nR)/angbracketrightbig\n,(C32)\nsee Eq. (76) of Ref. 44, and the term βloc,1can be written\nin the present formalism as\nβloc,1={ˆχ12(ˆ1−ˆw12ˆχ12)−1ϑ12;ϑ21},(C33)\nwhich corresponds to Eq. (74) of Ref. 44. The definitions\nof ˆw12and ˆχ12aregivenby(C15)and(C16), respectively,\nand ofϑ12andϑ21by (C10).\nThe quantity βloc,0\nR(C32) gives rise to four terms,\nβloc,0\nR=QR,A+QR,B+QR,C+QR,D, (C34)\nQR,A= Tr/an}bracketle{t¯g1(1−t1\nR¯g1)P1\nRK¯g2(1−t2\nR¯g2)P2\nRK/an}bracketri}ht,\nQR,B= Tr/an}bracketle{tP1\nR¯g1(1−t1\nR¯g1)KP2\nR¯g2(1−t2\nR¯g2)K/an}bracketri}ht,\nQR,C=−Tr/an}bracketle{tP1\nR¯g1(1−t1\nR¯g1)P1\nRK¯g2(1−t2\nR¯g2)K/an}bracketri}ht,\nQR,D=−Tr/an}bracketle{t¯g1(1−t1\nR¯g1)KP2\nR¯g2(1−t2\nR¯g2)P2\nRK/an}bracketri}ht,14\nwhich will be treated separately. The term QR,Acan\nbe simplified by employing the identities (C6) and the\nCPA-selfconsistency conditions. This yields:\nQR,A=UR,A+VR,A, (C35)\nUR,A= Tr{¯g1P1\nRK¯g2P2\nRK},\nVR,A= Tr/an}bracketle{t¯g1t1\nR(1−¯g1P1\nR)K¯g2t2\nR(1−¯g2P2\nR)K/an}bracketri}ht\n=VR,A1+VR,A2+VR,A3+VR,A4,\nVR,A1= Tr/an}bracketle{t¯g1t1\nRK¯g2t2\nRK/an}bracketri}ht,\nVR,A2= Tr/an}bracketle{t¯g1t1\nR¯g1P1\nRK¯g2t2\nR¯g2P2\nRK/an}bracketri}ht,\nVR,A3=−Tr/an}bracketle{t¯g1t1\nR¯g1P1\nRK¯g2t2\nRK/an}bracketri}ht,\nVR,A4=−Tr/an}bracketle{t¯g1t1\nRK¯g2t2\nR¯g2P2\nRK/an}bracketri}ht.\nA similar procedure applied to QR,Byields:\nQR,B=UR,B+VR,B, (C36)\nUR,B= Tr{P1\nR¯g1KP2\nR¯g2K},\nVR,B= Tr/an}bracketle{t(1−P1\nR¯g1)t1\nR¯g1K(1−P2\nR¯g2)t2\nR¯g2K/an}bracketri}ht\n=VR,B1+VR,B2+VR,B3+VR,B4,\nVR,B1= Tr/an}bracketle{tt1\nR¯g1Kt2\nR¯g2K/an}bracketri}ht,\nVR,B2= Tr/an}bracketle{tP1\nR¯g1t1\nR¯g1KP2\nR¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,B3=−Tr/an}bracketle{tt1\nR¯g1KP2\nR¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,B4=−Tr/an}bracketle{tP1\nR¯g1t1\nR¯g1Kt2\nR¯g2K/an}bracketri}ht.\nThe term QR,Crequires an auxiliary relation\nP1\nR¯g1(1−t1\nR¯g1)P1\nR=P1\nR(¯g1P1\nR−1)\n+P1\nR−(1−P1\nR¯g1)t1\nR(1−¯g1P1\nR),(C37)\nthat follows from a repeated use of the identities (C6).\nThis relation together with the CPA-selfconsistency lead\nto the form:\nQR,C=UR,C+VR,C, (C38)\nUR,C= Tr{P1\nR(1−¯g1P1\nR)K¯g2K}\n−Tr/an}bracketle{tP1\nRK¯g2(1−t2\nR¯g2)K/an}bracketri}ht,\nVR,C=−Tr/an}bracketle{t(1−P1\nR¯g1)t1\nR(1−¯g1P1\nR)K¯g2t2\nR¯g2K/an}bracketri}ht\n=VR,C1+VR,C2+VR,C3+VR,C4,\nVR,C1=−Tr/an}bracketle{tt1\nRK¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,C2=−Tr/an}bracketle{tP1\nR¯g1t1\nR¯g1P1\nRK¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,C3= Tr/an}bracketle{tt1\nR¯g1P1\nRK¯g2t2\nR¯g2K/an}bracketri}ht,\nVR,C4= Tr/an}bracketle{tP1\nR¯g1t1\nRK¯g2t2\nR¯g2K/an}bracketri}ht.\nA similar procedure applied to QR,Dyields:\nQR,D=UR,D+VR,D, (C39)\nUR,D= Tr{¯g1KP2\nR(1−¯g2P2\nR)K}\n−Tr/an}bracketle{t¯g1(1−t1\nR¯g1)KP2\nRK/an}bracketri}ht,\nVR,D=−Tr/an}bracketle{t¯g1t1\nR¯g1K(1−P2\nR¯g2)t2\nR(1−¯g2P2\nR)K/an}bracketri}ht\n=VR,D1+VR,D2+VR,D3+VR,D4,\nVR,D1=−Tr/an}bracketle{t¯g1t1\nR¯g1Kt2\nRK/an}bracketri}ht,\nVR,D2=−Tr/an}bracketle{t¯g1t1\nR¯g1KP2\nR¯g2t2\nR¯g2P2\nRK/an}bracketri}ht,\nVR,D3= Tr/an}bracketle{t¯g1t1\nR¯g1KP2\nR¯g2t2\nRK/an}bracketri}ht,\nVR,D4= Tr/an}bracketle{t¯g1t1\nR¯g1Kt2\nR¯g2P2\nRK/an}bracketri}ht,Let us focus now on U-terms in Eqs. (C35 – C39). The\nsecond terms in UR,C(C38) and UR,D(C39) do not con-\ntribute to the sum over four pairs ( p,q) in Eq. (C1),\nso that the original UR,CandUR,Dcan be replaced by\nequivalent expressions\n˜UR,C= Tr{P1\nR(1−¯g1P1\nR)K¯g2K},\n˜UR,D= Tr{¯g1KP2\nR(1−¯g2P2\nR)K}.(C40)\nThe sum of all U-terms for the site Ris then equal to\n˜UR=UR,A+UR,B+˜UR,C+˜UR,D\n= Tr{P1\nRK¯g2K}+Tr{¯g1KP2\nRK}\n+Tr{¯g1γ12\nR¯g2γ21\nR}, (C41)\nwhereγ12\nRandγ21\nRare defined in (C8), and the lattice\nsum of all U-terms can be written as\n/summationdisplay\nR˜UR= Tr{P1K¯g2K}+Tr{¯g1KP2K}\n+/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}. (C42)\nThe summation of V-terms in Eqs. (C35 – C39) can be\ndone in two steps. First, we obtain\nVR,1=VR,A1+VR,B1+VR,C1+VR,D1\n= Tr/an}bracketle{tt1\nRf12t2\nRf21/an}bracketri}ht,\nVR,2=VR,A2+VR,B2+VR,C2+VR,D2\n= Tr/an}bracketle{tt1\nR¯g1γ12\nR¯g2t2\nR¯g2γ21\nR¯g1/an}bracketri}ht,\nVR,3=VR,A3+VR,B3+VR,C3+VR,D3\n= Tr/an}bracketle{tt1\nR¯g1γ12\nR¯g2t2\nRf21/an}bracketri}ht,\nVR,4=VR,A4+VR,B4+VR,C4+VR,D4\n= Tr/an}bracketle{tt1\nRf12t2\nR¯g2γ21\nR¯g1/an}bracketri}ht, (C43)\nwhere the operators f12andf21have been defined in\n(C7). Second, one obtains the sum of all V-terms for the\nsiteRas\nVR=VR,1+VR,2+VR,3+VR,4 (C44)\n= Tr/an}bracketle{tt1\nR(f12+ ¯g1γ12\nR¯g2)t2\nR(f21+ ¯g2γ21\nR¯g1)/an}bracketri}ht.\nThe lattice sums of all U- andV-terms lead to an expres-\nsion equivalent to the original quantity βloc,0(C32):\n˜βloc,0=/summationdisplay\nR˜UR+/summationdisplay\nRVR\n= Tr{P1K¯g2K}+Tr{¯g1KP2K}\n+/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}\n+/summationdisplay\nRTr/angbracketleftbig\nt1\nR(f12+¯g1γ12\nR¯g2)\n×t2\nR(f21+ ¯g2γ21\nR¯g1)/angbracketrightbig\n,(C45)\nwhere the tildes mark omission of terms not contributing\nto the summation over ( p,q) in Eq. (C1).\nLet us turn now to the contribution βloc,1(C33). It\ncan be reformulated by expressing the quantity ϑ12(and15\nϑ21) in terms of the quantities γ12andζ12(andγ21and\nζ21) from the sum rule (C17) and by using the identities\n(C20). The resultingformcan be written compactlywith\nhelp of an auxiliary operator ̺12(and̺21) defined as\n̺12= ˆχ12γ12+ζ12. (C46)\nThe result is\nβloc,1=βnl,vc+{ˆχ12γ12;γ21}\n−{ˆw12̺12;̺21}, (C47)\nwhere the first term has been defined in (C23). For the\nsecond term in (C47), we use the relation\nˆχ12γ12=/summationdisplay\nRΠR¯g1(γ12−γ12\nR)¯g2ΠR,(C48)\nwhich follows from the site-diagonal nature of the opera-\ntorγ12(C8) and from the definition of the superoperator\nˆχ12(C16). This yields:\n{ˆχ12γ12;γ21}= Tr{¯g1γ12¯g2γ21}\n−/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}.(C49)\nFor the third term in (C47), only the site-diagonalblocks\nof the operator ̺12(and̺21), Eq. (C46), are needed be-\ncauseofthesite-diagonalnatureofthesuperoperator ˆ w12\n(C15). These site-diagonal blocks are given by\nΠR̺12ΠR= ΠR/bracketleftbig\n¯g1(γ12−γ12\nR)¯g2+ζ12/bracketrightbig\nΠR\n=−ΠR(f12+ ¯g1γ12\nR¯g2)ΠR,(C50)\nwhichfollowsfromthepreviousrelations(C48)and(C9).This yields:\n{ˆw12̺12;̺21}=/summationdisplay\nRTr/angbracketleftbig\nt1\nR(f12+ ¯g1γ12\nR¯g2)\n×t2\nR(f21+ ¯g2γ21\nR¯g1)/angbracketrightbig\n.(C51)\nThe term βloc,1(C47) is then equal to\nβloc,1=βnl,vc+Tr{¯g1γ12¯g2γ21}\n−/summationdisplay\nRTr{¯g1γ12\nR¯g2γ21\nR}\n−/summationdisplay\nRTr/angbracketleftbig\nt1\nR(f12+¯g1γ12\nR¯g2)\n×t2\nR(f21+ ¯g2γ21\nR¯g1)/angbracketrightbig\n.(C52)\nThe total quantity βloc(C31) is thus equivalent to the\nsum of (C45) and (C52):\n˜βloc=˜βloc,0+βloc,1\n= Tr{P1K¯g2K}+Tr{¯g1KP2K}\n+Tr{¯g1γ12¯g2γ21}+βnl,vc,(C53)\nwhere the tildes mark omission of terms irrelevant for the\nsummation over ( p,q) in Eq. (C1).\n5. 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Tosi (North-Holland, New York, 1985) p. 59."    },    {        "title": "2305.13564v1.Current_driven_motion_of_magnetic_topological_defects_in_ferromagnetic_superconductors.pdf",        "content": "Current-driven motion of magnetic topological defects in ferromagnetic\nsuperconductors\nSe Kwon Kim1,∗and Suk Bum Chung2, 3,†\n1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea\n2Department of Physics and Natural Science Research Institute,\nUniversity of Seoul, Seoul 02504, Republic of Korea\n3School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea\n(Dated: May 24, 2023)\nRecent years have seen a number of instances where magnetism and superconductivity intrinsically\ncoexist. Our focus is on the case where spin-triplet superconductivity arises out of ferromagnetism,\nand we make a hydrodynamic analysis of the effect of a charge supercurrent on magnetic topological\ndefects like domain walls and merons. We find that the emergent electromagnetic field that arises out\nof the superconducting order parameter provides a description for not only the physical quantities\nsuch as the local energy flux density and the interaction between current and defects but also\nthe energy dissipation through magnetic dynamics of the Gilbert damping, which becomes more\nprominent compared to the normal state as superconductivity attenuates the energy dissipation\nthrough the charge sector. In particular, we reveal that the current-induced dynamics of domain\nwalls and merons in the presence of the Gilbert damping give rise to the nonsingular 4 πand 2 π\nphase slips, respectively, revealing the intertwined dynamics of spin and charge degrees of freedom\nin ferromagnetic superconductors.\nI. INTRODUCTION\nWhile magnetism has traditionally been regarded as\ninimical to superconductivity, recent years have seen ob-\nservation of ferromagnetism and superconductivity co-\nexisting or cooperating in varieties of materials which\nincludes uranium heavy-fermion compounds [1–3] and\ntwo-dimensional moir´ e materials such as twisted bilayer\ngraphene [4–6]. It has been known that such coexistence\ncan be naturally accommodated by the Cooper pairing\nof spin-polarized electrons [7]. In such cases, it is natural\nto question what effect, if any, ferromagnetism may have\non superconductivity and vice versa.\nIt is well established in magnetism and spintronics\nthat the current-induced motions of spin textures such\nas domain walls in magnetic metals give rise to the spin\nand energy dissipation into the baths of quasiparticles or\nphonons, commonly known as the Gilbert damping [8, 9].\nThe conservation of energy dictates that the dissipated\nenergy should be externally supplied by the input power.\nIn the case of normal metals, however, resistivity-induced\nenergy dissipation is present regardless of the presence\nor the absence of any spin textures. Hence the Gilbert\ndamping gives rise to only an additional term in the en-\nergy dissipation and, in this sense, its presence can be\ndifficult to confirm solely through charge transport.\nCharge transport detection of the Gilbert damping in\nferromagnetic superconductors may be more straightfor-\nward despite involving a feature unconventional for su-\nperconductors. To maintain a steady-state motion of spin\ntextures in the presence of the Gilbert damping, a fer-\nromagnetic superconductor needs the finite input power\n∗sekwonkim@kaist.ac.kr\n†sbchung0@uos.ac.kr\n(a)(b)\nFIG. 1. (a) The illustration of the mutually orthogonal unit\nvectors ˆs,ˆu, and ˆvthat describe the directional degrees of\nfreedom of the order parameter of a ferromagnetic supercon-\nductor. (b) The configuration of the triad {ˆs,ˆu,ˆv}for a do-\nmain wall in a ferromagnetic superconductor with easy-axis\nspin anisotropy along the zdirection.\nthat goes out of the superconductor solely in the form of\nthe Gilbert damping. This indicates voltage arising in-\nside the superconductor in the direction of the current by\nthe dynamics of spin textures. The mechanisms by which\na superconductor acquires a finite voltage difference be-\ntween two points is referred to as phase slips [10, 11].\nIn conventional superconductors, these phase slips gen-\nerally accompany the singularities, i.e.the vanishing of\nthe order parameter at a certain time during the phase\nslips.\nIn this paper, we show that this is not necessar-\nily the case for ferromagnetic superconductors by us-\ning the concrete example of the current-induced mo-\ntions of two types of magnetic defects, domain walls and\nmerons, which are schematically illustrated in Fig. 1(b)\nand Fig. 3, respectively. To this end, we begin by ex-\namining the order parameter of the spin-polarized super-\nconductor and show how the Cooper pair spin rotation\naround the spin polarization direction is actually equiv-\nalent to the twisting of the overall phase. This gives rise\nto a channel for the interaction between ferromagnetism\nand superconductivity, namely the coupling of CooperarXiv:2305.13564v1  [cond-mat.supr-con]  23 May 20232\npairs to the effective gauge field arising from spin tex-\nture [7, 12, 13].\nWe then proceed to show how such formalism can be\nused to obtain the current-induced motion of topologi-\ncal spin defects such as a domain wall and a meron in\npresence of a background superflow. First, for a domain\nwall, we show that the current-induced motion of do-\nmain walls in the presence of the Gilbert damping ac-\ncompanies the precessional dynamics of the local spin\npolarization and this in turn gives rise to the nonsingu-\nlar 4πphase slips through the generation of an emergent\nelectric field. The induced phase slip opens a channel\nthrough which the ferromagnetic superconductor can ac-\nquire input power, which is shown to be dissipated by the\nspin dynamics entirely via the Gilbert damping. Also,\na current-induced motion of a meron is shown to give\nrise to the nonsingular 2 πphase slips perpendicular to\nits motion, engendering a channel for the input power\nthat is dissipated via the Gilbert damping. The gener-\nation of the 2 πphase slips can be understood from the\nemergent electromagnetic field associated with the meron\ndynamics. For ferromagnetic metals, the emergent elec-\ntromagnetic fields associated with spin textures and their\ndynamics have been discussed theoretically [14–17] and\nconfirmed experimentally [18–21]. However, their man-\nifestations in the dynamics of magnetic defects in fer-\nromagnetic superconductors and the resultant nonsingu-\nlar phase slips have not been discussed yet. Our work\nreveals that the current-induced dynamics of magnetic\ndefects exemplify the intertwined dynamics of spin and\ncharge degrees of freedom in ferromagnetic superconduc-\ntors, where the emergent electromagnetic fields play cru-\ncial roles.\nThe paper is organized as follows. The general formal-\nism for the order parameter and its dynamics of ferro-\nmagnetic superconductors is developed phenomenologi-\ncally in Sec. II. The current-induced dynamics of a do-\nmain wall and its relation to the nonsingular 4 πphase\nslips are discussed in Sec. III. Section IV concerns the\ncurrent-induced dynamics of a meron and its relation to\nthe nonsingular 2 πphase slips. We conclude the paper\nin Sec. V with discussions.\nII. GENERAL FORMALISM\nA. Order parameter\nThe order parameter of a fully spin-polarized triplet su-\nperconductor provides a starting point for understanding\nhow superconductivity and magnetism are intertwined\nthrough the emergent gauge field. In the d-vector for-\nmalism defined by i(d·σσy)s,s′≡∆s,s′, it is given\nby [7, 12, 13]\nd=√ρ\n2eiϕ(ˆu+iˆv) =rρ\n2eiϕ(ˆu+iˆv)√\n2≡rρ\n2ˆd,(1)where ρ= 2d∗·dis the number density of the Cooper\npairs, ˆuandˆvare perpendicular unit vectors, and ˆd∗·ˆd=\n1; the simplest example would be ˆu=ˆx,ˆv=ˆywhich\ngives ∆ s,s′= 0 except for ∆ ↑↑(see Appendix A for the\ndetails). There is an ambiguity here in defining ϕas\nthe above order parameter remains invariant under the\nfollowing simultaneous change of ϕandˆuandˆv:\neiϕ(ˆu+iˆv) =ei(ϕ+δϕ)\u0002\ne−iδϕ(ˆu+iˆv)\u0003\n≡ei(ϕ+δϕ)(ˆu′+iˆv′),\n(2)\nwhere ˆu′,ˆv′are obtained by rotating ˆu,ˆvby +δϕaround\nˆu׈v. As the spin density in units of ℏcan be written\nas\ns= 2id×d∗=ρˆu׈v≡ρˆs, (3)\nEq. (2) denotes the U(1) ϕ+sorder parameter redundancy\n[7, 12], i.e.the invariance of the order parameter when\nthe angle of the spin rotation around ˆsequals the change\ninϕ. Such redundancy implies the existence of an effec-\ntive gauge field arising from the spin degrees of freedom.\nSee Fig. 1(a) for the illustration of the three mutually\northogonal unit vectors ˆs,ˆu, and ˆv, which are depicted\nby red, green, and blue arrows, respectively.\nFor deriving the vector potential and magnetostatics\nof this effective gauge field, the above order parameter\nsuffices. From the spin rotation angle around ˆsdefined\nasα, the effective vector gauge can be written as [14]\nai≡ℏ\nq∂iα=ℏ\nqˆs·(ˆu×∂iˆu) ; (4)\nhence the emergent gauge is a direct consequence of the\nU(1) ϕ+sorder parameter redundancy of Eq. (2). Indeed,\nthe emergent gauge field of spatial curvature in the chi-\nral superconductor has been attributed to the analogous\norder parameter redundancy there [22–25]. Here, while\nwe have kept the charge, q, generic, q=−2e <0 holds in\nsuperconductors. From this emergent vector potential, it\nis straightforward to obtain the emergent magnetic field,\nbi=ϵijk∂jak=−ℏϵijk\n2qˆs·(∂jˆs×∂kˆs) ; (5)\nnote that this is in the same form as the well-known\nMermin-Ho relation between the orbital angular momen-\ntum texture and superfluid velocity in the 3He-A super-\nfluid [26, 27]. Yet this discussion does not include any\ndynamics, for which we shall adopt a two-step approach\nof first formulating the simplest free energy for the order\nparameter [Eq. (2)] and then use its Lagrangian to obtain\nthe equations of motion.\nB. Free energy\nGiven that we seek results relevant to wide-ranging su-\nperconductors whose common attributes may not extend\nbeyond the spin-polarized Cooper pairing [1–6] we will\nconsider for our free energy the simplest minimal model3\nthat includes the spin anisotropy and the Zeeman cou-\npling:\nF[d] =Z\ndVF′\n0[d] +Z\ndVU\n2\u0000\n2|d|2−ρ0\u00012,(6)\nwhere\nF′\n0=A′\nd\n2|(∇−iq\nℏA)d|2+ρ\u0014A′\ns\n2|∇ˆs|2−D\n2(ˆs·ˆz)2−Hsz+qV\u0015\n,\nwhere A′\nsrepresents the excess spin stiffness; note that,\nin contrast to previous analysis [12], our treatment will\nencompass both the easy-axis anisotropy D > 0 and the\neasy-plane anisotropy D < 0. As we will focus on the\ncases where the fluctuation of the condensate density ρ≡\n2|d|2is strongly suppressed, it is convenient to separatelygroup together terms dependent on ρfluctuations [12, 13]\nF=Z\ndV ρF0+Z\ndV\u0014A′\nd\n16ρ(∇ρ)2+U\n2(ρ−ρ0)2\u0015\n,\nwhere\nF0=A′\nd\n2|(∇−iq\nℏA)ˆd|2+A′\ns\n2|∇ˆs|2−D\n2(ˆs·ˆz)2−Hsz+qV\nis the free energy density per unit density. The gauge\ntransformation is implemented as\nA7→A+∇Λ,d7→eiqΛ/ℏd.\nThe free energy can be recast into the following form:\nF=Z\ndV ρ\u001aAc\n2h\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)i2\n+As\n2(∂iˆs)2−D\n2(ˆs·ˆz)2−Hsz+qV\u001b\n+Z\ndV\u0014Ac\n8ρ(∇ρ)2+U\n2(ρ−ρ0)2\u0015\n,\n(7)\nwhere Ac=A′\ndandAs=A′\ns+A′\nd/2. Here, Acand\nAsrepresent the charge stiffness and the spin stiffness,\nrespectively. The similar expression without the sec-\nond and the third terms can be found in Eq. (2.3) and\nEq. (2.5) of Ref. [28].\nAccordingly, the charge supercurrent density is modi-\nfied to\nJi=−δF\nδAi=q\nℏρAc(∂iϕ−q\nℏAi−q\nℏai), (8)\nwith the velocity field is given by\nvi=Ji\nqρ=Ac\nℏ(∂iϕ−q\nℏAi−q\nℏai). (9)\nIt satisfies the following equation (by assuming a nonsin-\ngular θ):\n∇×v=−qAc\nℏ2(B+b). (10)\nThe free energy formalism provides a convenient\nspringboard for extending our analysis to dynamics as\nwell. In particular, such analysis helps us understand\nhow emergent electric field would arise, in analogy with\nthe standard electrodynamics. This is accomplished by\nconsidering the Langrangian for this minimal model.\nC. Equations of motion\nFor the dynamics analysis, we now obtain the classical\nequation of motion for both the charge and the spin com-ponent of the order parameter through considering the\nLagrangian of the spin-polarized superconductor. This\ncan be written as\nL=Z\ndV2iℏd∗·∂td−F\n=−Z\ndVℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)]−F . (11)\nThe first term of the above Lagrangian arises from 2 iℏd∗\nbeing the conjugate variable to d; the detailed derivation\nof its relation to ρ, ϕ,ˆscan be found in Appendix B.\nThe low-energy dynamics of the order parameter can be\ndescribed by the three Euler-Lagrange equations for ϕ, ρ,\nandˆs.\nThe equations for ρandϕare basically analogous to\nthose of the conventional superconductors. The equation\nof motion for the density ρ,\n˙ρ=−1\nℏ∂in\nρAch\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)io\n=−1\nq∂iJi,\n=−∇·(ρv) (12)\nis obtained from δL/δϕ = 0 and is none other than the\ncontinuity equation for the Cooper pair density. Simi-\nlarly, the equation of motion for the phase ϕ\n−ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =F0+U(ρ−ρ0)−Ac\n4ρ∇2ρ(13)\nobtained from δL/δρ = 0 (where only terms constant or\nlinear in ρare retained) comes out to be the Josephson\nrelation. We, however, want to obtain a hydrodynamic\nequation of motion for Cooper pairs, for which purpose\nwe take the spatial derivative of the Josephson relation:4\n−ℏh\n∂t∂iϕ−q\nℏ(Ei+ei)−q\nℏ∂t(Ai+ai)i\n=ℏ2\nAcv·∂iv+∂i\u0014\nρ′\ne+U(ρ−ρ0)−Ac\n4ρ∇2ρ\u0015\n,\nwhere\nei=−ℏ\nqˆs·(∂iˆs×∂tˆs) (14)\nis the emergent electric field and\nρ′\ne=As(∂iˆs)2\n2−Ds2\nz\n2−Hsz (15)\nis the magnetic energy density (per unit density). By using\nv·∂iv=1\n2∂i(v2) = (v·∇)vi+ϵijkvj(∇×v)k= (v·∇)vi−qAc\nℏ2ϵijkvj(Bk+bk)\nand defining the material derivative Dt≡∂t+v·∇and the effective mass of a Cooper pair, m≡ℏ2/Ac, we obtain\nmDtv=q(E+e) +qv×(B+b)−∂i\u0014\nρ′\ne+U(ρ−ρ0)−Ac\n4ρ∇2ρ\u0015\n. (16)\nThe novelty in the ferromagnetic superconductor is the\nequation of motion for the spin direction ˆsthat is derived\nfrom δL/δˆs= 0 (see Appendix C for details):\nℏρ∂tˆs=−ℏJi\nq∂iˆs+∂i[ρAs(ˆs×∂iˆs)]+ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z,\n(17)\nwhich is identical to the Landau-Lifshitz equation [29]\naugmented by the adiabatic spin-transfer torque [9, 30,\n31]. This can be also written as\nℏρDtˆs=∂i[ρAs(ˆs×∂iˆs)] +ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z.\nBy using ˙ ρ=−∂iJi/q, we can obtain the spin continuity\nequation:\n∂t(ℏρˆs) =−∂iJs\ni+ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z,(18)\nwhere\nJs\ni=ℏJi\nqˆs−ρAs(ˆs×∂iˆs),\nis the spin current density. The first term and the second\nterm on the right-hand side are longitudinal spin currents\nproportional to the charge current and the transverse\nspin current that is carried by a spin texture, respec-\ntively.\nA complete set of equations describing the hydrody-\nnamics of a ferromagnetic superconductor in the absence\nof external fields ( E= 0 and B= 0) can now be given;\nthe analogous equations have been written down for a\nspinor BEC [32]. It is convenient to measure energy in\nthe unit of the anisotropy energy absolute value |D|andlength in the unit derived from the combination of |D|\nwith the spin stiffness As,i.e.\nl=s\nAs\n|D|, ϵ=|D|.\nAlso, we will use ˜ ρ≡ρ/ρ0, Uρ/D ≡η. This gives us the\ndimensionless equations\n−Dt˜ρ= ˜ρ(∇·v),\n˜m(∇×v) =−b,\n˜mDtv=e+v×b\n−∂i[ρe+η(˜ρ−1)− ∇2˜ρ/4],\n˜ρDtˆs=∂i[˜ρ(ˆs×∂iˆs)] + ˜ρν(ˆs·ˆz)ˆs׈z+ ˜ρhˆs׈z,\n(19)\nwhere ν≡sgn(D), ˜m≡As/Acthe dimensionless mass\nwhich is on the order of unity, h≡H/D the dimension-\nless external field, and\nρe=1\n2(∂iˆs)2−ν1\n2(ˆs·ˆz)2−hsz\nthe dimensionless magnetic energy density; for zero ex-\ncess spin stiffness ˜ m= 1/2. The emergent electromag-\nnetic fields are now re-defined as\nei=−ˆs·(∂iˆs×∂tˆs), b i=−ϵijk\n2ˆs·(∂jˆs×∂kˆs),\nwhere the charge qis absorbed into the fields.5\nD. Gilbert damping\nDue to the inevitable nonconservation of spin angu-\nlar momentum in solids, it is reasonable to expect the\ndamping of spin dynamics and the associated energy dis-\nsipation, which are not included in the hydrodynamics\nequations [Eq. (19)], to play an important role in spin\ndynamics of the ferromagnetic superconductors as in any\nother solid-state systems. The spin sinks can be quasi-\nparticles, phonons, and any other excitations that can\npossess angular momentum [33–36]. The spin dissipation\ncan be treated phenomenologically with the addition of\nthe Gilbert damping term α˜ρˆs×∂tˆs[8] to the spin equa-\ntion of motion in Eq. (19),\n˜ρDtˆs+α˜ρˆs×∂tˆs=∂i[˜ρ(ˆs×∂iˆs)]+ ˜ρν(ˆs·ˆz)ˆs׈z+ ˜ρhˆs׈z.\n(20)\nIn the incompressible limit η→ ∞ where ˜ ρis uniform and\nconstant, this gives us the energy continuity equation,\n∂tρe+∇·je=−v·e−α(∂tˆs)2, (21)\nwhere\nje=−∂tˆs·∇ˆs (22)\nis the magnetic energy flux density (per unit density).\nThis continuity equation, which has been previously\nnoted in literature [37, 38], can be derived by taking the\nproduct of both sides of Eq. (20) with ˆs×∂tˆs. One can\nnote that the first term on the right-hand side of Eq. (21)\n(−v·e) is the power dissipated (supplied) by the su-\nperflow vflowing parallel (antiparallel) to the direction\nof the emergent electric field e, while the second term\n(−α(∂tˆs)2) is the energy dissipation through the Gilbert\ndamping.\nEquation (21) implies that, in the incompressible limit,\nthe energy dissipated by the Gilbert damping is equal to\nthe work done by the emergent electric field when spin\ntexture is transported without any distortion. This is\nbecause the total magnetic energy should be unchanged\nin this process and hence the left-hand side of Eq. (21)\nintegrated over the whole system should be zero.\nIII. DOMAIN WALL\nFor the supercurrent-driven motion of topological de-\nfects, we first consider an easy-axis ferromagnetic super-\nconductor with spin-anisotropy sign ν= sgn( D) = +1.\nA domain wall is a generically stable topological defect\nbetween two different ground states for easy-axis spin\nsystems and intrinsically has no skyrmion density, i.e.\nno emergent magnetic field. Therefore for the rest of this\nsection, we will take the Cooper pair velocity to be ir-\nrotational, i.e.∇×v= 0 [Eq. (19)]. In addition, we\nalso set the applied magnetic field to be zero and hence\nh= 0.A. Deriving dynamics from a static solution\nThe solution for a domain-wall motion in the back-\nground of the constant and uniform background super-\nflow can be straightforwardly constructed from the static\ndomain wall solution in absence of any background su-\nperflow for the incompressible limit η→ ∞ . For this\ncase, the absence of the emergent magnetic field allows\nus to consider only the spin equation of motion [40]\n(∂t+v·∇+αˆs×∂t)ˆs=∂i[(ˆs×∂iˆs)] + (ˆs·ˆz)ˆs׈z; (23)\nfrom the hydrodynamic equations Eq. (19); others are\neither irrelevant due to b= 0 or merely provides the\nconstraint ˜ ρ= 1 in the incompressible limit. Given the\nintrinsically quasi-one-dimensional nature of the domain\nwall, we can set the boundary condition\nˆs(x→ ±∞ ) =±ˆz,\nfor any domain-wall configuration. For the static domain\nwall at x= 0 in absence of background superflow, the\nsolution is given by the following Walker ansatz [41]:\nˆs0= (ˆxcosφ0+ˆysinφ0)sech x+Qˆztanhx, (24)\nwhere Q=±1 represents the domain wall type, satisfies\nthe static domain-wall equation\n0 =∂x[(ˆs0×∂xˆs0)] + ( ˆs0·ˆz)ˆs0׈z, (25)\nderived from Eq. (23), for an arbitrary domain-wall angle\nφ0; it is important to note here that Eq. (25) is sufficient\nas Eq. (24) gives us v= 0 and b= 0 everywhere.\nFrom Eq. (25), it can be shown that the general solu-\ntion can be obtained by applying to the static solution\nboth giving boost in the spatial direction and precession\naround the easy-axis:\nˆs= (ˆxcos Ω t+ˆysin Ωt)sech( x−V t) +Qˆztanh( x−V t).\n(26)\nIn deriving the domain-wall velocity ˆxVand the preces-\nsion rate Ω, it is convenient to note that Eq. (26) also\nsatisfies Eq. (25) as the latter equation involves no time\nderivatives. Also, as Eq. (26) is obtained from boost and\nprecession,\n∂tˆs= (−V ∂x+ˆzΩ×)ˆs.\nThe velocity Vand the precession rate Ω therefore can\nbe obtained from\n[v∂x+ (1 + αˆs×)(−V ∂x+ˆzΩ×)]ˆs= 0, (27)\nwhere we set the background superflow to be perpendic-\nular to the domain wall without any loss of generality,\nv=vˆx. By taking the scalar product of the above equa-\ntion with ˆzandˆz׈swe obtain\nv−V=QαΩ,Ω =−QαV6\n(a)<latexit sha1_base64=\"mvV1IC9/vAkApJWzfWyUx3QadJw=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVpiZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBSnJC1</latexit>\n(b)<latexit sha1_base64=\"Lyp4rKMmNObzzWYxIHNaXwltWlk=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVoKzibdQtEtuzPQZeJlpEgy1LqFr04v4kkICrlkxrQ9N0Y/ZRoFlzDJdxIDMeMjNoC2pYqFYPx0dvGEnlqlR/uRtqWQztTfEykLjRmHge0MGQ7NojcV//PaCfYv/VSoOEFQfL6on0iKEZ2+T3tCA0c5toRxLeytlA+ZZhxtSHkbgrf48jJpnJe9SvnqrlKsXmdx5MgxOSEl4pELUiW3pEbqhBNFnskreXOM8+K8Ox/z1hUnmzkif+B8/gBUIpC2</latexit>\n(c)<latexit sha1_base64=\"eT4ibbwfOOWkpDCTAUUKdbMXNmE=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVriZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBVqJC3</latexit>\nFIG. 2. (a) A series of snapshots of a precessing domain wall moving to the right, where xandtare the spatial and the\ntemporal coordinates, respectively. The red arrows, blue arrows, and green arrows represent ˆs,ˆu, and ˆv, respectively. The\ndomain-wall position is denoted by the gray dot. The domain-wall angle, which is the azimuthal angle of ˆsat the center of the\ndomain wall, changes from φ0= 0 to φ0=−2πgradually with increasing time from bottom to top. On the left end, ˆu(green\narrow) rotates by −2πabout ˆs(red arrow), whereas on the right end, ˆurotates by 2 πabout ˆs. This process of a domain-wall\nprecession can be considered as a nonsingular 4 πphase slip since these opposite 2 πrotations of ˆuaround ˆsat the left and\nthe right ends induces a finite voltage across the wire. (b) Mapping of the instantaneous configuration of the two vectors ˆs\n(red arrows) and ˆu(green arrows) onto the unit sphere with the normal vector identified with ˆs. The yellow line represents\na spatial dimension of the system. (c) Collection of the mapping of the configuration of ˆsandˆuonto the unit sphere with ˆs\nidentified with the normal vector for all the snapshots shown in (a). Note that the unit tangent vector field ˆuis not uniquely\ndetermined at the north and the south poles as dictated by the Poincar´ e-Hopf theorem [39]. Rather, ˆurotates once around\nˆscounterclockwise (counterclockwise) at the north (south) pole as the domain wall completes one cycle of rotation, which is\nconsistent with the Euler number 2 of the sphere. See the main text for further detailed discussions.\nrespectively, giving us\nV=1\n1 +α2v ,Ω =−Qα\n1 +α2v; (28)\nNote that in absence of the Gilbert damping, α= 0, there\nwould have been no precession and the domain wall would\nhave remained static with respect to the background su-\nperflow. See Fig. 2 for the illustration of the domain-wall\ndynamics with precession.\nFrom the above solution, it is straightforward to con-\nfirm that all work done by the emergent electric field is\ndissipated through the Gilbert damping. The work done\nby the emergent electric field is\n−v·e=vjˆs·(∂jˆs×∂tˆs) =QvΩ[1−(ˆs·ˆz)2],(29)\nwhich gives the total energy input of\nW=Z\ndx(−v·e) = 2 QvΩ. (30)\nThe energy dissipation rate per unit density is given by\nα(∂tˆs)2=α(V2+ Ω2)[1−(ˆs·ˆz)2]. (31)\nThis energy dissipation through the spin dynamics and\nthe work rate done by the emergent electric field on the\nsuperflow are the same since\nQvΩ =α\n1 +α2v2(32)\nand\nα(V2+ Ω2) =α\n1 +α2v2. (33)B. 4πphase slips from a domain-wall dynamics\nDue to the U(1) ϕ+sorder parameter redundancy, the\nenergy dissipation from the damping-induced precession\nin the domain-wall motion can be regarded as an equiva-\nlent of 4 πphase slips. As shown in Fig. 2, ˆuandˆvrotate\naround ˆsby±2πby adiabatically following the dynamics\nof the local spin direction ˆsin one cycle of precession. To\nsee this, note that if we adopt the condition ˆv·ˆz= 0 in\ndefining ˆv, Eq. (26) will give us ˆv=−ˆxsin Ωt+ˆycos Ω t.\nBut given the U(1) ϕ+sredundancy, this is equivalent to\nthe±2πphase twist on the left and the right end, re-\nspectively.\nThe voltage arising from this precession can be under-\nstood either as arising from the emergent electric field, or\nequivalently, arising from the constant rate of 4 πphase\nslips. When φincreases at the rate ˙ φ= Ω, the emergent\nscalar potential at the two ends of the wire, x=±∞, is\ngiven by\n˜Ve=−ˆs·(ˆu×∂tˆu) =(\n−QΩ at x=−∞,\nQΩ at x=∞,(34)\nwhich is exactly the Josephson voltage for the 4 Qπphase\nslip occurring at the rate of Ω /2π. Then, to maintain the\nfinite superflow, there must be work given by\nW=v[˜Ve(x=∞)−˜Ve(x=−∞)] = 2 QvΩ,(35)\nby external reservoirs on the system, matching the work7\n[Eq. (30)]. Figure 2(a) shows the time evolution of the\ntriad{ˆs,ˆu,ˆv}associated with the domain-wall that both\nmoves and precesses. Note that ˆu(green arrow) at the\nleft end ( x→ −∞ ) and the right end ( x→ ∞ ) ro-\ntates clockwise and counterclockwise, respectively, about\nˆs(red arrow), engendering the nonsingular phase slips\nacross the xdirection. In a nutshell, the domain-wall an-\ngular dynamics produces the nonsingular 4 πphase slips\nthat give rise to a finite voltage difference between the\ntwo ends of the superconducting wire, which constitutes\nour first main result.\nThere is a topological reason why one precession of a\nmagnetic domain wall induces a 4 πphase slip, which can\nbe derived from the Poincar´ e-Hopf theorem or Poincar´ e-\nBrower theorem [39]. For concrete discussion on this, we\nwill consider in the following the case with Q= 1 as\nshown in Fig. 2(b) and (c). At a given time, the instan-\ntaneous configuration of ˆsandˆucan be mapped onto a\nline connecting the north pole [ ˆs(x→ ∞ )] and the south\npole [ ˆs(x→ −∞ )] on the unit sphere by identifying ˆs\nwith the surface normal as shown in Fig. 2(b). When we\nconsider the collection of the configuration of {ˆs,ˆu}onto\nthe unit sphere during one complete precession of the do-\nmain wall, i.e., φ0→φ0+2π,ˆucan be regarded as a unit\ntangent vector field on the sphere since it is perpendic-\nular to ˆs, i.e., the surface normal as shown in Fig. 2(c).\nHere, note that ˆuis not uniquely determined at the north\npole and the south pole, which is consistent with the well-\nknown topological property of the sphere that the unit\ntangent vector field cannot be defined without a singu-\nlarity on it. Instead of being uniquely determined, ˆu\nrotates by 2 πaround the north pole and rotates by −2π\naround the south pole as the domain wall completes one\ncycle of precession, which gives rise to a 4 πphase slip\nacross the wire as discussed above [Eq. (34)]. This can\nbe understood by applying the Poincar´ e-Hopf theorem to\nthe unit tangent vector field ˆuon the sphere. The Euler\nnumber of the unit sphere is 2, meaning that the sum of\nthe indices of the isolated singularities of the unit tan-\ngent vector field on the sphere must be 2. In our case,\nthe indices of the north pole and the south pole associ-\nated with ˆuare both 1, adding up to 2, agreeing with the\nEuler number of the unit sphere.\nIV. MERON\nWe now consider ferromagnetic superconductors with\neasy-plane spin anisotropy ( ν= sgn( D) =−1). For easy-\nplane spin systems, a meron, with its one-half skyrmion\ncharge, is a generically stable topological defect [42–\n45]. Therefore, we consider the rotational Cooper pair\nvelocity in absence of the applied magnetic field i.e.\n∇×v=−b/˜mwith h= 0. See Fig. 3 for the schematic\nillustration of a meron.\n(a)<latexit sha1_base64=\"mvV1IC9/vAkApJWzfWyUx3QadJw=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVpiZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBSnJC1</latexit>\n(b)<latexit sha1_base64=\"Lyp4rKMmNObzzWYxIHNaXwltWlk=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVoKzibdQtEtuzPQZeJlpEgy1LqFr04v4kkICrlkxrQ9N0Y/ZRoFlzDJdxIDMeMjNoC2pYqFYPx0dvGEnlqlR/uRtqWQztTfEykLjRmHge0MGQ7NojcV//PaCfYv/VSoOEFQfL6on0iKEZ2+T3tCA0c5toRxLeytlA+ZZhxtSHkbgrf48jJpnJe9SvnqrlKsXmdx5MgxOSEl4pELUiW3pEbqhBNFnskreXOM8+K8Ox/z1hUnmzkif+B8/gBUIpC2</latexit>\n(c)<latexit sha1_base64=\"eT4ibbwfOOWkpDCTAUUKdbMXNmE=\">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVriZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBVqJC3</latexit>\nFIG. 3. (a) An illustration of the meron with polarity p= 1\nand vorticity n= 1, which is a nonsingular topological de-\nfect of ferromagnetic superconductors with easy-plane spin\nanisotropy. The red, the green, and the blue arrows repre-\nsentˆs,ˆu, and ˆv, respectively. The meron core is denoted by\nthe gray dot. The local spin direction ˆsrotates by 2 πcoun-\nterclockwise about the zaxis when we follow the infinitely\ndistant trajectory encircling the meron core counterclockwise\n(and thus the vorticity n= 1). Note that ˆuandˆvalso change\nspatially to keep their orthonormality to ˆs; they rotate by 2 π\nclockwise about the local spin direction ˆswhen we enclose\nthe meron center counterclockwise. (b) Mapping of the con-\nfiguration of ˆs(red arrows) and ˆu(green arrows) along the\ninfinitely distant circle in (a) onto the equator with the surface\nnormal identified with ˆs. (c) Mapping of the configuration of\nˆsand ˆuof the entire system onto the northern hemisphere\nwith ˆsidentified with the surface normal. Note that ˆuis a\nwell-defined unit tangent vector field on the northern hemi-\nsphere without any singularity. Since the Euler number of\nthe hemisphere is 1, if the unit tangent vector field defined on\nthe northern hemisphere has no singularity, it should rotate\naround the surface normal by 2 πalong the equator according\nto the Poincar´ e-Hopf theorem [39], which is exactly what ˆu\n(unit tangent vector) does around ˆs(surface normal).\nA. Static solution\nAnalogous to the case of the domain wall in the previ-\nous section, a straightforward construction of the meron\nmotion solution here in the background of the constant\nand uniform background superflow is possible from the\nstatic meron solution in absence of any background su-\nperflow for the incompressible limit, η→ ∞ [32, 46].\nThe static solution can be obtained from the following8\ntwo equations;\n˜m(∇×v) =−b,\n(v·∇)ˆs=∂i[(ˆs×∂iˆs)]−(ˆs·ˆz)ˆs׈z.(36)\nIt is important to note here that while we are dealing\nwith a static configuration we still have v̸= 0 due to\nthe intrinsic emergent magnetic flux of the meron. In\naddition, this solution would possess the axial symmetry,\ni.e.\nˆs0= (sin θcosφ,sinθsinφ,cosθ), (37)\nwith θ=θ(r) and φ=nχ+ Φ where ( r, χ) are polar\ncoordinates for the two-dimensional system, and follow\nthe universal boundary conditions for merons are given\nby\nθ(r= 0) = (1 −p)π\n2, θ(r→ ∞ ) =π\n2;\np=±1 here is the polarity, which is the z-component of\nthe local spin direction ˆsat the meron center, and n∈Z\nthe vorticity, which counts how many times ˆswinds in\nthe easy plane along the closed trajectory encircling the\nmeron center. For our purpose, obtaining the differential\nequation for θ(r) is sufficient for showing Eq. (37) to be\nthe solution of Eq. (36). We start by noting that the\nemergent magnetic field is aligned entirely along the z-\naxis and\nbz=−ˆs0·(∂xˆs0×∂yˆs0) =−nsinθ\nrdθ\ndr(38)\nis function only for θ, which gives us the well-known re-\nsult\nZ\ndxdyb z=Z\nrdχdr\u0012\n−nsinθ\nrdθ\ndr\u0013\n=−2πpn (39)\nfor the total emergent magnetic flux. With this emergent\nmagnetic field, the first equation of Eq. (36) requires the\ncirculating velocity v=ˆφv(r) around the meron with\n1\nrd(rv)\ndr=nsinθ\n˜mrdθ\ndr.\nInserting this relation into the second equation of\nEq. (36) gives us\nsinθ(1−cosθ)1\n˜mr2=1\nrd\ndr\u0012\nrdθ\ndr\u0013\n+cos θsinθ\u0012\n1−1\nr2\u0013\n,\nfrom which θ(r) can be obtained numerically.\nTo find explicit expressions for ˆuandˆv, note that the\nfollowing three unit vectors form an orthonormal triad:\nˆs0=ˆer≡(sinθcosφ,sinθsinφ,cosθ),\nˆeθ≡(cosθcosφ,cosθsinφ,−sinθ),\nˆeφ≡(−sinφ,cosφ,0),which gives us\nˆu0(r, χ) = cos φ(χ)ˆeθ(r, χ)−sinφ(χ)ˆeφ(r, χ),(40)\nˆv0(r, χ) = sin φ(χ)ˆeθ(r, χ) + cos φ(χ)ˆeφ(r, χ).(41)\nThe local configuration of the triad ( ˆs0,ˆu0,ˆv0) for a\nmeron with p= 1 and n= 1 is shown in Fig. 3. Note\nthat it is nonsingular, differing from a conventional vor-\ntex of a s-wave superconductor [10]. An analogous non-\nsingular topological defects that give rise to 4 πnonsin-\ngular phase slips has been discussed by Anderson and\nToulouse [47] and has been termed the skyrmion solu-\ntion in the more recent literature [28, 32]. By contrast,\nour solution {ˆs0,ˆu0,ˆv0}[Eqs. (37,40,41)] represents an\nexplicit solution for the nonsingular topological defect\nin the easy-plane case that gives rise to 2 πnonsingular\nphase slip, as can be seen from Eqs (36) and (39): it\nharbors the emergent magnetic flux −2πpnand thus its\nmotion gives rise to the emergent electric field, i.e., phase\nslips perpendicular to its motion.\nThe non-trivial emergent gauge field ai, and thus the\nquantized non-zero emergent magnetic flux, of our non-\nsingular topological defect can be understood from the\nrotation of ˆuaround ˆsas stated in Eq. (4). There exists\na topological constraint dictating that a meron texture\nofˆsshould trap a quantized non-zero emergent magnetic\nflux, which can be understood by invoking the Poincar´ e-\nHopf theorem [27, 39] as follows. For the given meron\nconfiguration with p= 1, let us consider the mapping of\ntwo unit vectors ˆsandˆuonto the northern hemisphere\nsuch that ˆsis identified with the surface normal. Then,\nˆubecomes a unit tangent vector field on the hemisphere\nsince it is perpendicular to ˆs, i.e., the surface normal as\nshown in Fig. 3(c). The Euler number of the hemisphere\nis 1, and thus the Poincar´ e-Hopf theorem dictates that, if\nthe unit tangent vector field is nonsingular on the north-\nern hemisphere, it should rotate exactly one time about\nthe surface normal while traversing the equator, which is\nexactly what ˆudoes around ˆsin Fig. 3(b). Therefore, the\none-time rotation of ˆuaround ˆsalong the closed loop that\ncontains, but is also infinitely far from, the meron core\nas shown in Fig. 3(a), which gives rise to the quantized\nemergent magnetic flux, can be regarded as the physical\nmanifestation of the topological constraint that should\nbe satisfied by the nonsingular unit tangent vector field\ndefined on the hemisphere.\nB. Dynamics with a background superflow\nAnalogous to the domain wall motion, it is straight-\nforward to work out the spin equation of motion for the\nmeron motion driven by a uniform constant background\nsuperflow v0if we assume the meron motion to be rigid,\ni.e.ˆs(r, t) =ˆs0(r−Vt) (the same holds for ˆuandˆv). The\nCooper pair velocity would then be given by v=vm+v0,\nwhere vmis the Cooper pair velocity around a static\nmeron. We can therefore employ the collective coordi-\nnate approach to describe the dynamics of a meron, and9\nFIG. 4. Snapshots of a meron moving in the ydirection in the increasing time from (a) to (d), where the meron core is depicted\nby a gray dot. At x→ −∞ ,ˆu(green arrow) rotates about ˆs(red arrow) counterclockwise, whereas at x→ ∞ ,ˆurotates about\nˆsclockwise, inducing phase slips and thereby generating a finite voltage in the xdirection.\nuse the fact that ˆs0(r−Vt) should satisfy the original\nspin equation of motion of Eq. (20) with, as we are in\nthe incompressible limit, the constant ˜ ρwhile ˆs0(r) also\nsatisfies the equation for the static meron of Eq. (36).\nSubtraction between the two equations, together with\n∂tˆs0(r−Vt) =−V·∇ˆs0(r−Vt) gives us\n−(1 +αˆs0×)V·∇ˆs0=−v0·∇ˆs0.\nTaking the scalar product with s0×∂is0on both sides\ngives us\n−Vj[s0·(∂iˆs0×∂jˆs0)]−αVj(∂iˆs0·∂jˆs0)\n=−v0,j[s0·(∂iˆs0×∂jˆs0)].(42)\nStrictly speaking, this result shows that whereas we have\nan exact rigid motion solution with V=v0in absence\nof damping, no rigid motion solution can be exact in\npresence of damping. Yet to the zeroth order in v0and\nalso in the spirit of collective coordinate, we can average\nout the effect of the spin texture, i.e.defining\nGij=Z\ndxdys0·(∂iˆs0×∂jˆs0), D ij=Z\ndxdy∂ iˆs0·∂jˆs0,\nknown respectively as the gyrotropic coefficients and the\ndissipation coefficients [48–50] ( Dij=DδijandGxy=\n−Gyx≡G= 2πpnfor a meron). By integrating Eq. (42),\nwe have\n(αD+Gˆz×)V=Gˆz×v0, (43)\nwhich yields the following solution for the velocity V:\nV=G\nG2+α2D2(G+αDˆz×)v0. (44)\nTo consider a concrete example, we will hereafter restrict\nthe discussion to the case where the background super-\nflow flows in the xdirection: v0= (v0,0). In this case,we have\n\u0012\nVx\nVy\u0013\n=G\nG2+α2D2\u0012\nGv0\n−αDv 0\u0013\n. (45)\nNote that the presence of damping gives rise to the com-\nponent of the meron velocity transverse to the uniform\nsuperflow in proportion to the skyrmion number G/2πof\nthe meron, Vy=−α[GD/(G2+α2D2)]v0, exhibiting the\nso-called skyrmion Hall effect [16, 51–53]. This transverse\nmotion of the meron with respect to the superflow gives\nrise to the finite voltage in the direction of the superflow\nvia the 2 πphase slips, which we turn our attention now.\nC. 2πphase slips from a meron motion\nAgain analogous to the domain wall motion, the\nU(1) ϕ+sorder parameter redundancy allows the energy\ndissipation due to the meron motion as equivalent to the\n2πphase slips. This can be seen from the emergent elec-\ntric field\ne=−ˆs·(∇ˆs×∂tˆs)\n=ˆs0·(∇ˆs0×V·∇ˆs0) =−V×b, (46)\nwhich is in the same form as the Josephson electric field\narising from the vortex motion [10]. The same can natu-\nrally be said about the input power density required for\ndriving the uniform constant background superflow\n−v0·e=v0,xˆs·(∂xˆs×∂tˆs),\n=−v0,xVyˆs·(∂xˆs×∂yˆs).\nIt can be checked explicitly that the total energy rate for\ndriving the superflow\n−Z\ndxdyv0·ˆe=−v0,xVyG=αG2D\nG2+α2D2v2\n0,10\nis equal to the energy dissipation rate\nZ\ndxdyα (∂tˆs)2=Z\ndxdyαV iVj(∂iˆs0·∂jˆs0)\n=αDV2=αG2D\nG2+α2D2v2\n0.\nThis energy must come from external reservoirs\nthrough the boundary of the system, meaning that there\nshould be a development of a finite voltage across the\nsystem in the xdirection. This can be explicitly seen\nfrom\n˜Ve=−ˆs·(ˆu×∂tˆu) = (1 −cosθ)∂tφ . (47)\nTo see how a finite voltage is generated by the motion\nof a vortex, let us assume that a vortex moves in the y\ndirection, V=Vyˆy. Then for a given point at large x,\nZ∞\n−∞dtˆs·(ˆu×∂tˆu) =−Z∞\n−∞dt∂tφ=nπ , forx→+∞.\n(48)\nThe vector ˆu(x→ ∞ ) rotates by nπaround the ˆs. Also,\nZ∞\n−∞dtˆs·(ˆu×∂tˆu) =−Z∞\n−∞dt∂tφ=−nπ , forx→ −∞ .\n(49)\nThe vector ˆu(x→ −∞ ) rotates by −nπaround the\nˆs. This indicates that the motion of a meron in the\nydirection induces a nonsingular 2 πphase slips across\nthexdirection. To keep the superflow in the xdirec-\ntion constant, we need to counteract the effect of these\nphase slips. The corresponding work on the system is\ndissipated to external baths such as quasiparticles or\nphonons through the Gilbert damping. Figure 4 shows\nthe schematic illustration of the process. As the vor-\ntex moves in the positive ydirection from Fig. 4(a) to\nFig. 4(d), ˆuatx→ ∞ andx→ −∞ rotates about the\nlocal spin direction ˆscounterclockwise and clockwise, re-\nspectively, producing phase slips in the xdirection. This\nis our second main result: The dynamics of a meron en-\ngenders the nonsingular 2 πphase slips perpendicular to\nits motion through the generation of the emergent elec-\ntric field, showcasing the intertwined dynamics of spin\nand charge degrees of freedom of ferromagnetic super-\nconductors.\nV. DISCUSSION\nWithin the phenomenological framework for the dy-\nnamics of the order parameter of ferromagnetic super-\nconductors, we have shown that the current-induced dy-\nnamics of magnetic defects in the presence of the spin\ndissipation, i.e., the Gilbert damping, give rise to nonsin-\ngular phase slips via the emergent electromagnetic fields.\nThe input power, which is the product of the applied\ncurrent that drives the magnetic defects and the voltage\ngenerated by the phase slip, is shown to be equivalent tothe dissipated energy in the form the Gilbert damping\ninto the baths of quasiparticles or phonons. Our work on\nthe dynamics of magnetic defects showcases the intrinsic\ninterplay of spin and charge dynamics of ferromagnetic\nsuperconductors.\nA few remarks are on order about the limitations of\nour work. First, we did not include the effects of the\nnon-adiabatic spin-transfer torque by a supercurrent on\nthe dynamics of magnetic defects, which is expected to be\npresent on general grounds whenever the Gilbert damp-\ning is present [9, 40, 54, 55]. While we believe that the\ninclusion of the non-adiabatic spin-transfer torque in our\nmodel would not qualitatively change the relations that\nwe have found between the dynamics of magnetic defects\nand nonsingular phase slips, it will certainly enrich the\nphysics of the interplay of spin and charge dynamics in\nferromagnetic superconductors. Secondly, in this work,\na ferromagnetic superconductor is assumed to be a fully\nspin-polarized triplet superconductor as in Ref. [12], by\nleaving the generalization to a partially spin-polarized\ncase as future work. Thirdly, our analysis shows that\nthe assumption of rigid motion is not exact for merons\nin presence of damping. The coupling between current\nand magnons bound to meron cores may be a relevant\ntopic for future study. Lastly, the dynamics of magnetic\ndefects has been discussed in the incompressible limit,\nwhere the dynamics of the order-parameter amplitude\nis frozen. Releasing this assumption would allow us to\nstudy the interplay of spin dynamics and longitudinal\norder-parameter dynamics, which is beyond the scope of\nthe current work.\nACKNOWLEDGMENTS\nWe thank Mike Stone, Grigori Volovik, Daniel\nAgterberg and Jim Sauls for useful discussions.\nS.K.K. was supported by Brain Pool Plus Program\nthrough the National Research Foundation of Korea\nfunded by the Ministry of Science and ICT (NRF-\n2020H1D3A2A03099291) and by the National Research\nFoundation of Korea funded by the Korea Government\nvia the SRC Center for Quantum Coherence in Con-\ndensed Matter (NRF-RS-2023-00207732). S.B.C. was\nsupported by the National Research Foundation of Korea\n(NRF) grants funded by the Korea government (MSIT)\n(NRF-2023R1A2C1006144, NRF-2020R1A2C1007554,\nand NRF-2018R1A6A1A06024977).11\nAppendix A: Details of the order parameter\nThe multicomponent superconducting gap is given by\nˆ∆ =\u0012\n∆↑↑∆↑↓\n∆↓↑∆↓↓\u0013\n≡\u0012\n−˜dx+i˜dy˜dz\n˜dz˜dx+i˜dy\u0013\n=i(d·σ)σy. (A1)\nThen,\nˆ∆ˆ∆†=\u0012\n∆↑↑∆↑↓\n∆↓↑∆↓↓\u0013\u0012∆∗\n↑↑∆∗\n↓↑\n∆∗\n↑↓∆∗\n↓↓\u0013\n=\u0012|∆↑↑|2+|∆↑↓|2∆↑↑∆∗\n↓↑+ ∆↑↓∆∗\n↓↓\n∆↓↑∆∗\n↑↑+ ∆↓↓∆∗\n↑↓|∆↓↓|2+|∆↓↑|2\u0013\n=\u0012\n−˜dx+i˜dy˜dz\n˜dz˜dx+i˜dy\u0013\u0012−˜d∗\nx−i˜d∗\ny˜d∗\nz\n˜d∗\nz˜d∗\nx−i˜d∗\ny\u0013\n=\u0012|˜dx|2+|˜dy|2+|˜dz|2+i(˜dx˜d∗\ny−˜d∗\nx˜dy) ( −˜dx˜d∗\nz+i˜dy˜d∗\nz) + c.c.\n(−˜d∗\nx−i˜d∗\ny)˜dz) + c.c. |˜dx|2+|˜dy|2+|˜dz|2−i(˜dx˜d∗\ny−˜d∗\nx˜dy)\u0013\n=|d|2σ0+i(d×d∗)·σ, (A2)\nwhere d= (˜dx,˜dy,˜dz). The total number density of the Cooper pairs is given by\n|∆↑↑|2+|∆↑↓|2+|∆↓↑|2+|∆↓↓|2= Tr[ ˆ∆ˆ∆†] = 2d·d∗= 2|d|2. (A3)\nThe expected spin angular momentum is given in units of ℏby\nTr[ˆ∆ˆ∆†σ] = Tr[ i(d×d∗)·σσ] = 2id×d∗. (A4)\nFor a fully spin-polarized triplet superconductor, we can use\nd=deiϕ(ˆu+iˆv), (A5)\nwith ˆu⊥ˆv. Then, we have d·d∗= 2d2. Therefore, d=√ρ/2. The spin polarization is then given by\ns= 2id×d∗= 4d2(ˆu׈v)\n≡4d2ˆs=ρˆs. (A6)\nTo see if the result makes sense, let us consider ˆs=ˆzwith ˆu=ˆxandˆv=ˆy. Then, d=deiϕ(ˆx+iˆy). Then,\n˜dx=deiϕand˜dy=deiϕi. Then, ∆ ↑↑=−2d,∆↓↓= ∆↑↓= ∆↓↑= 0. The condensate density (of the Cooper pairs) is\ngiven by ρ=|∆↑↑|2= 4d2and the spin density is given by s= 4d2ˆs=ρˆs.\nAppendix B: Kinetic term of the Lagrangian\nThe kinetic term of the Lagrangian density is given by\nLK= 2iℏd∗·∂td\n=iℏ\n2√ρe−iϕ(ˆu−iˆv)·∂t\u0002√ρeiϕ(ˆu+iˆv)\u0003\n=iℏ\n2√ρe−iϕ(ˆu−iˆv)·\u0014∂tρ\n2√ρeiϕ(ˆu+iˆv) + (i∂tϕ)√ρeiϕ(ˆu+iˆv) +√ρeiϕ(∂tˆu+i∂tˆv)\u0015\n=iℏ\n2√ρ(ˆu−iˆv)·\u0014∂tρ\n2√ρ(ˆu+iˆv) + (i∂tϕ)√ρ(ˆu+iˆv) +√ρ(∂tˆu+i∂tˆv)\u0015\n=iℏ\n2\u0014∂tρ\n22 + (i∂tϕ)ρ2 +ρ(iˆu·∂tˆv−iˆv·∂tˆu)\u0015\n=−ℏρ(∂tϕ+ˆu·∂tˆv)\n=−ℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)]. (B1)12\nThe factor of 2 in front is to ensure the commutation relation, [ d∗\ni(r), dj(r′)] = 2 iℏδ(r−r′). Then, the total Lagrangian\ndensity is given by\nL=−ℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)]\n−ρ\u001aAc\n2h\n∂iϕ+q\nℏAi+ˆs·(ˆu×∂iˆu)i2\n+As\n2(∂iˆs)2−D\n2(ˆs·ˆz)2−Hsz+qV\u001b\n−\u001aAc\n16(∇ρ)2+U\n2(ρ−ρ0)2\u001b\n.(B2)\nFrom this, we can read the emergent scalar potential\nVe=−ℏ\nqˆs·(ˆu×∂tˆu), (B3)\nand the emergent vector potential\nai=ℏ\nqˆs·(ˆu×∂iˆu). (B4)\nAppendix C: Equations of motion\nThe dynamics of the order parameter can be uniquely characterized by the dynamics of three variables, the con-\ndensate number density of the Cooper pairs ρ, the phase of the order parameter ϕ, and the spin direction ˆs.\nFirst, the equation of motion for ϕcan be obtained by\nδL\nδρ= 0,\n⇒ −ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =δF\nδρ,\n⇒ −ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =\u001aAc\n2h\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)i2\n+As\n2(∂iˆs)2−D\n2(ˆs·ˆz)2−Hsz+qV\u001b\n−Ac\n8∇2ρ+U(ρ−ρ0).\n(C1)\nSecond, the equation of motion for ρcan be obtained by\nd\ndt\u0012δL\nδ˙ϕ\u0013\n−δL\nδϕ= 0,\n⇒ −ℏ˙ρ=−δF\nδϕ,\n⇒ −ℏ˙ρ=∂in\nρAch\n∂iϕ−q\nℏAi−ˆs·(ˆu×∂iˆu)io\n,\n⇒˙ρ=−1\nq∂iJi, (C2)\nwhere Jiis the charge current density. This is nothing but the continuity equation.\nThird, to obtain the equation of motion for ˆs, by considering infinitesimal variations of the three vectors that\nmaintain the orthonormality conditions,\nˆs=ˆs0+δˆs=ˆs0+aˆu0+bˆv0,\nˆu=ˆu0−aˆs0,\nˆv=ˆv0−bˆs0, (C3)\nwe observe that, to zeroth order in aandb,\nδ\nδˆs(ˆs·(ˆu×∂tˆu)) = ˆu0∂\n∂a(ˆs·(ˆu×∂tˆu)) +ˆv0∂\n∂b(ˆs·(ˆu×∂tˆu))\n=−ˆu0(ˆs0·(ˆu0×∂tˆs0)) +ˆv0(ˆv0·(ˆu0×∂tˆu0))\n=ˆs0×∂tˆs0. (C4)13\nAlso, by the analogous steps,\nδ\nδˆs(ˆs·(ˆu×∂iˆu)) =ˆs×∂iˆs. (C5)\nThen, from the Lagrangian, we obtain\nδL\nδˆs= 0,\n⇒ℏρˆs×∂tˆs=δF\nδˆs,\n⇒ℏρ∂tˆs=−ℏJi\nq∂iˆs+∂i[ρAs(ˆs×∂iˆs)] +ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z. (C6)\nBy using ˙ ρ=−(∂iJi)/q, the last equation can be recast into\n∂t(ℏρˆs) =−∂i\u0014ℏJi\nqˆs−[ρAs(ˆs×∂iˆs)]\u0015\n+ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z. (C7)\nThe left-hand side is the spin density, s=ℏρˆs. 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Predictions of\nthe LLS equations depend sensitively on the ratio between th e dimensionless material parameter\nβwhich characterizes non-adiabatic spin-transfer torques and the Gilbert damping parameter α.\nThis ratio has been variously estimated to be close to 0, clos e to 1, and large compared to 1. By\nidentifying βas the influence of a transport current on α, we derive a concise, explicit and relatively\nsimple expression which relates βto the band structure and Bloch state lifetimes of a magnetic\nmetal. Using this expression we demonstrate that intrinsic spin-orbit interactions lead to intra-\nband contributions to βwhich are often dominant and can be (i) estimated with some co nfidence\nand (ii) interpreted using the “breathing Fermi surface” mo del.\nPACS numbers:\nI. INTRODUCTION\nAn electric current can influence the magnetic state of\na ferromagnet by exerting a spin transfer torque (STT)\non the magnetization.1,2,3This effect occurs whenever\ncurrents travel through non-collinear magnetic systems\nand is therefore promising for magnetoelectronic appli-\ncations. Indeed, STT’s have already been exploited in a\nnumber of technological devices.4Partly for this reason\nand partly because the quantitative description of order\nparameter manipulation by out-of-equilibrium quasipar-\nticles poses great theoretical challenges, the study of the\nSTT effect has developed into a major research subfield\nof spintronics.\nSpin transfer torques are important in both magnetic\nmultilayers, where the magnetization changes abruptly,5\nand in magnetic nanowires, where the magnetization\nchanges smoothly.6Theories of the STT in systems with\nsmooth magnetic textures identify two different types of\nspintransfer. Ononehand,theadiabaticorSlonczewski3\ntorque results when quasiparticle spins follow the under-\nlying magnetic landscape adiabatically. It can be math-\nematically expressed as ( vs· ∇)s0, wheres0stands for\nthe magnetization and vsis the “spin velocity”, which\nis proportional to the charge drift velocity, and hence to\nthe current and the applied electric field. The micro-\nscopic physics of the Slonczewski spin-torque is thought\nto be well understood5,6,7, at least8in systems with weak\nspin-orbitcoupling. Asimpleangularmomentumconser-\nvation argument argues that in the absence of spin-orbit\ncoupling vs=σsE/es0, wheres0isthe magnetization, σs\nis the spin conductivity and Eis the electric field. How-\never, spin-orbit coupling plays an essential role in real\nmagnetic materials and hence the validity of this sim-\nple expression for vsneeds to be tested by more rigorous\ncalculations.\nThe second spin transfer torque in continuous media,βs0×(vs·∇)s0, acts in the perpendicular direction and\nis frequently referred to as the non-adiabatic torque.9\nUnfortunately, the name is a misnomer in the present\ncontext. There are two contributions that have the pre-\nceeding form. The first is truly non-adiabatic and occurs\nin systems in which the magnetization varies too rapidly\nin space for the spins of the transport electrons to fol-\nlow the local magnetization direction as they traverse\nthe magnetization texture. For wide domain walls, these\neffects are expected to be small.10The contribution of\ninterest in this paper is a dissipative contribution that\noccurs in the adiabatic limit. The adiabatic torque dis-\ncussed aboveis the reactive contribution in this limit. As\nwe discuss below, processes that contribute to magnetic\ndamping, whether they derive from spin-orbit coupling\nor spin-dependent scattering, also give a spin-transfer\ntorque parameterized by βas above. In this paper, we\nfollow the common convention and refer to this torque as\nnon-adiabatic. However, it should be understood that it\nis a dissipative spin transfer torque that is present in the\nadiabatic limit.\nThe non-adiabatic torque plays a key role in current-\ndriven domain wall dynamics, where the ratio between\nβand the Gilbert parameter αcan determine the veloc-\nity of domain walls under the influence of a transport\ncurrent. There is no consensus on its magnitude of the\nparameter β.6,11Although there have a few theoretical\nstudies12,13,14of the STT in toy models, the relationship\nbetween toy model STT’s and STT’s in either transition\nmetal ferromagnets or ferromagnetic semiconductors is\nfar from clear. As we will discuss the toy models most\noften studied neglect spin-orbit interactions in the band-\nstructure of the perfect crystal, intrinsic spin-orbit inter-\nactions, which can alter STT physics qualitatively.\nThe main objectives of this paper are (i) to shed new\nlight on the physical meaning of the non-adiabatic STT\nby relating it to the change in magnetization damping\ndue to a transport current, (ii) to derive a concise for-2\nmula that can be used to evaluate βin real materials\nfrom first principles and (iii) to demonstrate that αand\nβhave the same qualitative dependence on disorder (or\ntemperature), even though their ratio depends on the\ndetails of the band structure. As a byproduct of our the-\noretical study, we find that the expression for vsin terms\nof the spin conductivity may not always be accurate in\nmaterials with strong spin-orbit coupling.\nWe begin in Section II by reviewing and expanding\non microscopic theories of α,βandvs. In short, our\nmicroscopic approach quantifies how the micromagnetic\nenergy of an inhomogeneous ferromagnet is altered in\nresponse to external rf fields and dc transport currents\nwhich drive the magnetization direction away from lo-\ncal equilibrium. These effects are captured by the spin\ntransfertorques,dampingtorques,andeffectivemagnetic\nfields that appear in the LLS equation. By relating mag-\nnetization dynamics to effective magnetic fields, we de-\nrive explicit expressions for α,βandvsin terms of mi-\ncroscopic parameters. Important contributions to these\nmaterials parameters can be understood in clear physical\nterms using the breathing Fermi surface model.15Read-\ners mainly interested in a qualitative explanation for our\nfindings may skip directly to Section VIII where we dis-\ncuss of our main results in that framework. Regardless\nof the approach, the non-adiabatic STT can be under-\nstood as the change in the Gilbert damping contribution\nto magnetization dynamics when the Fermi sea quasi-\nparticle distribution function is altered by the transport\nelectric field. The outcome of this insight is a concise an-\nalytical formula for βwhich is simple enough that it can\nbe conveniently combined with first-principles electronic\nstructure calculations to predict β-values in particular\nmaterials.16\nIn Sections III, IV and V we apply our expressionfor β\nto model ferromagnets. In Section III we perform a nec-\nessary reality check by applying our theory of βto the\nparabolic band Stoner ferromagnet, the only model for\nwhich more rigorous fully microscopic calculations13,14\nofβhave been completed. Section IV is devoted to\nthe study of a two-dimensional electron-gas ferromag-\nnet with Rashba spin-orbit interactions. Studies of this\nmodel provide a qualitative indication of the influence\nof intrinsic spin-orbit interactions on the non-adiabatic\nSTT. We find that, as in the microscopic theory17,18\nforα, spin-orbit interactions induce intra-band contri-\nbutions to βwhich are proportional to the quasiparticle\nlifetimes. These considerations carry over to the more\nsophisticated 4-band spherical model that we analyze in\nSectionV;thereourcalculationistailoredto(Ga,Mn)As.\nWe show that intra-band (conductivity-like) contribu-\ntions are prominent in the 4-band model for experimen-\ntally relevant scattering rates.\nSectionVIdiscussesthephenomenologicallyimportant\nα/βratio for real materials. Using our analytical results\nderived in Section II (or Section VIII) we are able to re-\nproduce and extrapolate trends expected from toy mod-\nels which indicate that α/βshould vary across materialsin approximately the same way as the ratio between the\nitinerant spin density and the total spin density. We also\nsuggest that αandβmay have the opposite signs in sys-\ntems with both hole-like and electron-like carriers. We\npresent concrete results for (Ga,Mn)As, where we obtain\nα/β≃0.1. This is reasonable in view of the weak spin\npolarizationand the strong spin-orbit coupling of valence\nband holes in this material.\nSection VII describes the generalization of the torque-\ncorrelation formula employed in ab-initio calculations of\nthe Gilbert damping to the case of the non-adiabatic\nspin-transfer torque. The torque correlation formula in-\ncorporates scattering of quasiparticles simply by intro-\nducing a phenomenological lifetime for the Bloch states\nand assumes that the most important electronic transi-\ntions occur between states near the Fermi surface in the\nsame band. Our ability to make quantitative predictions\nbased on this formula is limited mainly by an incomplete\nunderstanding of Bloch state scattering processes in real\nferromagnetic materials. These simplifications give rise\nto ambiguitiesandinaccuraciesthat wedissect in Section\nVII. Our assessment indicates that the torque correlation\nformula for βis most accurate at low disorder and rela-\ntively weak spin-orbit interactions.\nSection VIII restates and complements the effective\nfield calculation explained in Section II. Within the adi-\nabatic approximation, the instantaneous energy of a fer-\nromagnet may be written in terms of the instantaneous\noccupation factors of quasiparticle states. We determine\nthe effect ofthe external perturbationson the occupation\nfactors by combining the relaxation time approximation\nand the master equation. In this way we recover the re-\nsultsofSectionII andareableto interprettheintra-band\ncontributions to βin terms of a generalized breathing\nFermi surface picture.\nSection IX contains a brief summary which concludes\nthis work.\nII. MICROSCOPIC THEORY OF α,βANDvs\nThe Gilbert damping parameter α, the non-adiabatic\nspin transfer torque coefficient βand the “spin velocity”\nvsappear in the generalized Landau-Lifshitz-Gilbert ex-\npression for collective magnetization dynamics of a fer-\nromagnet under the influence of an electric current:\n(∂t+vs·∇)ˆΩ−ˆΩ×Heff=−αˆΩ×∂tˆΩ−βˆΩ×(vs·∇)ˆΩ.\n(1)\nIn Eq. (1) Heffis an effective magnetic field which we\nelaborate on below and ˆΩ =s0/s0≃(Ωx,Ωy,1−(Ω2\nx+\nΩ2\ny)/2)isthedirectionofthemagnetization.19Eq.(1)de-\nscribes the slow dynamics of smooth magnetization tex-\nturesinthepresenceofaweakelectricfieldwhichinduces\ntransport currents. It explicitly neglects the dynamics of\nthemagnetizationmagnitudewhichisimplicitlyassumed\nto be negligible. For small deviations from the easy di-3\nrection (which we take to be the ˆ z-direction), it reads\nHeff,x= (∂t+vs·∇)Ωy+(α∂t+βvs·∇)Ωx\nHeff,y=−(∂t+vs·∇)Ωx+(α∂t+βvs·∇)Ωy(2)\nThe gyromagnetic ratio has been absorbed into the units\nof the field Heffso that this quantity has inverse time\nunits. We set /planckover2pi1= 1 throughout.\nIn this section we relate the α,βandvsparameters\nto microscopic features of the ferromagnet by consider-\ning the transverse total spin response function. For a\ntechnically more accessible (yet less rigorous) theory ofαandβwe refer to Section VIII. The transverse spin re-\nsponse function studied here describes the change in the\nmicromagneticenergyduetothedepartureofthemagne-\ntization away from its equilibrium direction, where equi-\nlibrium is characterized by the absence of currents and\nexternal rf fields. This change in energy defines an ef-\nfective magnetic field which may then be identified with\nEq. (2), thereby allowingus to microscopicallydetermine\nα,βandvs. To first order in frequency ω, wave vector q\nand electric field, the transverse spin response function\nis given by\nS0ˆΩa=/summationdisplay\nbχa,bHext,b≃/summationdisplay\nb/bracketleftBig\nχ(0)\na,b+ωχ(1)\na,b+(vs·q)χ(2)\na,b/bracketrightBig\nHext,b (3)\nwherea,b∈ {x,y},Hextis the external magnetic field with frequency ωand wave vector q,S0=s0Vis the total\nspin of the ferromagnet ( Vis the sample volume), and χis the transverse spin-spin response function in the presence\nof a uniform time-independent electric field:\nχa,b(q,ω;vs) =i/integraldisplay∞\n0dt/integraldisplay\ndrexp(iωt−iq·r)∝an}b∇acketle{t/bracketleftbig\nSa(r,t),Sb(0,0)/bracketrightbig\n∝an}b∇acket∇i}ht. (4)\nIn Eq. (3), χ(0)=χ(q=0,ω= 0;E=0) de-\nscribes the spin response to a constant, uniform ex-\nternal magnetic field in absence of a current, χ(1)=\nlimω→0χ(q=0,ω;E=0)/ωcharacterizes the spin re-\nsponse to a time-dependent, uniform external mag-\nnetic field in absence of a current, and χ(2)=\nlimq,vs→0χ(q,ω= 0;E)/q·vsrepresents the spin re-\nsponsetoaconstant, non-uniformexternalmagneticfield\ncombined with a constant, uniform electric field E. Note\nthat first order terms in qare allowed by symmetry in\npresence of an electric field. In addition, ∝an}b∇acketle{t∝an}b∇acket∇i}htis a ther-\nmal and quantum mechanical average over states that\ndescribe a uniformly magnetized, current carrying ferro-\nmagnet.\nThe approach underlying Eq. (3) comprises a linear\nresponse theory with respect to an inhomogeneous mag-\nnetic field followed by a linear response theory with re-\nspect toan electricfield. Alternatively, onemaytreatthe\nelectric and magnetic perturbations on an equal footing\nwithout predetermined ordering; for further considera-\ntions on this matter we refer to Appendix A.\nIn the following we emulate and appropriately gen-\neralize a procedure outlined elsewhere.17First, we rec-\nognize that in the static limit and in absence of a cur-\nrent the transverse magnetization responds to the exter-\nnal magnetic field by adjusting its orientation to min-\nimize the total energy including the internal energy\nEintand the energy due to coupling with the exter-\nnal magnetic field, Eext=−S0ˆΩ· Hext. It follows\nthatχ(0)\na,b=S2\n0[∂2Eint/∂ˆΩa∂ˆΩb]−1and thus Hint,a=−(1/S0)∂Eint/∂ˆΩa=−S0[χ(0)]−1\na,bˆΩb, whereHintis the\ninternal energy contribution to the effective magnetic\nfield. Multiplying Eq. (3) on the left by [ χ(0)]−1and\nusingHeff=Hint+Hextwe obtain a formal equation for\nHeff:\nHeff,a=/summationdisplay\nb/bracketleftBig\nL(1)\na,b∂t+L(2)\na,b(vs·∇)/bracketrightBig\nˆΩb,(5)\nwhere\nL(1)=−iS0[χ(0)]−1χ(1)[χ(0)]−1\nL(2)=iS0[χ(0)]−1χ(2)[χ(0)]−1. (6)\nIdentifying of Eqs. (5) and (2) results in concise micro-\nscopic expressions for αandβandvs:\nα=L(1)\nx,x=L(1)\ny,y\nβ=L(2)\nx,x=L(2)\ny,y\n1 =L(2)\nx,y=⇒vs·q=iS0/bracketleftBig\n(χ(0))−1χ(χ(0))−1/bracketrightBig\nx,y.(7)\nIn the third line of Eq. (7) we have combined the second\nline of Eq. (6) with χ(2)=χ/(vs·q).\nWhen applying Eq. (7) to realistic conducting fer-\nromagnets, one must invariably adopt a self-consistent\nmean-field (Stoner) theory description of the magnetic\nstate derived within a spin-density-functional theory\n(SDFT) framework.20,21In SDFT the transverse spin\nresponse function is expressed in terms of Kohn-Sham4\nquasiparticleresponsetoboth externalandinduced mag-\nnetic fields; this allows us to transform17Eq. (7) into\nα=1\nS0lim\nω→0Im[˜χQP\n+,−(q= 0,ω,E= 0)]\nω\nβ=−1\nS0lim\nvs,q→0Im[˜χQP\n+,−(q,ω= 0,E)]\nq·vs\nvs·q=−1\nS0Re[˜χQP\n+,−(q,ω= 0,E)], (8)\nwhere we have used22χ(0)\na,b=δa,bS0/¯∆ and\n˜χQP\n+,−(q,ω;E) =1\n2/summationdisplay\ni,jfj−fi\nǫi−ǫj−ω−iη\n∝an}b∇acketle{tj|S+∆0(r)eiq·r|i∝an}b∇acket∇i}ht∝an}b∇acketle{ti|S−∆0(r)e−iq·r|j∝an}b∇acket∇i}ht\n(9)\nis the quasiparticle response to changes in the direction\nof the exchange-correlation effective magnetic field.23To\nestimate βthis response function should be evaluated\nin the presence of an electric current. In the derivation\nof Eq. (8) we have made use of the fact that χ(1)\nx,xand\nχ(2)\nx,xare purely imaginary, whereas χ(2)\nx,yis purely real;\nthis can be verified mathematically through S±=Sx±\niSy. Physically, “Im” and “Re” indicate that the Gilbert\ndampingandthenon-adiabaticSTTaredissipativewhile\nthe adiabatic STT is reactive. Furthermore, in the third\nline it is implicit that we expand Re[˜ χQP] to first order\ninqandE.\nIn Eq. (9), S±is the spin-rising/loweringoperator, |i∝an}b∇acket∇i}ht,\nǫiandfiare the Kohn-Sham eigenstates, eigenenergies\nand Fermi factors in presence of spin-dependent disorder,\nand ∆ 0(r) is the difference in the magnetic ground state\nbetween the majority spin and minority spin exchange-\ncorrelation potential - the spin-splitting potential. This\nquantity is alwaysspatially inhomogeneous at the atomic\nscale and is typically larger in atomic regions than in\ninterstitial regions. Although the spatial dependence of\n∆0(r) plays a crucial role in realistic ferromagnets, we\nreplace it by a phenomenological constant ∆0in the toy\nmodels we discuss below.\nOur expression of vsin terms of the transverse spin\nresponse function may be unfamiliar to readers familiar\nwith the argument given in the introduction of this pa-\nper in which vsis determined by the divergence in spincurrent. This argument is based on the assumption that\nthe (transverse) angular momentum lost by spin polar-\nized electrons traversing an inhomogeneous ferromagnet\nis transferred to the magnetization. However, this as-\nsumption fails when spin angular momentum is not con-\nserved as it is not in the presence of spin-orbit coupling.\nIn general, part of the transverse spin polarization lost\nby the current carrying quasiparticles is transferred to\nthe lattice rather than to collective magnetic degrees of\nfreedom8when spin-orbit interactions are present. It is\noften stated that the physics of spin non-conservation is\ncaptured by the non-adiabatic STT; however, the non-\nadiabatic STT per seis limited to dissipative processes\nand cannot describe the changes in the reactive spin\ntorque due to spin-flip events. Our expression in terms\nof the transverse spin response function does not rely on\nspin conservation, and while it agrees with the conven-\ntional picture24in simplest cases (see below), it departs\nfrom it when e.g.intrinsic spin-orbit interactions are\nstrong.\nIn this paper weincorporatethe influence ofan electric\nfield by simply shifting the Kohn-Sham orbital occupa-\ntion factors to account for the energy deviation of the\ndistribution function in a drifting Fermi sea:\nfi≃f(0)(ǫi+Vi)≃f(0)(ǫi)+Vi∂f(0)/∂ǫi(10)\nwhereViis the effective energy shift for the i-th eigenen-\nergy due to acceleration between scattering events by an\nelectric field and f(0)is the equilibrium Fermi factor.\nThis approximation to the steady-state induced by an\nexternal electric field is known to be reasonably accurate\nin many circumstances, for example in theories of electri-\ncal transport properties, and it can be used24to provide\na microscopic derivation of the adiabatic spin-transfer\ntorque. As we discuss below, this ansatzprovidesa result\nforβwhich is sufficiently simple that it can be combined\nwith realistic ab initio electronic structure calculations\nto estimate βvalues in particular magnetic metals. We\nsupport this ansatzby demonstrating that it agrees with\nfull non-linear response calculations in the case of toy\nmodels for which results are available.\nUsing the Cauchy identity, 1 /(x−iη) = 1/x+iπδ(x),\nand∂f(0)/∂ǫ≃ −δ(ǫ) we obtain\nIm[˜χQP\n+,−]≃π\n2/summationdisplay\ni,j[ω−Vj,i]|∝an}b∇acketle{tj|S+∆0(r)eiq·r|i∝an}b∇acket∇i}ht|2δ(ǫi−ǫF)δ(ǫj−ǫF)\nRe[˜χQP\n+,−]≃ −1\n2/summationdisplay\ni,j|∝an}b∇acketle{tj|S+∆0(r)eiq·r|i∝an}b∇acket∇i}ht|2Vjδ(ǫj−ǫF)−Viδ(ǫi−ǫF)\nǫi−ǫj(11)\nwhere we have defined the difference in transport devia- tion energ ies by\nVj,i≡Vj−Vi. (12)5\nIn the first line of Eq. (11), the two terms within the\nsquare brackets correspond to the energy of particle-\nhole excitations induced by radio frequency magnetic\nand static electric fields, respectively. The imaginarypart selects scattering processes that relax the spin of\nthe particle-hole pairs mediated either by phonons or by\nmagnetic impurities.25Substituting Eq. (11) into Eq. (8)\nwe can readily extract α,βandvs:\nα=π\n2S0/summationdisplay\ni,j|∝an}b∇acketle{tj|S+∆0(r)|i∝an}b∇acket∇i}ht|2δ(ǫi−ǫF)δ(ǫj−ǫF)\nβ= lim\nq,vs→0π\n2S0q·vs/summationdisplay\ni,j|∝an}b∇acketle{tj|S+∆0(r)eiq·r|i∝an}b∇acket∇i}ht|2Vj,iδ(ǫi−ǫF)δ(ǫj−ǫF)\nvs·q=1\n2S0/summationdisplay\ni,j|∝an}b∇acketle{tj|S+∆0(r)eiq·r|i∝an}b∇acket∇i}ht|2Vjδ(ǫj−ǫF)−Viδ(ǫi−ǫF)\nǫi−ǫj(13)\nwhere we have assumed a uniform precession mode for\nthe Gilbert damping.\nEq. (13) and Eq. (11) identify the non-adiabatic STT\nas acorrection to the Gilbert damping in the presence\nof an electric current; in other words, the magnetiza-\ntion damping at finite current is given by the sum of\nthe Gilbert damping and the non-adiabatic STT. We feel\nthat this simple interpretation of the non-adiabatic spin-\ntransfertorquehasnot receivedsufficientemphasisin the\nliterature.\nStrictly speaking the influence of a transport current\non magnetization dynamics should be calculated by con-\nsidering non-linear response of transverse spin to both\neffective magnetic fields and the external electric field\nwhich drives the transport current. Our approach, in\nwhich we simply alter the occupation probabilities which\nappear in the transverse spin response function is admit-\ntedly somewhat heuristic. We demonstrate below that\nit gives approximately the same result as the complete\ncalculation for the case of the very simplistic model for\nwhich that complete calculation has been carried out.\nIn Eq. (13), the eigenstates indexed by iare not Bloch\nstates of a periodic potential but instead the eigenstates\nof the Hamiltonian that includes all of the static dis-\norder. Although Eq. (13) provides compact expressions\nvalid for arbitrary metallic ferromagnets, its practical-\nity is hampered by the fact that the characterization of\ndisorder is normally not precise enough to permit a reli-able solution of the Kohn-Sham equations with arbitrary\nimpurities. An approximate yet more tractable treat-\nment of disorder consists of the following steps: (i) re-\nplace the actual eigenstates of the disordered system by\nBloch eigenstates corresponding to a pure crystal, e.g.\n|i∝an}b∇acket∇i}ht → |k,a∝an}b∇acket∇i}ht, where kis the crystal momentum and a\nis the band index of the perfect crystal; (ii) switch Vito\nVa=τk,avk,a·eE, whereτis the Bloch state lifetime and\nvk,a=∂ǫk,a/∂kis the quasiparticle group velocity, (iii)\nsubstitute the δ(ǫk,a−ǫF) spectral function of a Bloch\nstate by a broadened spectral function evaluated at the\nFermi energy: δ(ǫk,a−ǫF)→Aa(ǫF,k)/(2π), where\nAa(ǫF,k) =Γk,a\n(ǫF−ǫk,a)2+Γ2\nk,a\n4(14)\nand Γ a,k= 1/τa,kis the inverse of the quasiparticle\nlifetime. This minimal prescription can be augmented\nby introducing impurity vertex corrections in one of the\nspin-flip operators, which restores an exact treatment of\ndisorder in the limit of dilute impurities. This task is\nfor the most part beyond the scope of this paper (see\nnext section, however). The expression for αin Eq. (13)\nhas already been discussed in a previous paper;17hence\nfrom here on we shall concentrate on the expression for\nβwhich now reads\nβ(0)= lim\nq,vs→01\n8πs0/summationdisplay\na,b/integraldisplay\nk|∝an}b∇acketle{tk+q,b|S+∆0(r)|k,a∝an}b∇acket∇i}ht|2Aa(ǫF,k)Ab(ǫF,k+q)(vk+q,bτk+q,b−vk,aτk,a)·eE\nq·vs(15)\nwhere we have used/summationtext\nk→V/integraltext\ndDk/(2π)D≡V/integraltext\nkwithDas the dimensionality, Vas the volume and\nq·vs=1\n2s0/summationdisplay\na,b/integraldisplay\nk|∝an}b∇acketle{tk+q,b|S+∆0(r)|k,a∝an}b∇acket∇i}ht|2evk+q,bτk+q,bδ(ǫF−ǫk+q,b)−evk,aτk,aδ(ǫF−ǫk,a)\nǫk,a−ǫk+q,b.(16)6\nIn Eq. (15) the superscript “0” is to remind of the absence of impur ity vertex corrections; . In addition, we recall that\ns0=S0/Vis the magnetization of the ferromagnet and |ak∝an}b∇acket∇i}htis a band eigenstate of the ferromagnet withoutdisorder.\nIt is straightforward to show that Eq. (16) reduces to the usual expression vs=σsE/(es0) for vanishing intrinsic\nspin-orbit coupling. However, we find that in presence of spin-orbit interaction Eq. (16) is no longer connected to\nthe spin conductivity. Determining the precise way in which Eq. (16) d eparts from the conventional formula in real\nmaterials is an open problem that may have fundamental and practic al repercussions. Expanding the integrand in\nEq. (15) to first order in qand rearranging the result we arrive at\nβ(0)=−1\n8πs0q·vs/summationdisplay\na,b/integraldisplay\nk/bracketleftbig\n|∝an}b∇acketle{ta,k|S+∆0(r)|b,k∝an}b∇acket∇i}ht|2+|∝an}b∇acketle{ta,k|S−∆0(r)|b,k∝an}b∇acket∇i}ht|2/bracketrightbig\nAa(ǫF,k)A′\nb(ǫF,k)(vk,a·eE)(vk,b·q)τa\n−1\n4πs0q·vs/summationdisplay\na,b/integraldisplay\nkRe/bracketleftbig\n∝an}b∇acketle{tb,k|S−∆0(r)|a,k∝an}b∇acket∇i}ht∝an}b∇acketle{ta,k|S+∆0(r)q·∂k|b,k∝an}b∇acket∇i}ht\n+(S+↔S−)/bracketrightbig\nAa(ǫF,k)Ab(ǫF,k)(vk,a·eE)τa\n(17)\neVa,b eVa,ba,k;ω\na,k;ωeVa,b b, ω+k+q;S−\nS−S+\nS+ω(a)\n(b)b,k;ω +ωn\nn\nnnω\nFIG. 1: Feynman diagrams for (a) αand (b) β(q·vs), the\nlatter with a heuristic consideration of the electric field ( for a\nmore rigorous treatment see Appendix A). Solid lines corre-\nspond to Green’s functions of the band quasiparticles in the\nBorn approximation, dashed lines standfor themagnon offre -\nquencyωand wavevector q,ωnis the Matsubara frequency\nandeVa,bis the difference in the transport deviation energies.\nwhereA′(ǫF,k)≡2(ǫF−ǫk,a)Γa//bracketleftbig\n(ǫF−ǫk,a)2+Γ2\na/4/bracketrightbig2\nstands for the derivative of the spectral function and we\nhave neglected ∂Γ/∂k. Eq. (17) (or Eq. (15)) is the cen-\ntral result of this work and it provides a gateway to eval-\nuate the non-adiabatic STT in materials with complex\nband structures;16for a diagrammatic interpretation see\nFig. (1). An alternativeformula with a similar aspiration\nhas been proposed recently,26yet that formula ignores\nintrinsic spin-orbit interactions and relies on a detailed\nknowledge of the disorder scattering mechanisms. In the\nfollowing three sections we apply Eq. (17) to three differ-ent simplified models of ferromagnets. For a simpler-to-\nimplement approximate version of Eq. (15) or Eq. (17)\nwe refer to Section VI.\nIII. NON-ADIABATIC STT FOR THE\nPARABOLIC TWO-BAND FERROMAGNET\nThe model described in this section bears little resem-\nblance to any real ferromagnet. Yet, it is the only model\nin which rigorous microscopic results for βare presently\navailable, thus providing a valuable test bed for Eq. (17).\nThe mean-field Hamiltonian for itinerant carriers in a\ntwo-band Stoner model with parabolic bands is simply\nH(k)=k2\n2m−∆0Sz(18)\nwhere ∆ 0is the exchange field and Sz\na,b=δa,bsgn(a).\nIn this model the eigenstates have no momentum depen-\ndence and hence Eq. (17) simplifies to\n(vs·q)β(0)=−∆2\n0\n2πs0/summationdisplay\na/integraldisplay\nkAa(ǫF,k)A′\n−a(ǫF,k)\nk·q\nmk·eE\nmτk,a, (19)\nwherea= +(−) for majority (minority) spins, vk,±=\nk/m, andS±=Sx±iSywithSx\na,b=δa,b. Also, from\nhere on repeated indexes will imply a sum. Taking ∆ 0≤\nEFand ∆ 0>>1/τ, the momentum integral in Eq. (19)\nis performed in the complex energy plane using a keyhole\ncontour around the branch cut that stems from the 3D\ndensity of states:7\n(vs·q)β(0)=−∆2\n0\n2πs02eE·q\n3m/integraldisplay∞\n0ν(ǫ)Aa(ǫF,a−ǫ)A′\n−a(ǫF,−a−ǫ)ǫτk,a\n≃eE·q\n6m∆0s0sgn(a)νaǫF,aτaΓ−a\n=eE·q\n2m∆0s0(n↑τ↑γ↓−n↓τ↓γ↑) (20)\nwhereǫF,a=ǫF+ sgn(a)∆0,νais the spin-dependent\ndensity of states at the Fermi surface, na= 2νaǫF,a/3\nis the corresponding number density, and γa≡Γa/2.\nThe factor 1 /3 on the first line of Eq. (20) comes from\nthe angular integration. In the second line of Eq. (20)\nwe have neglected a term that is smaller than the one\nretainedbyafactorof∆2\n0/(12ǫ2\nF); suchextraterm(which\nwould have been absent in a two-dimensional version of\nthe model) appears to be missing in previous work.13,14\nThe simplicity of this model enables a partial incorpo-\nration of impurity vertex corrections. By adding to β(0)the contribution from the leading order vertex correction\n(β(1)), we shall recover the results obtained previously\nfor this model by a full calculation of the transverse spin\nresponse function. As it turns out, β(1)is qualitatively\nimportant because it ensures that only spin-dependent\nimpuritiescontributetothenon-adiabaticSTTintheab-\nsence of an intrinsic spin-orbit interaction. In Appendix\nB we derive the following result:\n(vs·q)β(1)=e∆2\n0\n4πs0/integraldisplay\nk,k′uiRe/bracketleftBig\nS+\na,bSi\nb,b′S−\nb′,a′Si\na′,a/bracketrightBigAa(ǫF,k)\n(ǫF−ǫk′,a′)/bracketleftbiggAb(ǫF,k+q)\n(ǫF−ǫk′+q,b′)Vb,a+Ab′(ǫF,k′+q)\n(ǫF−ǫk+q,b)Vb′,a/bracketrightbigg\n,(21)\nwhereui≡niw2\ni(i= 0,x,y,z),niis the density of scatterers, wiis the Fourier transform of the scattering potential\nand the overline denotes an average over different disorder config urations.13Also,Va,b= (τbvk+q,b−τavk,a)·eE.\nExpanding Eq. (21) to first order in q, we arrive at\n(vs·q)β(1)=−∆2\n0\n2πs0(u0−uz)/integraldisplay\nk,k′Aa(ǫF,k)\nǫF−ǫk′,a/bracketleftbiggA′\n−a(ǫF,k)\nǫF−ǫk′,−a+A−a(ǫF,k′)\n(ǫF−ǫk,−a)2/bracketrightbiggk·q\nmk·eE\nmτk,a (22)\nIn the derivation of Eq. (22) we have used S±=Sx±iSyand assumed that ux=uy≡ux,y, so that\nuiRe/bracketleftBig\nSx\na,bSi\nb,b′Sx\nb′,a′Si\na′,a/bracketrightBig\n=/parenleftbig\nu0−uz/parenrightbig\nδa,a′δb,b′δa,−b. In addition, we have used/integraltext\nk,k′F(|k|,|k′|)kik′\nj= 0. The first\nterm inside the square brackets of Eq. (22) can be ignored in the we ak disorder regime because its contribution is\nlinear in the scattering rate, as opposed to the second term, which contributes at zeroth order. Then,\n(vs·q)β(1)=−∆2\n0\nπs0(u0−uz)/integraldisplay\nk,k′Aa(ǫF,k)A−a(ǫF,k′)\n(ǫF−ǫk′,a)(ǫF−ǫk,−a)2k·q\nmk·eE\nmτk,a\n≃ −∆2\n0\nπs0(u0−uz)2eE·q\n3m/integraldisplay∞\n−∞dǫdǫ′ν(ǫ)ν(ǫ′)Aa(ǫF,a−ǫ)A−a(ǫF,−a−ǫ′)\n(ǫF−ǫ′a)(ǫF−ǫ−a)2ǫτa\n≃ −π(u0−uz)eE·q\n2m∆0s0sign(a)naτaν−a (23)\nCombining this with Eq. (20), we get\n(vs·q)β≃(vs·q)β(0)+(vs·q)β(1)\n=eE·q\n2ms0∆0/bracketleftbig\nn↑τ↑γ↓−n↓τ↓γ↑−π(u0−uz)(n↑τ↑ν↓−n↓τ↓ν↑)/bracketrightbig\n=πeE·q\nms0∆0[n↑τ↑(uzν↓+ux,yν↑)−n↓τ↓(uzν↑+ux,yν↓)] (24)\nwhere we have used γa=π/bracketleftbig\n(u0+uz)νa+2ux,yν−a/bracketrightbig\n. In\nthis model it is simple to solve Eq. (16) for vsanalyt-ically, whereupon Eq. (24) agrees with the results pub-8\nlished by other authors in Refs.[ 13,14] from full non-\nlinear response function calculations. However, we reit-\nerate that in order to reach such agreement we had to\nneglect a term of order ∆2\n0/ǫ2\nFin Eq. (20). This extra\nterm is insignificant in all but nearly half metallic ferro-\nmagnets.\nIV. NON-ADIABATIC STT FOR A\nMAGNETIZED TWO-DIMENSIONAL\nELECTRON GAS\nThe model studied in the previous section misses the\nintrinsic spin-orbit interaction that is inevitably present\nin the band structure of actual ferromagnets. Further-\nmore,sinceintrinsicspin-orbitinteractionisinstrumental\nfor the Gilbert damping at low temperatures, a similarly\nprominent role may be expected in regards to the non-\nadiabaticspin transfertorque. Hence, thepresentsection\nis devoted to investigatethe relativelyunexplored26,27ef-\nfect of intrinsic spin-orbit interaction on β. The minimalmodel for this enterprise is the two-dimensional electron-\ngas ferromagnet with Rashba spin-orbit interaction, rep-\nresented by\nH(k)=k2\n2m−b·S, (25)\nwhereb= (λky,−λkx,∆0),λis the Rashba spin-orbit\ncoupling strength and ∆ 0is the exchange field.\nThe eigenspinors of this model are |+,k∝an}b∇acket∇i}ht=\n(cos(θ/2),−iexp(iφ)sin(θ/2)) and |−,k∝an}b∇acket∇i}ht=\n(sin(θ/2),iexp(iφ)cos(θ/2)), where the spinor an-\ngles are defined through cos θ= ∆0//radicalbig\nλ2k2+∆2\n0\nand tan φ=ky/kx. The corresponding eigenen-\nergies are Ek±=k2/(2m)∓/radicalbig\n∆2\n0+λ2k2.\nTherefore, the band velocities are given by\nvk±=k/parenleftBig\n1/m∓λ2//radicalbig\nλ2k2+∆2\n0/parenrightBig\n=k/m±. Dis-\nregarding the vertex corrections, the non-adiabatic\nspin-torque of this model may be evaluated analytically\nstarting from Eq. (17). We find that (see Appendix C):\n(vs·q)β(0)≃∆2\n0eE·q\n8πs0/bracketleftbiggm2\n4m+m−/parenleftbigg\n1+∆2\n0\nb2/parenrightbigg1\nb2+1\n4λ2k2\nF∆2\n0\nb6/bracketrightbigg\n+∆2\n0eE·q\n8πs0/bracketleftbigg1\n2m2\nm2\n+λ2k2\nF\nb2/parenleftbigg\n1−δm+\nm∆2\n0\nb2/parenrightbigg\nτ2+1\n2m2\nm2\n−λ2k2\nF\nb2/parenleftbigg\n1−δm−\nm∆2\n0\nb2/parenrightbigg\nτ2/bracketrightbigg\n(26)\nwhereb=/radicalbig\nλ2k2\nF+∆2\n0(kF=√2mǫF), andδm±=\nm−m±. As we explain in the Appendix, Eq. (26) ap-\nplies forλkF,∆0,1/τ << ǫ F; for a more general analysis,\nEq. (17) must be solved numerically (e.g. see Fig. (2)).\nEq. (26) reveals that intrinsic spin-orbit interaction en-\nablesintra-band contributions to β, whose signature is\ntheO(τ2) dependence on the second line. In contrast,\ntheinter-band contributions appear as O(τ0). Since vs\nitself is linear in the scattering time, it follows that β\nis proportional to the electrical conductivity in the clean\nregime and the resistivity in the disordered regime, much\nlike the Gilbert damping α. We expect this qualitative\nfeature to be model-independent and applicable to real\nferromagnets.\nV. NON-ADIABATIC STT FOR (Ga,Mn)As\nInthis sectionweshallapplyEq.(17) toamoresophis-\nticated model which provides a reasonable description of\n(III,Mn)V magnetic semiconductors.28Since the orbitals\nat the Fermi energy are very similar to the states near\nthe top of the valence band of the host (III,V) semicon-\nductor, the electronic structure of (III,Mn)V ferromag-\nnets is remarkably simple. Using a p-d mean field theory\nmodelfortheferromagneticgroundstateandafour-bandspherical model for the host semiconductor band struc-\nture, Ga 1−xMnxAs may be described by\nH(k)=1\n2m/bracketleftbigg/parenleftbigg\nγ1+5\n2γ2/parenrightbigg\nk2−2γ3(k·S)2/bracketrightbigg\n+∆0Sz,\n(27)\nwhereSisthe spinoperatorprojectedontothe J=3/2to-\ntal angularmomentum subspace at the top of the valence\nband and {γ1= 6.98,γ2=γ3= 2.5}are the Luttinger\nparametersforthesphericalapproximationtothevalence\nbands of GaAs. In addition, ∆ 0=JpdsNMn=Jpds0is\ntheexchangefield, Jpd= 55 meVnm3isthep-dexchange\ncoupling, s= 5/2 is the spin of Mn ions, NMn= 4x/a3\nis the density of Mn ions and a= 0.565 nm is the lattice\nconstant of GaAs. We solve Eq. (27) numerically and\ninput the outcome in Eqs. (16), (17).\nThe results are summarized in Fig. (3). We find that\nthe intra-band contribution dominates as a consequence\nofthestrongintrinsicspin-orbitinteraction,muchlikefor\nthe Gilbert damping;18. Incidentally, βbarely changes\nregardless of whether the applied electric field is along\nthe easy axis of the magnetization or perpendicular to it.9\n0.00 0.05 0.10 0.15 0.20\n1/(εFτ)0.000.100.200.300.400.50β∆0=0.5εF  ;   λkF=0.05εF\nintra−band\ninter−band\ntotal\nFIG. 2: M2DEG : inter-band contribution, intra-band con-\ntribution and the total non-adiabatic STT for a magnetized\ntwo-dimensional electron gas (M2DEG). In this figure the ex-\nchange field dominates over the spin-orbit splitting. At hig her\ndisorder the inter-bandpart (proportional toresistivity ) dom-\ninates, while at low disorder the inter-band part (proporti onal\nto conductivity) overtakes. For simplicity, the scatterin g time\nτis taken to be the same for all sub-bands.\n0.00 0.10 0.20 0.30\n1/(εFτ)0.000.100.200.300.40βintra−band\ninter−band\ntotalx=0.08 ;  p=0.4 nm−3\nFIG. 3:GaMnAs :β(0)forEperpendicular to the easy axis\nof magnetization (ˆ z).xandpare the Mn fraction and the\nhole density, respectively. The intra-band contribution i s con-\nsiderably larger than the inter-band contribution, due to t he\nstrongintrinsicspin-orbitinteraction. Sincethe4-band model\ntypically overestimates the influence of intrinsic spin-or bit in-\nteraction, it is likely that the dominion of intra-band con-\ntributions be reduced in the more accurate 6-band model.\nBy evaluating βforE||ˆz(not shown) we infer that it does\nnot depend significantly on the relative direction between t he\nmagnetic easy axis and the electric field.\nVI.α/βIN REAL MATERIALS\nTheprecedingthreesectionshavebeenfocusedontest-\ning and analyzing Eq. (17) for specific models of ferro-magnets. In this section we return to more general con-\nsiderationsandsurveythephenomenologicallyimportant\nquantitative relationshipbetween αandβin realistic fer-\nromagnets, which always have intrinsic spin-orbit inter-\nactions. We begin by recollecting the expression for the\nGilbert damping coefficient derived elsewhere:17\nα=1\n8πs0/summationdisplay\na,b/integraldisplay\nk|∝an}b∇acketle{tb,k|S+∆0|a,k∝an}b∇acket∇i}ht|2Aa(ǫF,k)Ab(ǫF,k)\n(28)\nwhere we have ignored disorder vertex corrections. This\nexpression is to be compared with Eq. (15); for peda-\ngogical purposes we discuss intra-band and inter-band\ncontributions separately.\nStarting from Eq. (15) and expanding the integrand to\nfirst order in qwe obtain\nβintra=1\n8πs0/integraldisplay\nk|∝an}b∇acketle{ta,k|S+∆0|a,k∝an}b∇acket∇i}ht|2Aa(ǫF,k)2\neτaqi∂kivj\nk,aEj\nq·vs(29)\nwhere we have neglected the momentum dependence of\nthe scattering lifetime and a sum over repeated indices\nis implied. Remarkably, only matrix elements that are\ndiagonal in momentum space contribute to βintra; the\nimplicationsofthiswillbehighlightedinthenextsection.\nRecognizing that ∂kjvi\nk,a= (1/m)i,j\na, where (1 /m)ais\nthe inverse effective mass tensor corresponding to band\na, Eq. (29) can be rewritten as\nβintra=1\n8πs0/integraldisplay\nk|∝an}b∇acketle{ta,k|S+∆0|a,k∝an}b∇acket∇i}ht|2Aa(ǫF,k)2q·vd,a\nq·vs,\n(30)\nwhere\nvi\nd,a=eτa(m−1)i,j\naEj(31)\nis the “drift velocity” corresponding to the quasiparticles\nin band a. For Galilean invariant systems33vd,a=vs\nfor any ( k,a) and consequently βintra=αintra. At first\nglance, it might appear that vs, which (at least in ab-\nsence of spin-orbit interaction) is determined by the spin\ncurrent, must be different than vd,a. However, recall that\nvsis determined by the ratio of the spin current to the\nmagnetization. If the same electrons contribute to the\ntransport as to the magnetization, vs=vd,aprovided\nthe scattering rates and the masses are the same for all\nstates. These conditions are the conditions for an elec-\ntron system to be Galilean invariant.10\nThe interband contribution can be simplified by noting that\nτbvi\nk+q,b−τavi\nk,a= (τbvi\nk+q,b−τavi\nk+q,a)+(τavi\nk+q,a−τavi\nk,a). (32)\nThe second term on the right hand side of Eq.( 32) can then be manipu lated exactly as in the intra-band case to\narrive at\nβinter=1\n8πs0/summationdisplay\na,b(a/negationslash=b)/integraldisplay\nk|∝an}b∇acketle{tb,k|S+∆0|a,k∝an}b∇acket∇i}ht|2Aa(ǫF,k)Ab(ǫF,k)q·vd,a\nq·vs+δβinter (33)\nwhere\nδβinter=1\n8πs0/summationdisplay\na,b(a/negationslash=b)/integraldisplay\nk|∝an}b∇acketle{ta,k−q|S+∆0|b,k∝an}b∇acket∇i}ht|2Aa(ǫF,k−q)Ab(ǫF,k)(τbvk,b−τavk,a)·E\nq·vs. (34)\nWhen Galilean invariance is preserved the quasiparticle\nvelocity and scattering times are the same for all bands,\nwhich implies that δβ= 0 and hence that βinter=αinter.\nAlthough realistic materials are not Galilean invariant,\nδβis nevertheless probably not significant because the\nterm between parenthesis in Eq. (34) has an oscillatory\nbehavior prone to cancellation. The degree of such can-\ncellation must ultimately be determined by realistic cal-\nculations for particular materials.\nWith this proviso, we estimate that\nβ≃1\n8πs0/integraldisplay\nk|∝an}b∇acketle{tb,k|S+∆0|a,k∝an}b∇acket∇i}ht|2Aa(ǫF,k)Ab(ǫF,k)\nq·vd,a\nq·vs. (35)\nAs long as δβ≃0 is justified, the simplicity of Eq. (35)\nin comparisonto Eq. (15) or (17) makes ofthe former the\npreferred starting point for electronic structure calcula-\ntions. Even when δβ∝ne}ationslash= 0 Eq. (35) may be an adequate\nplatformfor ab-initio studiesonweaklydisorderedtransi-\ntion metal ferromagnets and strongly spin-orbit coupled\nferromagnetic semiconductors,29whereβis largely de-\ntermined by the intra-band contribution. Furthermore,\na direct comparison between Eq. (28) and Eq. (35) leads\nto the following observations. First, for nearly parabolic\nbands with nearly identical curvature, where the “drift\nvelocity” is weakly dependent on momentum or the band\nindex, we obtain β≃(vd/vs)αand thus β/αis roughly\nproportional to the ratio of the total spin density to the\nitinerant spin density, in concordance with predictions\nfrom toy models.12Second, if α/β >0 for a system with\npurely electron-like carriers, then α/β >0 for the same\nsystem with purely hole-like carriers because for a fixed\ncarrier polarization va\ndandvsreverse their signs under\nm→ −m. However, if both hole-like and electron-like\ncarriers coexist at the Fermi energy, then the integrand\nin Eq. (35) is positive for some values of aand negative\nfor others. In such situation it is conceivable that α/βbe\neither positive or negative. A negative value of βimplies\nadecrease in magnetization damping due to an applied\ncurrent.0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35\n1/(εFτ)0.120.220.320.42\n8 α\nβ\nFIG. 4: Comparison of αandβin (Ga,Mn)As for x= 0.08\nandp= 0.4nm−3. It follows that β/α≃8, with a weak\ndependence on the scattering rate off impurities. If we use th e\ntorque correlation formula (Section VII), we obtain β/α≃10.\nAs an illustration of the foregoing discussion, in\nFig. (4) we evaluate α/βfor (Ga,Mn)As. We find βto\nbe about an order of magnitude larger than α, which is\nreasonablebecause (i) the local moment magnetizationis\nlarger than the valence band hole magnetization, and (ii)\nthe spin-orbit coupling in the valence band decreases the\ntransportspin polarization. Accordingly βis of the order\nof unity, in qualitative agreement with recent theoretical\nwork30.\nVII. TORQUE-CORRELATION FORMULA FOR\nTHE NON-ADIABATIC STT\nThus far we have evaluated non-adiabatic STT us-\ning the bare vertex ∝an}b∇acketle{ta,k|S+|b,k+q∝an}b∇acket∇i}ht. In this section,\nwe shall analyze an alternative matrix element denoted\n∝an}b∇acketle{ta,k|K|b,k+q∝an}b∇acket∇i}ht(see below for an explicit expression),\nwhich may be better suited to realistic electronic struc-\nture calculations.16,31We begin by making the ap-11\n0.00 0.05 0.10 0.15 0.20\n1/(εFτ)0.000.100.200.300.400.50β\nS+\nK∆0=0.5 εF ;  λkF=0.05εF\nFIG. 5: M2DEG : comparing SandKmatrix element ex-\npressions for the non-adiabatic STT formula in the weakly\nspin-orbit coupled regime. Both formulations agree in the\nclean limit, where the intra-band contribution is dominant .\nIn more disordered samples inter-band contributions becom e\nmore visible and SandKbegin to differ; the latter is known\nto be more accurate in the weakly spin-orbit coupled regime.\nproximation that the exchange splitting can be writ-\nten as a constant spin-dependent shift Hex= ∆0Sz.\nThen, the mean-field quasiparticle Hamiltonian H(k)=\nH(k)\nkin+H(k)\nso+Hexcan be written as the sum of a spin-\nindependent part H(k)\nkin, the exchange term, and the spin-\norbit coupling H(k)\nso. With this approximation, we have\nthe identity:\n∝an}b∇acketle{ta,k|S+|b,k+q∝an}b∇acket∇i}ht\n=1\n∆0∝an}b∇acketle{ta,k|/bracketleftBig\nH(k),S+/bracketrightBig\n|b,k+q∝an}b∇acket∇i}ht\n−1\n∆0∝an}b∇acketle{ta,k|/bracketleftBig\nH(k)\nso,S+/bracketrightBig\n|b,k+q∝an}b∇acket∇i}ht.(36)\nThe last term in the right hand side of Eq. (36) is the\ngeneralization of the torque matrix element used in ab-\ninitiocalculations of the Gilbert damping:\n∝an}b∇acketle{ta,k|K|b,k+q∝an}b∇acket∇i}ht ≡1\n∆0∝an}b∇acketle{ta,k|/bracketleftBig\nH(k)\nso,S+/bracketrightBig\n|b,k+q∝an}b∇acket∇i}ht(37)\nEq. (36) implies that at q=0∝an}b∇acketle{tb,k|S+|a,k∝an}b∇acket∇i}ht ≃\n∝an}b∇acketle{tb,k|K|a,k∝an}b∇acket∇i}htprovided that ( Ek,a−Ek,b)<<∆0, which\nis trivially satisfied for intra-band transitions but less\nso for inter-band transitions.18Forq∝ne}ationslash=0the agreement\nbetween intra-band matrix elements is no longer obvi-\nous and is affected by the momentum dependence of\nthe band eigenstates. At any rate, Eq. (29) demon-\nstrates that only q=0matrix elements contribute to\nβintra; therefore βintrahas the same value for SandK\nmatrix elements. The disparity between the two formu-\nlations is restricted to βinter, and may be significant if\nthe most prominent inter-band matrix elements connect\nstates that are notclose in energy. When they disagree,0.00 0.05 0.10 0.15 0.20\n1/(εFτ)0.000.100.200.300.40βS+\nK∆0=0.5εF ; λkF=0.8εF\nFIG. 6: M2DEG : In the strongly spin-orbit coupled limit\nthe intra-band contribution reigns over the inter-band con tri-\nbution and accordingly SandKmatrix element expressions\ndisplay a good (excellent in this figure) agreement. Neverth e-\nless, this agreement does not guarantee quantitative relia bil-\nity, because for strong spin-orbit interactions impurity v ertex\ncorrections may play an important role.\n0.00 0.10 0.20 0.30\n1/(εFτ)0.00.20.40.6βS+\nKx=0.08 ; p=0.4 nm−3\nFIG. 7: GaMnAs : comparison between SandKmatrix\nelement expressions for E⊥ˆz. The disagreement between\nboth formulations stems from inter-band transitions, whic h\nare less important as τincreases. Little changes when E/bardblˆz.\nit is generally unclear32whether SorKmatrix elements\nwill yield a better estimate of βinter. The weak spin-orbit\nlimit is a possible exception, in which the use of Kap-\npearstoofferapracticaladvantageover S. Inthis regime\nSgenerates a spurious inter-band contribution in the ab-\nsence of magnetic impurities (recall Section III) and it is\nonlyafterthe inclusion ofthe leadingordervertexcorrec-\ntion that such deficiency gets remedied. In contrast, K\nvanishes identically in absence of spin-orbit interactions,\nthus bypassing the pertinent problem without having to\nintroduce vertex corrections.\nFigs. (5)- (7) display a quantitative comparison be-\ntween the non-adiabatic STT obtained from KandS,12\nboth for the M2DEG and (Ga,Mn)As. Fig. (5) reflects\nthe aforementioned overestimation of Sin the inter-band\ndominated regime ofweakly spin-orbitcoupled ferromag-\nnets. In the strong spin-orbit limit, where intra-band\ncontributions dominate in the disorder range of interest,\nKandSagree fairly well (Figs. (6) and (7)). Summing\nup, insofar as impurity vertex corrections play a minor\nroleandthedominantcontributionto βstemsfromintra-\nband transitions the torque-correlation formula will pro-\nvide a reliable estimate of β.\nVIII. CONNECTION TO THE EFFECTIVE\nFIELD MODEL\nAs explained in Section II we view the non-adiabatic\nSTT as the change in magnetization damping due to a\ntransport current. The present section is designed to\ncomplement that understanding froma different perspec-\ntive based on an effective field formulation, which pro-\nvides asimple physicalinterpretationforboth intra-band\nand inter-band contributions to β.\nAn effective field Heffmay be expressed as the varia-\ntion of the system energy with respect to the magnetiza-\ntion direction Heff\ni=−(1/s0)∂E/∂Ωi. Here we approxi-\nmate the energy with the Kohn-Sham eigenvalue sum\nE=/summationdisplay\nk,ank,aǫk,a. (38)\nThe variation of this energy with respect to the magne-\ntization direction yields\nHeff\ni=−1\ns0/summationdisplay\nk,a/bracketleftbigg\nnk,a∂ǫk,a\n∂Ωi+∂nk,a\n∂Ωiǫk,a/bracketrightbigg\n.(39)\nIt has previously been shown that, in the absence of cur-\nrent, the first term in the sum leads to intra-bandGilbert\ndamping15,35while the second term produces inter-band\ndamping.34In the following, we generalize these resultsbyallowingthe flowofan electricalcurrent. αandβmay\nbe extracted by identifying the the dissipative part of the\neffective field with −α∂ˆΩ/∂t−βvs·∇ˆΩ that appears in\nthe LLS equation.\nIntra-band terms : We begin by recognizing that as the\ndirection of magnetization changes in time, so does the\nshape of the Fermi surface, provided that there is an in-\ntrinsic spin-orbit interaction. Consequently, empty (full)\nstates appear below (above) the Fermi energy, giving rise\nto an out-of-equilibrium quasiparticle distribution. This\nconfiguration tends to relax back to equilibrium, but re-\npopulation requires a time τ. Due to the time delay,\nthe quasiparticle distribution lags behind the dynamical\nconfiguration of the Fermi surface, effectively creating a\nfriction (damping) force on the magnetization. From a\nquantitative standpoint, the preceding discussion means\nthat the quasiparticle energies ǫk,afollow the magnetiza-\ntion adiabatically, whereas the occupation numbers nk,a\ndeviate from the instantaneous equilibrium distribution\nfk,avia\nnk,a=fk,a−τk,a/parenleftbigg∂fk,a\n∂t+˙ra·∂fk,a\n∂r+˙k·∂fk,a\n∂k/parenrightbigg\n,\n(40)\nwhere we have used the relaxation time approximation.\nAs we explain below, the last two terms in Eq. (40) do\nnot contribute to damping in the absence of an electric\nfield and have thus been ignored by prior applications of\nthe breathing Fermi surface model, which concentrate on\nGilbert damping. It is customary to associate intra-band\nmagnetization damping with the torque exerted by the\npart of the effective field\nHeff\nintra=−1\ns0/summationdisplay\nk,ank,a∂ǫk,a\n∂ˆΩ(41)\nthat is lagging behind the instantaneous magnetization.\nPlugging Eq. (40) in Eq. (41) we obtain\nHeff\nintra,i=1\ns0/summationdisplay\nk,a/bracketleftbigg\n−fk,a∂ǫk,a\n∂Ωi+τa∂fk,a\n∂ǫk,a∂ǫk,a\n∂Ωi∂ǫk,a\n∂Ωj∂Ωj\n∂t+τa˙rl\na∂fk,a\n∂ǫk,a∂ǫk,a\n∂Ωi∂ǫk,a\n∂Ωj∂Ωj\n∂rl+τa˙kj∂fk,a\n∂ǫk,a∂ǫk,a\n∂kj∂ǫk,a\n∂Ωi/bracketrightbigg\n(42)\nwhere a sum is implied over repeated Latin indices. The\nfirst term in Eq. (42) is a contribution to the anisotropy\nfield; it evolves in synchrony with the dynamical Fermi\nsurfaceandisthusthereactivecomponentoftheeffective\nfield. The remaining terms, which describe the time lag\nof the effective field due to a nonzero relaxation time, are\nresponsible for intra-band damping. The last term van-\nishesincrystalswith inversionsymmetrybecause ˙k=eE\nand∂ǫ/∂kis an odd function of momentum. Similarly,if we take ˙r=∂ǫ(k)/∂kthe second to last term ought to\nvanish as well. This leaves us with the first two terms in\nEq. ( 42), which capture the intra-band Gilbert damping\nbut not the non-adiabatic STT. This is not surprising as\nthe latter involves the coupledresponse to spatial varia-\ntions of magnetization and a weak electric field, render-\ning linear order in perturbation theory insufficient (see\nAppendix A). In order to account for the relevant non-\nlinearity we use ˙r=∂ǫ(k−ev·Eτ)/∂kin Eq.( 42), where13\nv=∂ǫ(k)/∂k. The dissipative part of Heff\nintrathen reads\nHeff,damp\nintra,i=1\ns0/summationdisplay\nk,aτk,a∂fǫk,a\n∂ǫk,a∂ǫk,a\n∂Ωi∂ǫk,a\n∂Ωj/bracketleftbigg∂Ωj\n∂t+vl\nd,a∂Ωj\n∂rl/bracketrightbigg\n,\n(43)\nwherevi\nd,a=eτa(m−1)i,j\naEjis the “drift velocity” cor-\nresponding to band a. Eq. (43) may now be identified\nwith−αintra∂ˆΩ/∂t−βintravs· ∇ˆΩ that appears in the\nLLS equation. For an isotropic system this results in\nαintra=−1\ns0/summationdisplay\nk,a,iτk,a∂fk,a\n∂ǫk,a/parenleftbigg∂ǫk,a\n∂Ωi/parenrightbigg2\nβintra=−1\ns0/summationdisplay\nk,a,iτk,a∂fk,a\n∂ǫk,a/parenleftbigg∂ǫk,a\n∂Ωi/parenrightbigg2q·vd,a\nq·vs.(44)\nSince∝an}b∇acketle{t[Sx,Hso]∝an}b∇acket∇i}ht=∂φ∝an}b∇acketle{texp(iSxφ)Hsoexp(−iSxφ)∝an}b∇acket∇i}ht=\n∂ǫ/∂φfor an infinitesimal angle of rotation φaround\nthe instantaneous magnetization, βin Eq. (44) may be\nrewritten as\nβintra=∆2\n0\n2s0/summationdisplay\nk,aτk,a∂fk,a\n∂ǫk,a|∝an}b∇acketle{tk,a|K|k,a∝an}b∇acket∇i}ht|2q·vd,a\nq·vs(45)\nwhereK= [S+,Hso]/∆0is the spin-torque operator in-\ntroduced in Eq. ( 37) and we have claimed spin rota-\ntionalinvariancevia |∝an}b∇acketle{t[Sx,Hso]∝an}b∇acket∇i}ht|2=|∝an}b∇acketle{t[Sy,Hso]∝an}b∇acket∇i}ht|2. Using\n∂f/∂ǫ≃ −δ(ǫ−ǫF) and recalling from Section VII that\nKa,a=S+\na,a, Eq. (45) is equivalent to Eq. (30); note that\nthe product of spectral functions in the latter yields a\nfactor of 4 πτupon momentum integration. These obser-\nvations prove that βintradescribes the contribution from\natransportcurrenttothe“breathingFermisurface”type\nof damping. Furthermore, Eq. (44) highlights the impor-\ntance of the ratio between the two characteristic veloci-\nties of a current carrying ferromagnet, namely vsandvd.\nAs explained in Section VI these two velocities coincide\nin models with Galilean invariance. Only in these arti-\nficial models, which never apply to real materials, does\nα=βhold.\nInter-band terms : The Kohn-Sham orbitals are effec-\ntive eigenstates of a mean-field Hamiltonian where the\nspins are aligned in the equilibrium direction. As spins\nprecess in response to external rf fields and dc trans-\nport currents, the time-dependent part of the mean-field\nHamiltonian drives transitions between the ground-state\nKohn-Sham orbitals. These processes lead to the second\nterm in the effective field and produce the inter-band\ncontribution to damping.We thus concentrate on the second term in Eq. (39),\nHeff\ninter=−1\ns0/summationdisplay\nk,a∂nk,a\n∂ˆΩǫk,a. (46)\nMultiplying Eq. (46) with ∂ˆΩ/∂twe get\nHeff,damp\ninter·∂tˆΩ =−1\ns0/summationdisplay\nk,aǫk,a/bracketleftBig\n∂na,k/∂ˆΩ·∂ˆΩ/∂t/bracketrightBig\n=−1\ns0/summationdisplay\nk,aǫk,a∂na,k/∂t. (47)\nThe rate of change of the populations of the Kohn-\nShamstatescanbeapproximatedbythefollowingmaster\nequation\n∂na,k\n∂t=−/summationdisplay\nb,k′Wa,b(nk,a−nk′,b),(48)\nwhere\nWa,b= 2π|∝an}b∇acketle{tb,k′|∆0Sx|a,k∝an}b∇acket∇i}ht|2δk′,k+qδ(ǫb,k′−ǫa,k−ω)\n(49)\nis the spin-flip inter-band transition probability as dic-\ntated by Fermi’s golden rule. Eqs. (48) and (49)\nrely on the principle of microscopic reversibility36and\nare rather ad hocbecause they circumvent a rigorous\nanalysis of the quasiparticle-magnon scattering, which\nwould for instance require keeping track of magnon occu-\npation number. Furthermore, quasiparticle-phonon and\nquasiparticle-impurity scattering are allowed for simply\nby broadening the Kohn-Sham eigenenergies (see below).\nThe right hand side of Eq. (48) is now closely related\nto inter-band magnetization damping because it agrees37\nwith the netdecay rate of magnons into particle-hole\nexcitations, where the particle and hole are in different\nbands. Combining Eq. (47) and (48) and rearranging\nterms we arrive at\nHeff\ninter·∂tˆΩ =1\n2s0/summationdisplay\nk,k′,a,bWa,b(nk,a−nk′,b)(ǫk,a−ǫk′,b).\n(50)\nFor the derivation of αinterit is sufficient to approximate\nnk,aas a Fermi distribution in Eq. (50); here we ac-\ncountforatransportcurrentbyshiftingtheFermiseasas\nnk,a→nk,a−evk,a·Eτk,a∂nk,a/∂ǫk,a, which to leading\norder yields14\nHeff\ninter·∂tˆΩ =−πω\n2s0/summationdisplay\nk,a,b/vextendsingle/vextendsingle∝an}b∇acketle{tb,k+q|∆0S+|a,k∝an}b∇acket∇i}ht/vextendsingle/vextendsingle2δ(ǫb,k+q−ǫa,k−ω)∂nk,a\n∂ǫk,a(−ω+eVb,a)\n=ω\n8πs0/summationdisplay\nk,a,b/vextendsingle/vextendsingle∝an}b∇acketle{tb,k+q|∆0S+|a,k∝an}b∇acket∇i}ht/vextendsingle/vextendsingle2Aa(k,ǫF)Ab(k+q,ǫF)(−ω+eVb,a) (51)\nwhere we have used Sx= (S++S−)/2 and defined Vb,a=evk+q,b·Eτk+q,b−evk,a·Eτk,a. In the second line of\nEq.( 51) we have assumed low temperatures, and have introduced a finite quasiparticle lifetime by broadening the\nspectral functions of the Bloch states into Lorentzians with the c onvention outlined in Eq. (14): δ(x)→A(x)/(2π).\nIdentifying Eq.( 51) with ( −αinter∂tˆΩ−βinter(vs·∇)ˆΩ)·∂tˆΩ =−αinterω2+βinterω(q·vs) we arrive at\nαinter=1\n8πs0/summationdisplay\na,b/negationslash=a/summationdisplay\nk,a,b/vextendsingle/vextendsingle∝an}b∇acketle{tb,k+q|∆0S+|a,k∝an}b∇acket∇i}ht/vextendsingle/vextendsingle2Aa(k,ǫF)Ab(k+q,ǫF)\nβinter=1\n8πs0q·vs/summationdisplay\na,b/negationslash=a/summationdisplay\nk,a,b/vextendsingle/vextendsingle∝an}b∇acketle{tb,k+q|∆0S+|a,k∝an}b∇acket∇i}ht/vextendsingle/vextendsingle2Aa(k,ǫF)Ab(k+q,ǫF)Vb,a (52)\nin agreement with our results of Section II.\nIX. SUMMARY AND CONCLUSIONS\nStarting from the Gilbert damping αand including the\ninfluenceofanelectricfieldinthetransportorbitalssemi-\nclassically, we have proposed a concise formula for the\nnon-adiabatic spin transfer torque coefficient βthat can\nbe applied to real materials with arbitrary band struc-\ntures. Our formula for βreproduces results obtained\nby more rigorous non-linear response theory calculations\nwhen applied to simple toy models. By applying this ex-\npression to a two-dimensional electron-gas ferromagnet\nwith Rashba spin-orbit interaction, we have found that\nit implies a conductivity-like contributionto β, related to\nthecorrespondingcontributiontotheGilbertdamping α,\nwhich is proportionalto scattering time rather than scat-\ntering rate and arises from intra-band transitions. Our\nsubsequent calculations using a four-band model have\nshown that intra-band contributions dominate in ferro-\nmagnetic semiconductors such as (Ga,Mn)As. We have\nthen discussed the α/βratio in realistic materials and\nhave confirmed trends expected from toy models, in ad-\ndition to suggesting that αandβcan have the oppo-\nsite sign in systems where both hole-like and electron-like\nbands coexist at the Fermi surface. Afterwards, we have\nanalyzed the spin-torque formalism suitable to ab-initio\ncalculations, and have concluded that it may provide a\nreliable estimate of the intra-band contribution to β; for\nthe inter-band contribution the spin-torque formula of-fers a physically sensible result in the weak spin-orbit\nlimit but its quantitative reliability is questionable un-\nless the prominent inter-band transitions connect states\nthat are close in energy. Finally, we have extended the\nbreathing Fermi surface model for the Gilbert damping\nto current carrying ferromagnets and have accordingly\nfound a complementary physical interpretation for the\nintra-band contribution to β; similarly, we have applied\nthe master equation in order to offer an alternative inter-\npretation for the inter-band contribution to β. Possible\navenues for future research consist of carefully analyzing\nthe importance of higher order vertex corrections in β,\nbetter understanding the disparities between the differ-\nent approaches to vs, and finding real materials where\nα/βis negative.\nAcknowledgements\nWe acknowledge informative correspondence with\nRembert Duine and Hiroshi Kohno. In addition, I.G.\nis grateful to Paul Haney for interesting discussions and\ngenerous hospitality during his stay in the National In-\nstitute of Standards and Technology. This work was sup-\nported in part by the Welch Foundation, by the National\nScience Foundation under grant DMR-0606489, and by\nthe NIST-CNST/UMD-NanoCenter Cooperative Agree-\nment.\nAPPENDIX A: QUADRATIC SPIN RESPONSE TO AN ELECTRIC AND MAGNE TIC FIELD\nConsider a system that is perturbed from equilibrium by a time-depen dent perturbation V(t). The change in the\nexpectation value of an operator O(t) under the influence of V(t) can be formally expressed as\nδ∝an}b∇acketle{tO(t)∝an}b∇acket∇i}ht=∝an}b∇acketle{tΨ0|U†(t)O(t)U(t)|Ψ0∝an}b∇acket∇i}ht−∝an}b∇acketle{tΨ0|O(t)|Ψ0∝an}b∇acket∇i}ht (A1)15\nwhere|Ψ0∝an}b∇acket∇i}htis the unperturbed state of the system,\nU(t) =Texp/bracketleftbigg\n−i/integraldisplayt\n−∞V(t′)dt′/bracketrightbigg\n(A2)\nis the time-evolution operator in the interaction representation an dTstands for time ordering. Expanding the\nexponentials up to second order in Vwe arrive at\nδ∝an}b∇acketle{tO(t)∝an}b∇acket∇i}ht=i/integraldisplayt\n−∞dt′∝an}b∇acketle{t[O(t),V(t′)]∝an}b∇acket∇i}ht−1\n2/integraldisplayt\n−∞dt′dt′′∝an}b∇acketle{t[[O(t),V(t′)],V(t′′)]∝an}b∇acket∇i}ht. (A3)\nFor the present work, O(t)→Sa(a=x,y,z) and\nV(t) =−/integraldisplay\ndrj·A(r,t)+/integraldisplay\ndrS·Hext(r,t), (A4)\nwhereAis the vector potential, Hextis the external magnetic field, and jis the current operator. Plugging Eq. (A4)\ninto Eq. (A3) and neglecting O(A2),O(H2\next) terms we obtain\nδSa(x) =/summationdisplay\nb/integraldisplay\ndx′χa,b\nS,jAb(x′)+/summationdisplay\nb/integraldisplay\ndx′χa,b\nS,SHb\next(x′)+/summationdisplay\nb,c/integraldisplay\ndx′dx′′χa,b,c\nS,S,jAb(x′)Hc\next(x′′),(A5)\nwherex≡(r,t) and/integraltext\ndx′≡/integraltext∞\n−∞dt′/integraltext\ndr′. The linear and quadratic response functions introduced above ar e defined\nas\nχa,b\nS,j(x,x′) =i∝an}b∇acketle{t/bracketleftbig\nSa(x),jb(x′)/bracketrightbig\nΘ(t−t′)\nχa,b\nS,S(x,x′) =i∝an}b∇acketle{t/bracketleftbig\nSa(x),Sb(x′)/bracketrightbig\nΘ(t−t′)\nχa,b,c\nS,S,j(x,x′,x′′) =∝an}b∇acketle{t/bracketleftbig/bracketleftbig\nSa(x),jb(x′)/bracketrightbig\n,Sc(x′′)/bracketrightbig\nΘ(t−t′)Θ(t′−t′′)\n+∝an}b∇acketle{t/bracketleftbig/bracketleftbig\nSa(x),Sb(x′′)/bracketrightbig\n,jc(x′)/bracketrightbig\nΘ(t−t′′)Θ(t′′−t′) (A6)\nwhere we have used T[F(t)G(t′)] =F(t′)G(t′′)Θ(t′−t′′)+G(t′′)F(t′)Θ(t′′−t′), Θ being the step function. χS,jis the\nspin density induced by an electric field in a uniform ferromagnet, and it vanishes unless there is intrinsic spin-orbit\ninteraction. χS,Sis the spin density induced by an external magnetic field. χS,S,jis the spin density induced by the\ncombined action of an electric and magnetic field (see Fig. (8) for a dia grammatic representation); this quantity is\nclosely related to ( vs·q)χ(2), introduced in Section II.\nAPPENDIX B: FIRST ORDER IMPURITY VERTEX CORRECTION\nThe aim of this Appendix is to describe the derivation of Eq. (21). We s hall begin by evaluating the leading order\nvertex correction to the Gilbert damping. From there, we shall obt ain the counterpart quantity for the non-adiabatic\nSTT by shifting the Fermi occupation factors to first order in the e lectric field. The analytical expression for the\ntransverse spin response with one vertex correction is (see Fig. ( 9))\n˜χQP,(1)\n+,−=−V∆2\n0\n2T/summationdisplay\nωn/integraldisplay\nk,k′uiGa(iωn,k)S+\na,bGb(iωn+iω,k+q)Si\na,b′Gb′(iωn+iω,k′+q)S−\nb′,a′Ga′(iωn,k′)Si\na′,a.(B1)\nS+S−\nv.A\nFIG. 8: Feynman diagram for χS,S,j. The dashed lines correspond to magnons, whereas the wavy li ne represents a photon.16\nS+S−\nFIG. 9: Feynman diagram for the first order vertex correction . The dotted line with a cross represents the particle-hole\ncorrelation mediated by impurity scattering.\nwhereVis the volume of the system and the minus sign originates from fermion ic statistics. Using the Lehmannn\nrepresentation of the Green’s functions Gand performing the Matsubara sum we get\n˜χQP,(1)\n+,−=−V∆2\n0\n2/integraldisplay\nk,k′ui2 Re/bracketleftBig\nS+\na,bSi\nb,b′S−\nb′,a′Si\na′,a/bracketrightBig/integraldisplay∞\n−∞dǫ1dǫ′\n1dǫ2dǫ′\n2\n(2π)4Aa(ǫ1,k)Aa′(ǫ′\n1,k′)\n×Ab(ǫ2,k+q)Ab′(ǫ′\n2,k′+q)/bracketleftbiggf(ǫ1)\n(ǫ1−ǫ′\n1)(iω+ǫ1−ǫ2)(iω+ǫ1−ǫ′\n2)+/parenleftbigg\nǫ1↔ǫ2,ǫ′\n1↔ǫ′\n2,\nω↔ −ω/parenrightbigg/bracketrightbigg\n(B2)\nwhere twice the real part arose after absorbing two of the terms coming from the Matsubara sum. Next, we apply\niω→ω+i0+and take the imaginary part:\n˜χQP,(1)\n+,−=V∆2\n0\n22π/integraldisplay\nk,k′uiRe/bracketleftBig\nS+\na,bSi\nb,b′S−\nb′,a′Si\na′,a/bracketrightBig/integraldisplay∞\n−∞dǫ1dǫ′\n1dǫ2dǫ′\n2\n(2π)4Aa(ǫ1,k)Aa′(ǫ′\n1,k′)Ab(ǫ2,k+q)Ab′(ǫ′\n2,k′+q)\n×f(ǫ1)\nǫ1−ǫ′\n1/bracketleftbiggδ(ω+ǫ1−ǫ2)\nω+ǫ1−ǫ′\n2+δ(ω+ǫ1−ǫ′\n2)\nω+ǫ1−ǫ2−/parenleftbigg\nω→ −ω,\nq→ −q/parenrightbigg/bracketrightbigg\n(B3)\nwhere we used 1 /(x−iη) =PV(1/x) +iπδ(x), and invoked spin-rotational invariance to claim that terms with\nSx\na,bSi\nb,b′Sy\nb′,a′Si\na′,awill vanish. Integrating the delta functions we arrive at\n˜χQP,(1)\n+,−=V∆2\n0\n2/integraldisplay\nk,k′uiRe[...]/integraldisplay∞\n−∞dǫ′\n1dǫ2dǫ′\n2\n(2π)3f(ǫ2)Aa(ǫ2,k)Aa′(ǫ′\n1,k′)\n(ǫ2−ǫ′\n2)(ǫ2−ǫ′\n1)\n×/bracketleftbig\nAb(ǫ2+ω,k+q)Ab′(ǫ′\n2+ω,k′+q)+Ab(ǫ′\n2+ω,k+q)Ab′(ǫ2+ω,k′+q)/bracketrightbig\n−/parenleftbigg\nω→ −ω,\nq→ −q/parenrightbigg\n(B4)\nThe next step is to do the ǫ′\n1andǫ′\n2integrals, taking advantage of the fact that for weak disorder th e spectral\nfunctions are sharply peaked Lorentzians ( in fact at the present order of approximation one can take regard them as\nDirac delta functions). The result reads\n˜χQP,(1)\n+,−=V∆2\n0\n2/integraldisplay\nk,k′uiRe[...]/integraldisplay∞\n−∞dǫ2\n2πf(ǫ2)Aa(ǫ2,k)\nǫ2−ǫk′,a′/bracketleftbiggAb(ǫ2+ω,k+q)\nǫ2+ω−ǫk′+q,b′+Ab′(ǫ2+ω,k′+q)\nǫ2+ω−ǫk+q,b/bracketrightbigg\n−(ω→ −ω,q→ −q) (B5)\nBy making further changes of variables, this equation can be rewrit ten as\n˜χQP,(1)\n+,−=V∆2\n0\n2/integraldisplay\nk,k′uiRe[...]/integraldisplay∞\n−∞dǫ2\n2π(f(ǫ2)−f(ǫ2+ω))Aa(ǫ2,k)\nǫ2−ǫk′,a′/bracketleftbiggAb(ǫ2+ω,k+q)\nǫ2+ω−ǫk′+q,b′+Ab′(ǫ2+ω,k′+q)\nǫ2+ω−ǫk+q,b/bracketrightbigg\n(B6)\nThis is the first order vertex correction for the Gilbert damping. In order to obtain an analogous correction for the\nnon-adiabatic STT, it suffices to shift the Fermi factors in Eq. (B6) as indicated in the main text. This immediately\nresults in Eq. (21).\nAPPENDIX C: DERIVATION OF EQ. (26)\nLet us first focus on the first term of Eq. (17), namely\nEiqj/integraldisplay\nk/bracketleftbig\n|∝an}b∇acketle{ta,k|S+|b,k∝an}b∇acket∇i}ht|2+|∝an}b∇acketle{ta,k|S−|b,k∝an}b∇acket∇i}ht|2/bracketrightbig\nAaA′\nbvi\nk,avj\nk,bτk,a (C1)17\nWe shall start with the azimuthal integral. It is easy to showthat th e entire angle dependence comes from vivj∝kikj,\nfrom which the azimuthal integral vanishes unless i=j.\nRegarding the |k|integral, we assume that λkF,∆0,1/τ << ǫ F; otherwise the analytical calculation is complicated\nand must be tackled numerically. Such assumption allows us to use/integraltext\nk→N2D/integraltext∞\n−∞dǫ. For inter-band transitions\n(a∝ne}ationslash=b),AaA′\nbcontributes mainly thru the pole at ǫF,a, thus all the slowly varying factors in the integrand may be\nset at the Fermi energy. For intra-band transitions ( a=b),AaA′\nahas no peak at the Fermi energy; hence it is best\nto keep the slowly varying factors inside the integrand.\nThe above observations lead straightforwardly to the following res ult:\nEiqj/integraldisplay\nk/bracketleftbig\n|∝an}b∇acketle{ta,k|S+|b,k∝an}b∇acket∇i}ht|2+|∝an}b∇acketle{ta,k|S−|b,k∝an}b∇acket∇i}ht|2/bracketrightbig\nAaA′\nbvi\nk,avj\nk,bτk,a\n≃E·qm2\n8m+m−/parenleftbigg\n1+∆2\n0\nb2/parenrightbigg(ǫF,−τ−Γ+−ǫF,+τ+Γ−)\nb3\n−E·q/bracketleftbiggm2\nm2\n+1\n2λ2k2\nF\nb2/parenleftbigg\n1+∆2\n0\nb2/parenrightbigg\nτ2\n++m2\nm2\n−1\n2λ2k2\nF\nb2/parenleftbigg\n1+∆2\n0\nb2/parenrightbigg\nτ2\n−/bracketrightbigg\n(C2)\nThe second and third line in Eq. (C2) come from inter-band and intra- band transitions, respectively. The latter\nvanishes in absence of spin-orbit interaction, leading to a 2D version of Eq. (20). Since the band-splitting is much\nsmaller than the Fermi energy, one can further simplify the above e quation via τ+≃τ−→τ.\nLet us now move on the second term of Eq. (17), namely\nEiqj/integraldisplay\nkRe/bracketleftbig\n∝an}b∇acketle{tb,k|S−|a,k∝an}b∇acket∇i}ht∝an}b∇acketle{ta,k|S+∂kj|b,k∝an}b∇acket∇i}ht+(S+↔S−)/bracketrightbig\nAaAbvi\nk,aτk,a (C3)\nMost of the observations made above apply for this case as well. For instance, the azimuthal integral vanishes\nunlessi=j. This follows from a careful evaluation of the derivatives of the eige nstates with respect to momentum;\n∂kjθ= sin(θ)cos(θ)kj/k2(0≤θ≤π/2) is a useful relation in this regards, while ∂kjφplays no role. As for the |k|\nintegral, we no longer have the derivative of a spectral function, b ut rather a product of two spectral functions; the\nresulting integrals may be easily evaluated using the method of residu es. The final result reads\nEiqj/integraldisplay\nkRe/bracketleftbig\n∝an}b∇acketle{tb,k|S−|a,k∝an}b∇acket∇i}ht∝an}b∇acketle{ta,k|S+∂kj|b,k∝an}b∇acket∇i}ht+(S+↔S−)/bracketrightbig\nAaAbvi\nk,aτk,a\n≃ −E·q/bracketleftbiggm\n32m−λ2k2\nF∆2\n0\nb6/parenleftbigg\n1+τ−\nτ+/parenrightbigg\n+m\n32m+λ2k2\nF∆2\n0\nb6/parenleftbigg\n1+τ+\nτ−/parenrightbigg/bracketrightbigg\n+E·q/bracketleftbiggm\n4m+λ2k2\nF∆2\n0\nb4τ2\n++m\n4m−λ2k2\nF∆2\n0\nb4τ2\n−/bracketrightbigg\n(C4)\nThe first line in Eq. (C4) stems from inter-band transitions, wherea s the second comes from intra-band transitions;\nbothvanish in absence of SO. Once again we can take τ+≃τ−→τ. Combining Eqs. (C2) and (C4) one can\nimmediately reach Eq. (26).\n1L. Berger, J. Appl. Phys. 3, 2156 (1978); ibid.3, 2137\n(1979).\n2L. Berger, Phys. Rev. B 54, 9353 (1996).\n3J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n4H. Kubota, A. Fukushima, Y. Ootani, S. Yuasa, K.\nAndo, H. Maehara, K. Tsunekawa, D.D. Djayaprawira, N.\nWatanabe and Y. Suzuki, Jap. J. of Appl. Phys. 44, L1237\n(2005); J. Hayakawa, S. Ikeda, Y.M. Lee, R. Sasaki, T. Me-\nguro, F. Matsukura, H. Takahashi and H. Ohno, Jap. J.\nof Appl. Phys. 44, L1267 (2005); J. A. Katine and E. E.\nFullerton, J. Magn. Magn. Mater. 320, 1217 (2007).\n5For reviews of spin transfer torque in magnetic multilayerssee D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mater.\n320, 1190 (2007); M. D. Stiles and J. Miltat, Top. Appl.\nPhys.101, 225 (2006).\n6For reviews of spin transfer torque in continuously varying\nmagnetizations see P.M. Haney, R.A. Duine, A.S. Nunez\nand A.H. MacDonald, J. Magn. Magn. Mater. 320, 1300\n(2007); Y. Tserkovnyak, A. Brataas and G.E.W. Bauer,\nJ. Magn. Magn. Mater. 320, 1282 (2007); G. Tatara, H.\nKohno and J. Shibata, arXiv:0807.2894 (accepted to Phys.\nRep.).\n7M.D. Stiles and A. Zangwill, Phys. Rev. B 66,\n14407(2002); A. Shapiro, P. M. Levy, and S. Zhang, Phys.18\nRev. B, 67, 104430 (2003); J. Xiao, A. Zangwill, and M.\nD. Stiles, Phys. Rev. B 70, 172405 (2004); A. Brataas, G.\nE. W. Bauer, and P. J. Kelly, Phys. Rep. 427, 157 (2006).\n8A. S. Nunez and A. H. MacDonald, Solid State. Comm.\n139, 31 (2006).\n9S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n10J. Q. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B\n73, 054428 (2006).\n11M. Yamanouchi, D. Chiba, F. Matsukura and H. Ohno,\nPhys. Rev. Lett. 96, 96601 (2006).\n12Y. Tserkovnyak, H.J. Skadsem, A. Brataas and G.E.W\nBauer, Phys. Rev. B 74, 144405 (2006).\n13H. Kohno, G. Tatara and J. Shibata, J. Phys. Soc. Japan\n75, 113707 (2006).\n14R.A. Duine, A.S. Nunez, J. Sinova and A.H. MacDonald,\nPhys. Rev. B 75, 214420 (2007).\n15See for instance J. Kunes and V. Kambersky, Phys. Rev.\nB65212411 (2002) and references therein.\n16K. Gilmore, I. Garate, P.M. Haney, A.H. MacDonald and\nM.D. Stiles (in preparation).\n17I. Garate and A.H. MacDonald, arXiv:0808.1373.\n18I. Garate and A.H. MacDonald, arXiv:0808.3923.\n19Here we assume that the dependence of energy on mag-\nnetization direction which determines Heffis specified as\na function of Ω xand Ω yonly with Ω zimplicitly fixed by\nthe constraint Ω z= [1−Ω2\nx−Ω2\ny]1/2. If the free energy\nwas expressed in a form with explicit Ω zdependence we\nwould find Heff,x=−∂F/∂Ωx−(∂F/∂Ωz)(∂Ωz/∂Ωx) =\n−∂F/∂Ωx+(∂F/∂Ωz)Ωx, whereFis the free energy of the\nferromagnet. Similarly we would find Heff,y=−∂F/∂Ωy+\n(∂F/∂Ωz)Ωy. The terms which arise from the Ω zdepen-\ndence ofthe free energy would more commonly be regarded\nas contributions to Heff,z. The difference is purely a mat-\nter of convention since both results would give the same\nvalue for ˆΩ×Heff.\n20Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404\n(2002).\n21O. Gunnarsson, J. Phys. F 6, 587 (1976).\n22We assume that magnetic anisotropy and the external\nmagnetic fields are weak compared to the exchange-\ncorrelation splitting of the ferromagnet. ¯∆ is the spin-\ndensity weighted average of ∆( r) (see Ref. [17]).\n23For convenience in Eq. (8) we use /angbracketleftS+S−/angbracketrightresponse func-tions instead of /angbracketleftSxSx/angbracketrightand/angbracketleftSySy/angbracketright. They are related via\nSx= (S++S−)/2 andSy= (S+−S−)/2i.\n24J. Fernandez-Rossier, M. Braun, A. S. Nunez, A. H. Mac-\nDonald, Phys. Rev. B 69, 174412 (2004).\n25J.A.C. Bland and B. Heinrich (Eds.), Ultrathin Mag-\nnetic Structures III: Fundamentals of Nanomagnetism\n(Springer-Verlag, New York, 2005).\n26G. Tatara and P. Entel, Phys. Rev. B 78, 064429 (2008).\n27For a theoretical study on how Rashba spin-orbit interac-\ntion affects domain wall dynamics see K. Obata and G.\nTatara, Phys. Rev. B 77, 214429 (2008).\n28T. Jungwirth, J. Sinova, J. Masek, J. Kucera and A.H.\nMacDonald, Rev. Mod. Phys. 78, 809 (2006).\n29For actual ab-initio calculations it may be more con-\nvenient to substitute |/angbracketlefta,k|∆0S+|b,k/angbracketright|2in Eq. (35) by\n|/angbracketlefta,k|K|b,k/angbracketright|2, where Kis the spin-torque operator dis-\ncussed in Section VII. In either case we are disregarding\nimpurity vertex corrections, which may become significant\nin disordered and/or strongly spin-orbit coupled systems.\n30K.M.D. Hals, A.K. Nguyen and A. Brataas,\narXiv:0811.2235.\n31V. Kambersky, Phys. Rev. B 76, 134416 (2007); K.\nGilmore, Y.U. Idzerda and M.D. Stiles, Phys. Rev. Lett.\n99, 27204 (2007).\n32In order to gauge the accuracy of either matrix element,\none must obtain an exact evaluation of the non-adiabatic\nSTT, which entails a ladder-sum renormalization18ofS±.\nThis is beyond the scope of the present work.\n33S.E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204\n(2005).\n34K. Gilmore, Y.U. Idzerda and M.D. Stiles, J. Appl. Phys.\n103, 07D303 (2008).\n35D. Steiauf and M. Fahnle, Phys. Rev. B 72, 064450 (2005);\nD. Steiauf, J. Seib and M. Fahnle, Phys. Rev. B 78,\n02410(R) (2008).\n36This principle states that Wa,b=Wb,aexp((ǫa−ǫb)/T).\nSince the magnon energy is much smaller than the un-\ncertainty in the quasiparticle energies, we approximate\nWa,b≃Wb,a.\n37For an analogous observation in the context of electron-\nphonon interaction see e.g.D. Pines, Elementary Excita-\ntions in Solids (Benjamin, 1963)."    },    {        "title": "1907.04540v1.Temperature_dependence_of_magnetic_resonance_in_ferrimagnetic_GdFeCo_alloys.pdf",        "content": "1 \n Temperature dependence of magnetic resonance in ferrimagnetic \nGdFeCo alloys  \nTakaya  Okuno1, Se Kwon Kim2,3†, Takahiro  Moriyama1, Duck-Ho Kim1, Hayato  \nMizuno1,4, Tetsuya  Ikebuchi1, Yuushou  Hirata1, Hiroki  Yoshikawa5, Arata Tsukamoto5, \nKab-Jin Kim6, Yoichi  Shiota1, Kyung -Jin Lee7,8, Teruo Ono1,9† \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 611 -0011, Japan  \n2Department of Physics and Astronomy, University of California  Los Angeles, California \n90095, USA  \n3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, \nUSA \n4Institute for Solid State Physics, University of Tokyo, Kashiwa  277-8581 , Japan  \n5College of Science and Technology, Nihon University, Funabashi, Chiba 274 -8501, Japan  \n6Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea  \n7Department of Materials Science & Engineering, Korea University, Seoul 02841, \nRepublic of Korea  \n8KU-KIST Graduate School of Converging Science and Technology, Korea University, \nSeoul 02841, Republic of Korea  \n9Center for Spintronics Research Network (CSRN), Graduate School of Engineering \nScience, Osaka University, Osaka 560 -8531, Japan  \n†E-mail: kimsek@missouri.edu , ono@scl.kyoto -u.ac.jp  \n \n  2 \n We provide a macroscopic theory and experimental results for magnetic resonances of \nantiferromagnetically -coupled ferrimagnets. Our theory, which interpolates the dynami cs of \nantiferromag nets and ferromagnets smoothly, can describe ferrimagnetic resonance s across \nthe angular momentum compensation point . We also present experimental results for spin-\ntorque induced ferrimag netic resonance at several temperatures. The spectral analysis based \non our theory reveals that the Gilbert damping parameter, which has been considered to be \nstrongly temperature dependent, is insensitive to temperature. We env ision that our work \nwill facilitate further investigation of ferrimagnetic dynamics by providing a theoretical \nframework  suitable for a broad range of temperatures .  \n \n  3 \n Antiferromagnets have been gaining much attention in spintronics because of their \npotential utility for high -speed ultra -dense spintronic devices.1-4) Due to the antiparallel \nalignment of adjacent spins , their dynamics  is different  from that of ferromagnets.5) One \nemerging material platform for studying antiferromagnetic dynamics is \nantiferromagnetically -coupled ferrimagnets ,6-11) for which we can use conventional \ntechniques for ferromagnets owing to small but finite magnetizations . Indeed, r ecent \nexperiments in such ferrimagnets have found that both field -driven and current -driven \ndomain -wall dynamics are fastest at the angular momentum compensation point 𝑇A where \nthe magnetic dynamics are antiferromagnetic .12-15) However, the magnetic resonance \nphenomenon of ferrimagnets (FiMR) has not been fully clarified so far because of \ninsufficient experimental investigations. In the literature, Stanciu et al . have studied the \nlaser-induced precession and its decay to equilibrium in ferrimagnets and concluded that the \neffective Gilbert damping parameter 𝛼 , which governs the dissipation rate of angular \nmomentum,  is strongly temperature dependent and increases signif icantly  at 𝑇A.6) However, \nsome recent studies have provided a new perspective on 𝛼  of ferrimagnets by full y \nconsidering the antiferromagnetic dynamics in ferrimagnets . Kamra et al. have theoretically \nrevealed that the temperature dependence of FiMR occurs because of the temperature \ndependence of magnetic dynamics , not because of temperature dependence of  𝛼.16) Kim et \nal. have also reported the temperature -insensitive 𝛼 of ferrimagnets  through the  DW motion \nexperiment s.17) In this paper, we provide an additional evidence of temperature -insensitive \n𝛼 of ferrimagnets by performing the FiMR experiment analyzed by our macroscopic FiMR \ntheory.  \nFirst, we derive the equations for FiMR in a ferrimagnet consisting of two \nantiferromagnetically -coupled sublattices. Throughout the manuscript, we will focus on the \nregime where the ferrimagnet is away from the magnetization compensation temperature 𝑇M  \nso that th e magnetization is finite and well defined. Our experiments are also performed well \nwithin the considered regime  as detailed below.  To this end, we  expand the L andau -Lifshitz -\nGilbert -like (LLG -like) equation  for ferrimagnet films12,18-20) at the uniform ground state \nalong the positive in-plane z directio n to linear order in the small  fluctuations  |𝑛𝑥|,|𝑛𝑦|≪\n1, where the unit vector 𝒏 represents the N éel order parameter . The resultant equations are \ngiven by  \n𝑠net𝑛̇𝑥−𝛼𝑠total𝑛̇𝑦−𝜌𝑛̈𝑦=𝑀𝐻ext𝑛𝑦,     (1) 4 \n  \n𝑠net𝑛̇𝑦+𝛼𝑠total 𝑛̇𝑥+𝜌𝑛̈𝑥=−𝑀(𝐻ext+𝐻ani)𝑛𝑥,     (2) \n \nwhere 𝑠net=𝑠1−𝑠2 is the net spin density  of two sublattices  𝑠1>0 and 𝑠2>0, 𝛼 is the \nGilbert damping parameter , 𝑠total =𝑠1+𝑠2 is the sum of the magnitudes of the two spin \ndensities , 𝜌>0 is the  moment of inertia  for the dynamics  (which is inversely proportional \nto the microscopic exchange field between the two sublattices  and describes the \nantiferromagnetic dynamics of the magnet ),2) 𝐻ext is the external  field along the z direction,  \n𝐻ani  is the effective  anisotropy field along the x direction perpendicular to the film \n(including the effect of the demagnetizing field),  and 𝑀 is the magnetization . Here, we are \nneglecting the terms that are quadratic or higher order in 𝐻ext and the time derivative of the \norder parameter. The damping term is added by considering the Rayleigh dissipation \nfunction  𝑅=𝛼𝑠total ∫𝑑𝑉 𝒏̇2/2, which is the half of the ener gy dissipation rate through the \nmagnetic dynamics.20) Note that the Rayleigh function is defined in terms of 𝑠total, not in \nterms of 𝑠net, so that it is well defined even in the vicinity of 𝑇A where 𝑠net vanishes.17)  \nTo the zeroth -order in the damping paramete r 𝛼 , the resonance  frequencies for \nmonochromatic solutions to the above equations are given by  \n𝑓±2\n=𝑠net2+𝜌𝑀(2𝐻ext+𝐻ani)±√𝑠net4+2𝜌𝑀(𝑠net)2(2𝐻ext+𝐻ani)+𝜌2𝑀2𝐻ani2\n8𝜋2𝜌2,     (3) \n \nwhere 𝑓+ and 𝑓− are the frequencies for  higher and lower resonance  frequencies for the given \nfield. Far away from 𝑇A, where the  net spin density |𝑠net| is suf ficiently large, Eq. (1) and \nthe corresponding dynamics are dominated by the first -order time derivative term and thus \nwe can neglect the second -order term by setting 𝜌=0 . In that ferromagnetic limit, the \nexpression for the lower frequency is reduced to that for the ferromagnet resonance \nfrequency :21) \n         𝑓FiM= 𝑀\n2𝜋|𝑠net|√𝐻ext(𝐻ext+𝐻ani).     (4) \nNote that 𝑀/|𝑠net|  is the effective gyromagnetic ratio  𝛾eff  of the ferrimagnets. As the \ntemperature approaches 𝑇A , the net spin density | 𝑠net|  decreases and thus  the resonance \nfrequency  is expected to increase. However, this formula cannot be used in the vicinity of 5 \n 𝑇A, where 𝑠net vanishes and thus the second -order term cannot be neglected. Exactly at 𝑇A, \nthe net spin density vanishes 𝑠net=0, which reduces the obtained resonance  frequencies \n[Eq. (3)] to  \n   𝑓+= 1\n2𝜋√𝑀(𝐻ext+𝐻ani)\n𝜌,    𝑓−= 1\n2𝜋√𝑀𝐻ext\n𝜌.  (5) \nInclusion of the second -order time derivative term ∝𝜌 in the LLG -like equations [Eq. ( 1) \nand Eq. ( 2)] is necessary to obtain finite resonance  frequencies  at 𝑇A; otherwise, the LLG -\nlike equations lack in the reactive dynamic term ∝𝑠net at 𝑇A and becom e unable to describe \nthe ferrimagnetic dynamics properly therein.   \n Since our experimental results, which are presented below, are performed away from \n𝑇A, let us derive the resonance linewidth for ferrimagnets in the ferromagnetic regime. When \nwe include the Gilbert damping term, the resultant linewidth of  ferrimagnets ∆𝐻  (half -\nwidth -half-maximum) is given by  \n∆𝐻≈2𝜋𝛼\n𝛾eff 𝑠total\n|𝑠net| 𝑓FiM.     (6) \n \nTherefore 𝛼 in ferrimagnet is given by  \n𝛼FiM≈(𝛾eff\n2𝜋)|𝑠net|\n𝑠total (∆𝐻\n𝑓FiM).     (7) \nNote that b oth 𝑠total  and 𝑠net  appear in the lin ewidth expression because 1) the energy \ndissipation rate is proportional to 𝑠total since two lattices contribute additively and 2) the \nresonance frequency is inversely  proportional to 𝑠net. On the other hand, in conventional \nexpressions for ferromagnetic  resonance, the two spin -density parameters are assumed to be \nidentical , 𝑠total =𝑠net, and the corresponding expre ssion  𝛼FM≈(𝛾eff\n2𝜋)(∆𝐻\n𝑓FiM) was used to \nanalyze the magnetic resonance of ferrimagnets  in the previous reports.6,7) Below, these two \nexpressions for the Gilbert damping parameters, 𝛼FiM and 𝛼FM, will be compared based on  \nour experimental results.  \nWe experimentally  investigate d the FiMR in the GdFeCo compounds  by using  the \nhomodyne technique22-24) as shown in Fig 1 . For this study, we used a 5-nm SiN/10 -nm \nGd25.0Fe65.6Co9.4/5-nm Pt/100 -nm SiN /Si substrate  film. The film was patterned into a 10 -\nµm-wide  and 10 -µm-long strip pattern structure using optical lithography and Ar ion milling. \nA coplanar waveguide made of 100 -nm Au /5-nm Ti  were de posited at the ends of the strip.  6 \n The measurements were performed by sweeping an external magnetic field  𝐻ext at a fixed \nrf current  𝐼rf  (frequency 𝑓=4−18  GHz). 𝐻ext  was applied  in-plane  45°  away from the \nlong axis of the strip.  \nFigure 2a shows the FiMR  spectra at several temperatures 𝑇 between  220 K and  295 \nK. Although a single peak was clearly observed at 295 K, a second peak was also observed \nat 𝐻ext≈50 mT when 𝑇 is lower than 240 K. Note that the spontaneous magnetization lies \nin the sample plane at 𝑇=295  K while  it becomes perpendicular to the plane when 𝑇≤\n240  K . Thus, the two resonan ce peaks when 𝑇≤240  K  originat e from the magnetic \nresonance of perpendicular  (𝐻ext≈50 mT ) and in -plane (higher field) magnetization s, \nrespectively . Here we focus on the resonan ce peak originating from in -plane magnetization, \nso we cut off the low -field regime to exclude the resonance peak  from perpendicular \nmagnetization  and fit those spectra in Fig. 2a by the combination of symmetric and anti -\nsymmetric Lorentzian  function s, from which the resonan ce parameters are obtained .22,23)  \nFigure s 2b and 2c  show the resonance frequency  𝑓res as a function of  the resonance  \nfield 𝐻res and the spectral linewidth  ∆𝐻 (half -width -half-maximum) as a function of 𝑓res, \nrespectively . Firstly, we analyze th ese data using the conventional expressions of \nferromagnetic resonance,21,25) \n𝑓res=𝑔eff𝜇B\nℎ√𝐻res(𝐻res+𝐻ani),     (8) \n \n∆𝐻=𝛼FM\n(𝑔eff𝜇Bℎ⁄)𝑓res+∆𝐻0.     (9) \n \nHere, 𝑔eff  is the  effective  Landé  g-factor, 𝜇B  is the Bohr magneton,  ℎ  is the Planck ’s \nconst ant, 𝐻ani is the effe ctive  anisotropy field including  the demagnetization field, 𝛼FM is \nthe effective Gilbert damping parameter  defined as in Ref. 6, and ∆𝐻0  is a frequency -\nindependent linewidth known as the inhomogeneous broadening, which originates from \nmagnetic non -uniformity.25) Equation (8) can be matched with Eq. (4) once we identify \n𝑔eff𝜇B/ℏ as the effective gyrom agnetic ratio  𝑀/|𝑠net| (ℏ=ℎ2𝜋⁄ is the reduced Planck ’s \nconstant ) and 𝐻res  as 𝐻ext . The 𝐻res  vs 𝑓res  shown in Fig. 2b are well fitted b y Eq. (8), \nindicated by the solid lines, and  𝑔eff and 𝐻ani are obtained as the fitting parameters. Figures \n3a and 3b show 𝑔eff and 𝐻ani as a function of 𝑇, respectively.  It is found that 𝑔eff remarkably \nincreases as 𝑇 decreases. Since the 𝑇A of the device is estimated  to be 160 K (see below the 7 \n estimation method) , the result shows that 𝑔eff increases as 𝑇 approaches 𝑇A. Note that the \ndrastic decrease in 𝐻ani with decreasing 𝑇 (Fig. 3b) is attributed to the change in magnetic \nanisotropy from in -plane (295 K) to perpendicular (220 K) direction as mentioned above. \nThe 𝑓res vs ∆𝐻 show n in Fig. 2c are well fitted by Eq. (9), indicated by the solid lines, and  \n𝛼FM and ∆𝐻0 are obtained as the fitting parameters. Figures 3c and 3d show 𝛼FM and ∆𝐻0 \nas a function of 𝑇, respectively.  It is found that 𝛼FM increases significantly as 𝑇 decreases, \ni.e. as 𝑇 approaches 𝑇A. The 𝑇 dependences of 𝑔eff and 𝛼FM are in good agreement with the \nprevious paper s.6,7,2 7) According to the previous paper s,6,7) the 𝑇  dependences of 𝑔eff  and \n𝛼FM are understood in terms of that of the net angular momentum 𝑠net; both 𝑔eff𝜇Bℏ⁄=\n𝑀net 𝑠net⁄  and 𝛼FM [from Eq. (9)] are ill -defined  at 𝑇A where 𝑠net vanishes , which makes \nthe theory based on ferromagnets invalid therein . However, as shown in the discussion of \nour theory  for FiMR , by defining the Gilbert damping parameter in the Rayleigh dissipation \nfunction 𝑅=𝛼𝑠total ∫𝑑𝑉 𝒏̇2/2 in such a way that the damping parameter is always well -\ndefined, the resonance frequency  and the li newidth of FiMR can be described properly \nacross 𝑇A . In order to test whether our theory can explain the experimental results, we \nanalyze those data in Figs. 2b and 2c based on our theory.  \n As mentioned in the theory part, the ferrimagnetic resonance frequency  in the \nferromagnetic limit is reduced to the conventional ferromagnetic case, while the spectral \nlinewidth is modified by including the additional term 𝑠net 𝑠total⁄  . Therefore, the Gilb ert \ndamping parameter [Eq. (7)] in our theory [Eq. (1) and Eq. (2)] for the dynamics of \nferrimagnets can be obtained by the following expression:  \n𝛼FiM=𝛼FM|𝑠net\n𝑠total|.     (10) \nTo obtain 𝛼FiM from Eq. (7) and Fig. 2c, 𝑠net 𝑠total⁄  needs to be acquired.  Although the net \nspin density 𝑠net is easy to obtain from the effective gyromagnetic ratio, the total spin density \n𝑠total  is not straightforward to obtain.  To solve this problem , we perform the following \nanalysis . The effective net gyromagnetic ratio satisfies the following equation;6,7) \n𝑔eff𝜇B\nℏ=𝑀net\n𝑠net=𝑀FeCo −𝑀Gd\n𝑀FeCo\n(𝑔FeCo 𝜇Bℏ⁄)−𝑀Gd\n(𝑔Gd𝜇Bℏ⁄).     (11) \n \nHere, 𝑀FeCo (𝑀Gd) is the magnetizations of transition metal (rare -earth metal), and 𝑔FeCo \n(𝑔Gd) is the Landé  g-factor  of transition metal (rare -earth metal) sublattice. 𝑔eff is shown in 8 \n Fig. 3a, 𝑔FeCo  and 𝑔Gd  are obtained from literature  (𝑔FeCo ~2.2  and 𝑔Gd~2.0 ).28-30) Two \nquantities can be measured directly : 𝑀net is independently  measured  by SQUID  as shown \nin Fig. 4a  and 𝑠net  can be obtained from the effective gyromagnetic ratio  when the \nferrimagnet is well within the ferromagnetic regime.  With the measured values of 𝑀net  = \n𝑀FeCo −𝑀Gd and 𝑠net= 𝑀FeCo\n(𝑔FeCo 𝜇Bℏ⁄)−𝑀Gd\n(𝑔Gd𝜇Bℏ⁄), we can obtain the magnetizations of two \nsublattices,  𝑀FeCo  and 𝑀Gd , and also the spin densities of two sublattices,  𝑠FeCo  and 𝑠Gd .  \nFrom these results, we can obtain the total spin density 𝑠total =𝑠FeCo +𝑠Gd. Figures 4a and \n4b show 𝑀FeCo and 𝑀Gd, and 𝑠net=𝑠FeCo −𝑠Gd and 𝑠total =𝑠FeCo +𝑠Gd as a function of \n𝑇, respectively . Note that the 𝑇M (110 K ) determined by SQUID ( Fig. 4a ) and the 𝑇A (160 \nK) roughly estimated from the 𝑇  dependence of  𝑠net  (Fig. 4b) are clearly different,12,31) \nwhich supports the validity of this analysis. Finally, by substituting 𝑠net and 𝑠total into Eq. \n(10), the damping parameter  𝛼FiM is obtained as shown in Fig. 4c. It can be clearly seen  that \n𝛼FiM(≈0.01) is insensitive to 𝑇, in sharp contrast to 𝛼FM which significantly increases as 𝑇 \napproaches 𝑇A. Note that Eq. (10) is valid only in the ferromagnetic limit and, therefore,  it \nis necessary to confirm that the measured temperature range ( 220 – 295 K) is in the deep \nferromagne tic regime.  This would be guaranteed by the facts that 1) Fig. 4b shows \n𝑇A~160  K , which is  far below the lowest 𝑇  in our measurement s (220 K ), and 2) the \nresonance frequency  at 𝑇A is expected to  be similar to or larger than about 50  GHz  under \n300 mT ,6) which is much larger than the experimentally obtained resonance frequency  at 220 \nK (12 GHz under 300 mT).  \nThe observation that 𝛼FiM is insensitive to 𝑇 indicates that the 𝑇 dependence of the \nspectral linewidth in FiMR is attributed to the 𝑇  dependence of the net spin density 𝑠net \ninstead of that of the effective Gilbert damping parameter. This conclusion is consistent with \nsome recent papers,16,17) but it is in sharp contrast to the interpretation of the previous \nreport s,6,7) where the 𝑇 dependence of the spectral linewidth in FiMR was attributed to the \nchange of the effective Gilbert damping parameter. Our results provide an additional and \nclear evidence that properly defined Gilbert damping parameter 𝛼FiM  of ferrimagnets  is \ninsensitive to temperature, supporting the validity of these papers .16,17) Here, we would like \nto mention that, e ven though Fig. 4c is an evidence for the temperature -insensitive 𝛼FiM of \nferrimagnets, it lacks the information of 𝛼FiM in the vicinity of 𝑇A. However, obtaining  𝛼FiM \nin the vicinity of 𝑇A based on FiMR  experiment s is challenging  because 1) experimental \nobservation of ferrima gnetic resonance at 𝑇A which was measured to be larger than 5 0 GHz 9 \n for certa in ferrimagnets6) is expected to be difficult with homodyne detection technique (40 \nGHz at maximum in our measurement system) and 2) obtaining the necessary parameter \n𝑠total  is difficult because the net spin density 𝑠net  cannot be obtained from the effective \ngyromagnetic ratio in the vicinity of 𝑇A. Therefore, we believe that Fig. 4c serves as a good \nexperimental evidence to conclude that 𝛼FiM of ferrimagnets is insensitive to temperature.  \nIn conclusion, we have provided the macroscopic theoretical description of \nferrimagnetic resonance and experimental results that support it. Our theory shows that the \nresonance frequency  and the spectral linewidth of ferrimagnetic resonance can be described  \nwell across the angular momentum compensation point , by adding the antiferromagnetic -\nlike inertial term to the equations of motion and by defining the Gilbert damping parameter \nproperly through the Rayleigh dissi pation function. Moreover, we performed the spin -torque \ninduced ferrimagnetic resonances at various temperatures and successfully observed that the \nresonance frequency  and the linewidth depend on temperature. By analyzing the spectrum \nbased on our theory, we found that the Gilbert damping parameter in ferrimagnets is \ninsensitive to temperature, which has been considered to be strongly temperature -dependent. \nOur work introduces a new framework for studying ferrimagnetic resonance that allows us \nto interpret the ferrimagnetic dynamics for a wide range of temperatures .  \n \nAcknowledgments  \nThis work was supported by the JSPS KAKENHI (Grants  No. 15H05702,  No. 17H04924 , \nNo. 17H05181 , No. 26103002, and No. 26103004),  Collaborative Research Program of the \nInstitute for  Chemical Research, Kyoto University, and R & D project  for ICT Key \nTechnology of MEXT from the Japan Society  for the Promotion of Science (JSPS). This \nwork was  partly supported by The Cooperative Research Project  Program of the Researc h \nInstitute of Electrical  Communication, Tohoku University. S. K. K. w as supported by the \nstartup fund at the University of Missouri . D. H. K. was supported  as an Overseas \nResearcher under the Postdoctoral Fellowship  of JSPS (Grant No. P16314). K. J. L. w as \nsupported by the National Research Foundation of Korea  (2017R1A2B2006119) . K. J. K.  \nwas supported by  the KAIST -funded Global Singularity Research Program for 2019 .  10 \n  \nReferences  \n1) T. Jungwirth , X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol.  11, 231 \n(2016).  \n2) V . Baltz, A. Manchon , M. Tsoi , T. Moriyama , T. Ono , and Y . Tserkovnyak , Rev. Mod. \nPhys.  90, 015005 (2018).  \n3) J. Železn ý, P. Wadley, K. Olejník, A. Hoffmann, and H. Ohno , Nat. Phys.  14, 220 \n(2018).  \n4) M. B.  Jungfleisch, W. Zhang,  and A.  Hoffmann, Phys. Lett. A  382, 865 (2018).  \n5) O. Gomonay, V . Baltz, A. Brataas,  and Y . Tserkovnyak, Nat. Phys . 14, 213 (2018).  \n6) C. D.  Stanciu, A. V . Kimel, F. Hansteen, A. Tsukamoto, A. Itoh, A. Kirilyuk, and Th. \nRasing , Phys. Rev. B  73, 220402(R) (2006).  \n7) M. Binder, A. Weber, O. Mosendz, G. 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B  97, \n220403(R) (2018).  \n \n  12 \n  \nFigure Captions  \nFig. 1.  The schematic illustration of the device and the measurement  setup . The direction \nof the external magnetic field  𝐻ext and the AC current  𝐼rf are indicated. 𝐻ext was applied  \nin-plane  45° away from the long axis of the strip . \n \nFig. 2.  (a) The ferrimagnetic resonance spectra as a function of the external magnetic field \n𝐻ext at several temperatures from 220 -295 K. The emerging peak at 𝐻ext≈50 mT below \n240 K is attributed to the out -of-plane resonan ce peak and are neglected in this study. (b) \nThe resonance frequency  𝑓res as a function of the resonance  magnetic field  𝐻res. The solid \nlines are the fitting results by Eq. (8). (c) The spectral linewidth ∆𝐻 as a function of 𝑓res. \nThe solid lines are the fitting results by Eq. (9).  \n \nFig. 3. Resonance parameters as a function of temperature extracted by the fitting in Fig. \n(2). (a) The effective Landé  g-factor  𝑔eff. (b) The effective  anisotropy field 𝐻ani. (c) The \neffective Gilbert damping parameter 𝛼FM. (d) The f requency -independent linewidth ∆𝐻0. \n \nFig. 4. (a) The net magnetization 𝑀net and the magnetizations  of two sublattices  𝑀FeCo and \n𝑀Gd as function s of temperature . (b) The net spin density  𝑠net, the spin densities of two \nsublattices 𝑠FeCo and 𝑠Gd, and the sum of the magnitudes of the two spin densit ies 𝑠total as \nfunction s of temperature . (c) The effective Gilbert damping parameter 𝛼FM and the \nproperly defined Gilbert damping parameter of ferrimagnets  𝛼FiM as function s of \ntemperature . \n \n \n \n  13 \n  \nFig. 1 . \n  \n14 \n Fig. 2.  \n  \n0 5 10 15 200204060 295 K\n 260 K\n 240 K\n 220 KDH [mT]\nfres [GHz]数式 y = a + b*x\n重み 機械的\n残差平方和 6.89093\nピ ア ソ ン の r 0.99995\n--\n補正Ｒ 二乗 0.99989\n値 標準誤差\nΔH切片 38.04028 0.57739\n傾き 18.31533 0.07106\n数式 y = a + b*x\n重み 機械的\n残差平方和 3.53715\nピアソンの r 0.99996\n--\n補正Ｒ 二乗 0.9999\n値 標準誤差\nΔH切片 47.96301 0.69263\n傾き 21.8284 0.08274\n数式 y = a + b*x\n重み 機械的\n残差平方和 2.37794\nピ ア ソ ン の r 0.99992\n--\n補正Ｒ 二乗 0.9998\n値 標準誤差\nΔH切片 53.44711 1.23634\n傾き 24.77597 0.13079\n0 100 200 300 400 50005101520\n 295 K\n 260 K\n 240 K\n 220 Kfres [GHz]\nHres [mT]\n050100150200\n f = 4 GHz\n 6 GHz\n 10 GHz\n 14 GHz\n 18 GHzT = 295 K\n050100150200\nT = 280 K\n050100150200\nT = 260 K\n050100150200\nT = 240 K\n050100150200\nT = 230 K\n0 100 200 300 400 500 600 700 800050100150200V [mV]\nHext [mT]T = 220 K\n(a) \n(c) \n(b) 15 \n Fig. 3.  \n \n  \n220 240 260 280 3004681012DH0 [mT]\nT [K]\n220 240 260 280 300-120-80-400Hani [mT]\nT [K]\n220 240 260 280 3000.050.100.150.20\naFM\nT [K]\n220 240 260 280 3002.02.53.03.54.04.5geff\nT [K]\n(a) \n(c) \n(b) \n(d) 16 \n Fig. 4.  \n \n \n \n100 150 200 250 30001234Spin density [ ´10-6 J s/m3]\nT [K] snet\n sFeCo\n sGd\n stotal\nTA~160K\n0 50 100 150 200 250 300-101234Magnetization [ ´105 A/m]\nT [K] Mnet (by SQUID)\n MFeCo\n MGd\nTM » 110 K\n150 200 250 3000.000.050.100.150.20\na\nT [K] aFM\n aFiM\nTA~160K\n(a) \n(c) \n(b) "    },    {        "title": "1605.06578v1.Landau_Lifshitz_theory_of_the_magnon_drag_thermopower.pdf",        "content": "Landau-Lifshitz theory of the magnon-drag thermopower\nBenedetta Flebus,1, 2Rembert A. Duine,1, 3and Yaroslav Tserkovnyak2\n1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,\nUtrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands\n2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n3Department of Applied Physics, Eindhoven University of Technology,\nPO Box 513, 5600 MB, Eindhoven, The Netherlands\nMetallic ferromagnets subjected to a temperature gradient exhibit a magnonic drag of the electric\ncurrent. We address this problem by solving a stochastic Landau-Lifshitz equation to calculate the\nmagnon-drag thermopower. The long-wavelength magnetic dynamics result in two contributions to\nthe electromotive force acting on electrons: (1) An adiabatic Berry-phase force related to the solid\nangle subtended by the magnetic precession and (2) a dissipative correction thereof, which is rooted\nmicroscopically in the spin-dephasing scattering. The \frst contribution results in a net force pushing\nthe electrons towards the hot side, while the second contribution drags electrons towards the cold\nside, i.e., in the direction of the magnonic drift. The ratio between the two forces is proportional\nto the ratio between the Gilbert damping coe\u000ecient \u000band the coe\u000ecient \fparametrizing the\ndissipative contribution to the electromotive force.\nThe interest in thermoelectric phenomena in ferromag-\nnetic heterostructures has been recently revived by the\ndiscovery of the spin Seebeck e\u000bect [1]. This e\u000bect is now\nunderstood to stem from the interplay of the thermally-\ndriven magnonic spin current in the ferromagnet and the\n(inverse) spin Hall voltage generation in an adjacent nor-\nmal metal [2]. Lucassen et al. [3] subsequently proposed\nthat the thermally-induced magnon \row in a metallic\nferromagnet can also produce a detectable (longitudinal)\nvoltage in the bulk itself, due to the spin-transfer mecha-\nnism of magnon drag. Speci\fcally, smooth magnetization\ntexture dynamics induce an electromotive force [4], whose\nnet average over thermal \ructuations is proportional to\nthe temperature gradient. In this Letter, we develop\na Landau-Lifshitz theory for this magnon drag, which\ngeneralizes Ref. [3] to include a heretofore disregarded\nBerry-phase contribution. This additional magnon drag\ncan reverse the sign of the thermopower, which can have\npotential utility for designing scalable thermopiles based\non metallic ferromagnets.\nElectrons propagating through a smooth dynamic tex-\nture of the directional order parameter n(r;t) [such that\njn(r;t)j\u00111, with the self-consistent spin density given\nbys=sn] experience the geometric electromotive force\nof [4]\nFi=~\n2(n\u0001@tn\u0002@in\u0000\f@tn\u0001@in) (1)\nfor spins up along nand\u0000Fifor spins down. The resul-\ntant electric current density is given by\nji=\u001b\"\u0000\u001b#\nehFii=~P\u001b\n2ehn\u0001@tn\u0002@in\u0000\f@tn\u0001@ini;(2)\nwhere\u001b=\u001b\"+\u001b#is the total electrical conductivity, P=\n(\u001b\"\u0000\u001b#)=\u001bis the conducting spin polarization, and eis\nthe carrier charge (negative for electrons). The averaging\nh:::iin Eq. (2) is understood to be taken over the steady-\nstate stochastic \ructuations of the magnetic orientation.The latter obeys the stochastic Landau-Lifshitz-Gilbert\nequation [5]\ns(1 +\u000bn\u0002)@tn+n\u0002(Hz+h) +X\ni@iji= 0;(3)\nwhere\u000bis the dimensionless Gilbert parameter [6], H\nparametrizes a magnetic \feld (and/or axial anisotropy)\nalong thezaxis, and ji=\u0000An\u0002@inis the magnetic spin-\ncurrent density, which is proportional to the exchange\nsti\u000bnessA. ForH > 0, the equilibrium orientation is\nn!\u0000 z, which we will suppose in the following. The\nLangevin \feld stemming from the (local) Gilbert damp-\ning is described by the correlator [7]\nhhi(r;!)h\u0003\nj(r0;!0)i=2\u0019\u000bs~!\u000eij\u000e(r\u0000r0)\u000e(!\u0000!0)\ntanh~!\n2kBT(r);\n(4)\nupon Fourier transforming in time: h(!) =R\ndtei!th(t).\nAt temperatures much less than the Curie tempera-\nture,Tc, it su\u000eces to linearize the magnetic dynamics\nwith respect to small-angle \ructuations. To that end, we\nswitch to the complex variable, n\u0011nx\u0000iny, parametriz-\ning the transverse spin dynamics. Orienting a uniform\nthermal gradient along the xaxis,T(x) =T+x@xT,\nwe Fourier transform the Langevin \feld (4) also in real\nspace, with respect to the yandzaxes. Linearizing\nEq. (3) for small-angle dynamics results in the Helmholtz\nequation:\nA(@2\nx\u0000\u00142)n(x;q;!) =h(x;q;!); (5)\nwhere\u00142\u0011q2+[H\u0000(1+i\u000b)s!]=A,h\u0011hx\u0000ihy, and qis\nthe two-dimensional wave vector in the yzplane. Solving\nEq. (5) using the Green's function method, we substitute\nthe resultant ninto the expression for the charge current\ndensity (2), which can be appropriately rewritten in thearXiv:1605.06578v1  [cond-mat.mes-hall]  21 May 20162\nfollowing form (for the nonzero xcomponent):\njx=~P\u001b\n2eZd2qd!\n(2\u0019)3!\n\u0002Re(1 +i\f)hn(x;q;!)@xn\u0003(x;q0;!0)i\n(2\u0019)3\u000e(q\u0000q0)\u000e(!\u0000!0): (6)\nTedious but straightforward manipulations, using the\ncorrelator (4), \fnally give the following thermoelectric\ncurrent density:\njx=\u000bsP\u001b@xT\n4eA2kBT2Zd2qd!\n(2\u0019)3(~!)3\nsinh2~!\n2kBTRe [(1 +i\f)I];\n(7)\nwhereI(\u0014)\u0011\u0014=j\u0014j2(Re\u0014)2, having made the convention\nthat Re\u0014>0.\nTo recast expression (7) in terms of magnon modes, we\nincorporate the integration over qxby noticing that, in\nthe limit of low damping, \u000b!0,\nI=2\n\u0019Z\ndqx1 +iq2\nx=\u000b~!\n(~!\u0000q2x\u0000q2\u0000\u0018\u00002)2+ (\u000b~!)2:(8)\nHere, we have introduced the magnetic exchange length\n\u0018\u0011p\nA=H and de\fned ~ !\u0011s!=A . After approximating\nthe Lorentzian in Eq. (8) with the delta function when\n\u000b\u001c1, Eq. (7) can \fnally be expressed in terms of a\ndimensionless integral\nJ(a)\u0011Z1\na=p\n2dxx5p\n2x2\u0000a2\nsinh2x2; (9)\nas\nj=\u0012\n1\u0000\f\n3\u000b\u0013\nJ\u0012\u0015\n\u0018\u0013kBP\u001b\n\u00192e\u0012T\nTc\u00133=2\nrT: (10)\nHere,Tis the ambient temperature, kBTc\u0011A(~=s)1=3\nestimates the Curie temperature, and \u0015\u0011p\n~A=skBT\nis the thermal de Broglie wavelength in the absence of\nan applied \feld. We note that \u000b;\f\u001c1 while\u000b\u0018\f, in\ntypical transition-metal ferromagnets [8].\nFor temperatures much larger than the magnon gap\n(typically of the order of 1 K in metallic ferromagnets),\n\u0015\u001c\u0018and we can approximate J(\u0015=\u0018)\u0019J(0)\u0018\n1. This limit e\u000bectively corresponds to the gapless\nmagnon dispersion of \u000fq\u0011~!q\u0019~Aq2=s. Within\nthe Boltzmann phenomenology, the magnonic heat cur-\nrent induced by a uniform thermal gradient is given by\njQ=\u0000rTR\n[d3q=(2\u0019)3](@qx!q)2\u001c(!q)\u000fq@TnBE, where\n\u001c\u00001(!q) = 2\u000b!qis the Gilbert-damping decay rate of\nmagnons (to remain within the consistent LLG phe-\nnomenology) and nBE= [exp(\u000fq=kBT)\u00001]\u00001is the Bose-\nEinstein distribution function. By noticing that\n\u000fq@TnBE=kB\u0014~!q=2kBT\nsinh( ~!q=2kBT)\u00152\n; (11)\nrThydrodynamic\nr⌦<0geometricˆxˆzˆy\n⌦\ne\u0000e\u0000e\u0000e\u0000e\u0000e\u0000e\u0000e\u0000\nFIG. 1. Schematics for the two contributions to the electron-\nmagnon drag. In the absence of decay (i.e., \u000b!0), magnons\ndrifting from the hot (left) side to the cold (right) side drag\nthe charge carriers viscously in the same direction, inducing\na thermopower /\f. The (geometric) Berry-phase drag gov-\nerned by the magnon decay is proportional to \u000band acts in\nthe opposite direction. It is illustrated for a spin wave that is\nthermally emitted from the left. As the spin wave propagates\nto the right, the solid angle \n subtended by the spin preces-\nsion shrinks, inducing a force oriented to the left for spins\nparallel to n.\nit is easy to recast the second, /\fcontribution to\nEq. (10) in the form\nj(\f)=\f~P\u001b\n2eAjQ; (12)\nwhich reproduces the main result of Ref. [3].\nThe magnon-drag thermopower (Seebeck coe\u000ecient),\nS=\u0000@xV\n@xT\f\f\f\f\njx=0; (13)\ncorresponds to the voltage gradient @xVinduced under\nthe open-circuit condition. We thus get from Eq. (10):\nS=\u0012\f\n3\u000b\u00001\u0013\nJkBP\n\u00192e\u0012T\nTc\u00133=2\n= (\f\u00003\u000b)~P\u0014m\n2eA;\n(14)\nwhere\u0014m= (2=3\u00192)JkBA(T=Tc)3=2=\u000b~is the magnonic\ncontribution to the heat conductivity. Such magnon-drag\nthermopower has recently been observed in Fe and Co\n[9], with scaling/T3=2over a broad temperature range\nand opposite sign in the two metals. Note that the sign\ndepends on \f=\u000b and the e\u000bective carrier charge e.\nEquations (10) and (14) constitute the main results of\nthis paper. In the absence of Gilbert damping, \u000b!0,\nthe magnon-drag thermopower Sis proportional to the\nheat conductivity. This contribution was studied in\nRef. [3] and is understood as a viscous hydrodynamic\ndrag. In simple model calculations [8], \fP > 0 and this\nhydrodynamic thermopower thus has the sign of the ef-\nfective carrier charge e. WhenP > 0, so that the ma-\njority band is polarized along the spin order parameter3\nn, the/\u000bcontribution to the thermopower is opposite\nto the/\fcontribution. (Note that \u000bis always>0,\nin order to yield the positive dissipation.) The underly-\ning geometric meaning of this result is sketched in Fig. 1.\nNamely, the spin waves that are generated at the hot end\nand are propagating towards the cold end are associated\nwith a decreasing solid angle, @x\n<0. The \frst term\nin Eq. (1), which is rooted in the geometric Berry con-\nnection [10], is proportional to the gradient of this solid\nangle times the precession frequency, /!@i\n, resulting\nin a net force towards the hot side acting on the spins\ncollinear with n.\nNote that we have neglected the Onsager-reciprocal\nbackaction of the spin-polarized electron drift on the\nmagnetic dynamics. This is justi\fed as including the\ncorresponding spin-transfer torque in the LLG equation\nwould yield higher-order e\u000bects that are beyond our\ntreatment [11]. The di\u000busive contribution to the See-\nbeck e\u000bect,/T=EF, whereEFis a characteristic Fermi\nenergy, which has been omitted from our analysis, is ex-\npected to dominate only at very low temperatures [9].\nThe conventional phonon-drag e\u000bects have likewise been\ndisregarded. A systematic study of the relative impor-\ntance of the magnon and phonon drags is called upon in\nmagnetic metals and semiconductors.\nThis work is supported by the ARO under Contract\nNo. 911NF-14-1-0016, FAME (an SRC STARnet center\nsponsored by MARCO and DARPA), the Stichting voor\nFundamenteel Onderzoek der Materie (FOM), and the\nD-ITP consortium, a program of the Netherlands Orga-\nnization for Scienti\fc Research (NWO) that is funded by\nthe Dutch Ministry of Education, Culture, and Science\n(OCW).[1] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455,\n778 (2008); K. Uchida, J. Xiao, H. Adachi, J. Ohe,\nS. Takahashi, J. Ieda, T. Ota, Y. Kajiwara, H. Umezawa,\nH. Kawai, G. E. W. Bauer, S. Maekawa, and E. Saitoh,\nNat. Mater. 9, 894 (2010).\n[2] J. Xiao, G. E. W. Bauer, K. Uchida, E. Saitoh, and\nS. Maekawa, Phys. Rev. B 81, 214418 (2010).\n[3] M. E. Lucassen, C. H. Wong, R. A. Duine, and\nY. Tserkovnyak, Appl. Phys. Lett. 99, 262506 (2011).\n[4] R. A. Duine, Phys. Rev. B 77, 014409 (2008);\nY. Tserkovnyak and M. Mecklenburg, ibid.77, 134407\n(2008).\n[5] S. Ho\u000bman, K. Sato, and Y. Tserkovnyak, Phys. Rev. B\n88, 064408 (2013).\n[6] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[7] W. F. Brown, Phys. Rev. 130, 1677 (1963).\n[8] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J.\nMagn. Magn. Mater. 320, 1282 (2008).\n[9] S. J. Watzman, R. A. Duine, Y. Tserkovnyak, H. Jin,\nA. Prakash, Y. Zheng, and J. P. Heremans, \\Magnon-\ndrag thermopower and Nernst coe\u000ecient in Fe and Co,\"\narXiv:1603.03736.\n[10] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984);\nG. E. Volovik, J. Phys. C: Sol. State Phys. 20, L83\n(1987); S. E. Barnes and S. Maekawa, Phys. Rev. Lett.\n98, 246601 (2007); Y. Tserkovnyak and C. H. Wong,\nPhys. Rev. B 79, 014402 (2009).\n[11] The backaction by the spin-transfer torque would be ab-\nsent when the longitudinal spin current, ji=\u001b(PE i+\nFi=e)n, vanishes, where Eiis the electric \feld and Fi\nis the spin-motive force (1). Understanding Eq. (10) as\npertaining to the limit of the vanishing spin current ji\nrather than electric current ji=\u001b(Ei+PFi=e)nwould,\nhowever, result in higher-order (in T=T c) corrections to\nthe Seebeck coe\u000ecient (13). These are beyond the level\nof our approximations."    },    {        "title": "1501.07731v1.Head_to_Head_Domain_Wall_Structures_in_Wide_Permalloy_Strips.pdf",        "content": "Head-to-Head Domain Wall Structures in Wide Permalloy Strips\nVirginia Est\u0013 evez\u0003and 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.\nWe analyze the equilibrium micromagnetic domain wall structures encountered in Permalloy strips\nof a wide range of thicknesses and widths, with strip widths up to several micrometers. By per-\nforming an extensive set of micromagnetic simulations, we show that the equilibrium phase diagram\nof the domain wall structures exhibits in addition to the previously found structures (symmetric\nand asymmetric transverse walls, vortex wall) also double vortex and triple vortex domain walls\nfor large enough strip widths and thicknesses. Also several metastable domain wall structures are\nfound for wide and/or thick strips. We discuss the details of the relaxation process from random\nmagnetization initial states towards the stable domain wall structure, and show that our results are\nrobust with respect to changes of e.g. the magnitude of the Gilbert damping constant and details\nof the initial conditions.\nPACS numbers: 75.60.Ch, 75.78.Cd\nI. INTRODUCTION\nDuring the last decade, a lot of e\u000bort has been de-\nvoted to understand static and dynamic properties of\nmagnetic domain walls (DWs) in ferromagnetic nanos-\ntructures such as nanowires and -strips. These studies\nhave been largely driven by promising technological ap-\nplications based on domain walls and their dynamics, in\nparticular memory1,2and logic devices3{5. In typical ex-\nperiments DWs are driven by either applied magnetic\n\felds6,7or spin-polarized electric currents8{11. The re-\nsulting DW dynamics depends crucially on the micro-\nmagnetic DW structure, typically involving various in-\nternal degrees of freedom. These are essential e.g. for\nthe emergence of the Walker breakdown12, an instability\noccurring when the DW internal degrees of freedom get\nexcited by a strong external drive (a magnetic \feld H\nor a spin-polarized current Jexceeding the Walker \feld\nHWor currentJW, respectively), limiting the propaga-\ntion velocity of the DWs.\nTwo main classes of ferromagnetic materials have been\nextensively studied within the strip geometry. Ma-\nterials with a high perpendicular magnetic anisotropy\n(PMA13{16) exhibit simple and narrow DWs of the\nBloch and/or N\u0013 eel type. For H > H WorJ > J W,\nrepeated transitions between these two structures are\nobserved16. The second class of systems includes soft\n(low anisotropy) magnetic materials13,14such as Permal-\nloy, where in-plane domain magnetization along the long\naxis of the strip is induced by shape anisotropy. By us-\ning various experimental techniques17,18and micromag-\nnetic simulations, it has been established that the equi-\nlibrium DW structures separating these in-plane domains\nare more complex, and depend crucially on the sample\ngeometry19{22. Transverse DWs (TWs) and asymmetric\ntransverse DWs (ATWs) are observed for narrow and\nthin strips22{27, while in wider and thicker strips one\nencounters the vortex DW (VW)22,24{26,28,29. In addi-\ntion, various metastable DW structures with higher en-ergy may be found18,28{31. ForH >H WorJ >J W, the\nDW structures exhibit dynamical evolution: for TWs,\nrepeated nucleation and propagation of an antivortex\nacross the strip width takes place19. Similarly, in VWs\nthe vortex core performs oscillatory back and forth per-\npendicular motion19.\nIn Permalloy strips with even larger widths and/or\nthicknesses, one might expect also other, possibly more\ncomplicated equilibrium DW structures. For wider strips\nshape anisotropy is less important, implying that energy\nminima with more complex spin structures closing the\n\rux more e\u000eciently than TWs, ATWs or VWs may ap-\npear. Indeed, e.g. double and triple vortex DWs have\nbeen observed in experiments on wide strips28, but they\nhave been attributed to current-induced vortex nucle-\nation resulting in metastable DW structures. Conse-\nquently, a pertinent and fundamental question is what\nare the possible intermediate equilibrium DW structures\nobservable when the lateral Permalloy strip dimensions\nincrease from those corresponding to the typical nanos-\ntrip geometry (with TW, ATW or VW as the stable DW\nstructure) to strip widths of micrometers and beyond.\nIn this paper we present an extensive numerical study\nof the equilibrium and metastable micromagnetic DW\nstructures in Permalloy strips, with the strip widths up to\nan order of magnitude larger than before19{24. Contrary\nto previous studies focusing on comparing the energies\nof di\u000berent a priori known DW structures19{24, we per-\nform micromagnetic simulations of relaxation dynamics\nfrom random initial states towards the stable DW struc-\ntures. In addition to the previously observed TW, ATW\nand VW DWs, we \fnd also DWs with equilibrium double\nand triple vortex structures for wide and/or thick enough\nstrips. The last structure is encountered only in the very\nlargest system sizes we were able to simulate. Moreover,\nfor wide strips we \fnd a rich variety of metastable DWs\nwith even more complex micromagnetic structures. We\ndemonstrate that our results are robust with respect to\nchanges of the magnitude of the Gilbert damping con-arXiv:1501.07731v1  [cond-mat.mes-hall]  30 Jan 20152\nlength lwidth wthickness Δz\nlr(a)\n(b)\nFIG. 1. (color online) (a) Geometry of the Permalloy strip.\n(b) A top view of the magnetization in the initial state. Mag-\nnetization points along the long axis of the strip within the\ntwo domains (as indicated by the arrows) forming a head-\nto-head con\fguration. In between them, a region of random\nmagnetization (of length lr) has been included.\nstant or using di\u000berent initial conditions for the relax-\nation process. Our results underline the crucial role of\ntopological defects for physics of DWs in soft strips, and\nthat of micromagnetic simulations for \fnding the true\nequilibrium DW structure.\nII. MICROMAGNETIC SIMULATIONS\nThe system studied is a Permalloy strip of width wand\nthickness \u0001 z, satisfying \u0001 z\u001cw, see Fig. 1 (a). In the\nmicromagnetic simulations, magnetic charges are com-\npensated on the left and right ends of the strip, to mimic\nan in\fnitely long strip; the actual simulated length sat-\nis\fesl\u00154wfor all cases considered. The initial state\nfrom which the relaxation towards a stable DW struc-\nture starts is an in-plane head-to-head domain struc-\nture, with a region of random magnetization of length\nlrin the middle of the sample, see Fig. 1 (b). If not\nspeci\fed otherwise, we consider lr= 2w. Material pa-\nrameters of Permalloy are used, i.e. saturation magne-\ntizationMs= 860\u0002103A/m and exchange constant\nAex= 13\u000210\u000012J/m. The typical Gilbert damping\nconstant for Permalloy is \u000b= 0:01, but here we analyze\nalso the in\ruence of \u000bon the relaxation process, and thus\nconsider also other values. For simplicity, we set the tem-\nperatureTto zero, and focus on the ideal case of strips\nfree of any structural disorder or impurities.\nThe simulations are performed using the GPU-\naccelerated micromagnetic code MuMax332{34, o\u000bering\na signi\fcant speedup as compared to CPU codes for the\nlarge system sizes we consider here. To calculate the mag-\nnetization dynamics of the system, the Landau-Lifshitz-\nGilbert equation35,36,\n@m=@t=\rHe\u000b\u0002m+\u000bm\u0002@m=@t; (1)\nis solved numerically. Here, mis the magnetization, \r\nthe gyromagnetic ratio, and He\u000bthe e\u000bective \feld, with\n0 ns 0.2 ns\n1.0 ns 2.0 ns\n3.0 ns 4.0 ns\n6.0 ns 12.0 ns\nFIG. 2. (color online) An example of the temporal evolution\nof the relaxation, with w= 420 nm, \u0001 z= 10 nm,\u000b= 3 and\nlr= 2w. Relaxation towards the equilibrum DW structure\n(here, a VW) takes place via coarsening dynamics of the de-\nfect structure in the magnetic texture. The colorwheel in the\nmiddle shows the mapping between magnetization directions\nand colors.\ncontributions due to exchange, Zeeman, and demagne-\ntizing energies. The size of the discretization cell used\ndepends on the system size, but is always bounded by\nthe exchange length, \u0003 = (2 A=\u0016 0M2\ns)1=2\u00195 nm, in the\nin-plane directions, and equals \u0001 zin the the out-of-plane\ndirection.\nIII. RESULTS\nWe start by considering the e\u000bect of varying \u000bandlr\non the relaxation process. Fig. 2 shows an example of\nthe time evolution of m(r;t) forw= 420 nm, \u0001 z= 10\nnm,\u000b= 3 andlr= 2w. The initially random magne-\ntization evolves via coarsening of the defect structure of\nthe magnetization texture towards the stable DW (here,\na VW). During the relaxation, the total energy Eof the\nsystem decreases in a manner that for a given geometry\n(wand \u0001z) depends on both \u000bandlr, see Fig. 3 (a) and\n(b) wherew= 5120 nm and \u0001 z= 20 nm, is considered.\nFor instance, Edecreases faster for an intermediate \u000b\n[Fig. 3 (a)]. We attribute this behavior to the balance\nbetween inertial e\u000bects related to precession favored by\na small\u000b, helping to overcome energy barriers, and the\nhigher rate of energy dissipation due to a large \u000b. Thus,\nthe relaxation time to reach a (meta)stable DW struc-\nture depends on \u000b. Fig. 3 (b) illustrates that for a \fxed\n\u000b, systems with a larger lrrelax more slowly. Fig. 3\n(c) shows that on average, the early-time relaxation of\nEtowards its \fnal value Efexhibits temporal power-law\ndecay,hE\u0000Efi/t\u0000\fwith\f\u00191:3 for the\u000b= 0:3 case\nshown, possibly related to collective e\u000bects due to inter-\nactions between several topological defects during early\nstages of relaxation (Fig. 2).\nIn general, the \fnal (meta)stable DW structure may3\n02×10-9\nt [s]03×10-136×10-139×10-13E [J]α = 3\nα = 0.3\nα = 0.03\n02×10-9\nt [s]03×10-136×10-13\nE [J]lr = w\nlr = 2w\nlr = 3w\nlr = 2w(a) (b)\nα = 0.3\n10-1010-910-8\nt [s]10-310-210-1100<E(t) - Ef> / <E(0)>\nlr = w\nlr = 2w\nlr = 3w\nt-1.3(c)\nα = 0.3\nFIG. 3. (color online) The energy E(t) as a function of time\ntforw= 5120 nm and \u0001 z= 20 nm. (a) For di\u000berent values\nof\u000bandlr= 2w. (b) For di\u000berent values of lr, and\u000b= 0:3\n[resulting in the fastest relaxation in (a)]. (c) shows that on\naverage, the early time decay of E(t) towards its \fnal value\nEfobeyshE(t)\u0000Efi/t\u0000\f. For the\u000b= 0:3 case shown here,\n\f\u00191:3. Empty (\flled) symbols in (c) correspond to w= 420,\n\u0001z= 10 nm (w= 860, \u0001 z= 20 nm).\ndepend on the realization of the random initial state.\nThus, we consider 21 realizations of the initial random\nmagnetization for each wand \u0001z, and compare the en-\nergies of the resulting relaxed con\fgurations. The struc-\nture with the lowest energy is chosen as the equilibrium\nstructure, while others with higher energy are metastable\nstates. Although, as discussed above, the relaxation\ntimes depend on \u000bandlr, the equilibrium DW struc-\nture is found to be independent of \u000bandlrin the range\nconsidered, i.e. \u000b2[0:01;3] andlr2[w;3w]. Thus, in\nwhat follows, we will use \u000b= 3 andlr= 2w.\nThe main results of this paper are summarized in\nFigs. 4 and 5, showing the phase diagram of the equi-\nlibrium DW structures for wranging from 120 to 5120\nnm, and \u0001 zfrom 5 to 25 nm, and examples of these\nstructures, respectively. For small w, we recover the pre-\nvious results19{22, i.e. phases correspoding to TW, ATW\nand VW, shown in Fig. 5 (a), (b) and (c), respectively.\nFor larger strip widths ( wapproaching or exceeding\n1\u0016m, depending on \u0001 z, see Fig. 4), a new equilibrium mi-\ncromagnetic DW structure, a double vortex wall (DVW),\nis observed. This structure consists of two vortices with\nopposite sense of rotation of the magnetization around\nthe vortex core, see Fig. 5 (d). At the phase boundary\n(blue triangle symbols pointing left in Fig. 4), VW and\nDVW have the same energy. The DVW phase spans a\nrelatively large area within the ( w;\u0001z) space, highlight-\ning the robustness of our results.\nIn addition, a second new phase, with a triple-vortex\nwall (TVW) as the equilibrium structure [see Fig. 5 (e)],\n100 1000\nw [nm]510152025t [nm]V ortex wall\nTransverse wallDouble vortex wall\nAsymmetric transverse wallTriple vortex wallΔzFIG. 4. (color online) Phase diagram of the equilibrium\nDW structure in Permalloy strips of various thicknesses (from\n\u0001z= 5 to 25 nm) and widths ranging from w= 120 nm up to\n5120 nm. The symbols correspond to observations of the vari-\nous equilibrium DW structures, with phase boundaries shown\nas solid lines. Examples of the DW structures corresponding\nto the 5 di\u000berent phases are shown in Fig. 5.\nTransverse Wall\nAsymmetric Transverse Wall V ortex Wall\nDouble V ortex Wall Triple V ortex Wall(a)\n(b) (c)\n(d) (e)\nFIG. 5. (color online) Examples of the di\u000berent equilibrium\nmicromagnetic DW structures: (a) TW for w= 120 nm and\n\u0001z= 5 nm, (b) ATW for w= 160 nm and \u0001 z= 10 nm, (c)\nVW forw= 640 nm and \u0001 z= 15 nm, (d) DVW for w= 2560\nnm and \u0001 z= 20 nm, and (e) TVW for w= 5120 nm and\n\u0001z= 25 nm. The colorwheel (top left) shows the mapping\nbetween magnetization directions and colors.\nis found for the very largest system sizes we have been\nable to simulate. The middle vortex of the TVW has an\nopposite sense of rotation to the other two. For w= 5120\nand \u0001z= 25 nm, DVW and TVW have the same energy\n(the cyan square symbol in the top right corner of Fig. 4),\nsuggesting the presence of a phase boundary between the\ntwo structures. Indeed, by performing a set of 10 ad-\nditional simulations with w= 6144 and \u0001 z= 25 nm\n(i.e. outside the phase diagram in Fig. 4), suggests that\nTVW is the equilibrium DW structure for very large strip\nwidths. This structure has been observed in experiments4\n(a) (b)\n(c) (d)\n(e) (f)\n(g) (h)2V+A V\n3V+A V\n4V+A V\n4V+3A V3V+A V\n3V+2A V\n4V+2A V\n5V+2A V\nFIG. 6. (color online) Examples of metastable DW structures\nobserved for a system with w= 5120 nm and di\u000berent thick-\nnesses \u0001z: (a) Two vortices and an antivortex (2V+AV),\n\u0001z= 25 nm, (b) and (c) three vortices and an antivortex\n(3V+AV), \u0001 z= 5 nm, (d) three vortices and two antivortices\n(3V+2AV), \u0001 z= 10 nm, (e) four vortices and an antivortex\n(4V+AV), \u0001 z= 5 nm, (g) four vortices and three antivor-\ntices (4V+3AV), \u0001 z= 5 nm, and (h) \fve vortices and two\nantivortices (5V+2AV), \u0001 z= 5 nm.\nas a metastable state for smaller systems28,29. Notice also\nthat the middle part of the TVW [Fig. 5 (e)], exhibiting\nfour line-like 90\u000eDWs meeting at a vortex core in the\nmiddle of the TVW, resembles the typical Landau \rux-\nclosure magnetization patterns observed for rectangular\nPermalloy thin \flms37{39.\nFollowing the relaxation from a random magnetization\ninitial state, the system may in general end up into var-\nious metastable states with higher energy than that of\nthe equilibrium DW. Sometimes these metastable states\nhave even a higher probability than the equilibrium one,\nestimated here from the sample of 21 relaxed con\fgura-\ntions. Fig. 6 shows some of the metastable states found\nfor a large strip with w= 5120 nm and di\u000berent values of\n\u0001z; for strips with smaller lateral dimensions, di\u000berent\nmetastable states tend to be less numerous and have a\nsimpler structure. Despite their apparent complexity, all\nthe metastable DW structures shown in Fig. 6 respect the\nbasic principles of topology of DWs. Each of the DWs\nare composed of topological defects, with an associated\nwinding number: +1 for vortices, -1 for antivortices, and\n\u00061=2 for edge defects40. In a DW all the topological\ndefects have to be compensated, i.e. the total winding\nnumber is equal to zero. In the case of the DVW, the\ntwo topological vortex defects are compensated by four\nedge defects [Fig. 5 (d)]. For the metastable state of twovortices (with the same sense of rotation) and an antivor-\ntex [2V+AV, see Fig. 6 (a)], there are two vortices and\nonly two edge defects. Thus, in order to compensate the\ntopological defects, also an antivortex appears. In gen-\neral, we have observed that in a DW with Nvortices with\nthe same sense of rotation, there must be N\u00001 antivor-\ntices to get a zero total winding number, see Fig. 6 (a)\nand (d) for examples with 2V+AV and 3V+2AV con\fg-\nurations, respectively. When some of the vortices have\noppposite sense of rotation, more complex scenarios are\nencountered, with examples shown in Figs. 6 (b), (c), (e),\n(f), (g) and (h). Notice also that two DW structures with\nthe same elements can look very di\u000berent, see e.g. the\ntwo 3V+AV DWs shown in Figs. 6 (b) and (c). All the\nDW structures found, both the equilibrium ones in Fig. 5\nand the metastable states in Fig. 6 obey the principle of\ncompensation of topological defects to yield a total wind-\ning number of zero. The richness of the equilibrium phase\ndiagram and the large collection of metastable states in-\ndicate that for wide/thick strips in particular, the micro-\nmagnetic energy landscape is quite complex, with a large\nnumber of local minima. This is also in agreement with\nour observations of power-law energy relaxation.\nIV. SUMMARY AND CONCLUSIONS\nTo summarize, we have performed an extensive set\nof micromagnetic simulations to study the equilibrium\nand metastable DW structures in Permalloy strips of a\nwide range of widths and thicknesses, as well as the re-\nlaxation dynamics starting from random magnetization\ninitial states. The general trend of our results is that\nboth the equilibrium and metastable DW con\fgurations\nbecome increasingly complex (i.e. they consist of an in-\ncreasing number of topological defects) as the lateral strip\ndimensions increase. We note that somewhat analogous\nbehaviour - i.e. existence of equilibrium magnetization\ncon\fgurations with increasing complexity as the system\nsize increases - is observed also in some other systems\nsuch as three-dimensional cylindrical elements with per-\npendicular anisotropy41,42.\nSeveral remarks are in order: \frst, for strips with\neven larger lateral dimensions one may in principle ex-\npect more complex DW patterns - possibly with four or\nmore vortices with alternating sense of rotation. These,\nhowever, are currently beyond the reach of our available\ncomputing resources. Second, our phase diagram allows\none to check if experimental observations of the various\nDW structures in wider strips are equilibrium con\fgura-\ntions or metastable states. According to our review of\nthe experimental literature, most observations of DVWs\nand TVWs appear to be metastable states28,29. Third,\nwhile the equilibrium structures we \fnd are certainly sta-\nble in the absence of external perturbations such as ap-\nplied magnetic \felds, it remains to be seen how their \feld\ndriven dynamics is like, and whether wide strips with a\nrelatively weak shape ansitropy are able to support the5\nDWs as compact objects also when external perturba-\ntions are being applied43.ACKNOWLEDGMENTS\nWe thank Mikko J. Alava for a critical reading of\nthe manuscript. This work has been supported by\nthe Academy of Finland through its Centres of Excel-\nlence Programme (2012-2017) under project no. 251748,\nand an Academy Research Fellowship (LL, project no.\n268302). We acknowledge the computational resources\nprovided by the Aalto University School of Science\n\\Science-IT\" project, as well as those provided by CSC\n(Finland).\n\u0003virginia.esteveznuno@aalto.\f\n1S. S. P. Parkin, M. Hayashi and L. Thomas, Science 320,\n190 (2008).\n2S. E. Barnes, J. Ieda and S. Maekawa, Appl. Phys. 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Fullerton,3and A. D. Kent1\n1Department of Physics, New York University, 4 Washington Pl ace, New York, New York 10003, USA\n2Institut d’Electronique Fondamentale, UMR CNRS 8622,\nUniversit ´eParis Sud, 91405 Orsay Cedex, France\n3Center for Magnetic Recording Research, University of Cali fornia,\nSan Diego, La Jolla, California 92093-0401, USA\n(Dated: June 15, 2009)\nTransition metal ferromagnetic films with perpendicular ma gnetic anisotropy (PMA) have ferro-\nmagnetic resonance (FMR) linewidths that are one order of ma gnitude larger than soft magnetic\nmaterials, such as pure iron (Fe) and permalloy (NiFe) thin fi lms. A broadband FMR setup has\nbeen used to investigate the origin of the enhanced linewidt h in Ni |Co multilayer films with PMA.\nThe FMR linewidth depends linearly on frequency for perpend icular applied fields and increases sig-\nnificantly when the magnetization is rotated into the film pla ne. Irradiation of the film with Helium\nions decreases the PMA and the distribution of PMA parameter s. This leads to a great reduction\nof the FMR linewidth for in-plane magnetization. These resu lts suggest that fluctuations in PMA\nlead to a large two magnon scattering contribution to the lin ewidth for in-plane magnetization and\nestablish that the Gilbert damping is enhanced in such mater ials (α≈0.04, compared to α≈0.002\nfor pure Fe).\nPACS numbers: 75.47.-m,85.75.-d,75.70.-i,76.50.+g\nMagnetic materials with perpendicular magnetic\nanisotropy (PMA) are of great interest in information\nstorage technology, offering the possibility of smaller\nmagnetic bits [1] and more efficient magnetic random ac-\ncess memories based on the spin-transfer effect [2]. They\ntypically are multilayers of transition metals (e.g., Co |Pt,\nCo|Pd, Ni |Co) with strong interface contributions to the\nmagnetic anisotropy [3], that render them magnetically\nhard. In contrast to soft magnetic materials which have\nbeen widely studied and modeled [4, 5, 6, 7], such films\nare poorly understood. Experiments indicate that there\nare large distributions in their magnetic characteristics ,\nsuch as their switching fields [1]. An understanding of\nmagnetization relaxation in such materials is of particu-\nlar importance, since magnetization damping determines\nthe performance of magnetic devices, such as the time-\nscale for magnetization reversal and the current required\nfor spin-transfer induced switching [2, 8].\nFerromagnetic resonance (FMR) spectroscopy pro-\nvides information on the magnetic damping through\nstudy of the linewidth of the microwave absorption peak,\n∆H, when the applied field is swept at a fixed microwave\nfrequency. FMR studies of thin films with PMA show\nvery broad linewidths, several 10’s of mT at low frequen-\ncies (/lessorsimilar10 GHz) for polycrystalline alloy [9], multilayer\n[10] and even epitaxial thin films [11]. This is at least one\norder of magnitude larger than the FMR linewidth found\nfor soft magnetic materials, such as pure iron (Fe) and\npermalloy (FeNi) thin films [5]. Further, it has recently\nbeen suggested that the FMR linewidth of perpendicu-\nlarly magnetized CoCrPt alloys cannot be explained in\nterms of Landau-Lifshitz equation with Gilbert damping\n[12], the basis for understanding magnetization dynamicsin ferromagnets:\n∂M\n∂t=−γµ0M×Heff+α\nMsM×∂M\n∂t. (1)\nHereMis the magnetization and γ=|gµB//planckover2pi1|is the gy-\nromagnetic ratio. The second term on the right is the\ndamping term, where αis the Gilbert damping constant.\nThis equation describes precessional motion of the mag-\nnetization about an effective field Heff, that includes the\napplied and internal (anisotropy) magnetic fields, which\nis damped out at a rate determined by α. The absorp-\ntion linewidth (FWHM) in a fixed frequency field-swept\nFMR experiment is given by µ0∆H= 4παf/γ , i.e., the\nlinewidth is proportional to the frequency with a slope\ndetermined by α. This is the homogeneous or intrinsic\ncontribution to the FMR linewidth. However, experi-\nments show an additional frequency independent contri-\nbution to the linewidth:\n∆H= ∆H0+4πα\nµ0γf, (2)\nwhere ∆H0is referred as the inhomogeneous contribution\nto the linewidth.\nThe inhomogeneous contribution is associated with\ndisorder. First, fluctuations in the materials magnetic\nproperties, such as its anisotropy or magnetization, lead\nto a linewidth that is frequency independent; in a simple\npicture, independent parts of the sample come into res-\nonance at different applied magnetic fields. Second, dis-\norder can couple the uniform precessional mode ( k= 0),\nexcited in an FMR experiment, to degenerate finite- k\n(k/negationslash= 0) spin-wave modes. This mechanism of relaxation\nof the uniform mode is known as two magnon scattering2\n/s48 /s51/s48 /s54/s48 /s57/s48/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48 /s49/s40/s97/s41/s40/s98/s41\n/s72\n/s72/s32/s61/s57/s48/s111/s86/s105/s114/s103/s105/s110\n/s73/s114/s114/s97/s100/s105/s97/s116/s101/s100\n/s32/s32\n/s70/s105/s101/s108/s100/s32/s97/s110/s103/s108/s101/s32\n/s72/s32/s32/s40/s100/s101/s103/s46/s41/s50/s48/s32/s71/s72/s122/s32/s72\n/s114/s101/s115/s32/s32/s40/s32/s84/s32/s41/s120/s121/s122\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s48/s52/s48/s54/s48\n/s77\n/s115\n/s72/s32/s32/s102/s32/s32/s40/s71/s72/s122/s41\n/s48/s72\n/s114/s101/s115/s32/s32/s32/s32/s40/s32/s84/s32/s41/s32/s48/s72\n/s32/s32/s40/s84/s41/s32/s70/s77 /s82/s32/s115/s105/s103/s110/s97/s108\nFIG. 1: a) The frequency dependence of the resonance field\nwith the applied field perpendicular to the film plane. The\nsolid lines are fits using Eq. 4. Inset: FMR signal of the\nvirgin and irradated films at 21 GHz. b) The resonance field\nas a function of applied field angle at 20 GHz. The solid lines\nare fits to the experimental data points. The inset shows the\nfield geometry.\n(TMS) [13]. TMS requires a spin-wave dispersion with\nfinite-kmodes that are degenerate with the k= 0 mode\nthat only occurs for certain magnetization orientations.\nIn this letter we present FMR results on ultra-thin\nNi|Co multilayer films and investigate the origin of\nthe broad FMR lines in films with PMA. Ni |Co mul-\ntilayers were deposited between Pd |Co|Pd layers that\nenhance the PMA and enable large variations in the\nPMA with Helium ion irradiation [14]. The films are\n|3nm Ta |1nm Pd |0.3nm Co |1nm Pd |[0.8nm Ni |0.14nm\nCo]×3|1nm Pd |0.3nm Co |1nm Pd |0.2nm Co |3nm Ta |de-\nposited on a Si-SiN wafers using dc magnetron sputter-\ning and were irradiated using 20 keV He+ions at a flu-\nence of 1015ions/cm2. The He+ions induce interatomic\ndisplacements that intermix the Ni |Co interfaces lead-\ning to a reduction of interface anisotropy and strain in\nthe film. The magnetization was measured at room tem-\nperature with a SQUID magnetometer and found to be\nMs≃4.75×105A/m.\nFMR studies were conducted from 4 to 40 GHz at room\ntemperature with a coplanar waveguide as a function of\nthe field angle to the film plane. The inset of Fig. 1b\nshows the field geometry. The parameters indexed with\n‘⊥’ (perpendicular) and ‘ /bardbl’ (parallel) refer to the applied\nfield direction with respect to the film plane. The absorp-\ntion signal was recorded by sweeping the magnetic field\nat constant frequency [15]. FMR measurements were per-\nformed on a virgin film (not irradiated) and on an irra-\ndiated film.\nFig. 1a shows the frequency dependence of the reso-\nnance field when the applied field is perpendicular to the\nfilm plane. The x-intercept enables determination of the\nPMA and the slope is proportional to the gyromagneticratio. We take a magnetic energy density:\nE=−µ0M·H+1\n2µ0M2\nssin2φ\n−(K1+ 2K2)sin2φ+K2sin4φ.(3)\nThe first term is the Zeeman energy, the second the mag-\nnetostatic energy and the last two terms include the first\nand second order uniaxial PMA constants, K1andK2.\nTakingµ0Heff=−δE/δMin Eq. 1 the resonance con-\ndition is:\nf=γ\n2π/parenleftbigg\nµ0H⊥\nres−µ0Ms+2K1\nMs/parenrightbigg\n. (4)\nFrom thex-intercepts in Fig. 1a, K1= (1.93±0.07)×\n105J/m3for the virgin film and (1 .05±0.02)×105J/m3\nfor the irradiated film; Helium irradiation reduces the\nmagnetic anisotropy by a factor of two. Note that in\nthe irradiated film the x-intercept is positive ( µ0Ms>\n2K1/Ms). This implies that the easy magnetization di-\nrection is in the film plane. The angular dependence of\nHres(Fig. 1b) also illustrates this: the maximum res-\nonance field shifts from in-plane to out-of-plane on ir-\nradiation. The gyromagnetic ratio is not significantly\nchangedγ= 1.996±0.009×10111/(Ts) for the vir-\ngin film and γ= 1.973±0.004×10111/(Ts) for the\nirradiated film (i.e., g= 2.24±0.01). The second order\nanisotropy constant K2was obtained from the angular\ndependence of the resonance field, fitting HresversusφH\nfor magnetization angle φbetween 45oand 90o. For the\nvirgin film, K2= 0.11×105J/m3. Note that when K2\nis set to zero, χ2of the fit increases by a factor 30. For\nthe irradiated film, K2= 0.03×105J/m3. HenceK2de-\ncreases upon irradiation and remains much smaller than\nK1. The solid line in Fig. 1b is the resulting fit. When\nthe field approaches the in-plane direction, the measured\nresonance field is higher than the fit. The shift is of the\norder of 0.1 T for the virgin film and 0 .025 T for the ir-\nradiated film. It is frequency dependent: increasing with\nfrequency. This shift will be discussed further below.\nFig. 2a shows the frequency dependence of the\nlinewidth (FWHM) for two directions of the applied field.\n/s48/s49/s48/s48/s50/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48/s48/s53/s48/s49/s48/s48/s72/s32\n/s124/s124\n/s32/s32/s32 /s72/s32\n/s32\n/s32/s40/s97/s41 /s32/s86/s105/s114/s103/s105/s110/s72/s32 /s32/s40/s109/s84/s41\n/s32\n/s102/s32/s32/s40/s71/s72/s122/s41/s40/s98/s41 /s32/s73/s114/s114/s97/s100/s105/s97/s116/s101/s100\n/s72\n/s84/s77/s83\nFIG. 2: The frequency dependence of the FMR linewidth with\napplied field in-plane and perpendicular to the plane. The\nsolid black lines are linear fits that enable determination o fα\nand ∆ H0from Eq. 2. The dotted lines show the linewidth\nfrom TMS and the red lines is the total linewidth.3\n/s48/s49/s48/s48\n/s48 /s51/s48 /s54/s48 /s57/s48/s48/s49/s48/s48/s72 /s32/s32/s32/s32/s32 /s72\n/s105/s110/s104/s32\n/s72 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s72\n/s84/s77/s83/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s40/s98/s41\n/s32/s48/s72 /s32/s40/s32/s109/s84/s32/s41/s40/s97/s41\n/s32/s70/s105/s101/s108/s100/s32/s97/s110/s103/s108/s101/s32\n/s72/s32/s32/s40/s32/s100/s101/s103/s46/s32/s41\n/s32\nFIG. 3: Angular dependence of the linewidth at 20 GHz for\n(a) the virgin and (b) the irradiated film. The solid line\n(∆H) is a best fit of the data that includes the Gilbert damp-\ning (∆ Hα) and the inhomogeneous (∆ Hinh) contributions.\nLinewidth broadening from TMS (∆ HTMS) is also shown.\nThe total linewidth is represented by the red line.\n∆H⊥of the virgin film increases linearly with frequency\nconsistent with Gilbert damping. Fitting to Eq. 2, we\nfindα= 0.044±0.003 andµ0∆H⊥\n0= 15.6±3.6 mT.\nWhen the field is applied in the film plane, the linewidth\nis significantly larger. ∆ H/bardbldecreases with increasing\nfrequency for f≤10 GHz and then is practically inde-\npendent of frequency, at ≈140±20 mT. However, for\nthe irradiated film, the linewidth varies linearly with fre-\nquency both for in-plane and out-of-plane applied fields,\nwith nearly the same slope. The Gilbert damping is\nα= 0.039±0.004. Note that µ0∆H/bardbl\n0is larger than\nµ0∆H⊥\n0by about 15 mT.\nThe angular dependence of the linewidth at 20 GHz\nis shown in Fig. 3. The linewidth of the virgin film de-\ncreases significantly with increasing field angle up 30o,\nand then is nearly constant, independent of field angle.\nThe linewidth of the irradiated film is nearly indepen-\ndent of the field angle, with a relatively small enhance-\nment of ∼15 mT close to the in-plane direction. We\nfit this data assuming that the inhomogeneous broad-\nening of the line is associated mainly with spatial vari-\nations of the PMA, specifically local variation in K1,\n∆Hinh.(φH) =|∂Hres/∂K1|∆K1. ∆K1= 4×103J/m3\nfor the virgin film and 3 ×102J/m3for the irradiated film,\nwhich corresponds to a variation of K1of 2% and 0.3%\nrespectively. Including variations in K2and anisotropy\nfield direction do not significantly improve the quality\nof the fit. Such variations in K1produce a zero fre-\nquency linewidth in the perpendicular field direction,\nµ0∆H⊥\n0= 16.8 mT, in excellent agreement with linear\nfits to the data in Fig. 2. However, the combination\nof inhomogeneous broadening and Gilbert damping can-\nnotexplain the enhanced FMR linewidth observed for\nin-plane applied fields.\nThe enhanced linewidth observed with in-plane applied\nfields is consistent with a significant TMS contribution to\nthe relaxation of the uniform mode–the linewidth is en-\nhanced only when finite- kmodes equi-energy with theuniform mode are present. We derive the spin-wave dis-\npersion for these films following the approach of [16]:\nω2\nk=ω2\n0−1\n2γ2µ0Mskt(Bx0(cos2φ\n+ sin2φsin2ψk)−By0sin2ψk) +γ2Dk2(Bx0+By0),\n(5)\nwhere:\nBx0=µ0Hcos(φH−φ)−µ0Meffsin2φ\nBy0=µ0Hcos(φH−φ) +µ0Meffcos2φ\n+2K2\nMssin22φ.(6)\nThe effective demagnetization field is µ0Meff= (µ0Ms−\n2K1\nMs−4K2\nMscos2φ).ω0=γ/radicalbig\nBx0By0is the resonance\nfrequency of the uniform mode. Dis the exchange stiff-\nness andtis the film thickness. ψkis the direction of\npropagation of the spin-wave in the film plane relative\nto the in-plane projection of the magnetization. The in-\nset of Fig. 4 shows the dispersion relation for the virgin\nand the irradiated film for an in-plane applied field at\n20 GHz. For the virgin film, with the easy axis normal\nto the film plane ( M/bardbl\neff<0) there are degenerate modes\navailable in all directions in k-space. For the irradiated\nfilm (M/bardbl\neff>0) degenerate modes are only available when\nψk/lessorsimilar74o.\nThe spin waves density of states, determined from Eq.\n5, is shown as a function of field angle in Fig. 4 at 20\nGHz. The DOS of the virgin film is two times larger\nthan that of the irradiated film at φH= 0. Note that for\nboth films, the DOS vanishes at a critical field angle that\ncorresponds to a magnetization angle φ= 45o. For the\nvirgin film, the enhancement of ∆ Hoccurs atφH≃30o\n(Fig. 3a), at the critical angle seen in Fig. 4.\nThe TMS linewidth depends on the density of states\nand the disorder, which couples the modes:\n∆HTMS=/parenleftbigg∂Hres\n∂ω/parenrightbigg|A0|2\n2π/integraldisplay\nCk(ξ)δ(ωk−ω0)dk,(7)\nwhereA0is a scattering amplitude. Ck(ξ) = 2πξ2/(1 +\n(kξ)2)3/2is a correlation function, where ξis correlation\nlength, the typical length scale of disorder. Eq. 7 is valid\nin the limit of weak disorder.\nWe assume that the disorder of our films is associated\nwith spatial variations of the PMA, K1. Then the mag-\nnetic energy density varies as ∆ E(/vector r) =−k1(/vector r)M2\ny/M2\ns,\nand the scattering probability is [17]:\n|A0|2=γ4\n4ω2\n0(B2\nx0sin4φ+B2\ny0cos22φ\n−2(ω0/γ)2sin2φcos2φ)/parenleftbigg2∆k1\nMs/parenrightbigg2\n.(8)\n∆k1is the rms amplitude of the distribution of PMA,\nk1(r). Therefore the TMS linewidth broadening scales4\n/s48 /s51/s48 /s54/s48 /s57/s48/s48/s50/s48/s52/s48/s54/s48/s50/s48\n/s48 /s53/s50/s48\n/s32/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s115/s116/s97/s116/s101/s115/s32/s32/s40/s97/s46/s117/s41\n/s70/s105/s101/s108/s100/s32/s97/s110/s103/s108/s101\n/s72/s32/s32/s40/s100/s101/s103/s46/s41/s107/s61/s57/s48/s111\n/s107/s61/s48/s111/s32/s32 /s32/s32/s102/s32/s32 /s40/s71/s72/s122/s41\n/s32\n/s73/s114/s114/s97/s100/s105/s97/s116/s101/s100\n/s32\n/s107 /s32/s40/s49/s48/s53\n/s32/s114/s97/s100/s47/s99/s109/s41/s86/s105/s114/s103/s105/s110\nFIG. 4: The density of spin-waves states degenerate with the\nuniform mode as a function of field angle at 20 GHz for the\nvirgin film (solid line) and the irradiated film (dashed-dott ed\nline). Inset: Spin wave dispersion when the dc field is in the\nfilm plane.\nas the square of ∆ k1. Since the variations in PMA of the\nvirgin film are larger than that of the irradiated film the\nlinewidth broadening from the TMS mechanism is ex-\npected to be much larger in the virgin film, qualitatively\nconsistent with the data.\nA best fit of the linewidth data to the TMS model is\nshown in Fig. 3a. For the virgin film, we find ξ≈44 nm,\napproximately four times the film grain size, and ∆ k1=\n9×103J/m3. The exchange stiffness, D= 2A/µ0Mswith\nthe exchange constant A= 0.83×10−11J/m, is used in\nthe fittings. The cut-off field angle for the enhancement\nof the field linewidth agrees well with the data (Fig. 3a).\nFor the irradiated film, a similar analysis gives: ξ= 80±\n40 nm and ∆ k1= (4±2)×103J/m3.\nTMS is also expected to shift the resonance position\n[17]. For applied fields in-plane and f= 20 GHz we\nestimate the resonance field shift to be ≈33 mT. This is\nsmaller than what is observed experimentally ( ≈93 mT).\nThe deviations of the fits in Fig. 1b may be associated\nwith an anisotropy in the gyromagnetic ratio, i.e. a g\nthat is smaller for Min the film plane. Note that if we\nassume that the g-factor is slightly anisotropic ( ∼1%),\nwe can fit the full angular dependence of the resonance\nfield of the irradiated film.\nWe note that the TMS model cannot explain the en-\nhanced linewidth for small in-plane applied fields for the\nvirgin film (Fig. 2a). The FMR linewidth increases\ndramatically when the frequency and resonance field de-\ncreases. When the applied in-plane field is less than the\neffective demagnetization field ( −µ0M||\neff= 0.31 T) the\nmagnetization reorients out of the film plane. For fre-\nquencies less than about 8 GHz this leads to two resonant\nabsorption peaks, one with the magnetization having an\nout-of-plane component for Hres<−M||\neffand one with\nthe magnetization in-plane for Hres>−M||\neff. It may\nbe that these resonances overlap leading to the enhanced\nFMR linewidth.In sum, these results show that the FMR linewidth in\nNi|Co multilayer films is large due to disorder and TMS\nas well as enhanced Gilbert damping. The latter is an\nintrinsic relaxation mechanism, associated with magnon-\nelectron scattering and spin-relaxation due to spin-orbit\nscattering. As these materials contain heavy elements\nsuch as Pd and short electron lifetimes at the Fermi level,\nlarge intrinsic damping rates are not unexpected. The re-\nsults indicate that the FMR linewidth of Ni |Co multilay-\ners can be reduced through light ion-irradiation and fur-\nther demonstrate that the Gilbert damping rate is largely\nunaffected by irradiation. These results, including the re-\nduction of the PMA distribution at high irradiation dose,\nhave important implications for the applications of PMA\nmaterials in data storage and spin-electronic application s\nwhich require tight control of the anisotropy, anisotropy\ndistributions and resonant behavior.\nACKNOWLEDGMENTS\nWe thank Gabriel Chaves for help in fitting the data\nto the TMS model. This work was supported by NSF\nGrant No. DMR-0706322.\n[1] T. Thomson, G. Hu, and B. D. Terris, Phys. Rev. Lett.\n96, 257204 (2006).\n[2] S. Mangin et al., Nature Mater. 5, 210 (2006).\n[3] G. H. O. Daalderop, P. J. Kelly, and F. J. A. den Broeder,\nPhys. Rev. Lett. 68, 682 (1992).\n[4] B. Heinrich, Ultrathin Magnetic Structures III (Springer,\nNew York, 2005), p. 143.\n[5] C. Scheck et al., Phys. Rev. Lett. 98, 117601 (2007).\n[6] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n[7] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n[8] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[9] T. W. Clinton et al., J. Appl. Phys. 103, 07F546 (2008).\n[10] S. J. Yuan et al., Phys. Rev. B 68, 134443 (2003).\n[11] J. BenYoussef et al., J. Magn. Magn. Mater. 202, 277\n(2003).\n[12] N. Mo et al., Appl. Phys. Lett. 92, 022506 (2008).\n[13] M. Sparks, Ferromagnetic-Relaxation Theory (McGraw-\nHill, 1964).\n[14] D. Stanescu et al., J. Magn. Magn. Mater. 103, 07B529\n(2008).\n[15] J.-M. L. Beaujour et al., Eur. Phys. J. B 59, 475 (2007).\n[16] P. Landeros, R. E. Arias, and D. L. Mills, Phys. Rev. B\n77, 214405 (2008).\n[17] R. D. McMichael and P. Krivosik, IEEE Trans. Magn.\n40, 2 (2004)."    },    {        "title": "1510.01894v1.Tunable_damping__saturation_magnetization__and_exchange_stiffness_of_half_Heusler_NiMnSb_thin_films.pdf",        "content": "Tunable damping, saturation magnetization, and exchange sti\u000bness of half-Heusler\nNiMnSb thin \flms\nP. D urrenfeld,1F. Gerhard,2J. Chico,3R. K. Dumas,1, 4M. Ranjbar,1A. Bergman,3\nL. Bergqvist,5, 6A. Delin,3, 5, 6C. Gould,2L. W. Molenkamp,2and J. \u0017Akerman1, 4, 5\n1Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden\n2Physikalisches Institut (EP3), Universit at W urzburg, 97074 W urzburg, Germany\n3Department of Physics and Astronomy, Uppsala University, Box 520, 752 20 Uppsala, Sweden\n4NanOsc AB, 164 40 Kista, Sweden\n5Materials and Nano Physics, School of ICT, KTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden\n6Swedish e-Science Research Centre (SeRC), 100 44 Stockholm, Sweden\nThe half-metallic half-Heusler alloy NiMnSb is a promising candidate for applications in spin-\ntronic devices due to its low magnetic damping and its rich anisotropies. Here we use ferromagnetic\nresonance (FMR) measurements and calculations from \frst principles to investigate how the com-\nposition of the epitaxially grown NiMnSb in\ruences the magnetodynamic properties of saturation\nmagnetization MS, Gilbert damping \u000b, and exchange sti\u000bness A.MSandAare shown to have a\nmaximum for stoichiometric composition, while the Gilbert damping is minimum. We \fnd excellent\nquantitative agreement between theory and experiment for MSand\u000b. The calculated Ashows the\nsame trend as the experimental data, but has a larger magnitude. Additionally to the unique in-\nplane anisotropy of the material, these tunabilities of the magnetodynamic properties can be taken\nadvantage of when employing NiMnSb \flms in magnonic devices.\nI. INTRODUCTION\nInterest in the use of half-metallic Heusler and half-\nHeusler alloys in spintronic and magnonic devices is\nsteadily increasing,1{3as these materials typically exhibit\nboth a very high spin polarization4{8and very low spin-\nwave damping.9{12One such material is the epitaxially\ngrown half-Heusler alloy NiMnSb,13,14which not only has\none of the lowest known spin-wave damping values of\nany magnetic metal, but also exhibits an interesting and\ntunable combination of two-fold in-plane anisotropy15\nand moderate out-of-plane anisotropy,10all potentially\ninteresting properties for use in both nanocontact-\nbased spin-torque oscillators16{22and spin Hall nano-\noscillators23{27. To successfully employ NiMnSb in such\ndevices, it is crucial to understand, control, and tailor\nboth its magnetostatic and magnetodynamic properties,\nsuch as its Gilbert damping ( \u000b), saturation magnetiza-\ntion (MS), and exchange sti\u000bness ( A).\nHere we investigate these properties in Ni 1-xMn1+xSb\n\flms using ferromagnetic resonance (FMR) measure-\nments and calculations from \frst principles for compo-\nsitions of -0.1\u0014x\u00140.4.MSandAare shown experi-\nmentally to have a maximum for stoichiometric compo-\nsition, while the Gilbert damping is minimum; this is\nin excellent quantitative agreement with calculations of\nand experiment on MSand\u000b. The calculated Ashows\nthe same trend as the experimental data, but with an\noverall larger magnitude. We also demonstrate that the\nexchange sti\u000bness can be easily tuned over a wide range\nin NiMnSb through Mn doping, and that the ultra-low\ndamping persists over a wide range of exchange sti\u000b-\nnesses. This unique behavior makes NiMnSb ideal for\ntailored spintronic and magnonic devices. Finally, by\ncomparing the experimental results with \frst-principlescalculations, we also conclude that the excess Mn mainly\noccupies Ni sites and that interstitial doping plays only\na minor role.\nII. METHODS\nA. Thin Film Growth\nThe NiMnSb \flms were grown by molecular beam\nepitaxy onto InP(001) substrates after deposition of a\n200 nm thick (In,Ga)As bu\u000ber layer.15The \flms were\nsubsequently covered in situ by a 10 nm thick mag-\nnetron sputtered metal cap to avoid oxidation and sur-\nface relaxation.28The Mn content was controlled dur-\ning growth via the temperature, and hence the \rux, of\nthe Mn e\u000busion cell. Six di\u000berent samples (see table I)\nwere grown with increasing Mn concentration, sample 1\nhaving the lowest and sample 6 the highest concentra-\ntion of Mn. High-resolution x-ray di\u000braction (HRXRD)\nmeasurements give information on the structural proper-\nties of these samples, con\frming the extremely high crys-\ntalline quality of all samples with di\u000berent Mn concentra-\nSamplevertical\nlattice\nconstant ( \u0017A)thickness\n(nm)uniaxial\neasy axis2K1\nMS(Oe)\n1 5.94 38 [110] 170\n2 5.97 38 [110] 8.4\n3 5.99 40 [110] 0\n4 6.02 45 [1 \u001610] 9.0\n5 6.06 45 [1 \u001610] 14.2\n6 6.09 38 [1 \u001610] 25.5\nTable I. Overview of NiMnSb \flms investigated in this study.arXiv:1510.01894v1  [cond-mat.mtrl-sci]  7 Oct 20152\ntion, even in the far from stoichiometric cases (samples 1\nand 6).15The vertical lattice constant is found to increase\nwith increasing Mn concentration and, assuming a linear\nincrease,29we estimate the di\u000berence in Mn concentra-\ntion across the whole set of samples to be about 40 at. %.\nWe will thus represent the Mn concentration in the fol-\nlowing experimental results by the measured vertical lat-\ntice constant. Stoichiometric NiMnSb exhibits vertical\nlattice constants in the range of 5.96{6.00 \u0017A, leading to\nthe expectation of stoichiometric NiMnSb in samples 2\nand 3.15Finally, the layer thicknesses are also determined\nfrom the HRXRD measurements, giving an accuracy of\n\u00061 nm.\nB. Ferromagnetic Resonance\nBroadband \feld-swept FMR spectroscopy was per-\nformed using a NanOsc Instruments PhaseFMR system\nwith a coplanar waveguide for microwave \feld excitation.\nMicrowave \felds hrfwith frequencies of up to 16 GHz\nwere applied in the \flm plane, perpendicularly oriented\nto an in-plane dc magnetic \feld H. The derivative of\nthe FMR absorption signal was measured using a lock-\nin technique, in which an additional low-frequency mod-\nulation \feld Hmod<1 Oe was applied using a pair of\nHelmholtz coils parallel to the dc magnetic \feld. The\n\feld directions are shown schematically in Fig. 1(a) and\na typical spectrum measured at 13.6 GHz is given in the\ninset of Fig. 1(b). In addition to the zero wave vector\nuniform FMR mode seen at about H=2.1 kOe, an addi-\ntional weaker resonance is observed at a much lower \feld\nof about 500 Oe, and is identi\fed as the \frst exchange-\ndominated perpendicular standing spin wave (PSSW)\nmode. The PSSW mode has a nonzero wave vector point-\ning perpendicular to the thin \flm plane and a thickness-\ndependent spin-wave amplitude and phase.30,31This can\nbe e\u000eciently excited in the coplanar waveguide geome-\ntry due to the nonuniform strength of the microwave \feld\nacross the \flm thickness.32\nThe \feld dependence of the absorption spectra (inset\nof Fig. 1(b)) can be \ft well (red line) by the sum of a sym-\nmetric and an antisymmetric Lorentzian derivative:33,34\ndP\ndH(H) =\u00008C1\u0001H(H\u0000H0)\nh\n\u0001H2+ 4 (H\u0000H0)2i2\n+2C2\u0000\n\u0001H2\u00004(H\u0000H0)2\u0001\nh\n\u0001H2+ 4 (H\u0000H0)2i2; (1)\nwhereH0is the resonance \feld, \u0001 Hthe full width at\nhalf maximum (FWHM), and C1andC2\ftted param-\neters representing the amplitude of the symmetric and\nantisymmetric Lorentzian derivatives, respectively. Both\nthe FMR and the PSSW peaks can be \ftted indepen-\ndently, as they are well separated by the exchange \feld\n\u00160Hex/(\u0019=d)2, wheredis the thickness of the layer.\neasy axis\nH, Hmod\nhard axis\nhrf p = 0\n(FMR)p = 1\n(PSSW)z\nHexx\ny\nFMRPSSW(a)\n(b)\n0\n500\n1000\n1500\n2000\n2500\n0\n5\n10\n15\nf(GHz)\nField (Oe)\n400\n600\n2000\n2200\n-0.5\n0.0\n0.5\n1.0\n1.5\ndP/dH (a. u.)\nField (Oe)\nf= 13.6 GHzFigure 1. (a) Schematic diagram of the FMR measurement\nshowing \feld directions. In our setup, the FMR mode and the\n\frst PSSW mode are excited. (b) Frequency vs. resonance\n\felds of the PSSW (red) and uniform FMR (black) mode for\nsample 2. The solid lines are \fts to the Kittel equation, and\nboth modes are o\u000bset horizontally by Hex. Inset: Resonance\ncurves forf=13.6 GHz. The \frst PSSW mode on the left and\nthe FMR mode on the right were \ft with Eq. 1\nFor our chosen sample thicknesses, the di\u000berences in res-\nonance \felds are always much larger than the resonance\nlinewidths.\nThe \feld dependence of both resonances is shown in\nFig. 1(b) and can now be used to extract information\nabout the magnetodynamic properties and anisotropies\nof the \flms. The curves are \fts to the Kittel equation,\nincluding internal \felds from the anisotropy and the ex-\nchange \feld for the PSSW excitation:15,35\nf=\r\u00160\n2\u0019\u0014\u0012\nH0+2KU\nMS\u00002K1\nMS+Hex\u0013\n\u0002\u0012\nH0+2KU\nMS+K1\nMS+Hex+Me\u000b\u0013\u00151=2\n;(2)\nwhereH0is the resonance \feld, \r=2\u0019the gyromagnetic\nratio, and\u00160the permeability of free space. Me\u000bis the ef-\nfective magnetization, which has a value close to the satu-\nration magnetization MS. 2KU=MSand 2K1=MSstands\nfor the internal anisotropy \felds coming from the uniaxial\n(KU) and biaxial ( K1) anisotropy energy densities in the\nhalf-Heusler material. The e\u000bective magnetic \feld also\nincludes an exchange \feld \u00160Hex= (2A=M S)(p\u0019=d )2,3\nwhich is related to the exchange sti\u000bness A, the \flm\nthicknessd, and the integer order of the PSSW mode\np, wherep= 0 denotes the uniform FMR excitation and\np= 1 the \frst PSSW mode. This mode numbering re-\n\rects the boundary conditions with no surface pinning of\nthe spins, which is expected for the in-plane measurement\ngeometry.36\nWe stress that the expression for the anisotropy con-\ntribution in Eq. 2 is only valid for the case in which the\nmagnetization direction is parallel to the uniaxial easy\naxis and also parallel to the applied \feld. A full angular-\ndependent formulation of the FMR condition is described\nin Ref. 15. To ful\fll the condition of parallel alignment\nfor all resonances, we perform the FMR measurements\nwith the dc magnetic \feld being applied along the domi-\nnant uniaxial easy axis of each \flm, which changes from\nthe [110] crystallographic direction to the [1 \u001610]-direction\nwith increasing Mn concentration (see Table I).\nThe values of the biaxial anisotropy2K1\nMShave been de-\ntermined in a previous study by \fxed-frequency in-plane\nangular dependent FMR measurements,15and were thus\ntaken as constant values in the \ftting process for Eq. 2;\na simultaneous \ft of both contributions can yield arbi-\ntrary combinations of anisotropy \felds due to their great\ninterdependence. The values for the uniaxial anisotropy\n2KU\nMSobtained from the frequency-dependent \ftting are\nin very good agreement with the previously obtained val-\nues in Ref. 15. The gyromagnetic ratio was measured\nto be\r=2\u0019= (28.59\u00060.20) GHz/T for all investigated\nsamples, and was therefore \fxed for all samples to allow\nbetter comparison of the e\u000bective magnetization values.\nThe Gilbert damping \u000bof the \flms is obtained by\n\ftting the FMR linewidths \u0001 Hwith the linear depen-\ndence:37\n\u00160\u0001H=\u00160\u0001H0+4\u0019\u000b\n\rf; (3)\nwhere \u0001H0is the inhomogeneous linewidth broaden-\ning of the \flm. The parallel alignment between mag-\nnetization and external magnetic \feld ensures that the\nlinewidth is determined by the Gilbert damping process\nonly.38\nC. Calculations from First Principles\nThe electronic and magnetic properties of the NiMnSb\nhalf-Heusler system were studied via \frst-principles cal-\nculations. The material was assumed to be ordered in a\nface-centered tetragonal structure with an in-plane lat-\ntice parameter ak\nlat= 5.88 \u0017A, close to the lattice con-\nstant of the InP substrate, and an out-of-plane lattice\nconstant of a?\nlat= 5.99 \u0017A, matching the value for the\nstoichiometric composition. Fixed values for the lattice\nparameters were chosen since an exact relation between\nthe o\u000b-stoichiometric composition and the experimen-\ntally measured vertical lattice constants cannot be es-\ntablished. Moreover, calculations with a varying verticallattice parameter for a constant composition showed only\na negligible e\u000bect on M S,A, and\u000b. The calculations\nwere performed using the multiple scattering Korringa-\nKohn-Rostocker (KKR) Green's function formalism as\nimplemented in the SPRKKR package.39Relativistic ef-\nfects were fully taken into account by solving the Dirac\nequation for the electronic states, the shape of the poten-\ntial was considered via the Atomic Sphere Approximation\n(ASA), and the local spin density approximation (LSDA)\nwas used for the exchange correlation potential. The co-\nherent potential approximation (CPA) was used for the\nchemical disorder of the system.\nThe Gilbert damping \u000bof the material was calculated\nusing linear response theory40, including the temperature\ne\u000bects from interatomic displacements and spin \ructua-\ntions.41,42\nThe exchange interactions Jijbetween the atomic\nmagnetic moments were calculated using the magnetic\nforce theorem, as considered in the LKAG formalism.43,44\nThe interactions were calculated for up to 4.5 times the\nlattice constant in order to take into account any long-\nrange interactions. Given the interatomic exchange in-\nteractions, the spin-wave sti\u000bness Dcan be calculated.\nDue to possible oscillations in the exchange interactions\nas a function of the distance, it becomes necessary to in-\ntroduce a damping parameter, \u0011, to assure convergence\nof the summation. Dcan then be obtained by evaluating\nthe limit\u0011!0 of\nD=2\n3X\nijJijp\nMiMjr2\nijexp\u0012\n\u0000\u0011rij\nalat\u0013\n; (4)\nas described in [45]. Here, MiandMjare the local mag-\nnetic moments at sites iandj,Jijis the exchange cou-\npling between the magnetic moments at sites iandj,\nandrijis the distance between the atoms iandj. This\nformalism can be extended to a multisublattice system46.\nTo calculate the e\u000bect of chemical disorder on the ex-\nchange sti\u000bness of the system, the obtained exchange in-\nteractions were summed over a supercell with a random\ndistribution of atoms in the chemically disordered sub-\nlattice. The e\u000bect that distinct chemical con\fgurations\ncan have over the calculation of the exchange sti\u000bness\nwas treated by taking 200 di\u000berent supercells. The re-\nsults were then averaged and the standard deviation was\ncalculated. The cells were obtained using the atomistic\nspin dynamics package UppASD.47\nFinally, with the spin-wave sti\u000bness determined as de-\nscribed above, the exchange sti\u000bness Acan be calculated\nfrom:48\nA=DM S(T)\n2g\u0016B: (5)\nHere,gis the Land\u0013 e g-factor of the electron, \u0016Bthe Bohr\nmagneton, and MS(T) the magnetization density of the\nsystem for a given temperature T, which for T= 0 K\ncorresponds to the saturation magnetization.\nFrom the \frst-principles calculations, the magnetic\nproperties for ordered NiMnSb and chemically disordered4\n0.60.70.80.95\n.956 .006 .056 .100123(b) m0MS \nm0Meffm0M (T)(a)4  µB/u.f.KS (mJ/m2)v\nertical lattice constant (Å)\nFigure 2. (a) MSandMe\u000bas functions of vertical lattice\nconstant. The theoretical value of 4.0 \u0016B=u.f. is shown by\nthe blue dashed line. (b) The calculated surface anisotropy\ndensity follows from the di\u000berence between MSandMe\u000b.\nNi1-xMn1+xSb were studied. To obtain the values of the\nexchange sti\u000bness AforT= 300 K, the exchange interac-\ntions from the ab initio calculations were used in conjunc-\ntion with the value of the magnetization at T= 300 K\nobtained from Monte Carlo simulations.\nIII. RESULTS\nA. Magnetization\nThe values of \u00160Me\u000bare plotted in Fig. 2(a) as red\ndots. The e\u000bective magnetization is considerably lower\nthan the saturation magnetization \u00160MS, which was in-\ndependently assessed using SQUID measurements and al-\nternating gradient magnetometry (AGM). The values for\n\u00160MScorrespond to a saturation magnetization between\n3.5\u0016B=unit formula and 3.9 \u0016B=u.f., with the latter\nvalue being within the error bars of the theoretically ex-\npected value of 4.0 \u0016B=u.f. for stoichiometric NiMnSb.49\nA reduction of MSis expected in Mn-rich NiMnSb alloys,\ndue to the antiferromagnetic coupling of the Mn Nidefects\nto the Mn lattice in the C1 bstructure of the half-Heusler\nmaterial.29An even stronger reduction is observed for\nthe Ni-rich sample 1, which is in accordance with the\nformation of Ni Mnantisites.50\nWhile the measurement error for MSis comparatively\nlarge due to uncertainties in the volume determination,\nthe error bars for Me\u000b, as obtained from ferromagnetic\nresonance, are negligible. NiMnSb \flms have been shown\nto possess a small but substantial perpendicular mag-\nnetic anisotropy, which can arise from either interfacial\nanisotropy or lattice strain.10,12To quantify the di\u000ber-\nence observed between MSandMe\u000b, we assume a uniax-\nial perpendicular anisotropy due to a surface anisotropy\n2\n4\n6\n8\n10\n5.95\n6.00\n6.05\n6.10\n0\n2\n4\n(b)\nA (pJ/m)\n(a)\nα(10-3)\nvertical lattice constant (Å)\n2\n4\n6\n8\n10\n-0.1\n0.0\n0.1\n0.2\n0.3\n0.4\n0\n2\n4\n(d)\n(c)\nxantisitesFigure 3. (a) and (b) show respectively the exchange sti\u000bness\nand Gilbert damping constant obtained from FMR measure-\nments, plotted as a function of the vertical lattice constant.\n(c) and (d) show the corresponding values obtained from \frst-\nprinciple calculations for T= 300 K. Negative values for x\nimply the introduction of Ni Mnantisites and positive values\nare related to Mn Niantisite defects. The error bars in (c) are\nthe standard deviations from repeated \frst-principles calcu-\nlations with 200 randomized supercells.\nenergy density KS, which is known to follow the rela-\ntion:51\n\u00160Me\u000b=\u00160MS\u00002KS\nMSd: (6)\nTheKScalculated in this way has values between\n0.5mJ=m2and 1.5mJ=m2, as shown in Fig. 2(b); these\nare comparable to the surface anisotropies obtained in\nother crystalline thin \flm systems.52. Although the \flm\nthicknesses in our set vary unsystematically, we can ob-\nserve systematic behavior of KSwith the vertical lat-\ntice constant, with an apparent minimum under the con-\nditions where stoichiometric NiMnSb is expected|that\nis, for samples 2 and 3. The increasing values for o\u000b-\nstoichiometric NiMnSb can be thus attributed to the con-\ncomitant increase in lattice defects, and thus of surface\ndefects, in these \flms.\nB. Exchange Sti\u000bness and Gilbert Damping\nThe experimentally determined exchange sti\u000bness, as\na function of the vertical lattice constant, and the Gilbert\ndamping parameter are shown in Fig. 3(a) and (b), re-\nspectively. The minimum damping observed in our mea-\nsurements is 1 :0\u000210\u00003for sample 3, and so within sto-\nichiometric composition. Sample 1, with a de\fciency\nof Mn atoms, showed nonlinear linewidth behavior at\nlow frequencies, which vanished for out-of-plane measure-\nments (not shown). This is typical with the presence of\ntwo-magnon scattering processes.52However, the damp-\ning is considerably lower in all samples than in a permal-\nloy \flm of comparable thickness.\nThe exchange sti\u000bness and Gilbert damping ob-\ntained from the \frst-principles calculations are shown in5\nFig. 3(c) and (d), respectively. For both parameters, the\nexperimental trends are reproduced quantitatively, with\nAhaving a maximum and \u000ba minimum value at stoi-\nchiometry.\nAs the concentration of both Mn or Ni antisites in-\ncreases, the exchange sti\u000bness decreases. This behavior\ncan be explained by analyzing the terms in the expres-\nsion for the spin-wave sti\u000bness, Eq. 4. It turns out that\nthe new exchange couplings Jij, which appear when an-\ntisites are present, play a major role, whereas changes in\nthe atomic magnetic moments or the saturation magne-\ntization appear to be relatively unimportant. Mn anti-\nsites in the Ni sublattice (i.e., excess Mn) have a strong\n(2 mRy) antiferromagnetic coupling to the Mn atoms in\nthe adjacent Mn layers. This results in a negative contri-\nbution toDcompared to the stoichiometric case, where\nthis interaction is not present. On the other hand, Ni\nantisites in the Mn sublattice have a negative in-plane\nexchange coupling of 0.3 mRy to their nearest-neighbor\nMn atoms, with a frustrated antiferromagnetic coupling\nto the Ni atoms in the adjacent Ni plane. The net e\u000bect is\na decreasing spin-wave sti\u000bness as the composition moves\naway from stoichiometry. The calculated values of Aare\naround 30 % larger than the experimental results, which\nis the same degree of overestimation we recently observed\nin a study of doped permalloy \flms53. It thus seems to\nbe inherent in our calculations from \frst principles.\nThe calculated Gilbert damping also agrees well with\nthe experimental values. The damping has its minimum\nvalue of 1.0\u000210\u00003at stoichiometry and increases with\na surplus of Ni faster than with the same surplus of\nMn. Both Mn and Ni antisites will act as impurities and\nit is thus reasonable to attribute the observed increase\nin damping at o\u000b-stoichiometry to impurity scattering.\nWhile the damping at stoichiometry also agrees quanti-\ntatively, the increase in damping is underestimated in the\ncalculations compared to the experimental values.\nDespite the fact that the calculations here focus purely\non the formation of Mn Nior Ni Mnantisites, they are\nnonetheless capable of reproducing the experimental\ntrends well. However, interstitials|that is, Mn or Ni sur-\nplus atoms in the vacant sublattice|may also be a possi-\nble o\u000b-stoichiometric defect in our system.50We have cal-\nculated their e\u000bects and can therefore discuss about the\nexistence of interstitials in our samples. A large fraction\nof Mn interstitials seems unlikely, as an increase in the\nsaturation magnetization can be predicted through calcu-\nlations, contrary to the experimental trend; see Fig. 2(a).On the other hand, the existence of Ni interstitials may\nbe compatible with the observed experimental trend, as\nthey decrease the saturation magnetization|albeit at a\nslower rate than Ni antisites and slower than experimen-\ntally observed. Judging from the measured data, it is\ntherefore likely that excess Ni exists in the samples as\nboth antisites and interstitials.\nIV. CONCLUSIONS\nIn summary, we have found that o\u000b-stoichiometry in\nthe epitaxially grown half-Heusler alloy NiMnSb has a\nsigni\fcant impact on the material's magnetodynamic\nproperties. In particular, the exchange sti\u000bness can be\naltered by a factor of about 2 while keeping the Gilbert\ndamping very low ( \u00195 times lower than in permalloy\n\flms). This is a unique combination of properties and\nopens up for the use of NiMnSb in, e.g., magnonic cir-\ncuits, where a small spin wave damping is desired. At the\nstoichiometric composition, the saturation magnetization\nand exchange sti\u000bness take on their maximum values,\nwhereas the Gilbert damping parameter is at its mini-\nmum. These experimentally observed results are repro-\nduced by calculations from \frst principles. Using these\ncalculations, we can also explain the microscopic mecha-\nnisms behind the observed trends. We also conclude that\ninterstitial Mn is unlikely to be present in the samples.\nThe observed e\u000bects can be used to \fne-tune the mag-\nnetic properties of NiMnSb \flms towards their speci\fc\nrequirements in spintronic devices.\nACKNOWLEDGMENTS\nWe acknowledge \fnancial support from the G oran\nGustafsson Foundation, the Swedish Research Council\n(VR), Energimyndigheten (STEM), the Knut and Alice\nWallenberg Foundation (KAW), the Carl Tryggers Foun-\ndation (CTS), and the Swedish Foundation for Strate-\ngic Research (SSF). F.G. acknowledges \fnancial support\nfrom the University of W urzburg's \\Equal opportunities\nfor women in research and teaching\" program. This work\nwas also supported initially by the European Commission\nFP7 Contract ICT-257159 \\MACALO\". A.B acknowl-\nedges eSSENCE. 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B 92, 024427\n(2015)."    },    {        "title": "1605.04543v1.Propagation_of_Thermally_Induced_Magnonic_Spin_Currents.pdf",        "content": "arXiv:1605.04543v1  [cond-mat.mtrl-sci]  15 May 2016Propagation of Thermally Induced Magnonic Spin Currents\nUlrike Ritzmann, Denise Hinzke, and Ulrich Nowak\nFachbereich Physik, Universit¨ at Konstanz, D-78457 Konst anz, Germany\n(Dated: 19.12.2013)\nThe propagation of magnons in temperature gradients is inve stigated within the framework of an\natomistic spin model with the stochastic Landau-Lifshitz- Gilbert equation as underlying equation\nof motion. We analyze the magnon accumulation, the magnon te mperature profile as well as the\npropagation length of the excited magnons. The frequency di stribution of the generated magnons\nis investigated in order to derive an expression for the influ ence of the anisotropy and the damping\nparameter on the magnon propagation length. For soft ferrom agnetic insulators with low damping\na propagation length in the range of some µm can be expected for exchange driven magnons.\nPACS numbers: 75.30.Ds, 75.30.Sg, 75.76.+j\nI. INTRODUCTION\nSpin caloritronics is a new, emerging field in mag-\nnetism describing the interplay between heat, charge and\nspin transport1,2. A stimulation for this field was the dis-\ncovery of the spin Seebeck effect in Permalloy by Uchida\net al.3. Analog to the Seebeck effect, where in an elec-\ntric conductor an electrical voltage is created by apply-\ning a temperature gradient, in a ferromagnet a temper-\nature gradient excites a spin current leading to a spin\naccumulation. The generated spin accumulation was de-\ntected by measuring the spin current locally injected into\na Platinum-contact using the inverse spin Hall effect3,4.\nA first explanation of these effect was based on a spin-\ndependent Seebeckeffect, wherethe conductionelectrons\npropagate in two different channels and, due to a spin\ndependent mobility, create a spin accumulation in the\nsystem5.\nInterestingly, it was shown later on that this effect also\nappears in ferromagnetic insulators6. This shows that in\naddition to conduction-electron spin-currents, chargeless\nspin-currents exist as well, where the angular momentum\nis transported by the magnetic excitations of the system,\nso-called magnons. A first theoretical description of such\na magnonic spin Seebeck effect was developed by Xiao et\nal.7. With a two temperature model including the local\nmagnon (m) and phonon (p) temperatures the measured\nspin Seebeck voltage is calculated to be linearly depen-\ndent on the local difference between magnon and phonon\ntemperature, ∆ Tmp=Tm−Tp. This temperature dif-\nference decays with the characteristic lengthscale λ. For\nthe ferromagnetic material YIG they estimate the length\nscale in the range of several millimeters.\nThe contribution of exchange dominated magnons to\nthe spin Seebeck effect was investigated in recent experi-\nments by Agrawalet al.8. Using Brillouin lightscattering\nthe difference between the magnon and the phonon tem-\nperature in a system with a linear temperature gradient\nwas determined. They found no detectable temperature\ndifference and estimate a maximal characteristic length\nscale of the temperature difference of 470 µm. One pos-\nsible conclusion from this results might be be that in-\nstead of exchangemagnons, magnetostatic modes mainlycontribute to the spin Seebeck effect and are responsible\nfor the long-range character of this effect. Alternatively,\nphononsmightcontributetothemagnonaccumulationas\nwell viaspin-phonon drag9,10. A complete understanding\nof these different contributions to the spin Seebeck effect\nis still missing.\nIn this paper thermally excited magnonic spin currents\nand their length scale of propagation are investigated.\nUsingatomisticspinmodelsimulationwhichdescribethe\nthermodynamics of the magnetic system in the classical\nlimit including the whole frequency spectra of excited\nmagnons,wedescribespincurrentsbyexchangemagnons\nin the vicinity of a temperature step. After introducing\nour model, methods and basic definitions in Section II\nwe determine the magnon accumulation as well as the\ncorresponding magnon temperature and investigate the\ncharacteristic lengthscale of the decay of the magnon ac-\ncumulation in Section III. In Section IV we introduce\nan analytical description which is supported by our sim-\nulations shown in Section V and gives insight into the\nmaterial properties dependence of magnon propagation.\nII. MAGNETIZATION PROFILE AND\nMAGNON TEMPERATURE\nFor the investigationofmagnonic spin currentsin tem-\nperature gradients we use an atomistic spin model with\nlocalized spins Si=µi/µsrepresenting the normalized\nmagnetic moment µsof a unit cell. The magnitude of\nthe magnetic moment is assumed to be temperature in-\ndependent. Wesimulateathree-dimensionalsystemwith\nsimple cubic lattice structure and lattice constant a. The\ndynamics of the spin system are described in the classical\nlimit by solving the stochastic Landau-Lifshitz-Gilbert\n(LLG) equation,\n∂Si\n∂t=−γ\nµs(1+α2)Si×(Hi+α(Si×Hi)), (1)\nnumerically with the Heun method11withγbeing the\ngyromagnetic ratio. This equation describes a preces-\nsion of each spin iaround its effective field Hiand the2\ncoupling with the lattice by a phenomenological damp-\ning term with damping constant α. The effective field\nHiconsists of the derivative of the Hamiltonian and an\nadditional white-noise term ζi(t),\nHi=−∂H\n∂Si+ζi(t) . (2)\nThe Hamiltonian Hin our simulation includes exchange\ninteraction of nearest neighbors with isotropic exchange\nconstant Jand an uniaxial anisotropy with an easy axis\ninz-direction and anisotropy constant dz,\nH=−J\n2/summationdisplay\n<i,j>SiSj−dz/summationdisplay\niS2\ni,z. (3)\nThe additional noise term ζi(t) of the effective field Hi\nincludes the influence of the temperature and has the\nfollowing properties:\n/angbracketleftζ(t)/angbracketright= 0 (4)\n/angbracketleftBig\nζi\nη(0)ζj\nθ(t)/angbracketrightBig\n=2kBTpαµs\nγδijδηθδ(t) . (5)\nHerei,jdenote lattice sites and ηandθCartesian com-\nponents of the spin.\nWe simulate a model with a given phonon temperature\nTpwhich is space dependent and includes a temperature\nstep inz-direction in the middle of the system at z= 0\nfromatemperature T1\npinthehotterareato T2\np= 0K(see\nFig. 1). We assume, that this temperature profile stays\nconstant during the simulation and that the magnetic\nexcitationshavenoinfluence onthe phonontemperature.\nThe system size is 8 ×8×512, large enough to minimize\nfinite-size effects.\nAll spins are initialized parallel to the easy-axis in z-\ndirection. Due tothetemperaturestepanon-equilibrium\nin the magnonic density of states is created. Magnons\npropagate in every direction of the system, but more\nmagnons exist in the hotter than in the colder part of\nthe system. This leads to a constant net magnon current\nfrom the hotter towards the colder area of the system.\nDue to the damping of the magnons the net current ap-\npears around the temperature step with a finite length\nscale.After an initial relaxation time the system reaches\na steadystate. In this steady state the averagedspin cur-\nrent from the hotter towardsthe colderregionis constant\nand so the local magnetization is time independent. We\ncan now calculate the local magnetization m(z) depend-\ning on the space coordinate zas the time average over\nall spins in the plane perpendicular to the z-direction.\nWe use the phonon temperature T1\np= 0.1J/kBin the\nheated area, the anisotropy constant dz= 0.1Jand vary\nthe damping parameter α. The resulting magnetization\nversus the space coordinate zfor different damping pa-\nrameters in a section around the temperature step is\nshown in Fig. 1. For comparison the particular equi-\nlibrium magnetization m0of the two regions is also cal-\nculated and shown in the figure.m0α= 1α= 0.1α= 0.06Tp\nspace coordinate z/a\nphonon temperature kBTp/Jmagnetization m0.1\n0\n403020100-10-20-30-401\n0.995\n0.99\n0.985\n0.98\n0.975\nFIG. 1. Steady state magnetization mand equilibrium mag-\nnetization m0over space coordinate zfor a given phonon\ntemperature profile and for different damping parameters α\nin a small section around the temperature step.\nFar away from the temperature step on both sides the\namplitudes of the local magnetization m(z) converge to\nthe equilibrium values, only in the vicinity of the tem-\nperature step deviations appear. These deviations de-\nscribe the magnon accumulation, induced by a surplus\nof magnons from the hotter region propagating towards\nthe colder one. This leads to a less thermal excitation\nin the hotter area and the value of the local magneti-\nzation increases. In the colder area the surplus of in-\ncoming magnons decrease the value of the local magneti-\nzation. For smaller values of αthe magnons can propa-\ngate overlargerdistances before they are finally damped.\nThis leads to a damping-dependent magnon accumula-\ntion which increases with decreasing damping constant\nα.\nForafurtheranalysisinthecontextofthespin-Seebeck\neffect we define a local magnon temperature Tm(z) via\nthe magnetization profile m(z). For that the equilib-\nrium magnetization m0(T) is calculated for the same\nmodel but homogeneous phonon temperature Tp. In\nequilibrium magnon temperature Tmand the phonon\ntemperature Tpare the same and we can determine the\nfitequilibrium data\nmagnon temperature kBTm/Jmagnetization m0\n10.90.80.70.60.50.40.30.20.101\n0.95\n0.9\n0.85\n0.8\n0.75\n0.7\n0.65\n0.6\nFIG. 2. Equilibrium magnetization m0over the magnon\ntemperature Tm. Red points show the simulated equilibrium\nmagnetization and the black line shows a fit of the data.3\nT0\nmα= 1α= 0.1α= 0.06Tp\nspace coordinate z/a\nphonon temperature kBTp/Jmagnon temperature kBTm/J 0.1\n0\n403020100-10-20-30-400.1\n0.08\n0.06\n0.04\n0.02\n0\nFIG. 3. Magnon temperature Tmover the space coordinate\nzfor different damping parameters αcorresponding to the\nresults in Fig. 1.\n(magnon) temperature dependence of the equilibrium\nmagnetization m0(Tm) of the system. The magnetiza-\ntion of the equilibrium system decreases for increasing\nmagnontemperatureasshownin Fig. 2 andthe behavior\ncan be described phenomenologically with a function12\nm0(T) = (1−Tm/Tc)βwhereTcistheCurietemperature.\nFitting ourdatawefind Tc= (1.3326±0.00015)J/kBand\nfor the exponent we get β= 0.32984±0.00065. This fit\nof the data is also shown in Fig. 2 and it is a good\napproximation over the whole temperature range. The\ninverse function is used in the following to determine the\nmagnon temperature for a given local magnetization and\nwith that a magnon temperature profile Tm(z).\nThe resulting magnon temperature profiles are shown\nin Fig. 3. Far away from the temperature step the\nmagnon temperature Tm(z) coincides with the given\nphonon temperature Tp, and deviations — dependent on\nthe damping constant α— appear only around the tem-\nperature step. These deviations correspond to those of\nthe local magnetization discussed in connection with Fig.\n1.\nIII. MAGNON PROPAGATION LENGTH\nTo describe the characteristic lengthscale of the\nmagnon propagationaround the temperature step we de-\nfine the magnon accumulation ∆ m(z) as the difference\nbetween the relative equilibrium magnetization m0(z) at\nthe given phonon temperature Tp(z) and the calculated\nlocal magnetization m(z):\n∆m(z) =m0(z)−m(z) . (6)\nWeinvestigatethemagnonpropagationinthecolderpart\nof the system, where Tp(z) = 0. For a small magnon\ntemperature, the temperature dependence of the magne-\ntization can be approximated as\nm(Tm)≈1−β\nTcTm. (7)α= 1.00α= 0.50α= 0.10α= 0.08α= 0.06\nspace coordinate z/amagnon accumulation ∆m\n2502001501005001\n10−2\n10−4\n10−6\n10−8\n10−10\nFIG. 4. Magnon accumulation ∆ mover space coordinate z\nin the colder region of the system at Tp= 0K for different\ndamping constants αshows exponential decay with magnon\npropagation length ξ. The points show the data from our\nsimulation and the lines the results from an exponential fit.\nThese linear equation is in agreement with an analytical\nsolution for low temperatures presented by Watson et\nal.12. For low phonon temperatures one obtains for the\ndifference between phonon and magnon temperature\n∆T=Tm−Tp=β\nTc∆m. (8)\nNote, that the proportionality between magnon accumu-\nlation and temperature difference holds for higher tem-\nperatures as well as long as magnon and phonon temper-\nature are sufficiently close so that a linear approximation\napplies,thoughtheproportionalityfactorincreases. Note\nalso, that this proportionalitywas determined in theoret-\nical descriptions of a magnonic spin Seebeck effect7. Our\nresults for the magnon accumulation should hence be rel-\nevantfortheunderstandingofthemagnonicspinSeebeck\nwhere the temperature difference between the magnons\nin the ferromagnet and the electrons in the non-magnet\nplays a key role.\nWe further investigate our model as before with a tem-\nperature in the heated area of T1\np= 0.1J/kB, anisotropy\nconstant dz= 0.1Jand different damping parameters.\nThe magnon accumulation ∆ mversus the space coordi-\nnatezin the colder region of the system at Tp= 0K is\nshown in Fig. 4. Apart from a sudden decay close to the\ntemperature step the magnon accumulation ∆ m(z) then\ndecays exponentially on a length scale that depends on\nthe damping constant α. To describe this decay we fit\nthe data with the function\n∆m(z) = ∆m(0)·e−z\nξ. (9)\nWe define the fitting parameter ξas the propagation\nlength of the magnons. Here, the deviations from the\nexponential decay at the beginning of the system are ne-\nglected. The fits for the data are shown in Fig. 4 as\ncontinuous lines.\nThe propagation length dependence on the damping\nparameter αis shown in Fig. 5. The values of the prop-\nagation length from our simulations, shown as points,4\ndz= 0.01Jdz= 0.05Jdz= 0.10Jdz= 0.50J\ndamping constant αpropagation length ξ/a\n1 0.1100\n10\n1\nFIG. 5. Magnon propagation length ξover the damping con-\nstantαfor different anisotropy constant dz. Numerical data\nis shown as points and the solid lines are from Eq. (19).\nare inversely proportional to the damping constant α\nand, furthermore, show also a strong dependence on the\nanisotropy constant dz. This behavior will be discussed\nin the next two sections with an analytical analysisof the\nmagnon propagation and an investigation of the frequen-\ncies of the propagating magnons. A simple approxima-\ntion for the propagation length leads to Eq. (19) which\nis also shown as solid lines in Fig. 5.\nIV. ANALYTICAL DESCRIPTION WITH\nLINEAR SPIN-WAVE THEORY\nFor the theoretical description of the magnon accu-\nmulation, excited by a temperature step in the system,\nwe solve the LLG equation (Eq. (1)), analytically in\nthe area with Tp= 0K. We consider a cubical system\nwith lattice constant awhere all spins are magnetized in\nz-direction parallel to the easy-axis of the system. As-\nsuming only small fluctuations in the x- andy-direction\nwe have Sz\ni≈1 andSx\ni,Sy\ni≪1. In that case we can\nlinearize the LLG-equation and the solution of the re-\nsulting equation consists of a sum over spin waves with\nwavevectors qand the related frequency ωqwhich decay\nexponentially in time dependent on their frequency and\nthe damping constant αof the system,\nS±\ni(t) =1√\nN/summationdisplay\nqS±\nq(0)e∓iqri±iωqt·e−αωqt. (10)\nThe frequency ωqof the magnons is described by the\nusual dispersion relation\n¯hωq=1\n(1+α2)/parenleftBig\n2dz+2J/summationdisplay\nθ(1−cos(qθaθ))/parenrightBig\n. (11)\nThe dispersion relation includes a frequency gap due to\nthe anisotropy constant and a second wavevector depen-\ndent term with a sum over the Cartesian components13.\nConsidering now the temperature step, magnons from\nthe hotter area propagate towards the colder one. Weinvestigate the damping process during that propagation\ninordertodescribethe propagatingfrequenciesaswellas\nto calculate the propagation length ξof the magnons for\ncomparisonwiththeresultsfromsectionIII.Themagnon\naccumulation will depend on the distance to the temper-\nature step and — for small fluctuations of the SxandSy\ncomponents — can be expressed as\n∆m(z) = 1−/angbracketleftSz(z)/angbracketright ≈1\n2/angbracketleftbig\nSx(z)2+Sy(z)2/angbracketrightbig\n, (12)\nwhere the brackets denote a time average. We assume\nthat the local fluctuations of the SxandSycomponents\ncan be described with a sum over spin waves with differ-\nent frequencies and damped amplitudes aq(z),\nSx(z) =/summationdisplay\nqaq(z)cos(ωqt−qr) , (13)\nSy(z) =/summationdisplay\nqaq(z)sin(ωqt−qr) . (14)\nIn that case for the transverse component of the magne-\ntization one obtains\n/angbracketleftbig\nSx(z)2+Sy(z)2/angbracketrightbig\n=/angbracketleftBigg/summationdisplay\nqaq(z)2/angbracketrightBigg\n, (15)\nwhere mixed terms vanish upon time averaging. The\nmagnon accumulation can be written as:\n∆m(z) =1\n2/angbracketleftBigg/summationdisplay\nqaq(z)2/angbracketrightBigg\n. (16)\nThe amplitude aq(z) of a magnon decays exponentially\nas seen in Eq. (10) dependent on the damping constant\nand the frequency of the magnons. In the next step we\ndescribe the damping process during the propagation of\nthe magnons. In the one-dimensional limit magnons only\npropagate in z-direction with velocity vq=∂ωq\n∂q. Then\nthe propagation time can be rewritten as t=z/vqand\nwe can describe the decay of the amplitude with aq(z) =\naq(0)·f(z) with a damping function\nf(z) = exp/parenleftBig\n−αωqz\n∂ωq\n∂qz/parenrightBig\n. (17)\nThe amplitudes are damped exponentially during the\npropagation which defines a frequency dependent propa-\ngation length\nξωq=/radicalbigg\nJ2−/parenleftBig\n1\n2(1+α2)(¯hωq−2dz)−J/parenrightBig2\nα(1+α2)¯hωq,(18)\nwhere we used γ=µs/¯h. In the low anisotropy limit\nthis reduces to ξωq=λ/παwhereλ= 2π/qis the wave\nlength of the magnons.5\nThe total propagation length is then the weighted av-\nerage over all the excited frequencies. The minimal fre-\nquency is defined by the dispersion relation with a fre-\nquency gap of ωmin\nq= 1/(¯h(1 +α2))2dz. For small fre-\nquencies above that minimum the velocity is small, so\nthe magnons are damped within short distances. Due to\nthe fact that the damping process is also frequency de-\npendent higher frequencies will also be damped quickly.\nIn the long wave length limit the minimal damping is\nat the frequency ωmax\nq≈4dz/(¯h(1 +α2)) which can be\ndetermined by minimizing Eq. (17) .\nIn a three-dimensional system, besides the z-\ncomponent of the wavevector, also transverse compo-\nnents of the wavevector have to be included. The\ndamping of magnons with transverse components of\nthe wavevector is higher than described in the one-\ndimensionalcase,becausethe additionaltransverseprop-\nagationincreasethepropagationtime. Inoursimulations\nthe cross-section is very small, so that transverse compo-\nnents of the wave-vectors are very high and get damped\nquickly. Thisfact andthe highdamping forhighfrequen-\ncies described in Eq. (17) can explain the very strong\ndamping at the beginning of the propagation shown in\nFig. 4.\nV. FREQUENCIES AND DAMPING OF\nPROPAGATING MAGNONS\nIn this section we investigate the frequency distribu-\ntionofthemagnonicspincurrentwhilepropagatingaway\nfrom the temperature step. First we determine the fre-\nquencies of the propagating magnons in our simulations\nwith Fourier transformation in time to verify our as-\nsumptions from the last section. As before a system of\n8×8×512spins with a temperature step in the center of\nthe system is simulated with an anisotropy of dz= 0.1J.\nThe temperature of the heated area is T1\np= 0.1J/kBand\nthe damping constant is α= 0.1. After an initial relax-\nation to a steady-state the frequency distribution of the\npropagating magnons in the colder area is determined by\nFourier transformation in time of S±(i) =Sx(i)±iSy(i).\nThe frequency spectra are averaged over four points in\nthex-y-plane and analyzed depending on the distance z\nof the plane to the temperature step.\nThe results for small values of zare shown in Fig. 6(a)\nand for higher values of z, far away from the temper-\nature step, for the regime of the exponential decay, in\nFig. 6(b). For small values of z, near the temperature\nstep, the frequency range of the propagating magnons is\nvery broad. The minimum frequency is given by ωmin\nq=\n2dz/(¯h(1+α2)) and far away from the temperature step\nthe maximum peak is around ωmax\nq= 4dz/(¯h(1 +α2)).\nThese characteristic frequencies are in agreement with\nour findings in section IV.\nFurthermore, a stronger damping for higher frequen-\ncies can be observed. This effect corresponds to the\nstrong damping of magnons with wavevector compo-10864204·10−3\n3·10−3\n2·10−3\n1·10−3\n0z= 20az= 10az= 1a(a)\nfrequency ¯hωq/Jamplitude |S+(ωq)|\n1.41.210.80.60.40.208·10−5\n6·10−5\n4·10−5\n2·10−5\n0ωminz= 100az= 90az= 80a(b)\nfrequency ¯hωq/Jamplitude |S+(ωq)|\nFIG.6. Amplitude |S+(ωq)|versusthefrequency ωqfor asys-\ntem with 8 ×8×512 spins. (a):after propagation over short\ndistances form 1 to 20 lattice constants. (b): after propaga -\ntion over longer distances from 80 to 100 lattice constants.\nnents transverse to the z-direction and it explains the\nhigher initial damping, which was seen in the magnon\naccumulation in Fig. 4. A much narrower distribu-\ntion propagates over longer distances and reaches the\narea shown in Fig. 6(b). In that area the damping\ncan be described by one-dimensional propagation of the\nmagnons in z-direction with a narrow frequency distri-\nbution around the frequency with the lowest damping\nωmax\nq= 4dz/(¯h(1+α2)).The wavelength and the belong-\ning group velocity of the magnons depending on their\nfrequency in the one-dimensional analytical model are\nshown in Fig.7(a). In the simulated system magnons\nwith the longest propagation length have a wavelength\nofλ= 14a. Depending on the ratio dz/Jthe wave-\nlength increases for systems with lower anisotropy. As\ndiscussed in the last chapter, magnons with smaller fre-\nquencies are less damped in the time domain, but due to\ntheir smaller velocity the magnons very close to the min-\nimum frequency also have a smaller propagation length.\nTo investigate the frequency-dependent damping-\nprocess during the propagation of the magnons we calcu-\nlatetheratiooftheamplitudeofthemagnons |S+(ωq,x)|\nforz= 80aandz= 80a+∆ with ∆ = 10 a,20a,50aand\nnormalize it to a damping per propagation of one spin.\nThe resulting ratios ( |S+(ωq,x)|/|S+(ωq,x−∆)|)1/∆are\nshown in Fig. 7 in comparison with the frequency-6\n2\n1.5\n1\n0.5\n0\n43.532.521.510.50100\n80\n60\n40\n20\n0ωmin\nqvqλ(a)\nfrequency ¯hωq/J\nvelocityvq¯h/(Ja)wave length λ/a\ndamping function∆ = 10a∆ = 20a∆ = 50a(b)\nfrequency ω[J/¯h]damping ratio\n10.90.80.70.60.50.40.30.21\n0.98\n0.96\n0.94\n0.92\n0.9\nFIG. 7. (a): Wavelength λand group velocity vqof the\nmagnons in a one-dimensional model dependent on the fre-\nquencyωq. (b):Damping ratio as explained in the text versus\nthe frequency ωqfor different distances ∆ and compared to\nthe damping function (Eq. (17)).\ndependent damping-function (Eq. (17)). The figure\nshows a good agreement between simulation and our an-\nalytical calculations.\nThese results explain the dependence of the magnon\npropagation length on the model parameters. The fre-\nquency with the maximal amplitude is determined by\nthe anisotropy constant. Under the assumption that the\nfrequency with the lowest damping is dominant and the\ncontribution of other frequencies can be neglected the\npropagation length can be calculated as\nξ=a\n2α/radicalbigg\nJ\n2dz, (19)\nwhere the square-root term is the domain wall width of\nthe model. This formula is also plotted in Fig. 5.\nThe comparison with our simulations shows good\nagreement though the equation above gives only the\npropagation of those magnons with the smallest damp-\ningduringthe propagation.Inthe consideredsystemwith\nα= 0.1 anddz= 0.1Jwe get a propagation length of\naboutξ= 11aat a wavelength of the magnons λ= 14a.\nFor smaller values of the anisotropy and smaller damp-\ning parameters the frequency distribution of the thermal\nmagnons is broader and Eq. (19) is an overestimation\nof the real propagation length since the magnon accu-mulation is no longer exponentially decaying due to the\nbroader spectrum of propagating frequencies. However\nwe would expect for soft ferromagnetic insulators with a\nsmalldamping constantof10−4−10−3and ananisotropy\nconstant in the range of 10−3J−10−2Ja propagation\nlength of 103a−105awhich would be in the micrometer-\nrange.\nVI. SUMMARY AND DISCUSSION\nUsing the frameworkof an atomistic spin model we de-\nscribe thermally induced magnon propagationin a model\ncontaining a temperature step. The results give an im-\npression of the relevant length scale of the propagation\nof thermally induced exchange magnons and its depen-\ndence on system parameters as the anisotropy, the ex-\nchange and the damping constant. In the heated area\nmagnons with a broad frequency distribution are gener-\natedandbecauseoftheverystrongdampingformagnons\nwith high frequency, especially those with wave-vector\ncomponents transverse to the propagation direction in\nz-direction, most of the induced magnons are damped\non shorter length scales. Behind this region of strong\ndamping near the temperature step, the propagation of\nmagnons is unidirectional and the magnon accumula-\ntion decays exponentially with the characteristic prop-\nagation length ξ. This propagation length depends on\nthe damping parameter but also on system properties as\nthe anisotropy of the system, because of the dependence\non the induced frequencies.\nIn contrast to long range magnetostatic spin waves,\nwhich can propagate over distances of some mm14,15, we\nfind that for exchange magnons the propagation length\nis considerably shorter and expect from our findings for\nsoft ferromagnetic insulators with a low damping con-\nstant a propagation length in the range of some µm for\nthose magnons close to the frequency gap and the lowest\ndamping. These findings will contribute to the under-\nstanding of length scale dependent investigations of the\nspin Seebeck effect8,16–18.\nRecent experiments investigate the longitudinal spin\nSeebeck effect, where the generated spin current\nlongitudinal to the applied temperature gradient is\nmeasured19–22. In this configuration Kehlberger et al.\nshow that the measured spin current is dependent on the\nthickness of the YIG layer and they observe a saturation\nofthespincurrentonalengthscaleof100 nm16. Thissat-\nuration can be explained by the lengthscale of the prop-\nagation of the thermally excited magnons. Only those\nmagnons reaching the YIG/Pt interface of the sample\ncontribute to the measured spin current and — as shown\nhere — exchange magnons thermally excited at larger\ndistances are damped before they can reach the inter-\nface. In this paper, we focus on the propagation length\nof those magnons with the lowest damping, however the\nlengthscale of the magnon accumulation at the end of\na temperature gradient is dominated by a broad range7\nof magnons with higher frequencies which are therefore\ndamped on shorter length scales.\nACKNOWLEDGMENTS\nThe authors would like to thank the Deutsche\nForschungsgemeinschaft (DFG) for financial support viaSPP 1538 “Spin Caloric Transport” and the SFB 767\n“Controlled Nanosystem: Interaction and Interfacing to\nthe Macroscale”.\n1G. E. W. Bauer, E. Saitoh, and B. J. van Wees,\nNature Mater. 11, 391 (2012).\n2G. E. W. Bauer, A. H. MacDonald, and S. Maekawa,\nSolid State Commun. 150, 459 (2010).\n3K. Uchida, S. Takahashi, K. Harii, J. Ieda,\nW. Koshibae, K. Ando, S. Maekawa, and E. Saitoh,\nNature455, 778 (2008).\n4E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara,\nAppl. Phys. Lett. 88, 182509 (2006).\n5K. Uchida, S. Takahashi, J. Ieda, K. Harii,\nK. Ikeda, W. Koshibae, S. 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Althammer, F. D. Czeschka, H. Huebl,\nM. S. Wagner, M. Opel, I.-M. Imort, G. Reiss,\nA. Thomas, R. Gross, and S. T. B. Goennenwein,\nPhys. Rev. Lett. 108, 106602 (2012).\n21D. Qu, S. Y. Huang, J. Hu, R. Wu, and C. L. Chien,\nPhys. Rev. Lett. 110, 067206 (2013).\n22T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou,\nD. Tian, H. Nakayama, X.-F. Jin, and E. Saitoh,\nPhys. Rev. Lett. 110, 067207 (2013)."    },    {        "title": "0811.4118v1.The_quantum_mechanical_basis_of_an_extended_Landau_Lifshitz_Gilbert_equation_for_a_current_carrying_ferromagnetic_wire.pdf",        "content": "arXiv:0811.4118v1  [cond-mat.mtrl-sci]  25 Nov 2008The quantum-mechanical basis of an extended\nLandau-Lifshitz-Gilbert equation for a\ncurrent-carrying ferromagnetic wire\nD.M. Edwards1and O. Wessely1,2\n1 Department of Mathematics, Imperial College, London SW7 2BZ, U nited Kingdom\n2 Department of Mathematics, City University,London EC1V 0HB, Un ited Kingdom\nE-mail:d.edwards@imperial.ac.uk\nAbstract. An extended Landau-Lifshitz-Gilbert (LLG) equation is introduced to\ndescribe the dynamics of inhomogeneous magnetization in a current -carrying wire.\nThe coefficients of all the terms in this equation are calculated quant um-mechanically\nfor a simple model which includes impurity scattering. This is done by co mparing\nthe energies and lifetimes of a spin wave calculated from the LLG equa tion and from\nthe explicit model. Two terms are of particular importance since they describe non-\nadiabatic spin-transfer torque and damping processes which do no t rely on spin-orbit\ncoupling. It is shown that these terms may have a significant influenc e on the velocity\nof a current-driven domain wall and they become dominant in the cas e of a narrow\nwall.\nPACS numbers:An extended Landau-Lifshitz-Gilbert equation 2\n1. Introduction\nThe effect of passing an electric current down a ferromagnetic wire is of great current\ninterest. If the magnetization is inhomogeneous it experiences a sp in-transfer torque\ndue to the current [1, 2, 3, 4]. The effect is described phenomenolo gically by adding\nterms to the standard LLG equation [5, 6]. The leading term in the spin -transfer\ntorque is an adiabatic one arising from that component of the spin po larization of the\ncurrent which is in the direction of the local magnetization. However , in considering\nthe current-induced motion of a domain wall, Li and Zhang [3, 4] foun d that below a\nvery large critical current the adiabatic term only deforms the wall and does not lead\nto continuous motion. To achieve this effect they introduced [7] a ph enomenological\nnon-adiabatic term associated with the same spin non-conserving p rocesses responsible\nfor Gilbert damping. Subsequently Kohno et al[8] derived a torque of the Zhang-Li\nformquantum-mechanically using amodel ofspin-dependent scatt ering fromimpurities.\nThis may arise from spin-orbit coupling on the impurities. More recent ly Wessely et\nal[9] introduced two further non-adiabatic terms in the LLG equation in order to\ndescribe their numerical calculations of spin-transfer torques in a domain wall. These\nquantum-mechanical calculationsusingtheKeldyshformalismwerem adeintheballistic\nlimit without impurities and with spin conserved. Other terms in the LLG equation,\ninvolving mixed space and time derivatives, have been considered by S obolevet al[12],\nTserkovnyak et al[10], Skadsen et al[11] and Thorwart and Egger [13].\nThe object of this paper is to give a unified treatment of all these te rms in the LLG\nequation and to obtain explicit expressions for their coefficients by q uantum-mechanical\ncalculations for a simple one-band model with and without impurity sca ttering. The\nstrategy adopted is to consider a uniformly magnetized wire and to c alculate the effect\nof a current onthe energy andlifetime of a long wavelength spin wave propagating along\nthe wire. It is shown in section 2 that coefficients of spin-transfer t orque terms in the\nLLG equation are directly related to qandq3terms in the energy and inverse lifetime\nof a spin wave of wave-vector q. The Gilbert damping parameter is the coefficient of the\nωterm in the inverse lifetime, where ωis the spin-wave frequency. It corresponds to the\ndamping of a q= 0 spin wave while higher order terms ωqandωq2relate to damping\nof spin waves with finite wave-vector q. The relation between the qterm in the spin\nwave energy and the adiabatic spin-transfer torque has been not iced previously [2, 14].\nWe find that the qterm in the spin wave lifetime relates to the Zhang-Li non-adiabatic\nspin transfer torque. Our result for the coefficient of the Zhang- Li term is essentially the\nsame as that obtained by Kohno et al[8] and Duine et al[15] but our derivation appears\nsimpler. The q3terms in the spin wave energy and lifetime are related to the additiona l\nnon-adiabatic torques we introduced into the LLG equation [9], toge ther with an extra\none arising from spin non-conserving scattering. Explicit expressio ns for the coefficients\nof these terms are obtained in section 3. In section 4 we discuss brie fly the importance\nof the additional terms in our extended LLG equation for current- driven motion of a\ndomain wall. Some conclusions are summarized in section 5.An extended Landau-Lifshitz-Gilbert equation 3\n2. The LLG equation and spin waves\nWe write our extended LLG equation in the dimensionless form\n∂s\n∂t+αs×∂s\n∂t+α1s×∂2s\n∂z∂t−α′\n1s×/parenleftbigg\ns×∂2s\n∂z∂t/parenrightbigg\n−α′\n2s×∂3s\n∂z2∂t−α2s×/parenleftbigg\ns×∂3s\n∂z2∂t/parenrightbigg\n=s×∂2s\n∂z2−bexts×ez−a∂s\n∂z−fs×∂s\n∂z\n+a1/braceleftBigg\ns×/parenleftbigg\ns×∂3s\n∂z3/parenrightbigg\n+/bracketleftBigg\ns·∂2s\n∂z2−1\n2/parenleftbigg∂s\n∂z/parenrightbigg2/bracketrightBigg\n∂s\n∂z/bracerightBigg\n−f1s×/bracketleftbigg\ns×∂\n∂z/parenleftbigg\ns×∂2s\n∂z2/parenrightbigg/bracketrightbigg\n+g1s×∂3s\n∂z3. (1)\nHeres(z,t) is a unit vector in the direction of the local spin polarisation, time tis\nmeasured in units of ( γµ0ms)−1and the coordinate zalong the wire is in units of the\nexchange length lex= (2A/µ0m2\ns)1/2. The quantities appearing here are the gyroscopic\nratioγ= 2µB//planckover2pi1,thepermeabilityoffreespace µ0andtwopropertiesoftheferromagnetic\nmaterial, namely the saturation magnetisation msand the exchange stiffness constant\nA.ezis a unit vector in the zdirection along the wire. The equation expresses the\nrate of change of spin angular momentum as the sum of various torq ue terms, of which\ntheα1,α′\n1,a,f,a1,f1andg1terms are proportional to the electric current flowing.\nThe second term in the equation is the standard Gilbert term, with da mping factor\nα, while the α′\n1andα′\n2terms introduce corrections for spin fluctuations of finite wave-\nvector. Skadsem et al[11] point out the existence of the α′\n2term but do not consider\nit further. It was earlier introduced by Sobolev et al[12] within a microscopic context\nbased on the Heisenberg model. The α1andα2terms are found to renormalise the spin\nwave frequency, but for the model considered in section 3 we find t hatα1is identically\nzero. We shall argue that this result is model-independent. Tserko vnyaket al[10] and\nThorwart and Egger [13] find non-zero values of α1which differ from each other by a\nfactor 2; they attribute this to their use of Stoner-like and s−dmodels, respectively.\nThorwartandEgger[13]alsofindthe α′\n1termandtheyinvestigatetheeffectof α1andα′\n1\nterms on domain wall motion. Their results are difficult to assess beca use the constant\n|s|= 1 is not maintained during the motion. In eq.(1) we have omitted term s involving\nthe second order time derivatives, whose existence was pointed ou t by Thorwart and\nEgger [13]; one of these is discussed briefly in section 3.2.\nThe first term on the right-hand side of eq. (1) is due to exchange s tiffness and the\nnext term arises from an external magnetic field Bextezwith dimensionless coefficient\nbext=Bext/µ0ms. The third term is the adiabatic spin transfer torque whose coefficie nt\nais simple and well-known. In fact [3, 4]\na=1\n2/planckover2pi1JP\neµ0m2\nslex(2)An extended Landau-Lifshitz-Gilbert equation 4\nwhereJis the charge current density and eis the electron charge (a negative quantity).\nThe spin polarisation factor P= (J↑−J↓)/(J↑+J↓), where J↑, J↓are the current\ndensities for majority and minority spin in the ferromagnet ( J=J↑+J↓). Eq.(2) is\nvalid for both ballistic and diffusive conduction . The fourth term on th e right-hand\nside of eq.(1) is the Zhang-Li torque which is often characterised [8 ] by a parameter\nβ=f/a. The next term is the E1term of eq.(7) in ref. [9]. It is a non-adiabatic torque\nwhich is coplanar with s(z) ifs(z) lies everywhere in a plane. As shown in ref. [9] it\nis thezderivative of a spin current , which is characteristic of a torque occ urring from\nspin-conserving processes. In fact this term takes the form\na1∂\n∂z/bracketleftBigg\ns×/parenleftbigg\ns×∂2s\n∂z2/parenrightbigg\n−1\n2s/parenleftbigg∂s\n∂z/parenrightbigg2/bracketrightBigg\n. (3)\nThef1term may be written in the form\n−f1/parenleftbigg\ns·∂s\n∂z×∂2s\n∂z2/parenrightbigg\ns+f1∂\n∂z/parenleftbigg\ns×∂2s\n∂z2/parenrightbigg\n. (4)\nIfs(z) lies in a plane, the case considered in ref. [9], the first term vanishes and we\nrecover the F1term of eq.(9) in ref. [9]. Its derivative form indicate that it arises fr om\nspin-conserving processes so we conclude that the coefficient f1is of that origin. This\nis not true of the last term in eq.(1) and we associate the coefficient g1with spin non-\nconserving processes. For a spin wave solution of the LLG equation , where we work\nonly to first order in deviations from a state of uniform magnetisatio n, the last three\nterms of eq.(1) may be replaced by the simpler ones\n−a1∂3s\n∂z3+(f1+g1)s×∂3s\n∂z3. (5)\nApart fromadditional terms, eq.(1) looksslightly different fromeq.( 7) ofref. [9] because\nwe use the spin polarisation unit vector srather than the magnetisation vector mand\ns=−m. Furthermore the dimensionless coefficients will take different nume rical values\nbecause we have used different dimensionless variables zandtto avoid introducing the\ndomain wall width which was specific to ref. [9]. The torques due to an isotropy fields\nwere also specific to the domain wall problem and have been omitted in e q.(1).\nWe suppose that the wire is magnetised uniformly in the zdirection and consider a\nspin wave as a small transverse oscillation of the spin polarisation abo ut the equilibrium\nstate or, when a current flows, the steady state. Thus we look fo r a solution of eq.(1)\nof the form\ns=/parenleftbig\ncei(qz−ωt),dei(qz−ωt),−1/parenrightbig\n(6)\nwhere the coefficients of the xandycomponents satisfy c≪1,d≪1. This represents\na spin wave of wave-vector qand angular frequency ωpropagating along the zaxis.\nWhen (6) is substituted into eq.(1) the transverse components yie ld, to first order in c\nandd, the equations\n−iλc+µd= 0, µc+iλd= 0 (7)An extended Landau-Lifshitz-Gilbert equation 5\nwhere\nλ=ω−aq+a1q3−α2ωq2+iα′\n1qω\nµ=−iαω+bext+q2+ifq+i(f1+g1)q3+α1ωq−iωq2α′\n2. (8)\nOn eliminating canddfrom eq.(7) we obtain λ2=µ2. To obtain a positive real part\nfor the spin wave frequency, we take λ=µ. Hence\nω/parenleftbig\n1−α1q−α2q2/parenrightbig\n=bext+aq+q2−a1q3\n+i/bracketleftbig\nω/parenleftbig\n−α−α′\n1q−α′\n2q2/parenrightbig\n+fq+(f1+g1)q3/bracketrightbig\n. (9)\nThus the spin wave frequency is given by\nω=ω1−iω2 (10)\nwhere\nω1≃/parenleftbig\n1−α1q−α2q2/parenrightbig−1/parenleftbig\nbext+aq+q2−a1q3/parenrightbig\nω2≃/parenleftbig\n1−α1q−α2q2/parenrightbig−1/bracketleftbig\nω1/parenleftbig\nα+α′\n1q+α′\n2q2/parenrightbig\n−fq−(f1+g1)q3/bracketrightbig\n.(11)\nHere we have neglected terms of second order in α,α′\n1,α′\n2,f,f1andg1, the coefficients\nwhich appear in the spin wave damping. This form for the real and imag inary parts\nof the spin wave frequency is convenient for comparing with the qua ntum-mechanical\nresults of the next section. In this way we shall obtain explicit expre ssions for all the\ncoefficients in the phenomenological LLG equation. Coefficients of od d powers of qare\nproportional to the current flowing whereas terms in even powers ofqare present in the\nequilibrium state with zero current.\n3. Spin wave energy and lifetimes in a simple model\nAs a simple model of an itinerant electron ferromagnet we consider t he one-band\nHubbard model\nH0=−t/summationdisplay\nijσc†\niσcjσ+U/summationdisplay\nini↑ni↓−µBBext/summationdisplay\ni(ni↑−ni↓), (12)\nwherec†\niσcreates an electron on site iwith spin σandniσ=c†\niσciσ. We consider a simple\ncubic lattice and the intersite hopping described by the first term is r estricted to nearest\nneighbours. The second term describes an on-site interaction bet ween electrons with\neffective interaction parameter U; the last term is due to an external magnetic field. It\nis convenient to introduce a Bloch representation, with\nc†\nkσ=1√\nN/summationdisplay\niek·Ric†\niσ, nkσ=c†\nkσckσ, (13)\nǫk=−t/summationdisplay\nieik·ρi=−2t(coskxa0+coskya0+coskza0). (14)An extended Landau-Lifshitz-Gilbert equation 6\nThe sum in eq.(13) is over all lattice cites Riwhereas in eq.(14) ρi=\n(±a0,0,0),(0,±a0,0),(0,0,±a0) are the nearest neighbour lattice sites. Then\nH0=/summationdisplay\nkσǫknkσ+U/summationdisplay\nini↑ni↓−µBBext/summationdisplay\nk(nk↑−nk↓). (15)\nTo discuss scattering of spin waves by dilute impurities we assume tha t the effect of\nthe scattering from different impurity sites adds incoherently; hen ce we may consider\ninitially a single scattering center at the origin, We therefore introdu ce at this site a\nperturbing potential u+vl·σ, wherel= (sinθcosφ,sinφsinθ,cosθ) is a unit vector\nwhose direction will finally be averaged over. uis the part of the impurity potential\nwhich is indepndent of the spin σand the spin dependent potential vl·σis intended to\nsimulateaspin-orbit L·σinteractionontheimpurity. Itbreaksspinrotationalsymmetry\nin the simplest possible way. Clearly spin-orbit coupling can only be trea ted correctly\nfor a degenerate band such as a d-band, where on-site orbital angular momentum L\noccurs naturally. The present model is equivalent to that used by K ohnoet al[8] and\nDuineet al[15]. In Bloch representation the impurity potential becomes V=V1+V2\nwith\nV1=v↑1\nN/summationdisplay\nk1k2c†\nk1↑ck2↑+v↓1\nN/summationdisplay\nk1k2c†\nk1↓ck2↓\nV2=ve−iφsinθ1\nN/summationdisplay\nk1k2c†\nk1↑ck2↓+veiφsinθ1\nN/summationdisplay\nk1k2c†\nk1↓ck2↑ (16)\nandv↑=u+vcosθ,v↓=u−vcosθ. To avoid confusion we note that the spin\ndependenceoftheimpuritypotentialwhichoccursinthemany-bod yHamiltonian H0+V\nis not due to exchange, as would arise in an approximate self consiste nt field treatment\n(e.g. Hartree-Fock) of the interaction Uin a ferromagnet.\n3.1. Spin wave energy and wave function\nIn this section we neglect the perturbation due to impurities and det ermine expressions\nfor the energy and wave function of a long-wave length spin wave in t he presence of an\nelectric current. The presence of impurities is recognised implicitly sin ce the electric\ncurrent is characterised by a perturbed one-electron distributio n function fkσwhich\nmight be obtained by solving a Boltzman equation with a collision term. We consider\na spin wave of wave-vector qpropagating along the zaxis, which is the direction\nof current flow. Lengths and times used in this section and the next , except when\nspecified, correspond to actual physical quantities, unlike the dim ensionless variables\nused in section 2.\nWe first consider the spin wave with zero electric current and treat it, within\nthe random phase approximation (RPA), as an excitation from the H artree-Fock (HF)\nground state of the Hamiltonian (15). The HF one electron energies are given by\nEkσ=ǫk+U/angbracketleftn−σ/angbracketright−µBσBext (17)An extended Landau-Lifshitz-Gilbert equation 7\nwhereσ= 1,−1 for↑and↓respectively, and /angbracketleftn−σ/angbracketrightis the number of −σspin electrons\nper site. In a self-consistent ferromagnetic state at T= 0,/angbracketleftnσ/angbracketright=N−1/summationtext\nkfkσand\nn=/summationtext\nσ/angbracketleftnσ/angbracketright, where, fkσ=θ(EF−Ekσ),nis the number of electrons per atom, and\nEFis the Fermi energy. Nis the number of lattice sites and θ(E) is the unit step\nfunction. The spin bands Ekσgiven by eq.(17) are shifted relative to each other by an\nenergy ∆+2 µBBextwhere ∆ = U/angbracketleftn↑−n↓/angbracketrightis the exchange splitting. The ground state\nis given by |0/angbracketright=/producttext\nkσc†\nkσ|/angbracketrightwhere|/angbracketrightis the vacuum state and the product extends over\nall states kσsuch that fkσ= 1. Within the RPA, the wave function for a spin wave of\nwave-vector q, excited from the HF ground state, takes the form\n|q/angbracketright=Nq/summationdisplay\nkAkc†\nk+q↓ck↑|0/angbracketright (18)\nwhereNqis a normalisation factor. The energy of this state may be written\nEq=Egr+/planckover2pi1ωq=Egr+2µBBext+/planckover2pi1ω′\nq (19)\nwhereEgris the energy of the HF ground state and /planckover2pi1ωqis the spin wave excitation\nenergy. On substituting (18) in the Schr¨ odinger equation ( H0−Eq)|q/angbracketright= 0 and\nmultiplying on the left by /angbracketleft0|c†\nk′↑ck′−q↓, we find\nAk′/parenleftbig\nǫk′+q−ǫk′+∆−/planckover2pi1ω′\nq/parenrightbig\n=U\nN/summationdisplay\nkAkfk↑(1−fk+q↓). (20)\nHence we may take\nAk= ∆/parenleftbig\nǫk+q−ǫk+∆−/planckover2pi1ω′\nq/parenrightbig−1(21)\nand, for small q,/planckover2pi1ω′\nqsatisfies the equation\n1 =U\nN/summationdisplay\nkfk↑−fk+q↓\nǫk+q−ǫk+∆−/planckover2pi1ω′q. (22)\nThisistheequationforthepolesofthewell-knownRPAdynamicalsus ceptibility χ(q,ω)\n[16]. The spin wave pole is the one for which /planckover2pi1ω′\nq→0 asq→0.\nTo generalise the above considerations to a current-carrying sta te we proceed as\nfollows. We re-interpret the state |0/angbracketrightsuch that /angbracketleft0|...|0/angbracketrightcorresponds to a suitable\nensemble average with a modified one-electron distribution fkσ. When a current flows\nin thezdirection we may consider the ↑and↓spin Fermi surfaces as shifted by small\ndisplacement δ↑ˆkz,δ↓ˆkzwhereˆkzis a unit vector in the zdirection. Thus\nfkσ=θ(EF−Ek+δσˆkz,σ)\n≃θ(EF−Ekσ)−δσδ(EF−Ekσ)∂ǫk\n∂kz(23)\nand the charge current density carried by spin σelectrons is\nJσ=e\n/planckover2pi1Na3\n0/summationdisplay\nk∂ǫk\n∂kzfkσ=−eδσ\n/planckover2pi1Na3\n0/summationdisplay\nk/parenleftbigg∂ǫk\n∂kz/parenrightbigg2\nδ(EF−Ekσ)\n=−eδσ\n/planckover2pi1a3\n0/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\nσρσ(EF) (24)An extended Landau-Lifshitz-Gilbert equation 8\nwhere/angbracketleft(∂ǫk/∂kz)2/angbracketrightσis an average over the σspin Fermi surface and ρσ(EF) is the\ndensity of σspin states per atom at the Fermi energy. We shall also encounter the\nfollowing related quantities;\nKσ=1\nN∆2a3\n0/summationdisplay\nk∂ǫk\n∂kz∂2ǫk\n∂k2zfkσ\n=/planckover2pi1Jσ\n∆2e/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2∂2ǫk\n∂k2z/angbracketrightBigg\nσ/slashbigg/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\nσ(25)\nLσ=1\nN∆3a3\n0/summationdisplay\nk/parenleftbigg∂ǫk\n∂kz/parenrightbigg3\nfkσ\n=/planckover2pi1Jσ\n∆3e/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg4/angbracketrightBigg\nσ/slashbigg/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\nσ. (26)\nTo derive eqs.(25) and (26), δσhas been eliminated using eq.(24).\nTo solve eqn.(22) for /planckover2pi1ω′\nqwe expand the right-hand side of the equation in powers\nof (ǫk+q−ǫk−/planckover2pi1ω′\nq)/∆ and make the further expansions\nǫk+q−ǫk=q∂ǫk\n∂kz+1\n2q2∂2ǫk\n∂k2z+1\n6q3∂3ǫk\n∂k3z... (27)\n/planckover2pi1ω′\nq=Bq+Dq2+Eq3+... (28)\nin powers of q. We retain all terms up to q3except those involving B2; the coefficients\nBandEare proportional to the current and we keep only terms linear in the current.\nHence we find a solution of eq.(22) in the form (28) with\nB=1\nN↑−N↓/summationdisplay\nk(fk↑−fk↓)∂ǫk\n∂kz=Na3\n0\nN↑−N↓/planckover2pi1\ne(J↑−J↓) (29)\nD=1\nN↑−N↓/bracketleftBigg\n1\n2/summationdisplay\nk(fk↑+fk↓)∂2ǫk\n∂k2z−1\n∆/summationdisplay\nk(fk↑−fk↓)/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/bracketrightBigg\n(30)\nE=−a2\n0B\n6\n+B\n(N↑−N↓)∆/bracketleftBigg/summationdisplay\nk(fk↑+fk↓)∂2ǫk\n∂k2\nz−3\n∆/summationdisplay\nk(fk↑−fk↓)/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/bracketrightBigg\n−Ua3\n0/summationdisplay\nσ(Kσ−σLσ). (31)\nHereNσis the total number of σspin electrons so that Nσ=N/angbracketleftnσ/angbracketright.\nIn the absence of spin-orbit coupling the expression for Bin terms of spin current\nis a general exact result even in the presence of disorder, as show n in Appendix A. The\ncoefficient Dis the standard RPA spin-wave stiffness constant (e.g ref. [16]). W e note\nthat, in the limit ∆ → ∞, E takes the simple form −a2\n0B/6.\nOn restoring the correct dimensions (as indicated after eq.(1)) to the expression\nforω1in eq.(11) we may determine the coefficients aanda1by comparing with theAn extended Landau-Lifshitz-Gilbert equation 9\nequation\n/planckover2pi1ωq= 2µBBext+Bq+Dq2+Eq3. (32)\nFrom the coefficient of qwe have\na+α1bext=B/(2µBµ0mslex). (33)\naandBare both determined directly from the spin current JPindependently of a\nparticular model (see appendix A) so that bextshould not enter their relationship. We\nconclude quite generally that α1= 0. In this case we find that on combining eqs.(33)\nand (29), and noting that ms=−µB(N↑−N↓)/Na3\n0, eq.(2) is obtained as expected.\nIn section 3.2 we show explicitly for the present model that α1= 0. This conflicts\nwith the results of refs.[10] and [13]. From the coefficients of q2in eqs.(11) and (32) we\nfind 1+α2bext=D/(4µBA/ms). Thus an external field slightly disturbs the standard\nrelationA=Dms/4µB. However in the spirit of the LLG equation we take Aand\nms, which enter the units of length and time used in eq.(1), to be consta nts of the\nferromagnetic material in zero external field. The coefficients of q3in eqs.(11) and (32)\nyield the relation (taking α1= 0),\n−a1+α2a=E//parenleftbig\n2µBµ0msl3\nex/parenrightbig\n. (34)\nWe defer calculation of α2until section 3.2 and the result is given in eq.(44). Combining\nthis with eqs.(34) and (31) we find\n2µBµ0msl3\nexa1=a2\n0B\n6−2BD\n∆+Ua3\n0/summationdisplay\nσ(Kσ−σLσ). (35)\nWe have thus derived an explicit expression , for a simple model, for th e coefficient a1\nof a non-adiabatic spin torque term which appears in the LLG equatio n (1). We have\nneglected the effect of disorder due to impurities . In the absence o f spin-orbit coupling\nthe expression for the adiabatic torque coefficient a, given by eq.(2), is exact even in\npresence of impurities. In the next section we shall calculated furt her non-adiabatic\ntorque terms, with coefficients f1andg1, as well as damping coefficients α,α′\n1andα′\n2.\nIn the present model all these depend on impurity scattering for t heir existence.\n3.2. Spin wave lifetime\nThe solutions of eq.(22) are shown schematically in figure 1. They inclu de the spin\nwave dispersion curve and the continuum of Stoner excitations c†\nk+q↓ck↑|0/angbracketrightwith energies\nEk+q↓−Ek↑. The Zeeman gap2 µBBextinthe spin wave energy at q= 0 doesnot appear\nbecause we have plotted /planckover2pi1ω′\nqrather than /planckover2pi1ωq(see eq.(19)). Within the present RPA the\nspin wave in a pure metal has infinite lifetime outside the continuum and cannot decay\ninto Stoner excitations owing to conservation of the momentum q. However, when the\nperturbation V1due to impurities is introduced (see eqn.(16)), crystal momentum is no\nlonger conserved and such decay processes can occur. These ar e shown schematically\nby the dotted arrow in figure 1. If the bottom of the ↓spin band lies above the Fermi\nlevel there is a gap in the Stoner spectrum and for a low energy (sma llq) spin waveAn extended Landau-Lifshitz-Gilbert equation 10\nFigure 1. Spin-flip excitations from the ferromagnetic ground state. The do tted\narrow shows the mechanism of decay of a spin wave into Stoner excit ations which is\nenabled by the impurity potential V1.\nsuch processes cannot occur. However the spin-flip potential V2enables the spin wave\nto decay into single particle excitations c†\nk+qσckσ|0/angbracketrightabout each Fermi surface and these\ndo not have an energy gap.\nThe lifetime τ−1\nqof a spin wave of wave-vector qis thus given simply by the “golden\nrule” in the form\nτ−1\nq=2π\n/planckover2pi1Nimp(T1+T2) (36)\nwhereNinpis the number of impurity sites and\nT1=/summationdisplay\nkp/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\n0/vextendsingle/vextendsingle/vextendsinglec†\nk↑cp↓V1/vextendsingle/vextendsingle/vextendsingleq/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\nfk↑(1−fp↓)δ(/planckover2pi1ωq−Ep↓+Ek↑)\nT2=/summationdisplay\nkpσ/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\n0/vextendsingle/vextendsingle/vextendsinglec†\nkσcpσV2/vextendsingle/vextendsingle/vextendsingleq/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\nfkσ(1−fpσ)δ(/planckover2pi1ωq−ǫp+ǫk). (37)\nWe first consider T1and, using eqns.(16) and (18), we find\n/angbracketleftBig\n0/vextendsingle/vextendsingle/vextendsinglec†\nk↑cp↓V1/vextendsingle/vextendsingle/vextendsingleq/angbracketrightBig\n=Nq\nNfk↑(1−fp↓)[Akv↓(1−fp↓)−Ap−qv↑fp−q↑]\n=Nq\nNfk↑(1−fp↓)(Akv↓−Ap−qv↑) (38)\nfor small q. The last line follows from two considerations. Firstly, because of th eδ-\nfunction in eq.(37) we can consider the states k↑andp↓to be close to their respective\nFermi surfaces. Secondly the ↓spin Fermi surface lies within the ↑Fermi surface and q\nis small. Hence\nT1=N2\nq\nN2/summationdisplay\nkpfk↑(1−fp↓)δ(/planckover2pi1ωq−Ep↓+Ek↑)(Akv↓−Ap−qv↑)2.(39)An extended Landau-Lifshitz-Gilbert equation 11\nTo evaluate this expression in the case when a current flows we use t he distribution\nfunction fkσgiven by eq.(23). Thus, neglecting a term proportional to the squa re of the\ncurrent, we have\nT1=N2\nq\nN2/summationdisplay\nkpδ(/planckover2pi1ωq−Ep↓+Ek↑)(Akv↓−Ap−qv↑)2\n×/bracketleftbigg\nθ(EF−Ek↑)θ(Ep↓−EF)−δ↑θ(Ep↓−EF)δ(EF−Ek↑)∂ǫk\n∂kz\n+δ↓θ(EF−Ek↑)δ(EF−Ep↓)∂ǫp\n∂pz/bracketrightbigg\n. (40)\nWe wish to expand this expression, and a similar one for T2, in powers of qtoO(q3) so\nthat we can compare with the phenomenological expression (eq.(11 )) for the imaginary\npartofthespinwave frequency, which isgiven by τ−1\nq/2. Itisstraight-forwardtoexpand\nthe second factor in the above sum by using eqs.(21) and (28). We s hall show that the\ncontribution to T1of the first term in square brackets in eq.(40) leads to a contributio n\nproportional to spin wave frequency ωq. Together with a similar contribution to T2it\nyields the Gilbert damping factor αas well as the coefficients α′\n1,α′\n2of the terms in\neq.(11) which give the qdependence of the damping. The remaining terms in eq.(40)\nyield the spin-transfer torque coefficients f,f1andg1.\nThe normalisation factor N2\nqwhich appear in eq.(40) leads naturally to the factor\n(1−α1q−α2q2)−1which appears in eq.(11). From eq.(18) it is given by\n1 =/angbracketleftq|q/angbracketright=N2\nq\nN/summationdisplay\nk/parenleftbig\nA2\nkfk↑−A2\nk−qfk↓/parenrightbig\n. (41)\nBy expanding A2\nk−qin powers of q, and using eq.(23), we find to O(q2) that\nN−2\nq= (N↑−N↓)\n×/braceleftBigg\n1+q2\n∆2(N↑−N↓)/summationdisplay\nk/parenleftbigg∂ǫk\n∂kz/parenrightbigg2\n[θ(EF−Ek↑)−θ(EF−Ek↓)]/bracerightBigg\n.(42)\nWe deduce that\nα1= 0 (43)\nand\nα2=−1\nl2\nex∆2(N↑−N↓)/summationdisplay\nk/parenleftbigg∂ǫk\n∂kz/parenrightbigg2\n[θ(EF−Ek↑)−θ(EF−Ek↓)].(44)\nThe result α1= 0, which was predicted on general grounds in section 3.1 and in\nAppendix 1, arises here through the absence of a qterm, proportional to current,\nin the spin wave normalisation factor. In the derivation of eq.(42) th is occurs due\nto a cancellation involving the Bqterms in the spin energy, which appears in Ak.\nWithout this cancellation we would have α1= 2B/lex∆ which is of the form obtained\nby Tserkovnyak et al[10] and Thorwart and Egger [13].An extended Landau-Lifshitz-Gilbert equation 12\nWe now return to the programme for calculating the LLG coefficients\nα,α′\n1,α′\n2,f,f1,g1which was outlined after eq.(40). We have seen that the qdependence\nofN2\nqcorresponds to the prefactor in eq.(11). Hence to determine the coefficients listed\nabove we can take N2\nq=N2\n0= (N↑−N↓)−1inT1andT2when we expand terms in\npowers of qto substitute in eq.(36) and compare with eq.(11). We first consider the\ncaseq= 0 in order to determine the Gilbert damping factor α. Thus only the first term\nin square brackets in eq.(40) contributes, since ∂ǫk/∂kzis an odd function kz, and\nT1(q= 0) =4v2cos2θ\nN↑−N↓\n×N−2/summationdisplay\nkpδ(/planckover2pi1ω0−Ep↓+Ek↑)θ(EF−Ek↑)θ(Ep↓−EF) (45)\nwherecos2θis an average over the angle appearing in the impurity potential V(eq.(16))\nand we shall assume cosθ= 0. The summations in eq.(45) may be replaced by energy\nintegrals involving the density of states of per atom ρσ(ǫ) of the states Ekσ. Then, to\norder (/planckover2pi1ω0)2,\nT1(q= 0) =/bracketleftBigg\n4v2cos2θ\nN↑−N↓/bracketrightBigg/bracketleftbigg\n/planckover2pi1ω0ρ↑ρ↓+1\n2(/planckover2pi1ω0)2/parenleftbig\nρ↑ρ′\n↓−ρ′\n↑ρ↓/parenrightbig/bracketrightbigg\n(46)\nwhereρσ(ǫ) and its derivative ρ′\nσ(ǫ) are evaluated at ǫ=EF. Similarly\nT2(q= 0) =/bracketleftBigg\nv2sin2θ\nN↑−N↓/bracketrightBigg\n/planckover2pi1ω0/parenleftbig\nρ2\n↑+ρ2\n↓/parenrightbig\n(47)\nand noω2\n0terms appear. We have included the ω2\n0term in eq.(46) merely because it\ncorresponds to a term s×/parenleftBig\ns×∂2s\n∂t2/parenrightBig\nin the LLG equation whose existence was noted by\nThorwald and Egger [13]. We shall not pursue terms with second-ord er time derivatives\nany further. Since the imaginary part of the spin wave frequency is given by τ−1\nq/2 it\nfollows from eqs.(11), (36), (46) and (47) that\nα=πcv2\n/angbracketleftn↑−n↓/angbracketright/bracketleftBig\n4cos2θρ↑ρ↓+sin2θ/parenleftbig\nρ2\n↑+ρ2\n↓/parenrightbig/bracketrightBig\n, (48)\nwherec=Nimp/Nis the concentration of impurities, in agreement with Khono et al[8]\nand Duine et al[15]. If the direction of the spin quantisation axis of the impurities is\ndistributed randomly cos2θ= 1/3,sin2θ= 2/3 so that αis proportional to ( ρ↑+ρ↓)2.\nTo investigate the qdependence of Gilbert damping, and thus evaluate α′\n1andα′\n2\nin eq.(11), the second factor in the summation of eq.(40) must be ex panded in powers\nofq. All the terms which contribute to the sum are of separable form g(k)h(p). The\ncontribution to T1of interest here , proportional to ωq, again arises from the first term\nin square brackets in eq.(40), and similarly for T2. The summations required in eq.(40)\nare of the form\n/summationdisplay\nkpδ(/planckover2pi1ωq−Ep↓+Ek↑)θ(EF−Ek↑)θ(Ep↓−EF)g(k)h(p)\n=/angbracketleftg(k)/angbracketright↑/angbracketlefth(k)/angbracketright↓ρ↑ρ↓/planckover2pi1ωq (49)An extended Landau-Lifshitz-Gilbert equation 13\nwhere/angbracketleftg(k)/angbracketrightσ=N−1/summationtext\nkg(k)δ(EF−Ekσ) is an average over the Fermi surface, as used\npreviously in section 3.1. After some algebra we find\nα′\n1= 2Bα/∆lex (50)\nα′\n2=πc\n/angbracketleftn↑−n↓/angbracketrightl2\nex∆2/braceleftbigg\nρ↑ρ↓/parenleftBig\nu2+5v2cos2θ/parenrightBig/summationdisplay\nσ/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\nσ\n−2ρ↑ρ↓∆v2cos2θ/summationdisplay\nσσ/angbracketleftbigg∂2ǫk\n∂k2z/angbracketrightbigg\nσ\n−v2sin2θ/bracketleftBigg\n∆/summationdisplay\nσσρ2\nσ/angbracketleftbigg∂2ǫk\n∂k2z/angbracketrightbigg\nσ−3/summationdisplay\nσρ2\nσ/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\nσ/bracketrightBigg/bracerightbigg\n+2Dα\n∆l2ex. (51)\nWe note that, unlike αandα′\n1, the coefficient α′\n2is non-zero even when the spin-\ndependent part of the impurity potential, v, is zero. In this case the damping of a\nspin wave of frequency ωand small wave-vector qis proportional to ρ↑ρ↓u2ωq2. In\nzero external field ω∼q2so that the damping is of order q4. This damping due to\nspin-independent potential scattering by impurities was analysed in detail by Yamada\nand Shimizu [17]. One of the Fermi surface averages in eq.(51) is easily evaluated using\neqs.(14) and (17). Thus/angbracketleftbigg∂2ǫk\n∂k2\nz/angbracketrightbigg\nσ=−a2\n0\n3/angbracketleftǫk/angbracketrightσ=−a2\n0\n3(Ef−U/angbracketleftn−σ/angbracketright+σµBBext). (52)\nIn the spirit of the LLG equation we should take Bext= 0 in evaluating the coefficients\nα′\n2.\nWe now turn to the evaluation of the non-adiabatic spin-transfer t orque coefficients\nf,f1andg1. These arise from the second and third terms in square brackets in eq.(40),\nand in a similar expression for T2. The summations involved in these terms differ from\nthose in eq.(49) since one θ-function is replaced by a δ-function. This leads to the\nomission of the frequency factor /planckover2pi1ωq. The Fermi surface shifts δσare elininated in\nfavour of currents Jσby using eq.(24).\nBy comparing the coefficient of qin the expansion of eq.(36) with that in eq.(11)\nwe find the coefficient of the Zhang-Li torque in the form\nf=πcv2\nµ0m2s∆lex/planckover2pi1\ne/bracketleftBig\n2cos2θ(ρ↑J↓−ρ↓J↑)+sin2θ(ρ↓J↓−ρ↑J↑)/bracketrightBig\n.(53)\nThis is in agreement with Khono et al[8] and Duine et al[15]. In the “isotropic”\nimpurity case, with cos2θ= 1/3,sin2θ= 2/3, it follows from eqs.(53), (48) and (2) that\nβ=f\na=α2\nU(ρ↑+ρ↓). (54)\nIn the limit of a very weak itinerant forromagnet ρσ→ρ, the paramagnetic density of\nstates, and Uρ→1 by the Stoner criterion. Thus in this limit β=α. Tserkovnyak etAn extended Landau-Lifshitz-Gilbert equation 14\nal[10] reached a similar conclusion. For a parabolic band it is straightfor ward to show\nfrom Stoner theory that β/α >1 and may be as large as 1.5.\nAs discussed in section 2 the coefficient f1is associated with spin-conserving\nprocesses, and hence involves the spin independent potential u. The coefficient g1\nis associated with spin non-conserving processes and involves v. By comparing the\ncoefficient of q3in the expansion of eq.(36) with that in eq.(11) we deduce that\nf1=πc\n2µ0m2sl3exu2(K1+2L1+M1) (55)\nand\ng1=1\nl2ex/parenleftbigg3D\n∆−a2\n0\n6/parenrightbigg\nf\n+πcv2\n2µ0m2sl3ex/bracketleftBig\ncos2θ(5K1+6L1−M1)+sin2θ(3K2+4L2)/bracketrightBig\n. (56)\nHere\nK1=K↓ρ↑+K↑ρ↓, K2=K↓ρ↓+K↑ρ↑\nL1=L↓ρ↑−L↑ρ↓, L2=L↓ρ↓−L↑ρ↑\nM1=/planckover2pi1\ne∆3/summationdisplay\nσ/bracketleftBigg\n2σ/angbracketleftBigg/parenleftbigg∂ǫk\n∂kz/parenrightbigg2/angbracketrightBigg\n−σ+∆/angbracketleftbigg∂2ǫk\n∂k2z/angbracketrightbigg\n−σ/bracketrightBigg\nJσρ−σ. (57)\nThis complete the derivation of expressions for all the LLG coefficien ts of eq.(1) within\nthe present impurity model\n4. The extended LLG equation applied to current-driven doma in wall\nmotion\nIn a previous paper [9] we introduced the a1andf1terms of the extended LLG equation\n(cf. eqns.(1), (3) and (4)) in order to describe numerically-calcula ted spin-transfer\ntorques acting on a domain wall when it is traversed by an electric cur rent. In that\nwork the origin of the small f1term for a pure ferromagnetic metal was specific to\nthe domain wall problem; it was shown to be associated with those elec tronic states at\nthe bulk Fermi surface which decay exponentially as they enter the wall. The analytic\nderivation of f1in section 3 (see eqn.(55)) is based on impurity scattering in the bulk\nferromagnet and applies generally to any slowly-varying magnetizat ion configuration.\nFor a ferromagnetic alloy such as permalloy both mechanisms should c ontribute in the\ndomain wall situation but the impurity contribution would be expected to dominate.\nTo describe a domain wall we must add to the right-hand side of eqn.(1 ) anisotropy\nterms of the form\n−(s·ey)s×ey+b−1(s·ez)s×ez, (58)\nwhereeyis a unit vector perpendicular to the plane of the wire. The first term\ncorresponds to easy-plane shape anisotropy for a wire whose widt h is large compared\nwith its thickness and the second term arises from a uniaxial field Hualong the wire,An extended Landau-Lifshitz-Gilbert equation 15\nso thatb=ms/Hu. The solution of eqn.(1), with the additional terms (58), for a\nstationary N´ eel wall in the plane of the wire, with zero external fie ld and zero current,\nis\ns= (sech(z/b1/2),0,−tanh(z/b1/2)). (59)\nAs pointed out in ref. [9] there is no solution of the LLG equation of th e form\ns=F(z−vWt), corresponding to a uniformly moving domain wall, when the f1term\nis included. It is likely that the wall velocity oscillates about an average value, as\npredicted by Tatara and Kohno [18, 19] for purely adiabatic torque above the critical\ncurrent density for domain wall motion. However, we may estimate t he average velocity\nvWusing the method of ref. [9]. The procedure is to substitute the ap proximate form\ns=F(z−vWt) in the extended LLG equation (1), with the terms (58) added, tak e the\nscalar product with F×F′and integrate with respect to zover the range ( −∞,∞).\nThe boundary conditions appropriate to the wall are s→ ∓ezasz→ ±∞. Hence for\nbext= 0 we find the dimensionless wall velocity to be\nvW=f/integraltext∞\n−∞(F×F′)2dz+f1/integraltext∞\n−∞(F×F′′)2dz+g1/integraltext∞\n−∞(F′′)2dz\nα/integraltext∞\n−∞(F×F′)2dz+α′\n2/integraltext∞\n−∞(F′′)2dz.(60)\nTo estimate the integrals we take F(z) to have the form of the stationary wall s(z)\n(eqn.(59)) and, with the physical dimensions of velocity restored, the wall velocity is\ngiven approximately by\nvW=v0β\nα1+f1(3fb)−1\n1+α′\n2(αb)−1(61)\nwherev0=µBPJ/(mse). We have neglected g1here because, like fandα, it depends\non spin-orbit coupling but is a factor ( a0/lex)2smaller than f(cf. eqns.(53) and (56)).\nf1andα′\n2are important because they do not depend on spin-orbit coupling.\nIt is interesting to compare vWwith the wall velocity observed in permalloy\nnanowires by Hayashi et al[20]. We first note that v0is the velocity which one obtains\nvery simply from spin angular momentum conservation if the current -driven wall moves\nuniformly without any distortion such as tilting out of the easy plane a nd contraction\n[21]. This is never the case, even if f1= 0,α′\n2= 0, unless β=α. For a permalloy\nnanowire, with µ0ms= 1 T,v0= 110Pm/s forJ= 1.5·108A/cm2. Thus, from the\nstandard theory with f1= 0,α′\n2= 0,vW= 110Pβ/αm/s for this current density. In\nfact Hayashi et al[20] measure a velocity of 110 m/s which implies β > αsince the spin\npolarization Pis certainly less than 1. They suggest that βcannot exceed αand that\nsome additional mechanism other than spin-transfer torque is ope rating. However in the\ndiscussion following eqn.(54) we pointed out that in the model calculat ions it is possible\nto haveβ > α. Even if this is not the case in permalloy we can still have vW> v0if the\nlast factor in eqn.(61) is greater than 1 when f1andα′\n2are non-zero. We can estimate\ntermsinthisfactorusingtheobservationfromref. [20],that lW=lexb1/2= 23nm, where\nlWis the width of the wall. From eqns.(53) and (55) we find f1/(fb)∼(u/v)2(kFlW)−2,\nwherekFis a Fermi wave-vector. In permalloy we have Fe impurities in Ni so tha t inAn extended Landau-Lifshitz-Gilbert equation 16\nthe impurity potential u+v·σwe estimate u∼1 eV and v∼0.005 eV. The value for v\nis estimated by noting that the potential v·σis intended to model spin-orbit coupling\nof the form ξL·σwithξ/lessorsimilar0.1 eV and /angbracketleftLz/angbracketrightFe∼0.05,Lzbeing the component of orbital\nangular momentum in the direction of the magnetization [22]. Hence u/v∼200 and\nkFlW∼200 so that f1/(fb)∼1.α′\n2/(αb) is expected to be of similar magnitude. We\nconclude that the α′\n2andf1terms in the LLG equation (1) can be important in domain\nwall motion and should be included in micromagnetic simulations such as O OMMF\n[23]. For narrower domain walls these terms may be larger than the Gilb ert damping α\nand non-adiabatic spin- transfer torque fterms which are routinely included. Reliable\nestimates of their coefficients are urgently required using realistic m ultiband models of\nthe ferromagnetic metal or alloy.\n5. Conclusions\nThe coefficients of all the terms in an extended LLG equation for a cu rrent-carrying\nferromagnetic wire have been calculated for a simple model. Two of th ese (f1and\nα′\n2) are of particular interest since they do not rely on spin-orbit coup ling and may\nsometimes dominate the usual damping and non-adiabatic spin-tran sfer torque terms.\nOne term ( α1) which has been introduced by previous authors is shown rigorously to\nbe zero, independent of any particular model. Solutions of the exte nded LLG equation\nfor domain wall motion have not yet been found but the average velo city of the wall is\nestimated. It is pointed out that the f1andα′\n2terms are very important for narrow\nwalls and should be included in micrmagnetic simulations such as OOMMF. I t is shown\nthat there is no theoretical reason why the wall velocity should not exceed the simplest\nspin-transfer estimate v0, as is found to be the case in experiments on permalloy by\nHayashiet al[20]\nAcknowledgments\nWe are grateful to the EPSRC for financial support through the S pin@RT consortium\nand to other members of this consortium for encouragement and s timulation.\nAppendix A.\nThe simple single-band impurity model used in the main text is useful fo r obtaining\nexplicit expressions for all the coefficients in the LLG equation (1). H ere we wish to\nshow that some of these results are valid for a completely general s ystem. We suppose\nthe ferromagnetic material is described by the many-body Hamilton ian\nH=H1+Hint+Hext (A.1)\nwhereH1is a one-electron Hamiltonian of the form\nH1=Hk+Hso+V. (A.2)An extended Landau-Lifshitz-Gilbert equation 17\nHereHkis the total electron kinetic energy, Hsois the spin-orbit interaction, Vis a\npotential term, Hintis the coulomb interaction between electrons and Hextis due to an\nexternal magnetic field Bextin thezdirection. Thus\nHext=−2µBSz\n0Bext (A.3)\nwhereS0\nzis thezcomponent of total spin. Both HsoandVcan contain disorder. Since\nwe are interested in the energy and lifetime of a long-wavelength spin wave we consider\nthe spin wave pole, for small q, of the dynamical susceptibility.\nχ(q,ω) =/integraldisplay\ndt/angbracketleft/angbracketleftS−\nq(t),S+\n−q/angbracketright/angbracketrighte−iω−t(A.4)\n(ω−=ω−iǫ) whereS±\nq=Sx\nq±iSy\nqare Fourier components of the total transverse spin\ndensity. Here\n/angbracketleft/angbracketleftS−\nq(t),S+\n−q/angbracketright/angbracketright=i\n/planckover2pi1/angbracketleft/bracketleftbig\nS−\nq(t),S+\n−q/bracketrightbig\n/angbracketrightθ(t). (A.5)\nIngeneral we shall take theaverage /angbracketleft/angbracketrightina steady statein which a charge current density\nJis flowing in the qdirection. Following the general method of Edwards and Fisher\n[24] we use equations of motion to find that\nχ(q,ω) =−2/angbracketleftSz\n0/angbracketright\n/planckover2pi1(ω−bext)+1\n/planckover2pi12(ω−bext)2/braceleftbig\nχc(q,ω)−/angbracketleft/bracketleftbig\nC−\nq,S+\n−q/bracketrightbig\n/angbracketright/bracerightbig\n(A.6)\nwhere/planckover2pi1bext= 2µBBext,C−\nq= [S−\nq,H1] and\nχc(q,ω) =/integraldisplay\ndt/angbracketleft/angbracketleftC−\nq(t),C+\n−q/angbracketright/angbracketrighte−iωt. (A.7)\nFor small qandω,χis dominated by the spin wave pole, so that\nχ(q,ω) =−2/angbracketleftSz\n0/angbracketright\n/planckover2pi1(ω−bext−ωq)(A.8)\nwherebext+ωqis the spin wave frequency, in general complex corresponding to a fi nite\nlifetime. Following ref. [24] we compare (A.6) and (A.8) in the limit ωq≪ω−bextto\nobtain the general result\nωq=−1\n2/angbracketleftSz\n0/angbracketright/planckover2pi1/braceleftbigg\nlim\nω→bextχc(q,ω)−/angbracketleft/bracketleftbig\nC−\nq,S+\n−q/bracketrightbig\n/angbracketright/bracerightbigg\n. (A.9)\nEdwards and Fisher [24] were concerned with Reωqwhereas Kambersky [25] derived the\nabove expression for Imωqfor the case q= 0, and zero current flow. His interest was\nGilbert damping in ferromagnetic resonance. Essentially the same re sult was obtained\nearlier in connection with electron spin resonance, by Mori and Kawa saki [26], see also\nOshikawa and Affleck [27]. Since S−\nqcommutes with the potential term V, even in the\npresence of disorder, we have\nC−\nq=/bracketleftbig\nS−\nq,H1/bracketrightbig\n=/bracketleftbig\nS−\nq,Hk/bracketrightbig\n+/bracketleftbig\nS−\nq,Hso/bracketrightbig\n. (A.10)\nFor simplicity we now neglect spin-orbit coupling so that\nC−\nq=/bracketleftbig\nS−\nq,Hk/bracketrightbig\n=/planckover2pi1qJ−\nq (A.11)An extended Landau-Lifshitz-Gilbert equation 18\nwhere the last equation defines the spin current operator J−\nq. For a general system, with\nthenthelectron at position rnwith spin σnand momentum pn,\nS−\nq=/summationdisplay\nneiq·rnσ−\nn, Hk=/summationdisplay\nnp2\nn/2m. (A.12)\nHence, from eqns.(A.11) and (A.12),\n/angbracketleft/bracketleftbig\nC−\nq,S+\n−q/bracketrightbig\n/angbracketright=N/planckover2pi12q2\n2m+2/planckover2pi1/summationdisplay\nn/angbracketleftσz\nnvn/angbracketright·q (A.13)\nwhereNis the total number of electrons and vn=pn/mis the electron velocity, so\nthate/summationtext\nn/angbracketleftσz\nnvn/angbracketrightis the total spin current. Hence from eq.(A.9), we find\nωq=/planckover2pi1q2\n2/angbracketleftSz\n0/angbracketright/bracketleftbiggN\n2m−lim\nω→bextχJ(0,ω)/bracketrightbigg\n+Bq\n/planckover2pi1(A.14)\nwith\nB=/planckover2pi1µBPJ/em s. (A.15)\nThis expression for Bhas been obtained by Bazaliy et al[2] and Fern´ andez-Rossier et\nal[14] for simple parabolic band, s−dand Hubbard models. The derivation here is\ncompletelygeneralforanyferromagnet,eveninthepresenceof disorderduetoimpurities\nor defects, as long as spin-orbit coupling is neglected. Eqs.(2) and ( A.14) are both valid\nfor arbitrary bext, so that in eq. (33) we must have α1= 0.\nReferences\n[1] Berger L 1978 J.Appl.Phys. 492156\n[2] Bazaliy Y B, Jones B A and Zhang S C 1998 Phys. Rev. B 57R3213\n[3] Li Z and Zhang S 2004 Phys. Rev. B 70024417\n[4] Li Z and Zhang S 2004 Phys. Rev. Lett. 92207203\n[5] Landau L D, Lifshitz E M and Pitaevski L P 1980 Statistical Physics , part 2 (Oxford: Pergamon)\n[6] Gilbert T L 1955 Phys. Rev. 1001243\n[7] Zhang S and Li Z 2004 Phys. Rev. Lett. 93127204\n[8] Kohno H, Tatara G and Shibata J 2006 J. Phys. Soc. Japan 75113706\n[9] Wessely O, Edwards D M and Mathon J 2008 Phys. Rev. B 77174425\n[10] Tserkovnyak T, Skadsem H J, Brataas A and Bauer G E W 2006 Phys. Rev. B 74144405\n[11] Skadsem H J, Tserkovnyak T, Brataas A and Bauer G E W 2007 Phys. Rev. B 75094416\n[12] Sobolev V L, Klik I, Chang C R and Huang H L 1994 J. Appl. Phys. 755794\n[13] Thorwart M and Egger R 2007 Phys. Rev. B 76214418\n[14] Fern´ andez-Rossier J, Braun M, N´ u˜ nez A S and MacDonald A H 2004Phys. Rev. B 69174412\n[15] Duine R A, N´ u˜ nez A S, Sinova J and Macdonald A H 2007 Phys. Rev. B 75214420\n[16] Izuyama T, Kim D-J and Kubo R 1963 J. Phys. Soc. Japan 181025\n[17] Yamada H and Shimizu M 1971 J. Phys. Soc. Japan 311344\n[18] Tatara G and Kohno H 2004 Phys. Rev. Lett. 92086601\n[19] Tatara G and Kohno H 2005 J. Electron. Microsc. 54i69\n[20] Hayashi M, Thomas L, Rettner C, Moriya R, Bazaliy Y B and Parkin S S P 2007 Phys. Rev. Lett.\n98037204\n[21] Barnes S E and Maekawa S 2005 Phys. Rev. Lett. 95107204\n[22] Daalderop G H O, Kelly P J and Schuurmans M F H 1990 Phys. Rev. B 4111919\n[23] Donahue M and Porter D http://math.nist.gov/oommfAn extended Landau-Lifshitz-Gilbert equation 19\n[24] Edwards D M and Fisher B 1971 J. Physique 32C1 697\n[25] Kambersk´ y V 1976 Czech. J. Phys. B 261366\n[26] Mori H and Kawasaki K 1962 Prog. Theor. Phys. 27529\n[27] Oshikawa M and Affleck I 2002 Phys. Rev. B 65134410"    },    {        "title": "2102.03914v4.Spinterface_Induced_Modification_in_Magnetic_Properties_in_Co40Fe40B20_Fullerene_Bilayers.pdf",        "content": "Spinterface Induced Modification in Magnetic\nProperties in Co40Fe40B20/Fullerene Bilayers\nPurbasha Sharangi,†Esita Pandey,†Shaktiranjan Mohanty,†Sagarika Nayak,†\nand Subhankar Bedanta∗,†,‡\n†Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical\nSciences, National Institute of Science Education and Research (NISER), HBNI, P.O.-\nBhimpur Padanpur, Via Jatni, 752050, India\n‡Center for Interdisciplinary Sciences (CIS), National Institute of Science Education and\nResearch (NISER), HBNI, Jatni, 752050 India\nE-mail: sbedanta@niser.ac.in\nAbstract\nOrganic semiconductor/ferromagnetic bilayer thin films can exhibit novel properties\ndue to the formation of spinterface at the interface. Buckminsterfullerene (C 60) has\nbeen shown to exhibit ferromagnetism at the interface when it is placed next to a\nferromagnet (FM) such as Fe or Co. Formation of spinterface occurs due to the orbital\nhybridization and spin polarized charge transfer at the interface. In this work, we have\ndemonstrated that one can enhance the magnetic anisotropy of the low Gilbert damping\nalloy CoFeB thin film by introducing a C60layer. We have shown that anisotropy\nincreases by increasing the thickness of C 60which might be a result of the formation\nof spinterface. However, the magnetic domain structure remains same in the bilayer\nsamples as compared to the reference CoFeB film.\n1arXiv:2102.03914v4  [cond-mat.mtrl-sci]  7 Nov 2021Introduction\nOrganic spintronics has drawn immense research interest in the last few decades due to its\napplications in spin valve, magnetic tunnel junctions etc.1–3In organic spintronics, organic\nsemiconductors (OSCs) (e.g. C 60, Alq 3, ruberene etc.) are used to transport or control spin\npolarized signals.4–8The main advantage of OSCs are their low production cost, light weight,\nflexible and chemically interactive nature. Usually the spin orbit coupling is small in organic\nmaterials (e.g. C 60) as they consist of low Z (atomic number) materials (in particular carbon\n(C)). Moreover, the zero hyperfine interaction in C 60results in a longer spin relaxation\ntime.9–15As a consequence, spin of a carrier weakly interacts in organic environment and\nspin information is maintained for a long time. There are several reports on organic spin\nvalves, organic light emitting diodes (OLED) using C 60as a spacer layer.9–16It has been\nshown that C 60(∼2 nm) can be magnetized when it is placed next to a ferromagnetic (FM)\nlayer.9,17–19d−phybridization at the interface of FM/C 60modifies the density of states\n(DOS) and exhibits room temperature ferromagnetism. Such kind of interface is known as\nspinterface.20It has been shown that the fundamental magnetic properties like magnetic\nmoment, domain structure and magnetic anisotropy can be tuned by depositing C 60on top\nof a Fe, Co or Fe 4N layer.17–19,21Using first-principles calculations Han etal. have shown that\nmagnetic anisotropy energy (MAE) of Fe 4N system is changed from out-of-plane to in-plane\nafter inserting a C 60layer.21Their study indicates a strong d−phybridization between Fe\nand C atoms, which modifies the MAE of the system.21It has been found that ∼2 nm\nof C 60exhibits magnetic moment ∼3µB/cage at the epitaxial Fe/C 60interface.17There\nis a decrement in anisotropy in polycrystalline Fe/C 60system whereas for polycrystalline\nCo/C 60system anisotropy got enhanced. However, to the best of our knowledge no such\nbasic study has been performed on CoFeB system. For spintronic application a low damping\nmaterial is always desired as it directly affects the speed of a device. The main advantage\nof taking CoFeB as a ferromagnet is that it exhibits low Gilbert damping parameter and it\nis amorphous in nature.22It is very important to explore the effect of interface of such a\n2system (CoFeB/OSC) to enrich our fundamental knowledge of spinterface.\nIn this regard, we have prepared CoFeB/C 60bilayer films and compared the magnetic\nproperties to its reference CoFeB film. Also, we have varied the thickness of C 60layer\nto qualitatively define the extent of spinterface and study the modifications in the basic\nmagnetic properties. To study the qualitative nature of the interface, we have performed\nKerr microscopy and ferromagnetic resonance (FMR) measurements.\nMethods\nCoFeB reference film with 5 nm thickness and bilayer (CoFeB/C 60) samples have been de-\nposited on Si (100) substrate in a multi-deposition high vacuum chamber manufactured by\nMantis Deposition Ltd., UK. In the bilayer samples the thickness of CoFeB is fixed to 5 nm\nand the thickness of C 60has been varied between 1.1 to 15 nm. The composition of CoFeB\nconsidered here is 40:40:20. The base pressure in the chamber was 5 ×10−8mbar. CoFeB, C 60\nand MgO layers have been deposited using DC sputtering, thermal evaporation and e-beam\nevaporation techniques, respectively. The samples are named as S1, S2, S3, S4 and S5 for the\nthickness of C 60(tC60) taken as 0, 1.1, 2, 5, 15 nm, respectively. The schematic of the sample\nstructure is shown in Figure 1a (thicknesses not to scale). All the layers were deposited\nwithout breaking the vacuum to avoid oxidation and surface contamination. The deposition\npressure was 5×10−3mbar for CoFeB and 1 ×10−7mbar for C 60and MgO evaporation.\nThe deposition rates for CoFeB and C 60layers were 0.1 and ∼0.1 – 0.15 ˚A/s, respectively.\n2 nm of MgO has been deposited as a capping layer. C 60layer has been deposited normal to\nthe substrate whereas CoFeB plume was at 30◦w.r.t the substrate normal due to chamber’s\nin-built geometry.\nTo understand the growth of each layer and interfaces, cross-sectional TEM has been\nperformed on sample S4 using a high-resolution transmission electron microscope (HRTEM)\n(JEOL F200, operating at 200 kV and equipped with a GATAN oneview CMOS camera).\n3For the compositional analysis we have performed scanning transmission electron microscopy\n- energy dispersive X-ray spectroscopy (STEM - EDX). Selected area electron diffraction\n(SAED) has been performed on sample S4 to investigate the growth of the CoFeB and C 60\nlayers (supplementary information Figure S1). X-ray reflectivity (XRR) has been performed\non all the samples to know the exact thickness and roughness of all the layers (see Figure S2\nand Table S1 in the supplementary information ).\nWe have measured the hysteresis loop and magnetic domain images at room temperature\nby magneto-optic Kerr effect (MOKE) based microscopy manufactured by Evico magnetics\nGmbH, Germany. Longitudinal hysteresis loops are recorded for ±5 mT magnetic field by\nvarying the angle ( φ) between the easy axis (EA) and the applied magnetic field direction.\nTo measure the hysteresis loops along hard axis (HA), we have applied ±17.5 mT magnetic\nfield.\nIn order to determine the magnetic anisotropy constant and observe the anisotropy sym-\nmetry in the samples, angle dependent FMR measurements have been performed at a fre-\nquency of 7 GHz at 5◦interval. During the measurement the sample was kept on the wide\ncoplanar waveguide (CPW) in a flip-chip manner. An in-plane magnetic field (i.e, parallel\nto the sample plane) is applied to the sample (for the detailed measurement configuration,\nrefer to the figure S4 in the supplementary information). Frequency dependent FMR mea-\nsurements have been performed to calculate the Gilbert damping constant( α).\nResults and discussion\nHigh resolution TEM image is shown in Figure 1b and all the layers are marked separately.\nIt shows the amorphous growth of CoFeB and C60(see supplementary information figure S1).\nElement specific mapping has been shown in Figure 1c. Figure 1d shows the STEM image,\nin which the brighter part indicates the layer of the element having high atomic number(Z).\nPresence of Boron(B) is not properly visible as it is a lighter atom. Figure 1e-f represent the\n4Figure 1: (a)Schematic of the sample structure. The thicknesses shown in this schematic is\nnot to scale to the actual thicknesses of the samples. (b) Cross-sectional transmission electron\nmicroscopy (TEM) image of S4. (c) Elemental mapping for individual layers. (d)The region\nof the sample S4 where the STEM-EDX has been performed. (e) EDX line profile for each\nlayer of the sample S4. (f) EDX spectrum of sample S4 showing the presence of different\nelements.\nEDX line profile and EDX spectra, respectively. The position of the Co and Fe peak at the\nsame place indicates the formation of CoFeB alloy. EDX spectra shows the presence of C,\nMg, O, Fe and Co elements in the sample.\nFigure 2a-d show the in-plane angle ( φ) dependent hysteresis loops measured using longi-\ntudinal magneto optic Kerr effect (LMOKE) microscopy at room temperature for the samples\nS1, S2, S4 and S5, respectively. φis defined as the angle between the EA and the applied\nmagnetic field direction. The hysteresis loops along 90◦w.r.t. EA are shown in Figure 2e-h\nfor the samples S1, S2, S4 and S5, respectively. Angle dependent hysteresis loops show that\nthe magnetic HA of the samples is at 90◦w.r.t the EA, which marks the presence of uniaxial\nanisotropy in the system. The easy axis of the anisotropy lies in-plane at an angle of 90◦\nw.r.t the projection of the plume direction. The CoFeB target is at an angle of 30◦w.r.t. the\n5Figure 2: Hysteresis loops measured by magneto optic Kerr effect (MOKE) microscopy at\nroom temperature in longitudinal mode by varying the angle ( φ) between the EA and the\napplied magnetic field direction for the samples (a) S1, (b) S2, (c) S4 and (d) S5. (e) - (h)\nrepresent the hysteresis loops measured along 90◦w.r.t EA for S1, S2, S4 and S5, respectively.\nsubstrate normal due to the in-built geometry of the deposition system. Such kind of oblique\nangle deposition induced uniaxial anisotropy has been reported earlier.19,23–29It should be\nnoticed that there is a change in coercive field ( HC) in the bilayer samples as compared to\nthe single layer reference sample. The values of HCare 1.23, 0.71, 0.91 and 0.81 mT for\nthe samples S1, S2, S4 and S5, respectively. The HCfor the bilayer samples S2 to S5 are\ncomparable. However, the decrease in HCfrom single layer CoFeB to bilayers CoFeB/C 60\ncan be attributed to the formation of a spinterface between the CoFeB and C 60interface. In\nour previous study we have shown that the orbital hybridization at the FM (Fe or Co)/C 60\ninterface promote the change in anisotropy of the system.17–19\nThe square shaped loop along EA indicates the magnetization reversal is happening\nvia domain wall motion whereas, along HA the reversal occurs via coherent rotation. The\nmagnetization reversal is studied as a function of φ. By varying the angle ( φ) w.r.t easy axis\n(0◦), we have recorded the domain images near the HCatφ= 0◦, 30◦, 45◦, 60◦and 90◦.\nFigure 3a-e, Figure 3f-j, Figure 3k-o and Figure 3p-t show the magnetic domain images near\nHCfor the samples S1, S2, S4 and S5, respectively. Branched domains have been observed\n6Figure 3: Domain images near HCfor samples S1, S2, S4 and S5 are shown in (a) - (e), (f)\n- (j), (k) - (o) and (p) – (t), respectively. The scale bars of the domain images for all the\nsamples are same and shown in image (a). The applied field ( H) direction shown in image\n(a) was kept constant for all the measurements and the sample was only rotated to capture\nthe domain images at different φ.\nin all the samples due to the amorphous growth of CoFeB. Domain images captured at\ndifferent applied magnetic fields along EA for samples S1, S2, S4 and S5, are shown in\nFigure S3 in supplementary information. For the samples S2 and S5 the tilt of the domains\nare opposite to S1 and S4. This opposite tilt is due to the anti-clockwise (S2 and S5)\nand clockwise (S1 and S4) rotation of the sample stage w.r.t EA during measurement. It\nis noted that the HCis different between the reference CoFeB and the bilayer samples.\nHowever, the change in domain structure is not significant between the single layer CoFeB\nand the bilayer CoFeB/C 60samples. In our earlier reports it has been shown that the change\nin domain shape and size is significant in other ferromagnetic/OSC systems such as Co/C 60,\nFe/C 60.17–19In epitaxial Fe/C 60system the magnetization reversal process was different\nbetween the reference Fe and the Fe/C 60bilayer systems.17In case of polycrystalline Fe/C 60\n7system the domain size got reduced for the bilyaers as compared to the reference Fe film.18\nHowever, in case of polycrystalline Co/C 60, the domain size increased for the bilayers as\ncompared to the reference Co film.19In this study we considered CoFeB system and the\ndomain shape and size are comparable between the bilayers and the reference sample. The\norigin to this may be investigated theoretically in future work.\nFigure 4:fvsHresand ∆Hvsfplots for S1, S2, S3, S4 and S5 are shown in (a) and (b),\nrespectively. Solid circles represent the experimental data, while the solid lines are the best\nfits to the eqs. 2 and 3.\nWe have further invesigated the magnetization dynamics by performing the frequency\ndependent FMR measurement. The experimental data has been fitted using a Lorentzian\nfunction (eq. 1), where ∆ H,Hres, A1and A 2are linewidth, resonance field, anti-symmetric\nand symmetric components, respectively.30\nFMR signal =A14∆H(H−Hres)\n(4(H−Hres))2+ (∆H)2−A2(∆H)2−4(H−Hres)2\n(4(H−Hres))2+ (∆H)2+offset (1)\nThe plots of fvsHresand ∆Hvsfare shown in Figure 4a and Figure 4b, respectively.\nThe effective damping constant ( α) has been determined by fitting the eqs. 2 and 3:30–32\n8f=γ\n2π/radicalBig\n(HK+Hres)(HK+Hres+ 4πMeff) (2)\nwhere,γ(gyromagnetic ratio) = gµB/¯handg,µB, ¯h,HKare Lande-g factor, Bohr\nmagneton, reduced Planck’s constant and anisotropy field, respectively.\n∆H= ∆H0+4παf\nγ(3)\nwhere, ∆H0is the inhomogeneous line width broadening which depends on the mag-\nnetic inhomogeneity of the sample. αvalues for the samples S1, S2, S3, S4 and S5 are\n0.0095±0.0002, 0.0106±0.0002, 0.0110±0.0002, 0.0124±0.0003 and 0.0169 ±0.0006 , respec-\ntively. It has been observed that αincreases after introducing a C 60layer and it further\nincreased with increasing the C 60thickness. This increase in αmight be due to the interface\nroughness or other effects such as spin pumping at the interface33.\nTo quantify the change in anisotropy in all the samples we have performed in-plane angle\ndependent FMR measurements at a fixed frequency of 7 GHz. Resonance field ( Hres) has\nbeen measured by rotating the sample w.r.t the applied magnetic field in 5◦intervals.\nHresvsφplots have been shown in Figure 5 to calculate the anisotropy constants of the\nsystem. The open circles represent the raw data and the solid lines are the best fits. The\nexperimental data is fitted using Landau-Lifshitz-Gilbert (LLG) equation:34\nf=γ\n2π((H+2K2\nMSCos2φ)(H+ 4πMS+2K2\nMSCos2φ))1/2(4)\nwhere,K2is the in-plane uniaxial anisotropy constant, φis the in-plane angle between the\neasy axis w.r.t the applied magetic field direction and MSis the saturation magnetization.\nTheK2values extracted from the fitting are listed in Table 1. It has been observed\n9Figure 5: Angle dependent resonance field ( Hres) plot for all the five samples to calculate the\nanisotropy constants of the system. The measurement was performed at room temperature\nand a fixed frequency of 7 GHz. Open circles represent the experimental data, while the\nsolid lines are the best fits.\nthat by introducing a C60layer the anisotropy of the system increased. The possible reason\nbehind the enhancement in the magnetic anisotropy is the formation of spinterface at the\nCoFeB/C60interface. The anisotropy increases from 2.9 ×104to 3.1×104erg/cc when the\nC60thickness is varied from 1.1 to 2 nm. With further increase in C60thickness (at 5 nm),\nthe anisotropy become 4.1 ×104erg/cc. There is a small change in the anisotropy (4 .1×104\nto 4.3×104erg/cc) when C60thickness increases from 5 to 15 nm. After a certain thickness of\nTable 1: The value of K2for all the samples extracted from the fitting of LLG equation.\nSampleK2(erg/cc)\nS1 2.4×104\nS2 2.9×104\nS3 3.1×104\nS4 4.1×104\nS5 4.3×104\n10C60layer, the spinterface thickness remains almost constant. However, the exact thickness of\nthe spinterface for amorphous CoFeB/ C60system is not known. In future polarized neutron\nreflectivity (PNR) experiment may be carried out to evaluate the spinterface thickness.\nConclusion\nWe have studied the effect of C 60on the magnetization reversal and the magnetic anisotropy\nof a low damping amorphous CoFeB layer. In comparison to the single layer CoFeB sample\nthe magnetic anisotropy constant has been increased for the CoFeB/C 60bilayer samples. Fur-\nther from the Kerr microscopy measurements it is observed that there is a negligible change\nin the branch domain pattern in the samples. The enhancement in magnetic anisotropy\nmight be the result of d−phybridization between the CoFeB and C 60layer. This study\nreveals that one can enhance the anisotropy of a ferromagnetic CoFeB system by introducing\na C60layer, which can be suitable for future spintronics devices. Further in future, the nature\nof spinterface such as thickness, magnetic moment per atom etc. should be investigated by\nexperimental methods such as polarized neutron reflectometry. The results presented here\nmight bring interest to study similar system theoretically to elucidate the exact nature of\nspinterface and the origin behind it.\nSupporting Information\nSelected area electron diffraction (SAED) on sample S4 is shown in Figure S1. XRR data\nwith the best fits for samples S1 to S5 are shown in Figure S2. MOKE hysteresis loops\nwith corresponding domain images along the EA for samples S1, S2, S4 and S5 are shown\nin Figure S3. The schematic of the FMR measurement set-up and applied field direction is\nshown in Figure S4.\n11Acknowledgement\nWe sincerely thank Dr. Tapas Gosh and Mr. Pushpendra Gupta for helping in TEM imag-\ning. The authors also want to thank Dr. Ashutosh Rath for valuable discussion regarding\nthe SAED images. The authors also acknowledge Department of Atomic Energy, and De-\npartment of Science and Technology - Science and Engineering Research Board, Govt. of\nIndia (DST/EMR/2016/007725) for the financial support.\nReferences\n(1) Naber, W.; Faez, S.; van der Wiel, W. G. Organic Spintronics. J. Phys. D: Appl. Phy.\n2007 ,40, R205.\n(2) Stamps, R. L.; Breitkreutz, S.; ˚Akerman, J.; Chumak, A. V.; Otani, Y.; Bauer, G. E.;\nThiele, J.-U.; Bowen, M.; Majetich, S. A.; Kl¨ aui, M., et al. The 2014 Magnetism\nRoadmap. J. Phys. D: Appl. Phy. 2014 ,47, 333001.\n(3) Kuch, W.; Bernien, M. Controlling the Magnetism of Adsorbed Metal–Organic\nMolecules. J. Phys.: Condens. Matt. 2016 ,29, 023001.\n(4) Dediu, V. A.; Hueso, L. E.; Bergenti, I.; Taliani, C. Spin Routes in Organic Semicon-\nductors. Nat. Mater. 2009 ,8, 707–716.\n(5) Atodiresei, N.; Brede, J.; Lazi´ c, P.; Caciuc, V.; Hoffmann, G.; Wiesendanger, R.;\nBl¨ ugel, S. 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Hybridization-Induced Oscillatory Magnetic Polarization of C60 Orbitals\nat the C60/Fe (001) Interface. Appl. Phys. Lett. 2011 ,98, 222505.\n(11) Tran, T. L. A.; C+akır, D.; Wong, P. J.; Preobrajenski, A. B.; Brocks, G.; van der\nWiel, W. G.; de Jong, M. P. Magnetic Properties of bcc-Fe (001)/C60 Interfaces for\nOrganic Spintronics. ACS Appl. Mater. Interfaces 2013 ,5, 837–841.\n(12) Djeghloul, F.; Gruber, M.; Urbain, E.; Xenioti, D.; Joly, L.; Boukari, S.; Arabski, J.;\nBulou, H.; Scheurer, F.; Bertran, F., et al. High Spin Polarization at Ferromagnetic\nMetal–Organic Interfaces: A Generic Property. J. Phys. Chem. Lett. 2016 ,7, 2310–\n2315.\n(13) Gobbi, M.; Golmar, F.; Llopis, R.; Casanova, F.; Hueso, L. E. Room-Temperature Spin\nTransport in C60-Based Spin Valves. Adv. Mater. 2011 ,23, 1609–1613.\n(14) Zhang, X.; Mizukami, S.; Kubota, T.; Ma, Q.; Oogane, M.; Naganuma, H.; Ando, Y.;\nMiyazaki, T. Observation of a Large Spin-Dependent Transport Length in Organic Spin\nValves at Room Temperature. Nat. Commun. 2013 ,4, 1–7.\n(15) Nguyen, T. D.; Wang, F.; Li, X.-G.; Ehrenfreund, E.; Vardeny, Z. V. Spin Diffusion in\nFullerene-Based Devices: Morphology Effect. Phys. Rev. B 2013 ,87, 075205.\n13(16) Liu, H.; Wang, J.; Groesbeck, M.; Pan, X.; Zhang, C.; Vardeny, Z. V. Studies of Spin\nRelated Processes in Fullerene C 60 Devices. J. Mater. Chem. C 2018 ,6, 3621–3627.\n(17) Mallik, S.; Mattauch, S.; Dalai, M. K.; Br¨ uckel, T.; Bedanta, S. Effect of Magnetic\nFullerene on Magnetization Reversal Created at the Fe/C60 interface. Sci. Rep. 2018 ,\n8, 1–9.\n(18) Mallik, S.; Mohd, A. S.; Koutsioubas, A.; Mattauch, S.; Satpati, B.; Br¨ uckel, T.; Be-\ndanta, S. Tuning Spinterface Properties in Iron/Fullerene Thin Films. Nanotechnology\n2019 ,30, 435705.\n(19) Mallik, S.; Sharangi, P.; Sahoo, B.; Mattauch, S.; Br¨ uckel, T.; Bedanta, S. Enhanced\nAnisotropy and Study of Magnetization Reversal in Co/C60 Bilayer Thin Film. Appl.\nPhys. Lett. 2019 ,115, 242405.\n(20) Sanvito, S. Molecular Spintronics: The Rise of Spinterface Science. Nat. Phys 2010 ,6,\n562–564.\n(21) Han, X.; Mi, W.; Wang, X. Spin Polarization and Magnetic Properties at the C60/Fe4\nN (001) Spinterface. J. Mater. Chem. C 2019 ,7, 8325–8334.\n(22) Singh, B. B.; Jena, S. K.; Samanta, M.; Biswas, K.; Satpati, B.; Bedanta, S. Inverse\nSpin Hall Effect in Electron Beam Evaporated Topological Insulator Bi2Se3 Thin Film.\nPhys. Status Solidi RRL 2019 ,13, 1800492.\n(23) Smith, D.; Cohen, M.; Weiss, G. P. Oblique-Incidence Anisotropy in Evaporated\nPermalloy Films. J. Appl. Phys. 1960 ,31, 1755–1762.\n(24) Bubendorff, J.-L.; Zabrocki, S.; Garreau, G.; Hajjar, S.; Jaafar, R.; Berling, D.;\nMehdaoui, A.; Pirri, C.; Gewinner, G. Origin of the Magnetic Anisotropy in Ferro-\nmagnetic Layers Deposited at Oblique Incidence. EPL2006 ,75, 119 – 125.\n14(25) Chowdhury, N.; Mallick, S.; Mallik, S.; Bedanta, S. Study of Magnetization Relaxation\nin Co Thin Films Prepared by Substrate Rotation. Thin Solid Films 2016 ,616, 328–\n334.\n(26) Mallick, S.; Mallik, S.; Singh, B. B.; Chowdhury, N.; Gieniusz, R.; Maziewski, A.;\nBedanta, S. Tuning the Anisotropy and Domain Structure of Co Films by Variable\nGrowth Conditions and Seed Layers. J. Phys. D: Appl. Phys. 2018 ,51, 275003.\n(27) Mallik, S.; Bedanta, S. Study of Anisotropy, Magnetization Reversal and Damping in\nUltrathin Co Films on MgO (0 0 1) Substrate. J. Magn. Magn. Mater. 2018 ,446,\n270–275.\n(28) Mallik, S.; Mallick, S.; Bedanta, S. Effect of the Growth Conditions on the Anisotropy,\nDomain Structures and the Relaxation in Co Thin Films. J. Magn. Magn. Mater. 2017 ,\n428, 50–58.\n(29) Nayak, S.; Singh, B. B.; Mallick, S.; Bedanta, S. Tuning of Magnetic Properties by\nAlternating the Order of Hard/Soft Bilayers with Various Thicknesses. J. Phys. D:\nAppl. Phy. 2019 ,52, 305301.\n(30) Singh, B. B.; Jena, S. K.; Bedanta, S. Study of Spin Pumping in Co Thin Film vis-` a-vis\nSeed and Capping Layers Using Ferromagnetic Resonance Spectroscopy. J. Phys. D:\nAppl. Phys. 2017 ,50, 345001.\n(31) Kittel, C. On the Theory of Ferromagnetic Resonance Absorption. Phys. Rev. 1948 ,\n73, 155.\n(32) Heinrich, B.; Cochran, J.; Hasegawa, R. FMR Linebroadening in Metals due to Two-\nMagnon Scattering. J. Appl. Phys. 1985 ,57, 3690–3692.\n(33) Sharangi, P.; Singh, B. B.; Nayak, S.; Bedanta, S. Spin Pumping and Inverse Spin Hall\nEffect in CoFeB/C60 Bilayers. arXiv preprint arXiv:2106.06829 2021 ,\n15(34) Pan, S.; Seki, T.; Takanashi, K.; Barman, A. Role of the Cr Buffer Layer in the\nThickness-Dependent Ultrafast Magnetization Dynamics of Co2 Fe0.4Mn0.6 Si Heusler\nAlloy Thin Films. Phys. Rev. Appl. 2017 ,7, 064012.\n16Supporting Information\nSpinterface Induced Modification in Magnetic\nProperties in Co40Fe40B20/Fullerene Bilayers\nPurbasha Sharangi,†Esita Pandey,†Shaktiranjan Mohanty,†Sagarika Nayak,†\nand Subhankar Bedanta∗,†,‡\n†Laboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical\nSciences, National Institute of Science Education and Research (NISER), HBNI, P.O.-\nBhimpur Padanpur, Via Jatni, 752050, India\n‡Center for Interdisciplinary Sciences (CIS), National Institute of Science Education and\nResearch (NISER), HBNI, Jatni, 752050 India\nE-mail: sbedanta@niser.ac.in\nStructural information\nIn order to investigate the growth of deposited layers we have performed selected area electron\ndiffraction (SAED) on sample S4 (i.e., CoFeB(5nm)/C 60(5 nm)/MgO(2nm)). The SAED\nimage (Figure S1) shows the diffuse rings, which confirms the amorphous growth of CoFeB\nand C 60layers. Further, to know the structural information (thikness, roughness) ,we have\nperformed X-ray reflectivity (XRR) measurements on all the samples. We have fitted the\ndata by using GenX software. Figure S2a-e show the XRR data and best fits for samples S1,\nS2, S3, S4 and S5 ,respectively. The extracted thickness ( t) and roughness ( σ) of the layers\nare shown in Table S1.\nS1arXiv:2102.03914v4  [cond-mat.mtrl-sci]  7 Nov 2021Figure S1: SAED image of sample S4.\nFigure S2: XRR data and the best fits for samples S1, S2, S3, S4 and S5 are shown in (a),\n(b),(c), (d) and (e), respectively. The blue open circles are experimental data and the red\nsolid lines represent the best fit using GenX software. The parameters extracted from the\nbest fits are shown in Table S1.\nS2Table S1: Parameters obtained from XRR fits\nSample S1 Sample S2 Sample S3 Sample S4 Sample S5\nLayerst(nm)σ(nm)t(nm)σ(nm)t(nm)σ(nm)t(nm)σ(nm)t(nm)σ(nm)\nCoFeB 5.50 0.91 5.60 0.93 5.50 0.87 5.00 0.81 5.45 0.96\nC60 - - 1.10 0.20 2.00 0.60 5.00 0.52 15.00 1.80\nMgO 1.95 0.61 1.90 0.57 1.80 0.51 2.09 0.76 1.98 0.75\nFigure S3: (a), (b), (c) and (d) show the hysteresis loops for the samples S1, S2, S4 and S5,\nrespectively, along the easy axis (EA). The domain images at different applied fields (marked\ne to x in hysteresis loops) for samples S1, S2, S4 and S5 are shown in (e-i), (j-n), (o-s) and\n(t-x), respectively. The scale bars of the domain images for all the samples are same and\nshown in image (e).\nS3Hysteresis loop and domain imaging\nFigure S3a-d show the hysteresis loops for the samples S1, S2, S4 and S5, respectively, along\nthe easy axis (EA). The domain images are shown in Figure S3e-x are also marked in the\nhysteresis loops. Figure S3e, Figure S3j, Figure S3o and Figure S3t represent the domain\nimages for S1, S2, S4, S5 captured at positive saturation field. Similarly, Figure S3f, Figure\nS3k, Figure S3p, Figure S3u are the domain images near nucleation. Figure S3g, Figure\nS3l, Figure S3q, Figure S3v show the domain images which are captured near coercive field\nand Figure S3h, Figure S3m, Figure S3r, Figure S3w are captured near negative saturation\nfor samples S1, S2, S4 and S5, respectively. Further, domain images captured at negative\nsaturation field are shown in Figure S3i, Figure S3n, Figure S3s, Figure S3x for S1, S2, S4\nand S5, respectively.\nDuring FMR measurement we applied an in-plane magnetic field (i.e, parallel to the\nsample plane). Figure S4 shows the schematic of the FMR measurement set-up and the\napplied field direction.\nFigure S4: Schematic representation of FMR measurement set-up. Sample is placed on the\ncpw in a flip chip manner. The applied field ( Hext) is parallel to the sample plane and\nperpendicular to the rf field ( Hrf).\nS4"    },    {        "title": "1501.00444v1.Inertia__diffusion_and_dynamics_of_a_driven_skyrmion.pdf",        "content": "Inertia, diffusion and dynamics of a driven skyrmion\nChristoph Sch ¨utte,1Junichi Iwasaki,2Achim Rosch,1and Naoto Nagaosa2, 3,\u0003\n1Institut f ¨ur Theoretische Physik, Universit ¨at zu K ¨oln, D-50937 Cologne, Germany\n2Department of Applied Physics, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n3RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: January 5, 2015)\nSkyrmions recently discovered in chiral magnets are a promising candidate for magnetic storage devices\nbecause of their topological stability, small size ( \u00183\u0000100nm), and ultra-low threshold current density ( \u0018\n106A/m2) to drive their motion. However, the time-dependent dynamics has hitherto been largely unexplored.\nHere we show, by combining the numerical solution of the Landau-Lifshitz-Gilbert equation and the analysis of a\ngeneralized Thiele’s equation, that inertial effects are almost completely absent in skyrmion dynamics driven by\na time-dependent current. In contrast, the response to time-dependent magnetic forces and thermal fluctuations\ndepends strongly on frequency and is described by a large effective mass and a (anti-) damping depending\non the acceleration of the skyrmion. Thermal diffusion is strongly suppressed by the cyclotron motion and is\nproportional to the Gilbert damping coefficient \u000b. This indicates that the skyrmion position is stable, and its\nmotion responds to the time-dependent current without delay or retardation even if it is fast. These findings\ndemonstrate the advantages of skyrmions as information carriers.\nPACS numbers: 73.43.Cd,72.25.-b,72.80.-r\nI. INTRODUCTION\nMass is a fundamental quantity of a particle determining its\nmechanical inertia and therefore the speed of response to ex-\nternal forces. Furthermore, it controls the strength of quantum\nand thermal fluctuations. For a fast response one usually needs\nsmall masses and small friction coefficients which in turn lead\nto large fluctuations and a rapid diffusion. Therefore, usually\nsmall fluctuations and a quick reaction to external forces are\nnot concomitant. However a “particle” is not a trivial object\nin modern physics, it can be a complex of energy and mo-\nmentum, embedded in a fluctuating environment. Therefore,\nits dynamics can be different from that of a Newtonian parti-\ncle. This is the case in magnets, where such a “particle” can\nbe formed by a magnetic texture1,2. A skyrmion3,4is a rep-\nresentative example: a swirling spin texture characterized by\na topological index counting the number of times a sphere is\nwrapped in spin space. This topological index remains un-\nchanged provided spin configurations vary slowly, i.e., dis-\ncontinuous spin configurations are forbidden on an atomic\nscale due to high energy costs. Therefore, the skyrmion is\ntopologically protected and has a long lifetime, in sharp con-\ntrast to e.g. spin wave excitations which can rapidly decay.\nSkyrmions have attracted recent intensive interest because of\ntheir nano-metric size and high mobility5–14. Especially, the\ncurrent densities needed to drive their motion ( \u0018106A/m2)\nare ultra small compared to those used to manipulate domain\nwalls in ferromagnets ( \u00181011\u000012A/m2)15–19.\nThe motion of the skyrmion in a two dimensional film can\nbe described by a modified version of Newton’s equation. For\nsufficiently slowly varying and not too strong forces, a sym-\nmetry analysis suggests the following form of the equations\nof motion,\nG\u0002_R+\u000bD_R+mR+\u000b\u0000\u0002R=Fc+Fg+Fth:(1)\nHere we assumed translational and rotational invariance of\nthe linearized equations of motion. The ‘gyrocoupling’ G=G^e?is an effective magnetic field oriented perpendicular to\nthe plane,\u000bis the (dimensionless) Gilbert damping of a sin-\ngle spin,\u000bDdescribes the friction of the skyrmion, mits mass\nandRits centre coordinate. \u0000parametrizes a peculiar type of\ndamping proportional to the acceleration of the particle. We\nname this term ‘gyrodamping’, since it describes the damping\nof a particle on a cyclotron orbit (an orbit with R/G\u0002_R),\nwhich can be stronger ( \u0000parallel to G) or weaker (antipar-\nallel to G) than that for linear motion. Our main goal will\nbe to describe the influence of forces on the skyrmion arising\nfrom electric currents ( Fc), magnetic field gradients (Fg) and\nthermal fluctuations (Fth).\nBy analyzing the motion of a rigid magnetic structure\nM(r;t) =M0(r\u0000R(t))forstatic forces, one can obtain\nanalytic formulas for G;\u000bD;FcandFgusing the approach\nof Thiele19–22,24. In Ref. [25], an approximate value for the\nmass of a skyrmion was obtained by simulating the motion of\na skyrmion in a nanodisc and by estimating contributions to\nthe mass from internal excitations of the skyrmion.\nFor rapidly changing forces, needed for the manipulation of\nskyrmions in spintronic devices, Eq. (1) is however not suffi-\ncient. A generalized version of Eq. (1) valid for weak but also\narbitrarily time-dependent forces can be written as\nG\u00001(!)V(!) =Fc(!) +Fg(!) +Fth(!) (2)\n=Sc(!)vs(!) +Sg(!)rBz(!) +Fth(!)\nHereV(!) =R\nei!t_R(t)dtis the Fourier transform of the\nvelocity of the skyrmion, vs(!)is the (spin-) drift velocity\nof the conduction electrons, directly proportional to the cur-\nrent,rBz(!)describes a magnetic field gradient in frequency\nspace. The role of the random thermal forces, Fth(!), is spe-\ncial as their dynamics is directly linked via the fluctuation-\ndissipation theorem to the left-hand side of the equation, see\nbelow. The 2\u00022matrix G\u00001(!)describes the dynam-\nics of the skyrmion; its small- !expansion defines the terms\nwritten on the left-hand side of Eq. (1). One can expectarXiv:1501.00444v1  [cond-mat.str-el]  2 Jan 20152\nFIG. 1: When a skyrmion is driven by a time dependent external\nforce, it becomes distorted and the spins precess resulting in a de-\nlayed response and a large effective mass. In contrast, when the\nskyrmion motion is driven by an electric current, the skyrmion ap-\nproximately flows with the current with little distortion and preces-\nsion. Therefore skyrmions respond quickly to rapid changes of the\nelectric current.\nstrongly frequency-dependent dynamics for the skyrmion be-\ncause the external forces in combination with the motion of\nthe skyrmion can induce a precession of the spin and also ex-\ncite spinwaves in the surrounding ferromagnet, see Fig. 1.\nWe will, however, show that the frequency dependence of the\nright-hand side of the Eq. (2) is at least as important: not only\nthe motion of the skyrmion but also the external forces excite\ninternal modes. Depending on the frequency range, there is\nan effective screening or antiscreening of the forces described\nby the matrices Sc(!)andSg(!). Especially for the current-\ndriven motion, there will be for all frequencies an almost exact\ncancellation of terms from G\u00001(!)andSc(!). As a result\nthe skyrmion will follow almost instantaneously any change\nof the current despite its large mass.\nIn this paper, we study the dynamics of a driven skyrmion\nby solving numerically the stochastic Landau-Lifshitz-Gilbert\n(LLG) equation. Our strategy will be to determine the param-\neters of Eq. (2) such that this equation reproduces the results\nof the LLG equation. Section II introduces the model and\noutlines the numerical implementation. Three driving mecha-\nnisms are considered: section III studies the diffusive motion\nof the skyrmion due to thermal noise, section IV the skyrmion\nmotion due to time-dependent magnetic field gradient and sec-\ntion V the current-driven dynamics. We conclude with a sum-\nmary and discussion of the results in Sec. VI.\nII. MODEL\nOur study is based on a numerical analysis of the stochastic\nLandau-Lifshitz-Gilbert (sLLG) equations27defined by\ndMr\ndt=\rMr\u0002[Be\u000b+b\r(t)]\n\u0000\r\u000b\nMMr\u0002(Mr\u0002[Be\u000b+b\r(t)]):(3)\nHere\ris the gyromagnetic moment and \u000bthe Gilbert damp-\ning;Be\u000b=\u0000\u000eH[M]\n\u000eMris an effective magnetic field created by\nthe surrounding magnetic moments and b\r(t)a fluctuating,stochastic field creating random torques on the magnetic mo-\nments to model the effects of thermal fluctuations, see below.\nThe Hamiltonian H[M]is given by\nH[M] =\u0000JX\nrMr\u0001\u0000\nMr+aex+Mr+aey\u0001\n\u0000\u0015X\nr\u0000\nMr\u0002Mr+aex\u0001ex+Mr\u0002Mr+aey\u0001ey\u0001\n\u0000B\u0001X\nrMr (4)\nWe useJ= 1 ,\r= 1 ,jMrj= 1 ,\u0015= 0:18Jfor the\nstrength of the Dzyaloshinskii-Moriya interaction and B=\n(0;0;0:0278J)for all plots giving rise to a skyrmion with a\nradius of about 15lattice sites, see Appendix A. For this pa-\nrameter set, the ground state is ferromagnetic, thus the single\nskyrmion is a topologically protected, metastable excitation.\nTypically we simulate 100\u0002100spins for the analysis of dif-\nfusive and current driven motion and 200\u0002200spins for the\nforce-driven motion. For these parameters lattice effects are\nnegligible, see appendix B. Typical microscopic parameters\nused, areJ= 1meV (this yields Tc\u001810K) which we use to\nestimate typical time scales for the skyrmion motion.\nFollowing Ref. 27, we assume that the field bfl\nr(t)is gen-\nerated from a Gaussian stochastic process with the following\nstatistical properties\n\nbfl\nr;i(t)\u000b\n= 0\n\nbfl\nr;i(t)bfl\nr0;j(s)\u000b\n= 2\u000bkBT\n\rM\u000eij\u000err0\u000e(t\u0000s) (5)\nwhereiandjare cartesian components and h:::idenotes\nan average taken over different realizations of the fluctuating\nfield. The Gaussian property of the process stems from the in-\nteraction of Mrwith a large number of microscopic degrees\nof freedom (central limit theorem) which are also responsi-\nble for the damping described by \u000b, reflecting the fluctuation-\ndissipation theorem. The delta-correlation in time and space\nin Eq. (5) expresses that the autocorrelation time and length of\nbfl\nr(t)is much shorter than the response time and length scale\nof the magnetic system.\nFor a numerical implementation of Eq. (3) we follow\nRef. 27 and use Heun’s scheme for the numerical integration\nwhich converges quadratically to the solution of the general\nsystem of stochastic differential equations (when interpreted\nin terms of the Stratonovich calculus).\nFor static driving forces, one can calculate the drift veloc-\nity_Rfollowing Thiele20. Starting from the Landau-Lifshitz\nGilbert equations, Eq. (3), we project onto the translational\nmode by multiplying Eq. (3) with @iMrand integrating over\nspace21–23.\nG=~M0Z\ndr n\u0001(@xn\u0002@yn)\nD=~M0Z\ndr(@xn\u0001@xn+@yn\u0001@yn)=2\nFc=G\u0002vs+\fDvs;\nFg=MsrB; M s=M0Z\ndr(1\u0000nz) (6)3\nwhere nis the direction of the magnetization, M0the lo-\ncal spin density, vsthe (spin-) drift velocity of the conduc-\ntion electrons proportional to the electric current, and Ms\nis the change of the magnetization induced by a skyrmion\nin a ferromagnetic background. The ’gyrocoupling vector’\nG= (0;0;G)TwithG=\u0006~M04\u0019is given by the winding\nnumber of the skyrmion, independent of microscopic details.\nIII. THERMAL DIFFUSION\nRandom forces arising from thermal fluctuations play a de-\ncisive role in controlling the diffusion of particles and there-\nfore also the trajectories R(t)of a skyrmion. To obtain R(t)\nand corresponding correlation functions we used numerical\nsimulations based on the stochastic Landau-Lifshitz-Gilbert\nequation27. These micromagnetic equations describe the dy-\nnamics of coupled spins including the effects of damping\nand thermal fluctuations. Initially, a skyrmion spin-texture\nis embedded in a ferromagnetic background. By monitoring\nthe change of the magnetization, we track the center of the\nskyrmion R(t), see appendix A for details.\nOur goal is to use this data to determine the matrix G\u00001(!)\nand the randomly fluctuating thermal forces, Fth(!), which\ntogether fix the equation of motion, Eq. (2), in the presence\nof thermal fluctuations ( rBz=vs= 0). One might worry\nthat this problem does not have a unique solution as both the\nleft-hand and the right-hand side of Eq. (2) are not known\na priori. Here one can, however, make use of the fact that\nKubo’s fluctuation-dissipation theorem26constraints the ther-\nmal forces on the skyrmion described by Fthin Eq. (2) by\nlinking them directly to the dissipative contributions of G\u00001.\nOn averagehFth= 0i, but its autocorrelation is proportional\nto the temperature and friction coefficients. In general it is\ngiven by\nhFi\nth(!)Fj\nth(!0)i=kBT[G\u00001\nij(!) +G\u00001\nji(\u0000!)]2\u0019\u000e(!+!0):\n(7)\nFor small ! one obtainshFx\nth(!)Fx\nth(!0)i =\n4\u0019kBT\u000bD\u000e(!+!0)while off-diagonal correla-\ntions arise from the gyrodamping hFx\nth(!)Fy\nth(!0)i=\n4\u0019i!kBT\u000b\u0000\u000e(!+!0). Using Eq. (7) and demand-\ning furthermore that the solution of Eq. (2) reproduces\nthe correlation function h_Ri(t)_Rj(t0)i(or, equivalently,\nh(Ri(t)\u0000Rj(t0))2i) obtained from the micromagnetic\nsimulations, leads to the condition26\nGij(!) =1\nkBTZ1\n0\u0002(t\u0000t0)h_Ri(t)_Rj(t0)i (8)\nei!(t\u0000t0)d(t\u0000t0):\nWe therefore determine first in the presence of thermal fluc-\ntuations (rBz=vs= 0) from simulations of the stochastic\nLLG equation (3) the correlation functions of the velocities\nand use those to determine Gij(!)using Eq. (8). After a sim-\nple matrix inversion, this fixes the left-hand side of the equa-\ntion of motion, Eq. (2), and therefore contains all information\n0 5 10 15 20 25t ωp00.511.522.5 <ΔR2>α=0.01\nα=0.05\nα=0.1\nα=0.15\nα=0.2FIG. 2: Time dependence of the correlation function\n(Ri(t0+t)\u0000Ri(t0))2\u000b\nforT= 0:1Jand different values\nof the Gilbert damping \u000b(!p=B= 0:0278Jis the frequency for\ncyclotron motion).\non the (frequency-dependent) effective mass, gyrocoupling,\ndamping and gyrodamping of the skyrmion. Furthermore, the\ncorresponding spectrum of thermal fluctuations is given by\nEq. (7).\nFig. 2 showsh(\u0001R)2it=h(Rx(t0+t)\u0000Rx(t0))2i. As\nexpected, the motion of the skyrmion is diffusive: the mean\nsquared displacement grows for long times linearly in time\nh(\u0001R)2it= 2Dt, whereDis the diffusion constant. Usu-\nally the diffusion constant of a particle grows when the fric-\ntion is lowered26. For the skyrmion the situation is opposite:\nthe diffusion constant becomes small for the small friction,\ni.e., small Gilbert damping \u000b. This surprising observation has\nits origin in the gyrocoupling G: in the absence of friction\nthe skyrmion would be localized on a cyclotron orbit. From\nEq. (1), we obtain\nD=kBT\u000bD\nG2+ (\u000bD)2(9)\nThe diffusion is strongly suppressed by G. As in most materi-\nals\u000bis much smaller than unity while D\u0018G , the skyrmion\nmotion is characterized both by a small diffusion constant\nand a small friction. Such a suppressed dynamics has also\nbeen shown to be important for the dynamics of magnetic\nvortices28. For typical parameters relevant for materials like\nMnSi we estimate that it takes 10\u00006sto10\u00005sfor a skyrmion\nto diffusive over an average length of one skyrmion diameter.\nTo analyze the dynamics on shorter time scales we show in\nFig. 3 four real functions parametrizing G\u00001(!): a frequency-\ndependent mass m(!), gyrocouplingG(!), gyrodamping\n\u000b\u0000(!)and dissipation strength \u000bD(!)with\nG\u00001(!) =\u0012\n\u000bD(!)\u0000i!m(!)\u0000G(!) +i\u000b!\u0000(!)\nG(!)\u0000i!\u000b\u0000(!)\u000bD(!)\u0000i!m(!)\u0013\nFor!!0one obtains the parameters of Eq. (1). All pa-\nrameters depend only weakly on temperature, Gandmare ap-\nproximately independent of \u000b, while the friction coefficients4\n0 1 2 3 4 5\nω / ωp04812 α\nthermal diffusion\n0 1 2 3 4 5\nω / ωp00.51-G / 4 π\n0 1 2 3 4 5\nω / ωp0100200\nα Γ0 1 2 3 4 5\nω / ωp050100\nm\ncurrent driven motion\nforce driven motion\nFIG. 3: Dissipative tensor \u000bD, massm, gyrocouplingGand gyro-\ndamping\u000b\u0000as functions of the frequency !for the diffusive motion\natT= 0:1(solid lines). They differ strongly from the “apparent”\ndynamical coefficients (see text) obtained for the force driven (red\ndashed line) and current driven motion (green dot-dashed line). We\nuse\u000b= 0:2,\f= 0:1. The error bars reflect estimates of systematic\nerrors arising mainly from discretization effects, see appendix B.\n00.05 0.1 0.15 0.2α01234 αT=0.15\nT=0.2\n00.05 0.1 0.15 0.2α00.51-G / 4 π\n00.05 0.1 0.15 0.2α010203040\nα Γ00.05 0.1 0.15 0.2α0255075100\nmT=0.05\nT=0.1\nFIG. 4: Dissipative strength \u000bD, massm, gyrocouplingGand gy-\nrodamping\u000b\u0000as functions of the Gilbert damping \u000bfor different\ntemperatures T.\n\u000bDand\u000b\u0000are linear in \u000b, see Fig. 4. In the limit T!0,\nG(!!0)takes the value\u00004\u0019, fixed by the topology of the\nskyrmion15,20.\nBoth the gyrodamping \u0000and and the effective mass m\nhave huge numerical values. A simple scaling analysis of the\nLandau-Lifshitz-Gilbert equation reveals that both the gyro-\ncouplingGandDare independent of the size of the skyrmion,\nwhile \u0000andmare proportional to the area of the skyrmion,\nand frequencies scale with the inverse area, see appendix\nB. For the chosen parameters (the field dependence is dis-cussed in the appendix B), we find m\u00190:3N\ripm0and\n\u000b\u0000\u0019\u000b0:7N\ripm0, wherem0=~2\nJa2is the mass of a sin-\ngle flipped spin in a ferromagnet ( 1in our units) and we have\nestimated the number of flipped spins, N\rip, from the total\nmagnetization of the skyrmion relative to the ferromagnetic\nbackground. As expected the mass of skyrmions grows with\nthe area (consistent with an estimate29formobtained from the\nmagnon spectrum of skyrmion crystals), the observation that\nthe damping rate \u000bDis independent of the size of skyrmions\nis counter-intuitive. The reason is that larger skyrmions have\nsmoother magnetic configurations, which give rise to less\ndamping. For realistic system parameters J= 1meV (which\nyields a paramagnetic transition temperature TC\u001810K, but\nthere are also materials, i.e. FeGe, where the skyrmion lattice\nphase is stabilised near room-temperature16) anda= 5 ˚A and\na skyrmion radius of 200 ˚A one finds a typical mass scale of\n10\u000025kg.\nThe sign of the gyrodamping \u000b\u0000is opposite to that of the\ngyrocouplingG. This implies that \u000b\u0000describes not damp-\ning but rather antidamping: there is less friction for cyclotron\nmotion of the skyrmion than for the linear motion. The nu-\nmerical value for the antidamping turns out to be so large\nthatDm+ \u0000G<0. This has the profound consequence that\nthe simplified equation of motion shown in Eq. (1) cannot be\nused: it would wrongly predict that some oscillations of the\nskyrmion are not damped, but grow exponentially in time due\nto the strong antidamping. This is, however, a pure artifact\nof ignoring the frequency dependence of G\u00001(!), and such\noscillations do not grow.\nFig. 3 shows that the dynamics of the skyrmion has a strong\nfrequency dependence. We identify the origin of this fre-\nquency dependence with a coupling of the skyrmion coordi-\nnate to pairs of magnon excitations as discussed in Ref. 31.\nMagnon emission sets in for ! > 2!pwhere!p=Bis the\nprecession frequency of spins in the ferromagnet (in the pres-\nence of a bound state with frequency !b, the onset frequency\nis!p+!b, Ref. 31). This new damping channel is most ef-\nficient when the emitted spin waves have a wavelength of the\norder of the skyrmion radius.\nAs a test for this mechanism, we have checked that only\nthis high-frequency damping survives for \u000b!0. In Fig. 5\nwe show the frequency dependent damping \u000bD(!)for various\nbare damping coefficients \u000b. For small!it is proportional to\n\u000bas predicted by the Thiele equation. For !>2!p, however,\nan extra dampling mechanism sets in: the skyrmion motion\ncan be damped by the emission of pairs of spin waves. This\nmechanism is approximately independent of \u000band survives\nin the\u000b!0limit. This leads necessarily to a pronounced\nfrequency dependence of the damping and therefore to the ef-\nfective mass m(!)which is related by the Kramers-Kronig\nrelationm(!) =1\n!R1\n\u00001\u000bD(!0)\n!0\u0000!d!0\n\u0019to\u000bD(!). Note also that\nthe large\u000bindependent mass m(!!0)is directly related to\nthe\u000bindependent damping mechanism for large !. Also the\nfrequency dependence of m(!)andG(!)can be traced back\nto the same mechanism as these quantities can be computed\nfrom\u000bD(!)and\u000b\u0000(!)using Kramers-Kronig relations. For\nlarge frequencies, the effective mass practically vanishes and5\n0 1 2 3 4 5\nω/ωp05101520αD(ω)α=0.05\nα=0.1\nα=0.2\nFIG. 5: Effective damping, \u000bD(!)for\u000b= 0:2,0:1and0:05.\n0 1 2 3 4 5\nω / ωp-400-2000200400Mz\ntot600Re/Im SgRe Sg11\nRe Sg21\nIm Sg11\nIm Sg21\nFIG. 6: Dynamical coupling coefficients for the force driven motion\n(\u000b= 0:2). In the static limit everything but the real part of the diago-\nnal vanishes. R eS11\ng(!)however approaches the total magnetization\nMz\ntotas expected. The error bars reflect estimates of systematic er-\nrors, see appendix B.\nthe ‘gyrocoupling’ Gdrops by a factor of a half.\nIV . FORCE-DRIVEN MOTION\nNext, we study the effects of an oscillating magnetic field\ngradient rBz(t)in the absence of thermal fluctuations. As\nthe skyrmion has a large magnetic moment Mz\ntotrelative to the\nferromagnetic background, the field gradient leads to a force\nacting on the skyrmion. In the static limit, the force is exactly\ngiven by\nFg(!!0) =Mz\ntotrBz: (10)\nUsing G\u00001(!)determined above, we can calculate how the\neffective force Sg(!)rBz(!)(see Eq. 2) depends on fre-\nquency. Fig. 6 shows that for !!0one obtains the expectedresultSg(!!0) =\u000eijMz\ntot, while a strong frequency de-\npendence sets in above the magnon gap, for !&!p. This\nis the precession frequency of spins in the external magnetic\nfield.\nIn general, both the screening of forces (parametrized\nbySg(!)) and the internal dynamics (described by\nG\u00001(!)) determines the response of skyrmions, V(!) =\nG(!)Sg(!)rBz(!). Therefore it is in general not possi-\nble to extract, e.g., the mass of the skyrmion as described by\nG\u00001(!)from just a measurement of the response to field gra-\ndients. It is, however, instructive to ask what “apparent” mass\none obtains, when the frequency dependence of Sg(!)is ig-\nnored. We therefore define the “apparent” dynamics G\u00001\na(!)\nbyGa(!)Sg(!= 0) = G(!)Sg(!). The matrix elements\nofG\u00001\na(!)are shown in Fig. 3 as dashed lines. The appar-\nent mass for gradient-driven motion, for example, turns out\nto be more than a factor of three smaller then the value ob-\ntained from the diffusive motion clearly showing the impor-\ntance of screening effects on external forces. The situation\nis even more dramatic when skyrmions are driven by electric\ncurrents.\nV . CURRENT-DRIVEN MOTION\nCurrents affect the motion of spins both via adiabatic and\nnon-adiabatic spin torques30. Therefore one obtains two types\nof forces on the spin texture even in the static limit19–22,24.\nThe effect of a time-dependent, spin-polarized current on\nthe magnetic texture can be modelled by supplementing the\nright hand side of eq. (3) with a spin torque term TST,\nTST=\u0000(vs\u0001r)Mr+\f\nM[Mr\u0002(vs\u0001r)Mr]:(11)\nThe first term is called the spin-transfer-torque term and is\nderived under the assumption of adiabaticity: the conduction-\nelectrons adjust their spin orientation as they traverse the mag-\nnetic sample such that it points parallel to the local magnetic\nmoment Mrowing toJHandJsd. This assumptions is justi-\nfied as the skyrmions are rather large smooth objects (due to\nthe weakness of spin-orbit coupling). The second so called \f-\nterm describes the dissipative coupling between conduction-\nelectrons and magnetic moments due to non-adiabatic effects.\nBoth\u000band\fare small dimensionless constants in typical ma-\nterials. From the Thiele approach one obtains the force\nFc(!!0) =G\u0002vs+\fDvs: (12)\nFor a Galilei-invariant system one obtains \u000b=\f. In this\nspecial limit, one can easily show that an exact solution of the\nLLG equations in the presence of a time-dependent current,\ndescribed by vs(t)is given by M(r\u0000Rt\n\u00001vs(t0)dt0)pro-\nvided, M(~ r)is a static solution of the LLG equation for vs=\n0. This implies that for \u000b=\f, the skyrmion motion exactly\nfollows the external current, _R(t) =vs(t). Using Eq. (2),\nthis implies that for \u000b=\fone has G\u00001(!) =Sc(!). Defin-\ning the apparent dynamics, as above, Ga(!)Sc(!= 0) =\nG(!)Sc(!)one obtains a frequency independent G\u00001\na(!) =6\n0 1 2 3 4 5\nω / ωp-4 π-10-50β D(0)510Re/Im ScIm Sc11\nIm Sc21Re Sc11\nRe Sc21\nFIG. 7: Dynamical coupling coefficients (symbols) for the current-\ndriven motion ( \u000b= 0:2,\f= 0:1,J= 1,\u0015= 0:18J,B= 0:0278 ).\nThese curves follow almost the corresponding matrix elements of\nG\u00001(!)shown as dashed lines. A deviation of symbols and dashed\nline is only sizable for Re S11\nc.\n0 1 2 3 4 5\nω/ωp-5051015 mα=0.2,β=0\nα=0.2,β=0.1\n0 1 2 3 4 5\nω/ωp0510\nα Γα=0.2,β=0.15\nα=0.2,β=0.19\nα=0.2,β=0.3\nFIG. 8: Mass m(!)and gyrodamping \u000b\u0000(!)as functions of the\ndriving frequency !for the current-driven motion. Note that both M\nand\u0000vanish for\u000b=\f.\nSc(!= 0) =\fD 1\u0000i\u001byG: the apparent effective mass and\ngyrodamping are exactly zero in this limit and the skyrmion\nfollows the current without any retardation. For \u000b6=\f, the\nLLG equations predict a finite apparent mass. Numerically,\nwe find only very small apparent masses, ma\nc/\u000b\u0000\f, see\ndot-dashed line in upper-right panel of Fig. 3, where the case\n\u000b= 0:2,\f= 0:1is shown. This is anticipated from the anal-\nysis of the\u000b=\fcase: As the mass vanishes for \u000b=\f= 0,\nit will be small as long as both \u000band\fare small. Indeed\neven for\u000b6=\fthis relation holds approximately as shown\nin Fig. 7. The only sizable deviation is observed for Re S11\nc\nfor which the Thiele equation predicts Re S11\nc(!!0) =\fD\nwhile Re G\u0000111(!!0) =\u000bDas observed numerically.\nA better way to quantify that the skyrmion follows the cur-rent even for \u000b6=\falmost instantaneously is to calculate\nthe apparent mass and gyrodamping for current driven mo-\ntion, where only results for \u000b= 0:2and\f= 0:1have been\nshown. As these quantities vanish for \u000b=\f, one can ex-\npect that they are proportional to \u000b\u0000\fat least for small \u000b;\f.\nThis is indeed approximately valid at least for small frequen-\ncies as can be seen from Fig. 8. Interestingly, one can even\nobtain negative values for \f > \u000b (without violating causal-\nity). Most importantly, despite the rather large values for \u000b\nand\fused in our analysis, the apparent effective mass and\ngyrodamping remain small compared to the large values ob-\ntained for force-driven motion or the intrinsic dynamics. This\nshows that retardation effects remain tiny when skyrmions are\ncontrolled by currents.\nVI. CONCLUSIONS\nIn conclusion, we have shown that skyrmions in chiral mag-\nnets are characterised by a number of unique dynamical prop-\nerties which are not easily found in other systems. First, their\ndamping is small despite the fact that skyrmions are large\ncomposite objects. Second, despite the small damping, the\ndiffusion constant remains small. Third, despite a huge iner-\ntial mass, skyrmions react almost instantaneously to external\ncurrents. The combination of these three features can become\nthe basis for a very precise control of skyrmions by time-\ndependent currents.\nOur analysis of the skyrmion motion is based on a two-\ndimensional model where only a single magnetic layer was\nconsidered. All qualitative results can, however, easily be\ngeneralized to a film with NLlayers. In this case, all terms\nin Eq. (1) get approximately multiplied by a factor NLwith\nthe exception of the last term, the random force, which is en-\nhanced only by a factorpNL. As a consequence, the diffu-\nsive motion is further suppressed by a factor 1=pNLwhile\nthe current- and force-driven motion are approximately unaf-\nfected.\nAn unexpected feature of the skyrmion motion is the an-\ntidamping arising from the gyrodamping. The presence of\nantidamping is closely related to another important property\nof the system: both the dynamics of the skyrmion and the ef-\nfective forces acting on the skyrmion are strongly frequency\ndependent.\nIn general, in any device based on skyrmions a combination\nof effects will play a role. Thermal fluctuations are always\npresent in room-temperature devices, the shape of the device\nwill exert forces13,14and, finally, we have identified the cur-\nrent as the ideal driving mechanism. In the linear regime, the\ncorresponding forces are additive. The study of non-linear\neffects and the interaction of several skyrmions will be impor-\ntant for the design of logical elements based on skyrmions and\nthis is left for future works. As in our study, we expect that\ndynamical screening will be important in this regime.7\n 30\n 40\n 50\n 60\n 70  30 40 50 60 70 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004\nFIG. 9: Skyrmion density based on the normalized z-component of\nthe magnetization.\nAcknowledgments\nThe authors are greatful for insightful discussions with K.\nEverschor and Markus Garst. Part of this work was funded\nthrough the Institutional Strategy of the University of Cologne\nwithin the German Excellence Initiative” and the BCGS. C.S.\nthanks the University of Tokyo for hospitality during his re-\nsearch internship where part of this work has been performed.\nN.N. was supported by Grant-in-Aids for Scientific Research\n(No. 24224009) from the Ministry of Education, Culture,\nSports, Science and Technology (MEXT) of Japan, and by the\nStrategic International Cooperative Program (Joint Research\nType) from Japan Science and Technology Agency. J.I. is sup-\nported by Grant-in-Aids for JSPS Fellows (No. 2610547).\nAppendix A: Definition of the Skyrmion’s centre coordinate\nIn order to calculate the Green’s function, Eq. (3), one\nneeds to calculate the velocity-velocity correlation function.\nTherefore it is necessary to track the skyrmion position\nthroughout the simulation. Mostly two methods have been\nused so far for this25: (i) tracking the centre of the topological\ncharge and (ii) tracking the core of the Skyrmion (reversal of\nmagnetization).\nThe topological charge density\n\u001atop(r) =1\n4\u0019^ n(r)\u0001(@x^ n(r)\u0002@y^ n(r)) (A1)\nintegrates to the number of Skyrmions in the system. There-\nfore for our case of a single Skyrmion in the ferromagnetic\nbackground this quantity is normalized to 1. The center of\ntopological charge can therefore be defined as\nR=Z\nd2r\u001atop(r)r (A2)\nFor the case of finite temperature this method can, however,\nnot be used directly. Thermal fluctuations in the ferromagnetic\nbackground far away from the skyrmion lead to a large noise\nto this quantity which diverges in the thermodynamic limit.\nA similar problem arises when tracking the center using the\nmagnetization of the skyrmion.One therefore needs a method which focuses only on the\nregion close to the skyrmion center. To locate the skyrmion,\nwe use thez-component of the magnetization but take into ac-\ncount only points where Mz(r)<\u00000:7(the magnetization of\nthe ferromagnetic background at T= 0is+1). We therefore\nuse\n\u001a(r) = (1\u0000Mz(r)) \u0002[\u0000Mz(r)\u00000:7] (A3)\nwhere \u0002[x]is the theta function. A first estimate, Rest=RV,\nfor the radius is obtained from\nRA=R\nAr\u001a(r)d2rR\nA\u001a(r)d2r(A4)\nby integrating over the full sample volume V.Restis noisy\ndue to the problems mentioned above but for the system\nsizes simulated one nevertheless obtains a good first esti-\nmate for the skyrmion position. This estimate is refined by\nusing in a second step for the integration area only D=\b\nr2R2jjr\u0000Restj<r\t\nwhereris choosen to be larger\nthan the radius of the skyrmion core (we use r= 1:3p\nN<=\u0019,\nwhereN<is the number of spins with Mz<\u00000:7). Thus\nwe obtain a reliable estimate, R=RD, not affected by spin\nfluctuations far away from the skyrmion. From the resulting\nR(t), one can obtain the velocity-velocity correlation function\nhVi(t0+t)VJ(t0)i.\nAppendix B: Scaling invariance\nTo obtain analytic insight into the question of how param-\neters depend on system size and to check the numerics for\nlattice artifacts it is useful to perform a scaling analysis of the\nsLLG equations and the effective equations of motion for the\nskyrmion.\nWe investigate a scaling transformation, where the radius\nof the skyrmion is enlarged by a factor \u0011,M(r)!~M(r) =\nM(r=\u0011).\nFor the scaling analysis, we use the continuum limit of\nEq. (4) (setting a= 1)\nH[M] =Z\nd2r\u0014J\n2(rM)2+\u0015M\u0001r\u0002 M\u0000B\u0001M\u0015\nThe three terms scale with \u00110,\u0011and\u00112, respectively. To ob-\ntain a larger skyrmion, we therefore have to rescale \u0015!\u0015=\u0011\nandB!B=\u00112. This implies that the Be\u000bterm in the sLLG\nequation scales with 1=\u00112and therefore also the time axis has\nto be rescaled, t!\u00112t, implying that all time scales are a\nfactor of\u00112longer and all frequencies a factor 1=\u00112smaller.\nSimiliarly, the driving current, i.e. vsis reduced by a factor\n\u0011\u00001while the temperature remains unscaled. This implies that\nwhen M(r;t)is a solution for a given value of \u0015,Bandvs\nandG(!)the corresponding velocity-correlation function of\nthe skyrmion, then M(r=\u0011;t=\u00112)is a solution for \u0015=\u0011,B=\u00112,\nvs=\u0011with correlation function G(!\u00112).\nAccordingly, the !!0limit and therefore the gyrocou-\nplingG, the friction constant \u000bDand the diffusion constant of8\n0.03 0.035 0.04\nBz00.511.5 α D\n0.03 0.035 0.04\nBz00.51-G / 4 π\n0.03 0.035 0.04\nBz05101520\nα Γ0.03 0.035 0.04\nBz020406080\nm\nFIG. 10: Field dependence of the dissipative tensor \u000bD, the massm,\nthe gyrocouplingGand the gyrodamping \u000b\u0000forJ= 1,\u0015= 0:18J.\nThe main effect is that both mand\u000b\u0000shrink when the size of the\nskyrmion shrinks with increasing Bz.\nthe skyrmion are independent of\u0011, consistent with the analyt-\nical formulas eq. (6). In contrast, the mass of the skyrmion,\nm, and the gyrodamping \u000b\u0000scale with\u00112. They are therefore\nproportional to the number of spins constituting the skyrmion\nconsistent with our numerical findings. When one does, how-\never, change the external magnetic field for fixed strength\u0015\nof the DM interaction, the internal structure of the skyrmion\nand therefore also G(!)change quantitatively in a way not\npredictable by scaling. In Fig. 10 we therefore show the B-dependence of \u000bD,m,Gand\u000b\u0000. For increasing Bthe size of\nthe skyrmion shrinks. From our previous analysis, it is there-\nfore not surprising that both the effective mass mand the gy-\nrodamping\u000b\u0000shrink substantially while \u000bDandGare less\naffected . Quantitatively, this change of mand\u000b\u0000is how-\never not proportional to the number of spins constituting the\nskyrmion.\nWe have tested numerically the scaling properties for \u0011=\n1:2and find that all features are quantitatively reproduced.\nSmall variations on the level of a few percent do, however,\noccur reflecting the typical size of features arising from the\ndiscretization of the continuum theory. A conservative esti-\nmate of such systematic discretization effects for the diffusive\nmotion is given by the error bars in Figs. 3 (all statistical er-\nrors are smaller than the thickness of the line). For the field-\ndriven motion (Fig. 3 and Fig. 6) spatial discretization effects\nlead to a different source of errors. For very small field gradi-\nents and high frequencies the displacement of the skyrmion is\nmuch smaller than the lattice spacing and the response is af-\nfected by a tiny pinning of the skyrmion to the discreet lattice.\nFor larger gradients, however, nonlinear effects set in and for\nsmall frequencies the skyrmion starts to approach the edge of\nthe simulated area. In Fig. 3, we therefore used for the force-\ndriven motionrB= 0:0005 for!<2!pandrB= 0:0015\nfor! >2!p. Error bars have been estimated from variations\nof the numerical values when rBwas varied from 0:0001 to\n0:0015 . For the current-driven motion errors are so tiny that\nthey are not shown.\n\u0003Electronic address: nagaosa@riken.jp\n1Hubert, A. & Sch ¨afer, R. Magnetic Domains: The Analysis of\nMagnetic Microstructures (Springer, Berlin, 1998).\n2Malozemoff, A. P. & Slonczewski, J.C. Magnetic Domain Walls\nin Bubble Materials (Academic Press, New York, 1979).\n3Skyrme, T. H. R. A Non-Linear Field Theory. Proc. Roy. 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Nature 428, 539–542 (2004).\n19Schulz, T. et al. Emergent electrodynamics of skyrmions in a chi-\nral magnet. Nature Phys. 8,301–304 (2012).\n20Thiele, A. A. Steady-state motion of magnetic domains. Phys.\nRev. Lett. 30,230–233 (1973).\n21Everschor, K. et al. Current-induced rotational torques in the\nskyrmion lattice phase of chiral magnets. Phys. Rev. B 86,054432\n(2012).\n22Everschor, K. et al. Rotating skyrmion lattices by spin torques and\nfield or temperature gradients. Phys. Rev. B 86,054432 (2012).\n23He, J. et al. Current-driven vortex domain wall dynamics by mi-\ncromagnetic simulations. Phys. Rev. B 73,184408 (2006).\n24Iwasaki, J., Mochizuki, M. & Nagaosa, N. Universal current-9\nvelocity relation of skyrmion motion in chiral magnets. Nat. Com-\nmun. 4,1463 (2013).\n25Makhfudz, I., Kr ¨uger, B. & Tchernyshyov, O. Inertia and chiral\nedge modes of a skyrmion magnetic bubble. Phys. Rev. Lett. 109,\n217201 (2012).\n26Kubo, R. The fluctuation-dissipation theorem. Rep. Prog. Phys.\n29,255–284 (1966).\n27Garc ´ıa-Palacios, J. L. and L ´azaro, F. J. Langevin-dynamics study\nof the dynamical properties of small magnetic particles. Phys. Rev.\nB58,14937–14958 (1998).28Clarke, D. J. and Tretiakov, O. A. and Chern, G.-W. and Bazaliy,\nY . B. and Tchernyshyov, O., Phys. Rev. B 78, 134412 (2008).\n29Petrova, O. and Tchernyshyov O., Phys. Rev. B 84, 214433\n(2011).\n30Tatara, G., Kohno, H., & Shibata, J. Microscopic approach to\ncurrent-driven domain wall dynamics. Physics Reports 468, 213–\n301 (2008).\n31Sch¨utte, C. and Garst, M. Magnon-skyrmion scattering in chiral\nmagnets. arxiv 1405.1568 (2014)."    },    {        "title": "1901.05753v1.Spin_transport_parameters_of_NbN_thin_films_characterised_by_spin_pumping_experiments.pdf",        "content": "1 \n  \nSpin transport parameters of NbN thin films  characterised by spin \npumping experiments  \nK. Rogdakis1,*, A. Sud1, M. Amado2, C. M. Lee2, L. McKenzie -Sell2, K.R. Jeon2, \nM. Cubukcu1, M. G. Blamire2, J. W. A. Robinson2, L. F. Cohen3, and H. Kurebayashi1,† \n 1London Centre for Nanotechnology, University College London, London WC1H 0AH, United \nKingdom  \n2Department of Materials Science & Metallurgy, University of Cambridge, Cambridge CB3 0FS, \nUnited Kingdom  \n3The Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom  \n \nAbstract  \nWe present  measurements of ferro magnetic -resonance - driven  spin pumping and  inverse spin -\nHall effect in NbN/ Y3Fe5O12 (YIG) bilayers . A clear enhancement  of the (effective) Gilbert \ndamping constant of the thin -film YIG was observed due to  the presence of the NbN  spin sink . \nBy varying the NbN thickness and em ploying spin-diffusion theory , we have estimated the  \nroom temperature  values of the spin diffusion length and the spin Hall angle in NbN  to be  14 \nnm and -1.1×10-2, respectively.  Furthermore, we have determined  the spin-mixing conductance \nof the NbN/YIG  interface to be 10 nm-2. The experimental quantification of these spin transport  \nparameters is an important step towards  the development of  superconducti ng spintronic devices \ninvolving NbN  thin films . \n \nIntroduction  \nThe extraction of key functional materials parameters associated with electron transport \nis important for the development of new solid -state device schemes as well as testing \nprototypes. In the field of spintronics, the spin Hall angle ( θSH) represents the strength of spin -\nHall effect  (SHE)  [1] that converts charge currents into spin currents via the relativistic spin -\norbit interaction. T he spin diffusion le ngth (𝑙𝑆𝐷) [2] is a parameter that describes  the distance \nover which non-equilibrium spin currents  can diffuse  before dissipation and is crucial in \ndetermining the useful device dimensions  of future spintronic applications. Moreover, t he spin \nangular momentum  transfer  across  a ferromagnetic (FM) and non -magnetic (NM) interfac e can \nbe parameterised by the spin mixing conductance  (𝑔𝑟↑↓) which governs the spin current \ngeneration efficiency in spin pumping  process es [3]. These s pin transport parameters can be 2 \n determined by employing  different  measurement techniques. For example, it is possible to use \nlateral spin -valves to quantify 𝑙𝑆𝐷  and θSH in non -magnetic materials [ 4, 5, 6, 7]. Spin pumping  \n[3, 8, 9] is another established method to investigate spin transport parameters in a variety of \nmaterials, such as metals  [10], inorganic  [11, 12] and organic semiconductors  [13, 14], graphene  \n[15] and topological insulators  [16]. It should be noted that spin pumping  relies on the transfer \nof angular momentum from a ferromagnet with precessing moments into an adjacent non -\nmagnetic layer , and do es not suffer from the conductance mismatch problem which causes \ndifficulties in electric al spin injection through ohmic contacts  [11].  Using  a FM conductor as \nspin injector in a spin pumping experiment can potentially give rise to microwave  (MW) -\ninduced photo -voltages  [17] due to time -varying resistance changes produced by the magnetic \nprecession coupled with a time -varying current, as  well as the ISHE in the FM layers  [18, 19]. \nThe use of FM insulators such as Y3Fe5O12 (YIG) to conduct spin pumping experiments has the \nadvantage because these effects are negate d. In addition , YIG has a low bulk Gilbert damping \nconstant (α ≃ 6.7 × 10−5) and a high Curie temperature ( TC = 560 K) , enabling efficient spin \npumping at  room temperature ( RT) [20].  \nIn this paper, we report spin pumping in thin -film YIG/NbN bilayers with the aim of \nextracting multiple spin transport parameters of NbN thin films  in the normal state . NbN  is a \nkey material for superconducting (SC) spintronics  [21] with a  bulk TC of approximately 16.5 \nK, a SC energy gap of 2.5  meV, and a SC coherence length of 5  nm [22]. NbN is increasingly \nused in the field of SC spintronics , for example  in spin -filter Josephson junctions  [23, 24, 25] \nand to demonstrate spectroscopic evidence for odd frequency (spin -triplet) superconductivity \nat the interface with GdN [26]. Recently , Wakamura et al.  observed an unprecedented \nenhancement of the SHE  at 2K, interpreted in terms of  quasiparticle mediated transport [ 27]. \nQuasiparticle spin transport has also been investigated by spin pumping and by monitoring the \nspin Seebeck effect [ 28, 29]. To the best of our knowledge, s pin trans port parameters in NbN  \nsuch as 𝑙𝑆𝐷 and θSH have only been extracted  by Wakamura  et al.  [27] by the spin absorption \nmethod in lateral spin -valves , and it is vitally important to extract  these  parameters also by other \ncharacterisation techniques  and with NbN  grown by different growth method s. This can, for \nexample, help to understand whether spin transport parameters in NbN have a significant \ndependence on  the growth conditions . In our study, b y using high-quality epitaxial thin-film \nYIG it is possible to obs erve a modulation of the Gilbert damping constant  (α) with NbN \nthickness and therefore  extract 𝑙𝑆𝐷 of NbN (14 nm) and 𝑔𝑟↑↓ of the YIG/NbN interface (1 0 nm-\n2). Furthermore, we have investigated the NbN -thickness -dependence of the ISHE  voltage \n(VISHE) and have determined θSH of NbN ( -1.1 ×10-2) by the spin pumping technique . We 3 \n compare 𝑙𝑠𝑑 extracted by three independent methods , namely the thickness dependence of α and \nVISHE as well as Hanle spin precession , and we find  good agreement between them . Determining \nthe normal -state spin -transport parameters in NbN from spin -pumpi ng-induced ISHE is \nimportant, which enables the comparison between  parameters extract ed using various  \ntechniques from  different research groups [e.g. 27-29]. By accumulation of a body  of results , \nwe will then be able to understand the fundamental nature of SHE  and spin transport in  NbN  \nwhich can be useful and transferable to future spintronics research using SC NbN [ 21, 30].  \n \nMaterial growth   \n          Epitaxial  YIG thin films are grown on (111) -oriented GGG single crystal substrate s by \npulse laser depositio n (PLD) in a n ultra-high vacuum chamber (UHV) with a base pressure  \nbetter than 5×10-7 mbar. Prior to film growth, the GGG substrate are ultrasonically cleaned by \nacetone and isopropyl alcohol and  annealed ex-situ at 1000 oC in a constant O2 flow \nenvironment for 8 hours. The YIG is deposited from a stoichiometric ( polycrystalline ) target \nusing a KrF excimer laser (248 nm wavelength) , with a nominal energy of 450 mJ  and fluence \nof 2.2 W cm-2 in 0.12 mbar of O 2 at 680  oC, and pulse frequency of 4 Hz for 60 minutes. The \nYIG is p ost-anneal ed at 750  oC for 1.5 hours in 0.5 mb ar partial pressure of static O 2 and \nsubsequently cooled to RT at a rate of -10 K/min. Atomic force microscopy (AFM ) reveal s that \na root -mean -squared roughness of the YIG films is  less than 0.16 nm over 10 ×10 µm scan size  \n[Fig. 1(a) ]. The YIG films were characterised by a SC quantum interference device (SQUID) \nmagnetometer and have a saturation magnetisation ( MS) of 140  3 emu cm-3 [Fig. 1(b) ], which  \nmatches the bulk value  [31]. In Fig. 1(c) we have plotted a high -angle X -ray diffraction trace \nof the same film where Laue fringes indicate layer -by-layer growth of YIG  and good lattice -\nmatching with the substrate. Figure 1(d ) shows  low-angle X -ray reflectivity from a  YIG film  \nand from the  decay and angle separation of the Kiessig fringes , we determined a  nominal \nthickness tYIG = 60  2 nm.  Following the growth  of YIG,  films were  directly  transferred in air \nto a UHV  sputter deposition system with a base pressure  of 1×10-9 mbar. NbN is grown by \nreactive sputtering in a gas mixture of argon (72%) and nitrogen (28%) with the deposition rate \nof 85 nm  min-1. The growth temperature is RT,  giving  polycrystalline NbN  layers . We grew \nNbN with different thicknesses (tNbN) from  5 to 50  nm.  \n \nFerromagnetic resonance ( FMR ) setup and spin pumping measurements  4 \n FMR is performed using a broadband coplanar waveguide  (CPW)  and ac-field \nmodulation  technique as illustrated i n Fig. 2(a). The sample s are placed face down on top of \nthe CPW s where an insulator tape is used for electrical insulation. We generate dc ( H) and ac \n(hac) magnetic fields by electromagnets  and the absorbed power  at the modulation frequency  is \nmeasured by a MW power detector and a lock -in amplifier while  H is swept . An input MW \npower  (PMW) of 100  mW is used unless otherwise is stated . We kept t he modulation field  \namplitude  (hac) smaller than the  measured FMR  linewidth s of all samples tested , in order to \navoid artefacts by strong  modulation . The magnetic field  is applied along different in -plane and \nout-of-plane directions  related to the samples as shown  in Fig. 2(a ). The  FMR absorption ( VP) \nwas measured using a MW power detector for different frequencies  typically ranging from 2 -\n12GHz  as depicted  in Fig. 2(b) (for a sample  with tNbN = 10 nm ). For each scan, the resonance \nfield ( Hres) and the  half-width -at-half-maximum  linewidth ( ΔH) of the FMR signal are \ndetermined  by a fit using differential forms of symmetric and anti -symmetric Lorentzian \nfunctions  (Appendix A) . Figure 2(c) shows the frequency dependence of the extracted Hres for \ndifferent NbN thicknesses. The curves of the frequency dependen ce for all samples, including \ntNbN  = 0 nm, overlap suggesting no significant modification of the YIG magnetic anisotrop y due \nto the presence of NbN. We note here that the effective magnetisation (Meff) extracted from the \nfits for each sample shows larger values than the Ms value measured in the SQUID. This \nenhanced Meff has often been observed in other thin -film studies [ 32, 33] and a detailed \nunderstanding of this lies outside of the scope of the present  work . For spin transport analysis \ndiscussed later, we use the values extracted by SQUID measurements since it is a more direct \nmeasurement of magnetisati on, while we confirmed that the  discrepancy between Ms and Meff \ndoes not alter the calculated spin tran sport parameters significantly. Although the magnetic \nanisotropies of the YIG films are unchanged with or without the presence of NbN, the \nmagnetization  relaxation of YIG represented by ΔH shows a clear dependence on tNbN as shown \nin Fig. 3( a). With a linear fit to  the data for each thickness using  ΔH =ΔH0 + (4πα/γ)f where \nΔH0 and γ describe  the inhomogeneous broadening and  the gyromagnetic ratio respectively, we \nhave quantif ied α for each sample  as shown  in Fig. 3(b) . α = (5.4 ± 0.2) × 10−4 was obtained for \nbare YIG, which  compare s well to  previously reported values  [34, 35]. A gradual increase of  α \nis observed  with increasing  NbN thickness , in agreement with spin pumping through the \nYIG/NbN interface  where  the α dependence with tNbN is given by  [36]:  \n𝛼(𝑡𝑁𝑏𝑁)=𝛼0+(𝑔𝐿𝜇𝐵𝑔𝑟↑↓\n4𝜋𝑀𝑠𝑡𝑌𝐼𝐺)∙[1+𝑔𝑟↑↓𝜌𝑁𝑏𝑁𝑙𝑠𝑑𝑒2\n2πℏ tanh(𝑡NbN\n𝑙𝑠𝑑)]−1\n (1). 5 \n Here,  α0  is the Gilbert damping constant for tNbN= 0 nm and the second term represents the \ndamping enhancement by spin pumping  into NbN ; 𝑔𝐿 is the free electron Land é factor  which is \nassumed equal to  2,  𝑔𝑟↑↓ is the effective real -part spin -mixing conduct ance across the NbN/YIG \ninterface;  𝜌𝑁𝑏𝑁 is the resistivity of NbN  which was  measured for each sample  [see inset of Fig . \n3(b)], and 𝑒 is the electron charge. A best f it of the data in Fig. 3(b) using  Eq. (1) yield s 𝑔𝑟↑↓ = \n10 ±2 nm-2 and 𝑙𝑠𝑑 =14 ± 3 nm. The extracted 𝑙𝑠𝑑 can be well compared with the value  (7 nm)  \nby Wakamura et al.  [27] using the spin -absorption method in lateral spin -valve devices . We \nalso found that the spin couplin g of NbN/YIG  is as good as  heavy-metals/YIG interfaces since  \n𝑔𝑟↑↓ is comparable  to those of  YIG/Pt, YIG/Ta and YIG/W  [35]. We note from analytic \ncalculations (Appendix B)  that the additional damping expected from eddy currents cannot \nexplain t he observed NbN  thickness  dependence of α. \nWe now discuss the  ISHE  voltage (VISHE) measurements . In Figs. 4(a) and 4(b) we show \ntypical data set s for VISHE (for direct comparison we present also corresponding Vp data) for tNbN \n= 20 nm and f = 3 GHz . Note that , since we use d the lock-in ac field-modulation  method for \nboth detections , the curves represent the derivative of the signals  without the ac field -\nmodulation : for both VP and VISHE a symmetric Lorentzian lineshape is expected without the ac \nfield modulation. As expected  from spin pumping and ISHE , we observe a clear VISHE peak at \nthe YIG precession  frequency . By changing the sign of H [observe the sign of magnetic field \naxis for Figs. 4 (a) and 4 (b)], we observe  a sign change of VISHE  in agreement with the symmetry \nof spin pumping  [37]. Corresponding measurements for tNbN = 5 nm are depicted in Figs. 4(c) \nand 4(d). By using the known ac field modulation amplitude as well as differential forms of \nsymmetric and anti -symmetric Lorentzian fu nctions (Appendix A), we quantify the peak \namplitude of ISHE voltage defined as 𝑉𝐼𝑆𝐻𝐸∗. The PMW-dependence of 𝑉𝐼𝑆𝐻𝐸∗ shown in Figs. 5 (a) \nand 5 (b) suggests that 𝑉𝐼𝑆𝐻𝐸∗ is proportional to  PMW, consistent with standard spin -pumping \ntheory  [36].  \nWe have  also performed H - angular dependen t measurements of V*ISHE along in -plane \nand out-of-plane directions of the NbN/YIG  films . The in -plane angular dependence  of the  spin \npumping experiment  follows  the expres sion  𝑉𝐼𝑆𝐻𝐸∗∝ 𝝐𝒙∙(𝑱𝐬×𝝈)∙|𝝈×𝒉𝒓𝒇| where the first \npart is due to  the ISHE  symmetry , 𝐸𝐼𝑆𝐻𝐸∝(𝑱𝐬×𝝈) , multiplied by the amplitude of magnetic \ntorque generated by MW-induced magnetic field  |𝝈×𝒉𝒓𝒇|; here, 𝝐𝒙 is the unit vector along x \ndirection in the measurement ’s framework shown in Fig. 2(a). The first component gives  a cos \ndependence  whereas the second produces a |cos| dependence,  which combined nicely matches \nour expe rimental results shown in Fig. 6 (a).  The rationale to plot 𝑉𝐼𝑆𝐻𝐸∗𝑡𝑁𝑏𝑁/𝜌𝑁𝑏𝑁 against 𝑡𝑁𝑏𝑁 6 \n is to include the thickness dependence of 𝜌𝑁𝑏𝑁 allowing to fit the data points based on bare  \nNbN as well as those of the YIG/NbN bi-layers . In addition, this analysis can display the \nasymptotic behaviour of the data/fit -curves towards  the long thickness limit. The in -plane \nsymmetry re -confirm s that spin rectification effects are not a dominant mechanism in our \nmeasurements since in this case  a higher order sin  2θ component is expected in the voltage \nsymmetry  [17]. We also  measure d the out -of-plane angular dependence of 𝑉𝐼𝑆𝐻𝐸∗ as shown in \nFig. 6 (b) and moreover we applied the Hanle precession model  [38] to fit our data . In this case \nthe out -of-plane 𝑉𝐼𝑆𝐻𝐸∗  is given by:  \n𝑉𝐼𝑆𝐻𝐸∗(𝜙)∝{cos(𝜙)∙cos(𝜙−𝜙𝑀)+sin𝜙∙sin(𝜙−𝜙𝑀)∙[1\n1+(𝜔𝐿∙𝜏𝑠)2]}    (2) \n𝜔𝐿=𝑔𝐿𝜇𝐵∙(𝜇0𝐻)/ℏ is the Larmor frequency  and 𝜏𝑠 is the  spin relaxation time  in NbN ;  𝜙 and \n𝜙𝑀 represent the angle of between  the z -axis and  H and the equilibrium magnetic moment \ndirection, respectively. By minimizing the total magnetic energy of the FM layer consisting of \nthe Zeeman and demagnetization energy, the following equation is derived to determine the \nvalue of 𝜙𝑀 with respect to 𝜙: 𝜙𝑀=\n𝜙−arctan\n[    \nsgn(𝜙).√(cos(2𝜙)+(𝜇0𝐻𝑟𝑒𝑠\n𝜇0𝑀𝑒𝑓𝑓)\nsin(2𝜙))2\n+1−(cos(2𝜙)+(𝜇0𝐻𝑟𝑒𝑠\n𝜇0𝑀𝑒𝑓𝑓)\nsin(2𝜙))\n]    \n [39]. After spin currents are \ninjected  inside NbN,  they start precessing due to  the external ly applied  H. This is described by \nthe well -known  Hanle  precession model which is the basis of Eq. (2). The equilibrium spin \norientation depends on the precession rate ( 𝜔𝐿) and the spin relaxation rate (1/ 𝜏𝑠), both of which \ncontribute  in the equation. When 𝜏𝑠 is much shorter than 1/ 𝜔𝐿, the injected spins do not precess \nand instead generate 𝑉𝐼𝑆𝐻𝐸 with spin orientation along M (𝜙M). This is the case for the red curve \nin Fig. 6(b). In the opposite extreme condition ( depicted as blue curve in Fig. 6(b)), spins \nprecess many times and dephase along the H orientation (𝜙), resulting in an approximately  \ncos(𝜙) angle dependence . Fitting the data in Fig. 6(b) using  Eq. (2) allows us to  estimate 𝜏𝑠. In \nparticular, the  best fit of the measured  𝑉𝐼𝑆𝐻𝐸∗(𝜙) was obtained  giving  an extracted  𝜏𝑠 = 11 ± 2 \nps. This value quantified by the Hanle model can be compared with 𝜏𝑠 independently calculated \nfrom the spin diffusion model as already discussed above, i.e. 𝜏𝑠 =(𝑙𝑠𝑑)2/𝐷 where  D is the \nEinstein diffusion coefficient (its  value equal to  0.4−0.56 cm2/s was taken from Ref. [ 40]). \nFollowing this approach and by using 𝑙𝑠𝑑=14 nm as extracted from the thickness dependence \nof damping modulation , we calculated 𝜏𝑠 = 3.6-5.9 ps which is  a fair agreement between the \ntwo different  𝜏𝑠 extraction methods .  7 \n In the following section,  the 𝜃SH of NbN is determined from the thickness dependence \nof 𝑉𝐼𝑆𝐻𝐸∗ as shown in Fig. 7. Using the spin transport parameters discussed above  and Eq. (3) , \nwe estimate the  spin current emitted at the NbN/ YIG interface,  js, as well as  the value of   𝜃SH \nextracted by fitting the thickness dependence  of 𝑉𝐼𝑆𝐻𝐸∗ [39]: \n𝑉𝐼𝑆𝐻𝐸∗=(𝑤𝑦𝜌𝑁𝑏𝑁\n𝑡𝑁𝑏𝑁)∙𝜃SH𝑙𝑠𝑑∙tanh(𝑡𝑁𝑏𝑁\n2𝑙𝑠𝑑)∙𝑗𝑠                                          (3) \nwhere 𝑗𝑠=(𝐺𝑟↑↓ℏ\n8𝜋)∙(𝜇0ℎ𝑟𝑓𝛾\nα)2\n∙[𝜇0𝑀𝑠𝛾 + √(𝜇0𝑀𝑠𝛾)2 + 16(𝜋𝑓)2 \n(𝜇0𝑀𝑠𝛾)2 + 16(𝜋𝑓)2]∙(2𝑒\nℏ) \nwith   𝐺𝑟↑↓≡𝑔𝑟↑↓∙[1+𝑔𝑟↑↓𝜌𝑁𝑏𝑁𝑙𝑠𝑑𝑒2\n2πℏ tanh(𝑡𝑁𝑏𝑁\n𝑙𝑠𝑑)]−1\n. \nHere w e assume that  YIG is a perfect insulator ; 𝜇0ℎ𝑟𝑓 is the amplitude of MW magnetic field \n(56 µT for 100 mW);  𝑤𝑦 is defined by the width of MW transmission line . For the data fitting \nprocedure  we use 𝜃SH and 𝑙𝑠𝑑 as free parameters , where the best fitting was achieved for 1.1 \n×10-2 and 14 nm, respectively. We also confirm ed the sign of 𝜃SH to be negative by comparing \nYIG/NbN data with a YIG/Pt control sample where Pt is known to have a positive 𝜃SH [1]. We \nemphasise that  the value of  𝑙𝑠𝑑 extracted by the thickness dependence of 𝑉𝐼𝑆𝐻𝐸∗ agrees very well \nwith the one extracted from the thickness dependence  of damping . The former approach \nincludes spin -orbit and spin -transport properties of NbN, whereas the latter is purely related \nwith magnetic propert ies of YIG . We found that t he value we extract by our spin pumping \nexperiments is similar to  θSH quantified by Wakamura et al.  using lateral spin -valve samples \n(θSH ~-1× 10-2) [27] for the temperature region between 20 to 200 K.  Although there is \ndifference in temperature between experiments by Wakamura et al.  and ours, an agreement of \nthe same sign and magnitude in  θSH quantified by different techniques ( i.e. spin pumping and \nspin-absorption) has been observed. T he value of θSH of the same material but grown and \nmeasured by different research groups can vary rather significantly , for example as  in the case s \nof Pt  [41] and some topological insul ators [ 42, 43, 44]. Such differences might result from \nvariation in  sample qualit y where the density of scattering impurities can particularly  influence  \nθSH via the extrinsic spin-Hall mechanisms  [1]. We note that the resistivity of NbN used in  the \nWakamura  et al.  study measured at 20 K  (220 μΩcm ) is roughly three times greater than our \nNbN  films at the same temperature (65  μΩcm ). This highlights  that the resistivity and mobility \nof NbN might be  highly growth -dependent, possibly due to the stoichiometry of Nb and N  as \nwell as the nitrogen vacancy concentration . The NbN spin-Hall resistivity of Wakamura  et al.  \nis 2.2 μΩ∙cm at 20 K  [27], whereas  our spin-Hall resistivity  at RT is calculated to be  0.5 μΩ∙cm \nwhich is smaller owing to the resistivity difference . For the relevance of SC spintronics, w e also 8 \n compare our θSH value with those of Nb thin films  reported in previous works . Morota et al.  \nmeasured θSH of several 4d and 5d transition metals by the spin absorption method in the lateral \nspin valve structures  [6] including Nb. They quantified θSH of Nb to be  -8.7 ×10-3 at 10K, which \nis close  to our  θSH in NbN at RT. There is recent work by Jeon et al.   who measured  θSH = -\n1×10-3 in Nb at RT  [39]. Direct comparison between θSH of Nb and NbN is not possible but they \nare within the same order, suggesting that there are similar atomistic spin-orbit contributions \nfrom Nb atoms both for Nb and NbN. Details of this will be further clarified when more realistic \ntheoretical studies  of SHE  in NbN become available .   \nAs a final remark, we also performed FMR measurements as a fun ction of temperature  to \ndetermining the low -temperature spin -pumping properties of NbN through the SC Tc. However, \na significant increase of magnetic damping was observed as the temperature was lowered  (this \nbehaviour is summarised in Appendix  C). This enhanced damping complica ted the \ninvestigation of VISHE across the SC Tc. \n \nConclusions  \nWe determined the spin transport parameters  of polycrystalline NbN thin -films  by the spin \npumping technique  using epitaxial YIG thin-films  at RT . We observe a modification of  the YIG \nGilbert damping  param eter as a function of the variation of the  NbN film thickness, confirming \nspin current injection  in the NbN  layer . By applying a spin-diffusion model , we have  estimate d \n𝑙𝑠𝑑 =14 ± 3 nm in NbN and 𝑔𝑟↑↓ = 10 ±2 nm-2 at the NbN/YIG interface . From the  NbN  thickness \ndependence of the ISHE  voltages , we determine  θSH to be equal to  -1.1 ×10-2. We also compare \n𝑙𝑠𝑑 of NbN extracted by  three different techniques (thickness dependence of both α and 𝑉ISHE \nas well as the Hanle measurements) and found good agreement between them . The measured \nparameters are a good reference to understand the NbN spin -orbit and spin transport properties  \nand to aid the design of  feasible spintronic experiments/ devices in the normal and  SC state.  \nAcknowledgment  This work was supported by the Engineering and Physical Sciences \nResearch Council  through the  Programme Grant “Superspin” ( Grant No. EP/N017242/ 1) and \nInternational Network Grant ( Grant No. EP/P026311/1 ). \n \nAppendix A: Derivation of FMR fit curves  \nIn normal dc FMR analysis, the measured dc voltage can be decomposed into symmetric and \nanti-symmetric Lorentzian functions with respect to μ0Hres, with weights of Asym and Aasy \nrespectively , where combined lead to the following general power absorption expression  \n[which is applicable both for  FMR absorption (V p) and ISHE voltage (V ISHE)]:  9 \n 𝑃𝑑𝑐(𝐻)=𝐴𝑠𝑦𝑚(𝐻)+𝐴𝑎𝑠𝑦(𝐻)+𝑉0=𝐴𝑠𝑦𝑚∆𝐻2\n(𝐻−𝐻𝑟𝑒𝑠)2+∆𝐻2+𝐴𝑎𝑠𝑦∆𝐻(𝐻−𝐻𝑟𝑒𝑠)\n(𝐻−𝐻𝑟𝑒𝑠)2+∆𝐻2+𝑉0,       (4)                                  \nwhere V0 is a background voltage. The first term gives  the symmetric lineshape and the second \nterm produces the anti -symmetric one. For FMR mea surements based on a c magnetic -field \nmodulation, where an additional pair of coils on electromagnets provide small ac magnetic field, \nPac has the following relationship with Pdc. \n𝑃𝑎𝑐=𝑑𝑃𝑑𝑐\n𝑑𝐻ℎ𝑎𝑐                                                          (5) \nwhere, hac is the amplitude o f ac magnetic field modulation. With these two equations, we can \ncalculate 𝑃𝑎𝑐 as: \n𝑃𝑎𝑐(𝐻)=−𝐴𝑠𝑦𝑚ℎ𝑎𝑐2(𝐻−𝐻𝑟𝑒𝑠)∆𝐻2\n{(𝐻−𝐻𝑟𝑒𝑠)2+∆𝐻2}2−𝐴𝑎𝑠𝑦ℎ𝑎𝑐∆𝐻{(𝐻−𝐻𝑟𝑒𝑠)2+∆𝐻2}\n{(𝐻−𝐻𝑟𝑒𝑠)2+∆𝐻2}2                  (6) \nThis equation was used to fit the ac field modulated signals, bot h Vp and VISHE, in our study. \nThe first term gives now the anti -symmetric lineshape and the second term pr oduces the \ndistorted symmetric one . Figure 8  (a) and (b) display typical FMR  data together with best fit \ncurves using Eq. ( 4) and (6 ), respectively, with corresponding extracted parameters presented \nin Fig 8 as legends . We also checked that there was no experimental artifact by doing our ac \nexperiments, by directly confirming that  ac (Fig. 8a) and dc (Fig. 8b) measurements for the \nsame experimental conditions generate the same fit parameters.  \n \nAppendix B: A simpl ified model for the eddy -current damping  \nWe consider a slab of magnet containing a  chain of distributed magnetic moments m as shown \nin Fig 9 (a).  In order to model the eddy -current damping in NbN, we first calculate the magnetic \nflux at point P where the distance between the point and the slab is x (Fig 9a). We can estimate \nthe magnetic field at point P generated by a moment at (0, y) using the Biot -Savart law, as: \n𝐵=𝜇0\n4𝜋𝑚\n(𝑥2+𝑦2)3/2                                                                  (7) \nwhere 𝜇0 is the permeability of free space. We assume that  the length of the chain is infinitely \nlong, which is a valid assumption by taking in consideration that the film thickness is much \nshorter than the sample lateral dimensions. By integrating the contribution of the individual \nmoments, we calculate the tota l magnetic field  𝐵𝑡𝑜𝑡𝑎𝑙 as: 10 \n 𝐵𝑡����𝑡𝑎𝑙= 2∫𝜇0\n4𝜋𝑚\n(𝑥2+𝑦2)3/2𝑑𝑦∞\n0=𝜇0\n2𝜋𝑚\n𝑥2                                            (8) \nUsing  this 𝐵𝑡𝑜𝑡𝑎𝑙 expression within this quasi -2D picture, we can calculate the magnetic flux Φ \nat point P. By definition, Φ  = ∬𝐵𝑡𝑜𝑡𝑎𝑙𝑑𝑠 , where the integration surface is defined by the  \nthickness 𝑡𝑁𝑏𝑁 and the width w of the NbN film. This reads : \nΦ = ∬𝐵𝑡𝑜𝑡𝑎𝑙𝑑𝑠=𝑤×∫𝜇0\n2𝜋𝑚\n𝑥2𝑡𝑌𝐼𝐺\n2+𝑡𝑁𝑏𝑁\n𝑡𝑌𝐼𝐺/2𝑑𝑥=𝜇0𝑤𝑚\n𝜋𝑡𝑁𝑏𝑁\n𝑡𝑌𝐼𝐺(𝑡𝑌𝐼𝐺+2𝑡𝑁𝑏𝑁)                         (9) \nFor the definition of the integration region,  we assume that the chain of the magnetic moments \nis locate d at the centre of the YIG film . \nAfter estimating the magnetic flux, we can calculate t he radiative dissipation power P as: \n𝑃=𝜔\n2𝑍𝑁𝑏𝑁 Φ2=𝜔\n2𝑍𝑁𝑏𝑁( 𝜇0𝑤𝑚\n𝜋𝑡𝑁𝑏𝑁\n𝑡𝑌𝐼𝐺(𝑡𝑌𝐼𝐺+2𝑡𝑁𝑏𝑁))2\n                                   (10) \nHere 𝑍𝑁𝑏𝑁 is the impedance of the NbN film and for simplification  we assume that the  real part  \n(resistance)  dominates, meaning that 𝑍𝑁𝑏𝑁≈𝑅𝑁𝑏𝑁=𝜌𝑁𝑏𝑁(𝑑/𝑤𝑡𝑁𝑏𝑁). Using the total non -\nequilibrium magnon energy generated during the experiments as ħ𝜔𝑁𝑉  (here, 𝑁 is the number \nof the non -equilibrium magnons and V is the volume of YIG), we can express t he rate of energy \ndissipation being:  \n1\n𝜏=𝑃\n𝐸=𝜔 𝑤𝑡𝑁𝑏𝑁\n2𝜌𝑁𝑏𝑁𝑑ħ𝜔𝑁( 𝜇0𝑤𝑚\n𝜋𝑡𝑁𝑏𝑁\n𝑡𝑌𝐼𝐺(𝑡𝑌𝐼𝐺+2𝑡𝑁𝑏𝑁))2\n                                       (11) \nFinally, the damping component caused by eddy currents generated by the time -dependent flux \nchange can be given by:  \n    𝛼𝑒𝑑𝑑𝑦=1\n2𝜔(1/𝜏)=𝑤𝑡𝑁𝑏𝑁\n4𝜌𝑁𝑏𝑁𝑑ħ𝜔𝑁𝑉( 𝜇0𝑤𝑚\n𝜋𝑡𝑁𝑏𝑁\n𝑡𝑌𝐼𝐺(𝑡𝑌𝐼𝐺+2𝑡𝑁𝑏𝑁))2\n                           (12) \nAs this model is a simplified one, we only discuss 𝛼𝑒𝑑𝑑𝑦 qualitatively. In particular, we c an \nextract the NbN thickness dependence of 𝛼𝑒𝑑𝑑𝑦 by using this expression and find that it is \nproportional to ( 𝑡𝑁𝑏𝑁3/2\n𝑡𝑌𝐼𝐺+2𝑡𝑁𝑏𝑁)2\n. We plot the dependence in F ig. 9 (b) which indicates  that the \ndamping based on this mechanism should monotonically increase with thickness. However,  this \ntrend is different from what we experimentally observed, where 𝛼 becomes constant for the \nlarger thickness limit. This suggest that the damping mechanism through the eddy current in \nthe NbN layers is no t significant and can be neglected for the examined NbN thicknesses . \nMoreover, in the  work by Flovik et al.  [45] they discuss the  eddy current effect on the lineshape \nof the FMR spectrum. They show ed that when  eddy currents exist in an FM/ NM bi-layer, the 11 \n FMR lineshape can be significantly affected, varying from a pure symmetric shape to a mix ture \nof symmetric and anti -symmetric ones. Experimentally, we have not observed strong 𝐴𝑎𝑠𝑦 \ncomponent, suggesting that the eddy current in our NbN film does not pla y a significant role in \nour measurements. In addition, similar eddy current and radiative damping mechanisms has \nalso been discussed by Schoen et al.  [46]. They demonstrated that when their sample is placed \n100 μm away from the waveguide, radiative damping with the waveguide is largely supressed. \nSince we also inserted an insulating tape be tween our samples and the waveguide , we believe \nthat the radiative damping is minor  in our experiments. Furthermore, Qaid et al.  [47] reported \nthat although eddy -current da mping can be observed in a weak spin -orbit material ( in their case \na conducting polymer) , this is not the case for a high spin orbit metal (Pt). For instance, they \nshowed that the damping enhancement  in a YIG/Pt structure can still be dominated by the spin-\npumping effect in Pt. Since our NbN is a suffici ently high spin -orbit material, w e believe that \nthe eddy -current component is much smaller (an order of magnitude at least) than that of spin -\npumping into NbN.  \n \nAppendix  C: Low temperature measurements of spin pumping in NbN/YIG samples  \nIt is widely reported that  YIG thin -films tend to show signif icant temperature dependent \nmagnetic damping  [32, 33 , 48, 49], where the superb damping character at RT is  lost when the \nfilms are cooled to lower  temperatures. The origin of this remains  under debate but enhanced \nlow temperature two -magnon scattering (due to interfac ial defects  in ultrathin films) [32] in \ncombination with  rare-earth or Fe2+ impurity scattering [ 50, 51] are likely mechanisms . Jermain \net al.  [33] discuss  that, if the FMR linewidth has a peaked temperature -dependence that \ndominates over the  proportionality expected with MS(T) increase , impurity scattering is the \nmore  likely  mechanism. Although  the nature of the impurities remains ambiguous, other reports \nof the high frequency characterisation of PLD -grown and sputtered YIG thin film s have pointed \nout the likely significan ce of Gd3+ diffusion from the GGG substrate [ 52, 53, 54]. \nOur own extensive FMR measurements of bare YIG on GGG ( of comparable \nthicknesses)  [55] show that, when Gd3+ impurities are concentrate d in a thin (1 -5 nm) layer near \nthe substrate  interface , they form a  ferromagnetic sublattice that, as its moment increases at low \ntemperatures, opposes the net YIG magnetisation  [50, 56], and also introduces magnetic \ndisorder and additional damping channels that dominate the film’s FMR response . \nHere  we describe the  low-temperature characterisatio ns of our YIG/NbN samples . \nFigure 10 summarises both FMR absorption spectra and ISHE voltages as a function of 12 \n temperature for the sample with NbN thickness of 10 nm. With decreasing temperature, there \nis a clear increase of  ΔH, leading to a corresponding reduction of the FMR absorption signal , \nas shown in Fig. 10(a). The FMR spectrum at 3K can be extracted by taking multiple scans to \nimprove the signal to noise ratio through data averaging.   Figure 10(b) shows  that ΔH increases \nby a factor of 5 between 300K and 3K, with a steep enhancement below 100 K.  For direct \ncomparison we present data in Fig. 10(b) both of  YIG/NbN (black points) and bare YIG samples \n(red points). It is clear that linewidth enhancement at low tempe ratures is due to YIG. In \ncomparison with the previous low temperature FMR studies on YIG, we can detect an FMR \nsignal down to 3K in the  MW transmission geometry, possibly owing to a relativ ely thick film . \nUnfortunately, the  ΔH enhancement significantly hindered our ISHE detection plotted in Fig s. \n10(c) and (d). 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Gomez -Perez, et al., arXiv:1803.05545 . \n[55] L. McKenzie -Sell, M. Amado, G. Kimbell, G. Divitini, C. Ciccarelli, and J. W. A. \nRobinson, to be submitted (n.d.).  \n[56] T. Yamagishi, J. Awaka, Y. Kawas hima, M. Uemura, S. Ebisu, S. Chikazawa, and S. \nNagata, Philos. Mag. 85, 1819 (2005).  \n \nFigure captions  \nFIG.1: Structural and magnetic properties of a bare (111) -oriented YIG film (nominally 60 -nm-\nthick) used in this work and deposited onto GGG. (a) 10×10 µm2 AFM topography scan \nshowing a root -mean -square roughness of less than 0.16  nm. (b) Magnetization hysteresis loops \ncharacterised by a superconducting quantum interference device magnetometer show ing a \nsaturation volume magnetization of 140   3 emu  cm-3. (c) High angle X -ray diffraction data \ndemonstrating (111) orientation with visible Laue fringes on the (444) and (888) diffraction \npeaks characteristic of layer -by-layer growth. (d) Low angle X -ray reflectometry data (black) \nwith a best -fit (red) curve fr om which we estimate a nominal thickness of 60   2 nm.  \n \n \nFIG.2: (a) A schematic of the spin -pumping setup. The lateral area of all samples is 5x5mm2. \nMW magnetic fields ( hrf) were generated by the transmission line to generate magnetic \ndynamics in the YIG  film. Spin currents ( js) were emitted at the YIG/NbN interface, which can \ninduce ISHE voltages detected through the two electrodes attached to the edges of the sample. \nWe simultaneously measured the FMR absorption signal as a voltage in a microwave power \nmeter ( VP) connected to the microwave line and the ISHE signal ( VISHE) using two lock -in \namplifiers. (b) FMR absorption measurements for different MW frequencies. (b) FMR \nabsorption measurements for different MW frequencies. Voltages in our MW power detect or \nwere measured while magnetic fields were swept. Dots in red, green, blue, cyan, pink, yellow \nand black represents measurement results for 3, 4, 5, 6, 8, 10 and 12 GHz respectively. (c) A \nplot of frequency versus FMR field ( Hres) for samples with differe nt NbN thicknesses. Dots \nrepresent experimental results and curves are produced by fitting using the Kittel formula.  \n \n \nFIG.3: (a) Frequency dependence of FMR linewidth of YIG/NbN samples with different NbN \nthicknesses. Experimental data (filled points ) is fitted by a linear line ΔH =ΔH 0 + (4πα/γ) f, 15 \n                                                                                                                                                                                               \nwhere ΔH 0 and γ describe the inhomogeneous broadening and the gyromagnetic ratio \nrespectively, from which the Gilbert damping coefficient , α, is extracted . (b) Plots of α for \ndifferent YIG/NbN  samples. Equation (1) was used to fit to the thickness dependence with the \nspin-diffusion length and the real part of mixing conductance as fitting parameters. The inset \ndepicts  the resistivity as a function of NbN thickness.  \n \n \nFIG.4: ISHE  measurements. Simultaneous measurements of FMR absorption and ISHE \nvoltages for positive (a) and negative (b) magnetic field values for a tNBN = 20 nm sample. \nCorresponding data for tNBN = 5 nm are depicted in (c) and (d), respectively. Both VP and VISHE \npeaks appear at the same magnetic field, confirming that the voltages were generated when YIG \nmagnetic moments were preccessing. The sign change in voltage peaks observed between the \npositive and negative magnetic field regions is consistent with the spin -pumping/ISHE picture.  \n \nFIG.5: Microwave power dependent measurements. (a) ISHE voltage measurements with \ndifferent insertion powers ( PMW). (b) A plot of ISHE voltage peak to peak amplitude ( V*ISHE) \nas a function of PMW. VISHE scales with PMW as expected from the spin pumping theory in the \nlinear regime.  \n  \nFIG.6: In -plane (a) and out -of-plane (b) angular dependences of VISHE signal peak to peak \namplitude ( V*ISHE). Fit curves in both angular dependences are discussed in the main text. We \nshow f it curves with four different spin -relaxation time ( 𝜏𝑠) in (b) to illustrate how the model \ncurve changes with 𝜏𝑠.  The best fit curve was produced with 𝜏𝑠 = 11 ± 2 ps. We define three \nangles ( ϕ, ϕM, θ) as depicted in Figure’s insets.  \n  \nFIG.7: 𝑉ISHE𝑡NbN/𝜌NbN as a function of NbN thickness. We plot 𝑉ISHE𝑡NbN/𝜌NbN to normalise  \n𝑉ISHE  with NbN thickness and resistivity. By using Eq. (3) in the main text, we extract the spin -\nHall angle ( θSH) and spin -diffusion length ( 𝑙𝑠𝑑) of NbN to be  1.1 ×10-2 and 14 nm . The best fit \ncurve is shown along with the experimental data.  \n \nFIG.8: Comparison of (a) ac and (b) dc  VP measurements. The extracted parameters using \nEquations in Appendix A for each measurement method are depicted in the legends of the \nfigur es. We can confirm that the extracted values are almost the same for both measurements.  \nFIG.9: Eddy -current damping contribution. (a) A schematic of our model for the eddy -current \ndamping. A chain of Magnetic moments (red arrow) lines up along the y direct ion and we \nconsider the magnetic field at Point P (x, 0) . (b) A plot of calculated eddy -current damping as \na function of the NbN thickness. The unit of the eddy -current damping is arbitrary in order to \ndiscuss them qualitatively. The thickness dependence i s clearly different from our experimental \nresults in Fig. 3(b), suggesting that this damping mechanism is not significant in our \nexperiments.  \n \nFIG.10: FMR absorption spectra and ISHE voltages as a function of temperature for tNbN=10 \nnm sample. (a) FMR abso rption spectra measured at 3 GHz, with temperature ranging from \n260K to 3K. (b) Linewidth evolution with temperature for the 3 GHz measurements. Black data \ncorresponds to an YIG/NbN sample and red data to a bare YIG sample. (c) ISHE voltages \nmeasured at 3 GHz for the temperature region of 50K -300K. We confirm that the peak height \nis below the signal -to-noise ratio around 50 K. (d) The normalised ISHE voltage amplitude as \na function of temperature. The inset represents our four point probe measurements of Nb N \nresistivity (for tNbN = 10 nm).  16 \n                                                                                                                                                                                               \n(a) (b) (c) (d)Figure 1:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 17 \n                                                                                                                                                                                               \nFigure 2:\n(a)\n-400 -200 0 200 400-2-1012\n \n  \nH (mT)VP (mV)\nf=2GHz\nf=12GHz\nISHE(b)\n(c)\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 18 \n                                                                                                                                                                                               \n(a)Figure 3:\n0 5 10 150.00.51.01.5\n 0nm \n 5nm \n 10nm\n 15nm  \n 20nm  \n 30nm  \n 50nm  H (mT)\nf (GHz)\n(b)\n0 10 20 30 40 500.00040.00060.00080.00100.00120.0014\na\ntNbN(nm) Data\n gr = 10.44 nm-2 , lsd = 14.46 nm \n gr = 8 nm-2 , lsd = 14 nm\n gr = 13 nm-2, lsd = 18 nm\n gr = 22 nm-2 , lsd = 20 nm\n gr = 9 nm-2 , lsd = 16 nm\n0 10 20 30 40 5030405060 ( - cm)\ntNbN (nm)\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 19 \n                                                                                                                                                                                               \n(a)\n-0.40.00.4\n35 40 45 50 550.450.600.75 VP (mV)20nm\nT=300K\nf = 3GHz VISHE (V)\nH (mT) (b)Figure 4:\n-101\n35 40 45 50 5501T=300K\nf=3GHzVP (mV)5nmVISHE (V)\n 0H (mT)\n-101\n-55 -50 -45 -4001VP (mV)5nmVISHE  (V)\n0H (mT)T=300K\nf=3GHz\n-0.40.00.4\n-55 -50 -45 -40 -350.450.600.75T = 300K VP (mV)20nm\nf = 3GHzVlSHE  (V)\nH (mT)\n(c) (d)\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 20 \n                                                                                                                                                                                               \n(a)\n40 45 500.40.60.8 VISHE (V)\nH (mT) 8mW\n 18mW\n 32mW\n 56mW\n 100mW\n 178mW 300K \n3GHz(b)\n0 50 100 150 2000.00.10.20.30.40.5V*\nISHE (V)\nP\nMW (mW) 300K\n3GHzFigure 5:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 21 \n                                                                                                                                                                                               \n(b)Figure 6:\n(a)\n0 90 180 270 360-1.0-0.50.00.51.0\n  \n 10nm \n  Fit\n 20nm \n  FitV*\nISHE (V)\n (deg)\n90 75 60 45 30 15 00.00.51.0V*\nISHE()/V*=\nISHE (-)\n (deg) VISHE/V=\nISHE\n s = 11 ps\n s = 53 ps ( s = 1)\n s << 1/ \n s >> 1/ \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 22 \n                                                                                                                                                                                               \n0 10 20 30 40 500123\ntNbN (nm) V*\nISHEtNbN/NbN (A)\n data\n fitFigure 7:\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 23 \n                                                                                                                                                                                               \nFigure 8\n(a)(b)\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 24 \n                                                                                                                                                                                               \nFigure 9:\n(b)\n (a)\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 25 \n                                                                                                                                                                                               \nFigure 10:\n(a)\n(c)(b)\n(d)\n50 100 150 200 250 3000.00.20.40.60.81.01.2V*\nISHE/ V*300 K\nISHE\nT (K)\n20 30 40-3-2-10123\n  \n30K\n3K210K\n170KVP (mV)\n H (mT)260K\n100K\n(x10)\n20 30 40-3.0-1.50.01.53.0\n  \n100K170K210K260K\n50KVISHE(V)\nH (mT)75K\n10 1000510152025R4p (ohm)\nT (K)\n0 50 100 150 200 250 3000.51.01.52.02.53.0\n0H (mT)\nT (K)\n "    },    {        "title": "0904.3150v2.Tensor_damping_in_metallic_magnetic_multilayers.pdf",        "content": "Tensor damping in metalli c magnetic multilayers \n \nNeil Smith \nSan Jose Research Center, Hitachi Globa l Storage Technologies, San Jose, CA 95135 \n \nThe mechanism of spin-pumping, described by Tserkovnyak et al. , is formally analyzed in the general \ncase of a magnetic multilayer consisting of two or more metallic ferromagnetic (FM) films separated \nby normal metal (NM) layers. It is shown that the spin-pumping-induced dynamic coupling between \nFM layers modifies the linearized Gilbert equations in a way that replaces the scalar Gilbert damping \nconstant with a nonlocal matrix of Cartesian dampi ng tensors. The latter are shown to be methodically \ncalculable from a matrix algebra solution of the Valet- Fert transport equations. As an example, explicit \nanalytical results are obtained for a 5-layer (spi n-valve) of form NM/FM/NM'/FM/NM. Comparisons \nwith earlier well known results of Tserkovnyak et al. for the related 3-layer FM/NM/FM indicate that \nthe latter inadvert ently hid the tensor character of the damping, and instea d singled out the diagonal \nelement of the local damping tens or along the axis normal to the plane of the two magnetization \nvectors. For spin-valve devices of  technological interest, the influen ce of the tensor components of the \ndamping on thermal noise or spin -torque critical currents are st rongly weighted by the relative \nmagnitude of the elements of the nonlocal, anisotropic stiffness-fiel d tensor-matrix, and for in-plane \nmagnetized spin-valves are generally more sensitive to the in-plane element of the damping tensor.  I. INTRODUCTION \n       For purely scientific r easons, as well as technological applica tions such as magnetic field sensors \nor dc current tunable microwave oscillator s, there is significant present interest1 in the magnetization \ndynamics in current-perpendicular-to-plane (CPP)  metallic multilayer devices comprising multiple \nferromagnetic (FM) films separated by normal meta l (NM) spacer layers. The phenomenon of spin-\npumping, described earlier by Tserkovnyak et al.2,3 introduces an additional source of dynamic \ncoupling, either between the magnetization of a single FM layer and its NM elect ronic environment, or \nbetween two or more FM layers as mediated through their NM spacers. In the former case,2 the effect \ncan resemble an enhanced magnetic damping of an individual FM layer, whic h has important practical \napplication for substantially increasing the spin-t orque critical currents of CPP spin-valves employed \nas giant-magnetoresistiv e (GMR) sensors for read head applications.4 Considered in this paper is a \ngeneral treatment in the case of two or more FM layers in a CPP stack, where it will be shown in Sec. \nII that spin-pumping modifies the linearized equations of motion in a way that replaces a scalar \ndamping constant with a nonlocal matrix of Cartesian damping tensors.5 Analytical results for the case \nof a 5-layer spin-valve stack of the form NM/FM/NM'/FM/NM are discussed in de tail in Sec. III, and \nare in Sec. IV compared and contrasted with the early well-know n results of Tserkovnyak et al..3, as \nwell as some very recent results of that author and colleagues.6 In the case of CPP-GMR devices of \ntechnological interest, the relativ e importance of the different elements of the damping tensor on \ninfluencing measureable thermal fluctuations or spin-t orque critical currents is shown to be strongly \nweighted by the  anisotropic nature of the stiffness-field tensor-matrix.  \n \nII. SPIN-PUMPING AN D TENSOR DAMPING \n       As discussed by Tserkovnyak et al,2,3 the spin current pumpI flowing into the normal metal (NM) \nlayer at an FM/NM interface (Fig. 1) due to the spin-pumping effect is  described the expression  \n \n⎥⎦⎤\n⎢⎣⎡− ×π=↑↓ ↑↓\ndtdgdtdgm mm IˆIm )ˆˆ( Re4pumph                                                     (1) \n \nwhere  is a dimensionless mixing conductance, and m is the unit magnetization vector. In this \npaper,  for any ferromagnetic (FM) layer is treated as  a uniform macrospin. A restatement of (1) in \nterms more natural to Valet-Fert↑↓g ˆ\nmˆ\n7form of transport equations is di scussed in Appendix A. With the \nnotational conversion , where A is the cross sectional area of the film stack, \nequation (1), for the case , simplifies to  pump pump) 2 / (J I A eh− →\n↑↓ ↑↓> > g gIm Re \ninterface NM/FM for \" \" interface, FM/NM for \" \"ReIm,ˆ ˆˆ) 2 / (\n22\npump\n+ −≡ ε⎟\n⎠⎞⎜\n⎝⎛ε + ×π≅↑↓↑↓\n↑↓rr\ndtd\ndtd\nre h e m mm Jm                                       (2) \n \nwhere  is the inverse mixing conductance (with  dimensions of resistance-area), and  \nis the well known inverse conductance quantum ↑↓ ↑↓≅ r rRe22 /e h\n) k 9 . 12 (Ω≅ . In the present notation, all spin current \ndensities  have the same dimensions as electron charge current density , and for conceptual \nsimplicity are defined with a parallel (i.e., ) rather than anti-parallel alignment with \nmagnetization . Positive J is defined as electrons flowing to the right (along  in Fig. 1.).  spinJeJ\nm Jˆ ˆspin+ =\nmˆ yˆ+\n       For a FM layer sandwiched by tw o NM layers in which the FM layer is the  layer  of a \nmultilayer  thj ) 0 (≥j\nfilm stack, spin-pumping contributions at the  interface, i.e., either left or right thi ) (j i= ) 1 (+=j i  \nFM-NM interfaces, (2) can be expressed as  \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ε + ×−=\n↑↓−\n+ =dtd\ndtd\nr ej\nij\nj\nij i\nj j im m\nm Jˆ ˆ\nˆ) 1 (\n2pump\n1 ,h                                           (3) \n \n      The physical picture to now be invoked is that of small  (thermal) fluctuations of m about \nequilibrium  giving rise to the  terms in (2). Since ˆ\n0ˆm dt d/ˆm 1ˆ≡m , the three vector components of \n and/or are not linearly independent. To remove this  interdependency, as well as higher order mˆ dt d/ˆmFig. 1 Cross section cartoon of an N-layer multilayer stack with N-1 interior nterfaces of FM-NM or NM-NM type, \nsuch as found in CPP-GMR pillars sandwiched between conductive leads of much larger cross section. In the \nexample shown, the jth layer is FM, sandwiched by NM layers, with spin pumping contributions at the ith (NM/FM) \nand ( i+1)st (FM/NM) interfaces located at iy y=  and 1+iy (with i=j  for the labeling scheme shown).  j=0j-1j=1\ni=0 i=1 i-1j\ni+1j+1-Jpump\ni Jpump\ni+1\ni=jNM NM FMmj\nj=N-1z\ny\nlead lead\ni=N N-1terms in (3) it thus is useful to work in a primed coordinate system where , through use of a \n Cartesian rotation matrix  such that 0ˆ ˆm z=′\n3 3× )ˆ(0mℜ m mˆ ˆ′⋅ℜ= .8 To first order in linearly independent  \nquantities , ) , (y xm m′ ′ m m m′⋅ ℜ + =~ˆ ˆ0 , where , and where ℜ ⎟⎠⎞⎜⎝⎛≡′\n′′′′\nymxmm~ denotes the  matrix from \nthe first two (i.e., x and y ) columns of 2 3×\nℜ. Replacing z m′→ˆ ˆ0 , and _ˆ×′z  with matrix multiplication, \nthe linearized  form of (3) becomes \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n′′\n⋅⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nε− ε⋅ ℜ−=\n′′−\n+ = ↑↓ dt m ddt m d\nr e2h\ny jx j\nii\nj\nij i\nj j i//\n11 ~ ) 1 ( pump\n1 ,J                                  (4) \n \n     Using the present sign convention, j i s j A t Mm S ˆ/ ) (γ = is the spin angular momentum of the  \nFM layer with saturation magnetization-thickness product , and  is the gyromagnetic ratio. \nTaking thj\nj st M) ( 0> γ\nsM=M  as constant, it follows by angu lar momentum conservation that3 \n \n∑+\n=−× × − = ⇔γ1\nˆ ˆ ) 1 (21 ˆ ) (NMj\nj ij i jj i j j j s\ne dtd\nA dtd t M\nm J mS mh                                         (5) \n \nis the contribution to  due to the net transverse  spin current entering the FM layer (Fig. 1). \nIn (5),  denotes the spin-current density in the NM layer at the  FM-NM interface. Taking the \ncross product  on both sides of (5), transforming to pr imed coordinates by matrix-multiplying by \n, and employing similar linearization as to obtain (4), one finds to first order that dt dj/ˆmthj\nNM\niJthi\n×mˆ\nTℜ = ℜ−1\n \n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n− ≡ Δ ℜ ⋅⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−=′\n×′ ∑+\n=−⋅1spin NM) 1 (~\n0 11 0\n21ˆj\nj iij i\nj jj\nje dtd\nAJ JS\nzT h                                     (6) \n \nwhere Tℜ~ is the  matrix transpose of ℜ 3 2×~. By definition, 0 ˆ~\n0= ⋅ ℜj jmT. \n      The quantities  in (6) are not known a priori , but must be determined after solution of the \nappropriate transport equations (e.g. , Appendix B). Even in the absence of charge current flow (i.e., \n as considered here, the  are nonzero due to the set of  in (4) which appear as \nsource terms in the boundary conditions (A 9) at each FM-NM interface. Given the linear relation of \n(4), one can now apply linear superposition to express spin\njJΔ\n) 0=eJspin\njJΔpump\niJ \n∑ ∑+\n=↑↓ ↑↓ ↑↓≡′\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nε− ε⋅ ℜ ⋅ = Δ1spin 1\n21 1,11 ~ 1\n2k\nk ii k kk\nk jk\nkjr r dtdC\nremJt h                                (7) \n \nin terms of the set of 3-D dimensionless Cartesian tensor jkCt\n. The jkCt\n are convenient for formal \nexpressions such as (9), or for analytical work in  algebraically simple cases, such as exampled in \nSec.III. However, they are also subject to met hodical computation. For the kth magnetic layer, the \ncolumns of each  are the dimensionless vectors simultaneously obtainable \nfor all magnetic layers j from a matrix solutionrd nd st3 or , 2 , 1jkCtspin\njJΔ\n9 of the Valet-Fert7 transport equatio ns with nonzero \ndimensionless spin-pump vectors )ˆor ,ˆ,ˆ)( / ( ) 1 (pump\n1 ,z y x J↑↓ ↑↓ −\n+ =− =i kk i\nk k ir r .  \n      To include spin currents via (5) into  the magnetization dynamics, the conventional Gilbert \nequations of motion for  can be amended as ) (ˆtm\n \ndtd\nA t M dtd\ndtdj\nj sj\nj j j ij S m\nm H mm 1\n) (ˆ\nˆ ) ˆ(ˆG eff γ+ × α + × γ − =                                     (8) \n \nwhere  is the usual (scalar) Gilbert damping paramete r. From (6) and (7), one can deduce that the \nrightmost term in  (8) will scale linearly with G\njα\ndt d/m′, as does the conventiona l Gilbert damping term. \nCombining these terms together after applying the analogous linearization procedure to (8) as was \ndone in going from (5) to (6), one obtains  \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nε− ε⋅ ℜ ⋅ ⋅ ℜ ⋅⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n− πγ= α′α′+ δ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nαα≡ α′′⋅ α′−′⋅ − ⋅ ℜ γ =′\n×′\n↑↓∑\n11 ~ ~\n0 11 0 2 /\n) 4 (00] ) ˆ(~[ ˆ\n2\npumppumpeff eff\nGG\nk jk j\nj j sjkjk jk\njj\njkk\nkjk j j j j jj\nj\nC\nre h\nt Mdtd\ndtd\nt h tt tt\nTT mm H m Hm\nz\n                      (9) \nwhere Kronecker delta k j k jjk jk ≠=δ= = δ if 0 and , if 1 . \n      In (9),  is a 2-dimensional Cartesian \"damping tensor \" expressed in a coordinate system where \n, while  is a \"nonlocal tensor\" spanning two such  coordinate systems. This formalism \nfollows naturally from the lineariza tion of the equations of motion for non-collinear macrospins, and is \nparticularly useful for describing the influence of \"tensor damping\" on the thermal fluctuations and/or j jα′t\nj jz m′=′ ˆ ˆ0 j k j≠α′tspin-torque critical currents of su ch multilayer film structures (e.g., as  described further in Sec. IV.). \nDue to the spin-pumping contribution pump\njkα′t, the four individual  (with v u\njk′ ′α′ y x v u′′=′ ′ or , ) are in \ngeneral nonzero with , reflecting the true tensor na ture of the damping in this \ncircumstance, which is additionally nonl ocal between magnetic layers (i.e., ). The are \nsomewhat arbitrary to the extent that one may replace y y\njkx x\njk′ ′ ′ ′α′≠ α′\n0≠ α′′ ′\n≠v u\nj jkv u\njk′ ′α′\n2~ ~ℜ ⋅ ℜ ↔ ℜ in (9), where 2ℜ is the \nmatrix representation of any rotation about the 2 2× z′ˆaxis. \n      It is perhaps tempting to contemplate an \"inverse linearization\" of (9 ) to obtain a 3D nonlinear \nGilbert equation with a fully 3D damping tensor T\nk jk j jkℜ ⋅ α′⋅ ℜ = αt t. However, (9) has a null zˆ′ \ncomponent, and contains no information rega rding the heretofore undefined quantities  or . \nFor local, isotropic/scal ar Gilbert damping, one can independen tly argue on spatial symmetry grounds \nthat . However, the analogous extension is  not so obviously available for z u\njk′ ′α′z z\njk′ ′α′\nG G G α = α′= α′′ ′ ′ ′u u z z pump\njkα′t, \ngiven the intrinsically nonlocal, anisotropic natu re of spin-pumping. The proper general equation \nremains that of (8), with the rightmost term  given by that in (5), or its equivalent.  \n \nIII. EXAMPLE: FIVE LAYER SYSTEM \n \n \n   Fig. 2 shows a 5-layer system with 2 FM layers resembling a CPP-GMR spin-valve, to be used as a Fig. 2. Cartoon of a prototypical 5-layer CPP-GMR stack (leads not shown) with two FM layers (#1 and #3),\nsandwiching a central NM spacer layer (#2 ), and with outer NM cap layers (#0 and #4). For discussion purpose\n  \nprototype. Although the full genera lization is straightforward, the material properties and layer s\ndescribed in text, the magnetization vectors 1ˆm and 3ˆm can be considered to lie in the film plane ( zx- plane). m1\nNMj=0 j=1 23 4\nFM NM' FM NMm3z\nxθ\ny\ny1y0 y2y3y4y5thickness will be assumed symmetric about the centr al (#2) normal metal spacer layer, which will \nadditionally be taken to have  a large spin-diffusion length  (with  the thickness of the  \nlayer), such that the \"ballistic\" approximation (B3) applies. The inverse mixing conductances \nwill also be assumed to be real. Using the outer boundary conditions described by (B5), one \nfinds for the FM-NM interfaces at  that 2 2t l> >>jtthj\n↑↓\n=4 1-ir\n, and4 1y y y=\n \n] )) / hyp( ( [ˆ\nNMFM NM\n1 1pump\n1\n3 , 1 4 , 1l t l r rrJi\nj i iρ + ≡ ′+ =\n↑↓ ↑↓↑↓\n= =Jm J                                  (10a) \n \nFM\nNMFM\n4 1 1 4 21] )) / hyp( ( [J l t l r r Vi ρ + ≡′= Δ −=                                        (10b) \n \nwhere , , and subscript \"NM\" refers to either outer layer 0 or 4. In  (10) and below, \n are used interchangeably. Inside FM la yer 3, (B1,2) have solution expressible as 4 1r r=↑↓↑↓=4 1r r\nj j0ˆ ˆm m↔\n \n3 1 1 33 3 3 3spin\n33 3 3 3 4 3 3\n] ) / tanh( ) /[( ] ) / tanh( ( [() / ) sinh(( ) / ) cosh(( [ ) /( 1 ) () / ) cosh(( 2 ) / ) sinh(( 2 ) (\nFM FM FMFM FM FMFM FM\nA l t r l l t l r Bl y y B l y y A l y Jl y y B l y y A y y y V\n′+ ρ ρ +′− =− + − ρ =− + − = ≤ ≤ Δ\n                   (11) \n \nwhere the expression for  follows from (10b). Subscript \"FM\" re fers to either layers 1 or 3. The \nboundary conditions (A5) and (A9) applied to the FM/NM boundary at 3B\n3y y= yield  \n \n) ( ˆ]ˆ ) /( )[ ( ) 2 (pump\n3spin\n2 2 3 3spin\n2 3 2 2 2 3 21\nFM J J m m J V − + ⋅ = ρ − = −↑↓ ↑↓r l A r r BΔ               (12) \n \nwhere , . The \"ballistic\" values 3 2r r=↑↓↑↓=3 2r r2VΔ  and  are constant inside  central layer 2. \nUsing (11) to eliminate coefficient  in (12), the latter may be rewritten as spin\n2J\n3B\n \n⎥⎦⎤\n⎢⎣⎡\nρ ′+ρ +′+ − ≡− ⋅ ⋅ + = −\n↑↓\n↑↓↑↓\nFMFM\n)] /( ) / [tanh( 1)) / tanh( (\n21] )ˆ ˆ 2 1 [(\n11\n2 2\n2pump\n3spin\n2 3 3 2 2 21\nl l t rl t l rr r\nrqq r J J m m VTt\nΔ\n                                    (13) \n \nwhere  is the 3-D identity tensor, and  denotes the 3-D tensor formed from the vector outer -\nproduct of  with itself.  1tT\n3 3ˆ ˆm m⋅\n3ˆm     Working through the equivalent comput ations applied now to the NM/FM interface at 2y y= , one \nfinds  the analogous result: \n \n] )ˆ ˆ 2 1 [(pump\n2spin\n2 1 1 2 2 21J J m m V − ⋅ ⋅ + = +↑↓ Tq rt\nΔ                                      (14) \n \nEliminating  between (13) and (14) derives the remaining needed result for : 2VΔspin\n2J\n \n1\n3 3 1 1pump\n3pump\n2 21 spin\n2)]ˆ ˆ ˆ ˆ ( 1 [ ), (−⋅ + ⋅ + ≡ + ⋅ = ⋅T Tm m m m J J J q Q Qtt t\n                      (15) \n \ntreating tensor Qt\n as the  matrix inverse of the [ ]-bracketed te nsor in (15). Using (10a) and (15) to \ncompute , then additional use of (4) and (6), allow computation of the 3 3×\nNM\n4 1-=iJjkCt\n defined in (7): \n \n) / 1 / 1 ( / 1 ; 2 / , /, 1\n2 1 212 131 13 33 11\n↑↓ ↑↓ ↑↓ ↑↓↑↓ ↑↓ ↑↓+ = ≡′≡− = = + = =\nr r r r r b r r aQ b C C Q b a C Cttttttt\n                               (16) \n \n      For explicit evaluation of pump\njkα′t, it is convenient to assume a choice of 3 , 1~\n=ℜj for which 3 1ˆ ˆy y′=′ , \nsuch that  and  lie in the  plane. To simplify the inte rmediate algebra to obtain Q03ˆm01ˆm z x′ ′-t\n from \n(15), one can consider \"in-plane\" magnetizations (Fig.2), taking z mˆ ˆ03=, and  in the x-z plane \n( ). This allows a particularly easy determination of 01ˆm\nθ = ⋅cosˆ ˆ01z mjℜ~ for which : y y yˆ ˆ ˆ3 1=′=′\n \n0 , ;0 1 0sin 0 cos ~\n3 1 3 , 1 = θ θ = θ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ θ − θ= ℜ=j j\njT                                       (17) \n \nUsing (16) and (17) with (9) allows explicit solution for the pump\njkα′t: \n \nθ + +θ + −= =\nθ + +θ + += =⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+ δ− δ + δ\nπγ= α′\n↑↓\n2 231 132 22\n33 112\npump\nsin 2 1cos ) 2 1 (,\nsin 2 1cos 100 ) 1 2 ( 2 /\n) 4 (\nq qqd d\nq qq qd dd b ab a\nre h\nt M jk jkjk jk\nj j sjkh t\n                              (18) \n \nTaking , (18) holds for arbi trary orientation of  and , provided the flexibility \nin choosing the 03 01ˆ ˆ cos m m⋅ = θ01ˆm03ˆm\n3 , 1~\n=ℜj is used to maintain 3 1ˆ ˆy y′=′ . However, for multilayer film stacks with three or \nmore magnetic layers with magnetizations  that do not all lie in a singl e plane, it wi ll generally be \nthe case that some of the off-diagonal elements of the j0ˆm\npump\njkα′t will be nonzero.   \n \n \nIV. DISCUSSION \n \n      It is perhaps instructive to compare and contrast the results of (9) and (18). with the prior results of \nRef. 3. The latter are for a a trilaye r stack, corresponding most  directly to taking ∞→ ρNM  in the \npresent model, whereby . It is also effectively equivalent to the 5-layer case with \ninsulating outer boundaries in the limit , whereby  but  due to \nperfect cancellation by the spin current reflected from the  boundaries without intervening spin-\nflip scattering. Either  way, it corresponds to  in (10) and  in (16) and (18).  0NM\n4 , 1pump\n4 , 1= == = i iJ J\n0 ) / (NM→ l t 0pump\n4 , 1≠=iJ 0NM\n4 , 1→=iJ\n5 , 0=iy\n∞ →′ ′↑↓\n1 1,r r 0→a\n      However, a more interesting difference is that Ref. 3 treats  as stationary (hence , \nand  as undergoing a perfectly circular  precession about  with a possibly large cone angle 3ˆm ) 0pump\n3= J\n1ˆm3ˆm θ. \nBy contrast, the present analysis treats  and  equally as quasi-stationary vectors which undergo \nsmall but otherwise random fluctuations about their equilibrium positions  and , with \n. To further elucidate this distinction, one can assume the aforementioned physical \nmodel of Ref. 3, and reanalyz e that situation in terms of the present formalism. With \n, and by explicitly inserting the condition (e.g., from (3)) that , an \nexplicit solution of (15) can be expressed in the form: 1ˆm3ˆm\n01ˆm03ˆm\nθ = ⋅cos ˆ ˆ03 03m m\npump\n3 3 0 /J m = = dt d 0 ˆ1pump\n2≡ ⋅m J\n \n]ˆ\ncos ) 1 (ˆ) 1 ( ˆ cos[3pump\n2 2 2 23 12\npump\n2 212NMm Jm mJ J ⋅\nθ − ++ − θ+ =\nq qq q q                                   (19) \n \nCombining (19) with the earlier re sult from (5) and then (3) (with ) 0=ε , it is readily found that  \n \ndtd\nq qq q\nre h\nt Mq qq q\nt M et M e dtd\nA t M dtd\nsss s\n1\n2 2 22\n22\n11 32 2 2pump\n2 3 pump\n2 1\n12 1\n11\n1\n11\n1\nˆ\ncos ) 1 (sin ) 1 (12 /\n) 8 (ˆ ˆ\ncos ) 1 () ˆ)( 1 (ˆ\n) ( 4ˆ\n) ( 21ˆ\n) (ˆˆNM\nmm mJ mJ mJ mSmmm\n⎥⎥\n⎦⎤\n⎢⎢\n⎣⎡\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nθ − +θ +−πγ− =⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n×\nθ − +⋅ ++ ×γ− =×γ− = ×γ⇔ ×\n↑↓hhh\n                       (20) \n The last result in (20) uses  from (3), and the fact that pump\n2J θ = ×sin ˆ ˆ1 3m m , and that  and \n are parallel  vectors in the case of steady circular  precession of  about a fixed . It is the \ndirect equivalent of Eq. (9) of Ref. 3 with the identification dt d/ˆ1m\n1 3ˆ ˆm m×1ˆm3ˆm\n) 1 /(+⇔ν q q . \n     Although the final expression in (20) is azimuthally invariant with vector orientation of , it is \nmost convenient to compare it with  (18) at that instant where  is \"in-plane\" as shown in Fig. 2. At \nthat orientation,1ˆm\n1ˆm\ndt m d dt dm dt dy y / / /ˆ1 1 1′= → m , and it is immediately confirmed from (9) and (18) \n(with ) that the [ ]-term in (20) is simply the tensor element 0→ay y′′α′11 of pump\n11α′t. It is now seen that \nthe analysis of Ref. 3 happens to mask the tensor  nature of the spin-pump damping by its restricting \nattention a specific form of the mo tion of the magnetization vectors, which in this case singles out the \nsingle diagonal element of the pump\n11α′t tensor along the axis perpendicular to the plane formed by \nvectors  and . The very recent results of Ref. 6 do addr ess this deficiency of generality, and \nreveal the tensor nature of 1ˆm3ˆm\npump\n11α′t with specific results for ππ=θ and , 2 / , 0 . The present Sec. III \nadditionally includes the nonlocal tensors pump\n31pump\n13α′= α′t t, as well as diagonal terms jkaδ in (18) \n(and the variation in parameter q) when it is not the case that )/ hyp( ) (NM NM NM FM NM l t l r ρ<<-  in \nboundary condition (B4). The latter condition will likely apply in the case of the technological \nimportant example of CPP-GMR spin-valves.  \n      Speaking of such, two important practical i ssues for these devices involve thermal magnetic noise \nand spin-torque induced oscillat ions. As described previously8, an explicit linearization of the effH  \nterm in (9) about equilibrium state  that is a minimum of the free energy 0ˆm E leads to the following \nmatrix form of the linearized Gilber t equation including spin-pumping (with : )0=eJ\n \nmA t M\npE\nmH H Hp p p\nGp p\nDt p t HdtdG D\nj s\nj\njjk jk j jk\nkj\njk j j jkkj k jk j\njkj\njkkj k jk j\njkj j j j\nkk jkk\nkjk jk\nΔ≡∂∂\nΔ−≡ℜ ⋅ ⋅ ℜ ≡ ′\n∂∂\n− δ ⋅ ≡⎥\n⎦⎤\n⎢\n⎣⎡\nγα′− α′\n+ δγ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−≡′\nγα′+ α′\n≡′⋅ ℜ ≡ ′= ⋅′⋅′+′⋅′+′ ∑ ∑\n) (\n,ˆ)ˆ( 1~ ~,ˆ1 ) ˆ(2 0 11 0,2) (~) ( ) (\n0 eff\n0eff\n0 eff\n0 0\nmmHmH\nH mh h mm\nt t t tt ttt ttt tt\nTT\n                           (21) \n where the  are small perturbation fields. The form of ) (tjhjkDt\n′ and jkG′t\n in (21) is chosen so that they \nretain the original delineation8 as symmetric and antisymmetic tensors regardless of the symmetry of \n. By use of a fixed  \"reference-moment\" jkα′tmΔ in the definition of , the \"stiffness-field\" tensor-\nmatrix  is symmetric positive-definite, and eff\njH\nv k u jv u\njk m m E H′ ′′ ′′∂′∂ ∂ ∝ ′ / ∑ ⋅ − =δj j j j sδ t M A E m h ) ( \n∑′⋅′Δ − =j j j m m h has the proper conjugate form so that (21) are now ready to directly apply \nfluctuation-dissipation expressions  specifically suited to such linear matrix equations of motion.8 \nTreating the fields  now as thermal fluctuat ion fields driving the ) (tjh′ ) (tjm′-fluctuations,  \n \nv u\njkB\nh hv u\njkB\nv k u j DmT kS DmT kh hv k u j′ ′\n′ ′′ ′\n′ ′ ′\nΔ= ω′⇔ τ δ′\nΔ= 〉′τ′〈′ ′2) ( ) (2) 0 ( ) (                                 (22) \n \nare the time-correlation or  cross power spectral density (PSD) F ourier transform pairs. Through their \nrelationship described in (21), the nonlocal, tensor nature of  the spin-pumping contribution pump\njkα′t to \njkα′t is directly translated into those of the FM FM2 2N N×  system \"damping tensor-matrix\" v u\njkD′ ′′↔′Dt\n, \nwhere  is the number of FM layers in the multila yer film stack. The cross-PSD tensor-matrix FMN\n) ( ) ( ω′↔ ω′′ ′′ ′ ′v k u jm mSm mSt\n for the m-fluctuations can then be expressed as′8  \n \n1)] ( [ ) () ( ) ( )] ( ) ( [ ) (\n−′ ′ ′ ′\n′+′ω −′≡ ω′ω′⋅′⋅ ω′→ ω′− ω′\nΔ ω= ω′\nD G HS Sh h m m\nttt tttt t t t\nim iT kB\nχχ χ χ χ@ @\n                           (23) \n \nwhere  is the complex susceptibility tensor-matrix for the ) (ω′χt} , {h m′′ system, and ) (ω′@χt its \nHermitian transpose. It has been theoretically argued10 that (22), and thus the second expression in (23), \nremain valid when , despite spin-torque contributions to  resulting in an asymmetric  0≠eJeff\njH H′t\n \n(e.g., see (25)) that violates the condition of therma l equilibrium implicitly a ssumed for the fluctuation \ndissipation relations.  \n     Since  is in general a fully nonlocal with anisotropic/tensor character, any additional tensor \nnature of H′t\nDt\n will likely be altered or muted as to the influence on the detectable -fluctuations. As an \nexample, one can again consider th e situation depicted in Fig. 2, applied to the case of a CPP-GMR \nspin-valve with typical in-plane  magnetization. The device's output noise PSD will reflect fluctuations m′in . Taking  to again play the simplifying role of an  ideal fixed (or pinned) reference layer \n(i.e., ), the PSD will be proportional to . As was also shown \npreviously,3 1ˆ ˆm m⋅3ˆm\n0 /ˆ3→dt dm ) ( sin1 12ω′θ′ ′′x xm mS\n11 it follows from (23) (and assuming azimuthal symmetry 011 11=′=′′ ′ ′′ x y y xH H ) that  \n \nx x y y y y x x x x y yy y x x y y x x\nsB\nm m\nH H H HH H\ntA MT kSx x\n′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′′ ′ ′′′′′′\n′ ′\n′α′+′α′= ω Δ′ ′ γ = ωω Δ ω + ω − ωω α′+ ω′ ′ α′γ≅ ω′′\n11 11 11 11 11 11 02 2 2\n022\n112\n0 11 11 11\n1 ) ( ) () / (\n) (2) (1 1                         (24) \n \ntreating . The tensor influence of the  is seen to be weighted by the relative size \nof the stiffness-field matrix elements . For the thin film geometries 111 11 < < < α′α′′ ′′ ′ y y x x u u′ ′α′11\nv vH′ ′′11 A t< <  typical of such \ndevices, out-of-plane demagnetization fi eld contribution typically result in  that are an order of \nmagnitude larger than . Since y yH′ ′′11\nx xH′ ′′11x x y y ′ ′ ′′α′≤ α′11 11 from (18), it follows that the linewidth ωΔ and the \nPSD  in the spectral range of practical interest will both be expected to be determined \nprimarily by . ) (01 1ω ≤ ω ′′ ′′x xm mS\nx x′ ′α′11\n     A similar circumstance also applies to the im portant problem of critical currents for spin-torque \nmagnetization excitation in CPP-GMR spin valves with 0≠eJ . Consider the same example as above, \nagain treating  as stationary, and seeking nontri vial solutions of (21) (with of the form \n. Summarizing results obtainable from (5), (8), and (21) 3ˆm ) 0 ) (=′th\nste t−∝′) (1m\n \n0 detˆ\n) (2 /\n11 11 1111 11 111 2\n10eff\n1eff\n1NM\n=⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nα′ ′−′ ′ −′′+′ α′ ′−′× + =\n′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′=\ny y y y x yy x x x x xsJ\ns H s Hs H s Ht Me\nem J H Hh\n                                           (25) \n \nwhere ,  as in (18),and where  in (25) is now the solution of the transport \nequations with  but . The cross-product form of the spin-torque contribution to  \nexplicitly yields an asymmetr ic/nonreciprocal contribution γ =′ /s sv u′ ′α′11 eJ∝NM\n2J\n0pump= J 0≠eJeff\n1H\nex y y xJ H H∝′−′′′′′\n11 11 to . The critical \ncurrent density is that value of  where  becomes negative. Given th e basic stability criterion \nthat , the spin-torque critical  condition from (25) can be expressed as  Ht\n′\neJ sRe\n0 det11>′Ht \nx y y x y y x x x x y yH H H H′′′′′′′ ′ ′ ′ ′ ′′−′=′α′+′α′11 11 11 11 11 11                                            (26) \n \nLike for thermal noise, the spin-torque critical  point should again be determined primarily by  for \nin-plane magnetized CPP-GMR spin-valves with typical x x′ ′α′11\nx x y yH H′ ′ ′′′> >′11 11. This simply reflects the fact \nthat the (quasi-uniform) modes of thermal fluctuati on or critical-point spin-torque oscillation tend to \nexhibit rather \"elliptical\", mostly in-plane motion when x x y yH H′ ′ ′′′> >′11 11. This is obviously different \nthan the steady, pure circular precession described in Ref. 3, which contrastingly highlights the \ninfluence of , along with its inte resting, additional y y′ ′α′11θ-dependence.  \n \nAPPENDIX A \n \n     The well known \"circuit theory\" formulation12 of the boundary conditions for the electron charge \ncurrent density  and the (dimensionally equiva lent) spin current density  at a FM/NM interface \ncan (taking ) be expressed as  eJspin\nNMJ\nm V ˆFM FM VΔ =Δ\n) ˆ )( ( ) )( (FM NM FM NM 21V G G V V G G Je Δ − ⋅ − + − + =↓ ↑ ↓ ↑m VΔ                        (A1) \n \n)ˆ ( Im ) ˆ ˆ( Reˆ)] ˆ )( ( ) )( [(\nNM NMFM NM FM NM NM 21 spin\nm V m V mm m V J\n× + × × +Δ − ⋅ + + − − =\n↑↓ ↑↓↓ ↑ ↓ ↑\nΔ ΔΔ\nG GV G G V V G G\n                      (A2) \nin terms of spin-indepe ndent electric potential V and accumulation VΔ (Δμe= ). Setting 0=eJ  in \n(A1) and substituting into (A2), one obtains in the limit the result 0 Im→↑↓G\n \n)ˆ ˆ( ˆ) ˆ (2\nNM FM NM NM0spinm V m m m V J × × + Δ − ⋅\n+=↑↓\n↓ ↑↓ ↑\n=Δ Δ G V\nG GG G\neJ                           (A3) \n \nComparing with Eq. (4) of Tserkovnyak et al.3 (with )sμΔ⇔V  and remembering the present \nconversion of , one immediately make s the identification spin 1 spin\nNM NM ) 2 / (I J−− ↔ e Ah\n \n↑↓ ↑↓= G e h A g) 2 / ( 22                                                      (A4) \n \nrelating dimensionless  in (1) to , the conventional mixing conductance (per area). ↑↓g↑↓G\n     The common approximations that  inside all FM layers, and that longitudinal  spin \ncurrent density is conserved at  FM/NM interfaces, yields the us ual interface boundary condition  m J ˆspin spin\nFM FM J=spin spin\nFM NMˆJ= ⋅m J                                                                     (A5) \nSolving for from (A2) then leads (with (A1)) to a second scalar boundary condition: m Jˆspin\nNM⋅\nspin\nFM FM NM4 4J\nG GG GJ\nG GG GV Ve ↓ ↑↓ ↑\n↓ ↑↓ ↑−−+= −                                               (A6) \nEquation (A6) is identical in form with the standard (collinear) Valet-Fert model,7 and immediately \nyields the following identifications \n↓ ↑↓ ↑\n↓ ↑↓ ↑\n+−= γ+=\nG GG G\nG GG Gr ,\n4                                                         (A7) \nfor the conventional Valet-Fert interface parameters .  γandr\n     The three vector terms on th e right of (A2) are mutually orthogona l. Working in a rotated (primed) \ncoordinate system where , (A1) and (A2) can be similarly inverted to solve for the three \ncomponents of the vector m z′=′ˆˆ\n)ˆ (FM NM m V ′Δ −′ VΔ  in terms of , , and . A final transformation \nback to the original (umprimed) coordinate s yields the vector interface-boundary condition spin\nNMJ′spin\nFMJeJ\n \n) / /( ) 2 / ( ) 2 /( 1ˆ Im Reˆ] ) Re [( ) ˆ (\n2spin spin spin\n21\nNM NM FM FM NM\nA g e h G rr r J r J r r Ve\n↑↓ ↑↓ ↑↓↑↓ ↑↓ ↑↓\n= ≡× + + γ − − = Δ − J m J m m VΔ\n          (A8) \n \nCombined with (A4), the last relation in (A8) yields (2). Equation (A8) is a generalization of Valet-\nFert to the non-collinear case.  \n     As noted by Tserkovnyak et al.,3 boundary conditions (A3) do not di rectly include spin-pumping \nterms, but instead involve only \"backflow\" terms  in the NM layer. With spin-pumping \nphysically present,  arises as the response to the spin accumulation back spin\nNM NM J J↔\nback\nNMJNMVΔ created by . It \nfollows that , where  is henceforth the total spin current in the NM layer. \nThus, including spin-pumping in Va let-Fert transport equations is then a matter of replacing \n in (A8). The modified form of (A 8), for a FM/NM interface, becomes: pumpJ\npump spin back\nNM NM J J J− =spin\nNMJ\npump spin spin\nNM NM J J J− →\n \n) (ˆ Im ) ( Reˆ] ) Re [( ) ˆ (\npump spin pump spinspin\n21\nNM NMFM FM NM\nJ J m J Jm m V\n− × + − +γ − − = Δ −\n↑↓ ↑↓↑↓\nr rJ r J r r Ve Δ\n                       (A9) \n \nFor an NM/FM interface, the sign is flippe d on the left sides of (A6) and (A9).  \n APPENDIX B \nFor 1-D transport (flow along the y-axis), the quasi-static Valet-Fert7 (drift-diffusion, quasi-static) \ntransport equations can be written as9 \n \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∂∂+∂∂βρ==⎥\n⎦⎤\n⎢\n⎣⎡\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∂∂⋅ β +∂∂\nρ=∂∂=\n∂∂\ny yVy yVJy l ye\nVm JVmV V\nΔΔ Δ Δ\n21ˆ1with along0 ˆ21 1,\nspin2 22\n                                      (B1) \n \nwhere = bulk resistivityρ13, l = spin diffusion length, and β = bulk/equilibrium spin current \npolarization in FM layers (  in NM layers). The solution for any one layer has the form 0≡ β\n \nm B m AB A V m V\nˆ ,ˆ : layers FM for,ˆ/ /\n21\nB Ae e C y J Vl y l y\ne\n= =+ = ⋅ β − + ρ =−Δ Δ                              (B2) \n \nIn the case where  film thickness, one may employ an al ternative \"ballistic\" approximation: > > >l\n \nC V , = = = ,spinB J A V Δ                                                        (B3) \n \nIt is not necessary to solve for the V and/or the C-coefficients using (A6) if only  and  are \nrequired. The remaining coefficients are determin ed by the interface boundary conditions (A5), (A6,7) \nand (A9), and external boundary conditions at the outer two surfaces of the film stack.  VΔspinJ\n     Regarding the latter, one approximation is to treat the external \"leads\" (with quasi-infinite cross \nsection) as equilibrium reservoirs and set 0 ) (, 0→==N i y y VΔ  at the outermost (i =0, N) lead-stack \ninterfaces of an N -layer stack (Fig. 1). The complimentary approximation is of an insulating boundary, \nwith . . For the case (such as in Sec. III) where the outer ( j=0, N-1) layers are \nNM, and the adjacent inner ( j=1, N-2) layers are FM, it is  readily found using (B1) and (B2) that  0 ) (, 0spin→==N i y y J\n \nNM NM) / hyp( ) ( 21 , 0 1 , 1 i j j N j N i l t l J V− = − =ρ ± = Δ                                       (B4) \n \nwhere hyp( ) = tanh( ) or coth( ) for equipotential,  or insulating boundaries, respectively. Combining \n(B4) with (A9), and neglecting ↑↓rIm , one finds for 0=eJ  that  \n ) / hyp( ) (ˆ)] / hyp( ) ( [\npump1 , 0 1 , 1 21\nFM NMFM FM\nj j j ii i\ni ii j j N j i N i\nl t l rrJJ l t l r V\nρ ++ =ρ + = Δ ±\n↑↓↑↓− = − =\nJm J                                 (B5) \n \nACKNOWLEDGMENT \n \nThe author would like to thank Y. Tserkovnya k for bringing Ref. 6 to his attention. \n \nREFERENCES \n \n1. D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mat. 320, 1190 (2008) and many re ferences therein. \n2. Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B, 66, 224403 (2002). \n3. Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B, 67, 140404 (2003). \n4. S. Maat, N. Smith, M. J. Carey, and J.  R. Childress, Appl. Phys. Lett., 93, 103506 (2008). \n5. J. Foros, A. Brataas, Y. Tserkovnyak,  and G. E. W.Bauer, Phys. Rev. B 78, 140402 (2008). This \npaper describes a damping mechanism distinct from  Refs. 6, or this work, where nonlocal/tensor \nproperties arise from a strong magnetization gr adient in a single FM film or wire. \n6. J. Foros, A. Brataas, G. E. W.Bauer,  and Y. Tserkovnyak, arXiv:con-mat/0902.3779.  \n7. T. Valet and A. Fert, Phys. Rev. B, 48, 7099 (1993). \n8. N. Smith, J. Appl. Phys. 92, 3877 (2002); N. Smith, J. Magn. Magn. Mater. 321, 531 (2009) \n9. N. Smith, J. Appl. Phys., 99, 08Q703, (2006). \n10. R. Duine, A.S. Nunez, J. Si nova, A.H. MacDonald, Phys. Rev. B 75, 214420 (2007) \n11. N. Smith, J. Appl. Phys. 90, 5768 (2001). \n12. A. Brataas, Y. V. Nazarov, and G.  E. W. Bauer, Eur. Phys. J. B 22, 99, (2001). \n13. Some poor choice of words in the appendix of Ref. 9 c onfused the bulk resistivity, ρ, with the \nValet-Fert7 parameter . *ρ\n \n \n "    },    {        "title": "2001.06217v1.Fermi_Level_Controlled_Ultrafast_Demagnetization_Mechanism_in_Half_Metallic_Heusler_Alloy.pdf",        "content": " Fermi Level Controlled Ultrafast Demagnetization Mechanism in Half -Metallic Heusler \nAlloy  \nSantanu Pan1, Takeshi Seki2,3, Koki Takanashi2,3,4, and Anjan Barman1,* \n1Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for \nBasic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India.  \n2Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan.  \n3Center for Spintronics Research Network, Tohoku University, Sendai 980 -8577, Japan.  \n4Center f or Science and Innovation in Spintronics, Core Research Cluster, Tohoku University, Sendai \n980-8577, Japan . \n*E-mail: abarman@bose.res.in  \n \n \n \nThe electronic band structure -controlled ultrafast demagnetization mechanism in Co2FexMn 1-\nxSi Heusler alloy is underpinned by systematic variation of composition. We find the spin-flip \nscattering rate controlled by spin density of states at Fermi level  is responsible for non-\nmonotonic variation  of ultrafast demagnetization time (τ M) with x with a maximum  at x = 0.4 . \nFurthermore, Gilbert damping constant exhibits an inverse relationship with  τM due to the \ndominance of inter -band scattering mechanism. This establishes a unified mechanism of \nultrafast spin dynamics based on Fermi level position.  \n \n \n \n \n \n \n \n \n \n \n  The tremendous application potential of spin -polarized Heusler alloys in advanced spintronic s \ndevices  ignites immense interest to investigate the degree and sustainability of their spin-\npolarization under various conditions [1-4]. However, interpreting spin -polarization from the \nconventional methods  such as photoemission, spin transport measurement, point contact \nAndreev reflection and spin-resolved positron annihilation are non -trivial [5-7]. In the quest of \ndeveloping alternative methods, Zhang et al . demonstrated that all -optical ultrafast \ndemagne tization measurement is a reliable technique for probing spin -polarization [8]. They \nobserved a very large ultrafast demagnetization time as a signature of high spin -polarization in \nhalf-metallic CrO 2. However, Co -based half -metallic Heusler alloys exhibit  a comparatively \nsmaller ultrafast demagnetization time (~ 0.3 ps) which raised a serious debate on the \nperception of ultrafast demagnetization mechanism in Heusler alloys [9-11]. A smaller \ndemagnetization time in Heusler alloys than in CrO 2 is explained d ue to the smaller effective \nband gap in the minority spin band  and enhanced spin-flip scattering  (SFS) rate [9]. However, \nfurther experimental evidence shows that the amount of band gap  in minority spin band  cannot \nbe the only deciding factors for SFS medi ated ultrafast demagnetization efficiency [10]. Rather, \none also has to consider the efficiency of optical excitation for majority and minority spin bands \nas well as the optical pump -induced hole dynamics below  Fermi energy  (EF). Consequently, a \nclear interpretation of spin -polarization from ultrafast demagnetization measurement requires \na clear and thorough understanding of its underlying mechanism. Since its inception in 1996 \n[12], several theoretical models and experimental evi dences based on different microscopic \nmechanisms, e.g. spin -flip scattering (SFS) and super -diffusive spin current have been put \nforward to interpret ultrafast demagnetization [13-20]. However, the preceding proposals are \ncomplex and deterring to each othe r. This complexity increases even more in case of special \nclass of material such as the Heusler alloys. The electronic band structure and the associated \nposition of Fermi level can be greatly tuned by tuning the alloy composition of Heusler alloy \n[21,22]. By utilizing this tunability, h ere, we experimentally demonstrate that the ultrafast \ndemagnetization mechanism relies on the spin density of states at Fermi level in case of half -\nmetallic Heusler alloy system. We extracted the value of ultrafast demagnetiz ation time using \nthree temperature modelling [23] and found its non -monotonic dependency on alloy \ncomposition ( x). We have further showed that the Gilbert damping and ultrafast \ndemagnetization time are inversely proportional in CFMS Heusler alloys suggesti ng the inter -\nband scattering as the primary mechanism behind the Gilbert damping in CFMS Heusler alloys . \nOur work has established a unified theory of ultrafast spin dynamics.   A series of Co 2FexMn 1-xSi (CFMS) thin films have been deposited using magnetron co -\nsputtering system for our investigation with  x = 0.00, 0.25, 0.40, 0.50, 0.60, 0.75 and 1.00 . The \nthickness of the CFMS layer was fixed at 30 nm. It is imperative to study the crystalline phase \nwhich is the most crucial parameter that determines other magnetic properties of Heusler alloy. \nPrior to the magnetization dynamics measurement, we invest igate both the crystalline phase as \nwell as growth quality of all the samples. Fig. 1A shows the ex-situ x-ray diffraction (XRD) \npattern for all the samples. The well -defined diffraction peak of CFMS (400) at 2θ = 66.50º \nindicates that the samples are well  crystalline having cubic symmetry. The intense superlattice \npeak at 2θ = 31.90º represents the formation of B2 phase. The presence of other crucial planes \nare investigated by tilting the sample x = 0.4 by 54.5º and 45.2º from the film plane to the \nnormal direction, respectively  and observed the presence of (111) superlattice peak along with \nthe (220) fundamental peak as shown in Fig. 1B and 1C. The presence of (111) superlattice \npeak confirms the best atomic site ordering in the desired L2 1 ordered phase, whereas the (220) \nfundamental peak results from the cubic symmetry.  The intensity ratios of the XRD peaks are \nanalysed to obtain the microscopic atomic site ordering which remain same for the whole range \nof x (given in Supplemental Materials). The epitaxia l growth of the thin films is ensured by \nobserving the in-situ reflection high -energy electron diffraction (RHEED) images. The square \nshaped hysteresis loops obtained using in -plane bias magnetic field shows the samples have in -\nplane magnetization. The nearly increasing trend of saturation magnetization with alloy \ncomposition ( x) follow the Slater -Pauling curve. In -depth details of sample deposition \nprocedure, RHEED pattern and the hysteresis loops are provided in the Supplemental Materials \n[24]. The ultrafast demagnetization dynamics measurements using  time-resolved magneto -\noptical Kerr effect (TRMOKE) magnetometer have been performed  at a fixed probe fluence of \n0.5 mJ/cm2, while the pump fluence have been varied over a large range . Details of the \nTRMOKE technique is provided in Supplemental Materials [24]. The experi mental data of \nvariation of Kerr rotation corresponding to the ultrafast demagnetization  measured for pump \nfluence = 9.5 mJ/cm2 is plotted in Fig. 2A for different values of x. The data points are then \nfitted with a phenomenological expression derived from  the three temperature  model -based \ncoupled rate equations in order to extract the ultrafast demagnetization time (\nMτ) and fast \nrelaxation (\nEτ) time [23], which is given below:  \n  \nME/τ - /τ- 1 2 E 1 M E 1 2\nk3 1/2\n0 E M E MA (A τ -A τ ) τ (A -A )-Δ {[ - e - e ]H( ) A δ( )} G( )( / t 1) ( τ -τ ) (τ -τ )ttθ t t tt= + +       (1)    where A1 represents the magnetization amplitude after equilibrium between electron, spin and \nlattice is restored,  A2 is proportional to the maximum rise in the electron temperature and  A3 \nrepresents the state filling effects during pump -probe temporal overlap described by a Dirac \ndelta function.  H(t) and δ(t) are the Heaviside step and Dirac delta functions , and G(t) is a \nGaussian function which corresponds to the laser pulse.  \nThe \nMτ  extracted from the fit s are plotted as a function of x in Fig. 2B, which shows  a slight \ninitial increment followed by a sharp  decrement with x. In addition, the ultrafast \ndemagnetization rate is found to be slower in the present Heusler alloys than in the 3d metals \n[9]. The theoretical calculation of electronic band structure of CFM S showed no discernible \nchange in the amount of energy gap  in minority spin band  but a change in position of EF with \nx, which lies at the two extreme ends of the gap for x = 0 and x = 1. Thus, the variation of \nMτ  \nwith x clearly indicates  that the composition dependent EF position is somehow  responsible for \nthe variation in  \nMτ . This warrants the investigation of ultrafast demagnetization with \ncontinuously varying x values between 0 and 1. However, a majority of earlier investigations \n[10,11,2 5], being focused on exploring the ultrafast demagnetization only of Co 2MnSi ( x = 0) \nand Co 2FeSi ( x = 1), lack a convincing conclusion about the role of  electronic band structure \non ultrafast demagnetization  mechanism .  \nIn case of 3d transition metal ferromagnets, Elliott -Yafet (EY) -based SFS mechanism is \nbelieved to be responsible for rapid rise in the spin temperature and ultrafast demagnetization \n[15]. In this theory it has been shown that a  scattering event of an excited electron with a \nphonon changes the probability to find that electron in one of the spin states, namely the \nmajority spin -up (\n ) or minority spin -down (\n ) state, thereby delivering angular momentum \nto the lattice from the electronic system. It arises from the band mixing of majority and minority \nspin states with similar energy value near the Fermi surface owing to the spin -orbit coupling \n(SOC). The spin mixing para meter (b2) from the EY theory [26,27] is given by:  \n                             \n2\nk k k k b min ( ψ ψ , ψ ψ )=                                                               (2) \nwhere \nkψ  represent the eigen -state of a single electron and the bar denotes a defined average \nover all electronic states involved in the EY scattering processes. This equation represents that \nthe spin-mixing due to SFS between spin -up and spin -down states depend o n the  number of  \nspin-up (\n ) and spin -down (\n ) states  at the Fermi level, which is already represented by D F.  A compact differential equation regarding rate of ultrafast demagnetization dynamics as \nderived by Koopmans et al. [27], is given below:  \n                                                  \np C\nCeT TR (1 coth( ))TTm dmmdt=−                                                         (3) \nwhere m = M/MS, and Tp, TC, and Te denote the phonon temperature, Curie temperature and \nelectronic temperature, respectively. R is a material specific scaling factor [28], which is \ncalculated to be:  \n                                                               \n2\nsf C ep\n2\nB D S8a T gRk T D=   ,                                                                  (4) \nwhere asf, gep, DS represent the SFS probability, coupling between electron and phonon sub -\nsystem and magnetic moment divided by the  Bohr -magneton (\nB ), whereas TD is the Debye \ntemperature and kB represents the Boltzmann constant. Further, the expression for gep is: \n22\nF P B D ep\nep3πD D k T λg2=\n, where  DP, and λep denote the number of polarization states of spins and \nelectron -phonon coupling constant, respectively , and ℏ is the reduced Planck’s constant. \nMoreover, the ultrafast demagnetization time at low fluence limit can be derived under various \napproximations as:  \n                                                         \n0C\nM 22\nF si B CC F( / T )τπD λ k TT=\n ,                                                               (5)  \nwhere C0 = 1/4, \nsiλ  is a factor scaling with impurity concentration, and F(T/TC) is a function \nsolely dependent on ( T/TC) [29].  \nEarlier, it has been shown that a negligible DF in CrO 2 is responsible  for large ultrafast \ndemagnetization time. The theoretical calculation  for CFMS  by Oogane et al.  shows that DF \ninitially decreases and then increases with x [30] having  a minima at x = 0.4. As DF decreases, \nthe number of effective minority spin states become less, reducing both SOC strength, as shown \nby Mavropoulos et al. [31], and the effective spin -mixing paramet er is given by Eq. (2), and \nvice versa. This will result in a reduced SFS probability  and rate of demagnetization. In \naddition, the decrease in DF makes gep weaker, which, in turn, reduces the value of R as evident \nfrom Eq. (4). As the value of R diminishes, it will slow down the rate of ultrafast \ndemagnetization which is clear from Eq. (3). In essence , a lower value of DF indicates a lower  value of R, i.e. slower demagnetization rate and larger  ultrafast  demagnetization time. Thus, \ndemagnetization time is highest for x = 0.4. O n both sides of x = 0.4, the value of R will increase \nand ultrafast demagnetization time will decline continuously.  Our experimental results, \nsupported by the existing theoretical re sults for  the CFMS samples with varying alloy \ncomposition, clearly show that the position of Fermi level is a crucial decisive factor for the \nrate of ultrafast demagnetization. This happens due to the continuous tunability of DF with x, \nwhich causes an ensuing variation in the number of scattering channels available for SFS.  To \ncapture the effect of pump fluence on the variation of  \nMτ, we have measured the ultrafast \ndemagnetization curves for various applied pump fluences. All the flu ence dependent ultrafast \ndemagnetization curves are fitted with Eq. (1) and the values of corresponding  \nMτ are \nextracted. The change in \nMτ  with fluence is shown in Fig. 2C. A slight change in \nMτ with \nfluence is observed which is negligible in comparison to the change of \nMτ  with x. However, \nthis increment can be explained using the enhanced spin fluctuations at much higher elevated \ntemperature of the spin sy stem [28].  \nAs the primary microscopic channel for spin angular momentum transfer is the same for both \nultrafast demagnetization and magnetic damping, it is expected to find a correlation between \nthem. We have measured the time -resolved Kerr rotation data corresponding to the \nmagnetization precession at an applied in -plane bias magnetic field  (Hb) of 3.5 kOe  as shown \nin Fig. 3A. The macrospin modelling is employed to analyse the time dependent precessional  \ndata obtained by solving the Landau -Lifshitz -Gilbert equation [32] which is given below:  \n                                                 \neffˆˆˆˆγ( ) α( )dm dmm H mdt dt=−  +                                                    (6) \nwhere \nγ  is the gyromagnetic ratio and is related to Lande g factor by  \n/μg=γB . Heff is the \ntotal effective magnetic field consisting of Hb, exchange field ( Hex), dipolar field ( Hdip) and \nanisotropy field (\nKH ). The experimental variation of precession frequency ( f) against Hb is \nfitted with the Kittel formula for uniform precession to extract HK values. The details of the fit \nare discussed in the Supplementa l Materials [24] . \nFor evaluation of  \nα, all the measured data representing single frequency oscillation are fitted \nwith a general damped sine -wave equation superimposed on a bi -exponential decay function, \nwhich is given as:                                  \nfast slow/τ /τ /τ\n12 ( ) A B e B e (0)e sin( ω ζ)tt tM t M t−− −= + + + −  ,                (7) \nwhere \nζ is the initial phase of oscillation and  \nτ is the precessional relaxation time . \nfastτ  and \nslowτ\n are the fast and slow relaxation times, representing the rate of energy transfer in between \ndifferent energy baths (electron, spin and lattice) following the ultrafast demagnetization and \nthe energy transfer rate between the lattice and surrounding, respec tively. A, B1 and B2 are \nconstant coefficients. The value of \nα  is extracted by further analysing \nτ  using  \n                                        \n( )122α[γτ 2 cos( H H ]=− + +bHδφ                                                         (8)  \nwhere \n22\n12\n1S\nS S S2K 2K sin K (2 sin (2 ))4πMM M MφφH⊥ −= + − +  and \n12\n2\nSS2K cos(2 ) 2K cos(4 )\nMMφφH=+ . Here \n\nand \n  represent the angles of Hb and in -plane equilibrium M with respect to the CFMS [110] \naxis [33]. The uniaxial, biaxial and out -of-plane magnetic anisotropies are denoted as K1, K2 \nand \nK⊥, respectively. In our case K2 has a reasonably large value while K1 and \nK⊥  are \nnegligibly small. Plugging in all parameters including the magnetic anisotropy constant K2 in \nEq. (8), we have obtained the values of \nα  to be 0.0041, 0.0035, 0.0046, 0.0055, 0.0061, and \n0.0075 for x = 0.00, 0.40, 0.50, 0.60, 0.75, and 1.00, respectively. Figure 3B shows the variation \nof \nα  with frequency for all the samples. For each sample, \nα  remains constant with frequency, \nwhich rules out the presence of extrinsic mechanisms contributing to the  \nα. Next, we focus on \nthe variation of \nα  with x. Our experimental results show a non -monotonic variation of  \nα with \nx with a minima at x = 0.4 , which  is exactly opposite to the variation of \nMτ  with x. On the basis \nof Kambersky’s SFS model [34], \nα is governed by the spin -orbit interaction and can be \nexpressed as:  \n                                                      \n22\nF\nSγ (δg)αD4ΓM=\n                                                           (9) \nwhere \ngδ and \n1− represent the deviation of g factor from free electron value (~2.0) and \nordinary electron -phonon collision frequency. Eq. (9) suggests that \nα  is directly proportional \nto DF and thus it become s minimum when DF is minimum [3 0]. This leads to the non -monotonic \nvariation of\nα , which agrees well with earlier observation [30]. To eliminate the possible effects  of \nγ  and \nSM , we have plotted the variation of relaxation frequency, \nSMαγ=G  with x which \nalso exhibits similar variation as  \nα (see the supplementary materials [24] ).  \nFinally , to explore the correlation between\nα , \nMτ and alloy composition, we have plotted these \nquantities  against x as shown in Fig. 4A. We observe that \nMτ  and \nα  varies in exactly opposite \nmanner with x, having their respective maxima and minima at x = 0.4. Although \nMτ  and \nα  \nrefer to two different time scales, both of them follow the trend of variation of  DF with x. This \nshows  that the alloy composition -controlled Fermi level tunability and the ensuing SFS is \nresponsible for both ultrafast demagnetization and Gilbert damping . Figure 4B represents the \nvariation of \nMτ  with inverse of  \nα, which establishes an inversely proportional relation between \nthem . Initially under the assumption of two different magnetic fields, i.e. exchange field and \ntotal effective magnetic field, Koopmans et al.  theoretically proposed that Gilbert damping \nparame ter and ultrafast demagnetization time are inversely proportional [29]. However, that \nraised intense debate and in 2010, Fahnle et al.  showed that \nα  can either be proportional or \ninversely proportional to \nMτ  depending upon the dominating microscopic contribution to the \nmagnetic damping [32]. The linear relation sustains when the damping is dominated by \nconductivity -like contribution, whereas the resistivity -like contribution leads to an inverse \nrelation. The basic difference between the conductivity -like and the resistivity -like \ncontribution s lies in the angular momentum transfer mechanism via electron -hole ( e-h) pair \ngeneration. The generation of e-h pair in the same band, i.e. intra -band mechanism leads to t he \nconductivity -like contribution. On the contrary, when e-h pair is generated in different bands \n(inter -band mechanism), the contribution is dominated by resistivity. Our observation of the \ninversely proportional relation between \nα  and \nMτ  clearly indicates that in case of the CFMS \nHeusler alloy  systems, the damping is dominated by resistivity -like contribution arising from \ninter-band e-h pair generation. This is in contrast to the case of Co, Fe and Ni, where the \nconductivity contribution dominates [35]. Typical resistivity (\nρ ) values for Co 2MnSi ( x = 0) \nare 5\ncm− at 5 K and 20 \ncm− at 300 K [36]. The room temperature value of \nρ\ncorresponds to an order of magnitude larger contribution of the inter -band e-h pair generation \nthan the intra -band generation [36]. This is in strong agreement with our experimental results \nand its conclusion. This firmly establishes that unlike convention al transition metal \nferromagnets, damping in CFMS Heusler alloys  is dominated by resistivity -like contribution , \nwhich results in an inversely proportional relation between \nα  and\nMτ .  In summary, we have investigated the ultrafast demagnetization and magnetic Gilbert damping \nin the CFMS Heusler alloy systems with varying alloy composition ( x), ranging from x = 0 \n(CMS) to x = 1 (CFS) and identified a strong correlation between \nMτ  and x, the latter \ncontrolling the position of Fermi level in the electronic band structure of the system. We have \nfound that \nMτ  varies non -monotonically with x, having a maximum value of ~  350 fs for x = \n0.4 corresponding to the lowest DF and highest degree of spin -polarization. In -depth \ninvestigation has revealed that the ultrafast demagnetization process in CFMS is primarily \ngoverned by the composition -controlled variation in spin -flip scattering rate  due to variable DF. \nFurthermore, we have systematically investigated the precessional dynamics with variation in \nx and extracted the value of \nα  from there. Our results have led to a systematic correlation in \nbetween\nMτ ,\nα and x and we have found an inversely proportional relationship between \nMτ  and \nα\n. Our thorough investigation across the alloy composition ranging from CMS to CFS have \nfirmly establishe d the fact that both ultrafast demagnetization and magnetic  Gilbert  damping \nin CFMS are strongly controlled by the spin density of states at Fermi level. Therefore, our \nstudy has enlighten ed a new path for qualitative understanding of spin -polarization from \nultrafast demagnetization time as well as magnetic  Gilbert  dampin g and led a step forward for \nultrafast magnetoelectronic device applications.  \nAcknowledgements  \nThis work was funded by: S. N. Bose National Centre for Basic Sciences under Projects No. \nSNB/AB/12 -13/96 and No. SNB/AB/18 -19/211.  \nReferences  \n[1] T. Kubota,  S. Tsunegi,  M. Oogane,  S. Mizukami, T. Miyazaki,  H. Naganuma, and Y. Ando,   \nAppl. Phys. Lett.  94, 122504 (2009).  \n[2] A. Hirohata , and K. Takanashi, J. Phys. D: Appl. Phys.  47, 193001 (2014).  \n[3] R. J. Soulen  et al., Science  282, 85 (1998).  \n[4] I. I. Mazin, Phys. Rev. Lett.  83, 1427 (1999).  \n[5] K. E. H. M. Hanssen, P. E.  Mijnarends, L. P. L. M. Rabou, and K. H. J. Buschow, Phys. Rev. B  \n42, 1533 (1990).  \n[6] L. Ritchie , Phys. Rev. B  68, 104430 (2003).  \n[7] D. T. Pierce , and F. Meier,  Phys. Rev. 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Lett.  95, 022509 \n(2009).    \n \n \n \n \n \n \n \n \n \n \n    \n \nFig. 1. (A)  X-ray diffraction (XRD) patterns of Co 2FexMn 1-xSi (CFMS) thin films for different alloy \ncomposition ( x) measured in conventional θ-2θ geometry. Both CFMS (200) superlattice and CFMS \n(400) fundamental peaks are marked along with Cr (200) peak. (B) The tilted XRD patterns reveal the \nCFMS (111) superlattice peak for L2 1 structure. (C) CFMS (220) fundamental peak together with Cr \n(110) peak.  \n \n \n \n \n \n \n \n \n  \nFig. 2. (A) Ultrafast demagnetization curves for the samples with different alloy composition ( x) \nmeasured using TRMOKE. Scattered symbols  are the experimental data and solid lines are fit using Eq. \n3. (B) Evolution of \nMτ  with x at pump fluence of 9.5 mJ/cm2. Symbols are experimental results and \ndashed line is guide to eye.  (C) Variation in \nMτ  with pump fluence.  \n \n \n \n \n  \nFig. 3. (A) Time -resolved Kerr rotation data  showing precessional dynamics  for samples with different \nx values . Symbols  are the experimental data and solid lines are fit with damped sine wave equation ( Eq. \n6). The extracted \nα  values are given  below  every curve. (B) Variation of \nα  with precession frequency \n(f) for all samples as shown by symbols, while  solid lines are linear fit.  \n \n \n \n \n \n \n \n \n \n  \n \nFig. 4. (A) Variation of \nMτ  and \nα  with x. Square and circular symbols  denote the experimental results , \nand dashed , dotted  lines are guide to eye.  (B) Variation of \nMτ  with \n1α− . Symbols represent the \nexperimentally obtained values and solid line refers to linear fit.  \n \n \n \n \n \n \n \n \n \n \n \n \n Supplementa l Material s \n \nI. Sample preparation method  \nA series of MgO Substrate /Cr (20 nm)/ Co 2FexMn 1-xSi (30 nm)/Al -O (3 nm) sample stacks \nwere deposited using an ultrahigh vacuum magnetron co -sputtering system. First a 20 -nm-thick \nCr layer was deposited on top of a single crystal MgO (100) substrate at room temperature \n(RT) followed by annealing it at 600 ºC for 1 h. Next, a Co 2FexMn 1-xSi layer of 30 nm thickness \nwas deposited on the Cr layer followed by an in -situ annealing process at 500 ºC for 1 h. \nFinally, each sample stack was capped with a 3 -nm-thick Al -O protective layer. A wide range \nof values of x is chosen, namely, x = 0.00, 0.25, 0.40, 0.50, 0.60, 0.75 and 1.00. To achieve the \ndesired composition of Fe and Mn precisely, the samples were deposited using well controlled \nco-sputtering of Co 2FeSi and Co 2MnSi. Direct deposition of Co 2FexMn 1-xSi on top of MgO \nproduces strain  due to lattice mismatch in the Co 2FexMn 1-xSi layer which alters its intrinsic \nproperties [1S]. Thus, Cr was used as a buffer layer to protect the intrinsic Co 2FexMn 1-xSi layer \nproperties [2S].  \nII. Details of measurement techniques  \nUsing ex-situ x-ray diffraction ( XRD ) measurement we investigated the crystal structure and \ncrystalline phase of the samples. The in-situ reflection high -energy electron diffraction  \n(RHEED ) images were observed after the layer deposition without breaking the vacuum \ncondition in order to investigate the epitaxial relation and surface morphology of Co 2FexMn 1-\nxSi layer. To quantify the values of M S and H C of the samples, we measured the  magnetization \nvs. in -plane magnetic field  (M-H) loops using a vibrating sample magnetometer ( VSM) at room \ntemperature with H directed along the [110] direction of Co 2FexMn 1-xSi. The ultrafast \nmagnetization dynamics for all the samples were measured by using a time-resolved magneto -\noptical Kerr effect ( TRMOKE ) magnetometer [ 3S]. This is a two -colour  pump -probe \nexperiment in non -collinear arrangement. The fundamental output (wavelength, λ = 800 nm, \npulse -width, \ntσ ~ 40 fs) from an amplified laser system (LIBRA, Coherent) acts as probe and \nits second harmonic signal (λ = 400 nm, \ntσ ~ 50 fs) acts as pump beam. For investigating both \nultrafast demagnetization within few hundreds of femtosecond s and p recessional \nmagnetization dynamics in few hundreds of picosecond time scale, we collected the time -\nresolved Kerr signal in two different time regimes. The time resolution during the  measurements was fixed at 50 fs in -0.5 To 3.5 ps and 5 ps in -0.1 ns to 1 .5 ns to trace both the \nphenomena precisely. The pump and probe beams were focused using suitable lenses on the \nsample surface with spot diameters of ~250 µm and ~100 µm, respectively. The reflected signal \nfrom the sample surface was collected and analysed  using a polarized beam splitter and dual \nphoto detector assembly to extract the Kerr rotation and reflectivity signals separately. A fixed \nin-plane external bias magnetic field ( Hb) of 1 kOe was applied to saturate the magnetization \nfor measurement of ult rafast demagnetization dynamics, while it was varied over a wide range \nduring precessional dynamics measurement.  \nIII. Analysis of XRD peaks  \nTo estimate the degree of Co atomic site ordering, one has to calculate the ratio of integrated \nintensity of (200) and (400) peak. Here, we fit the peaks with Lorentzian profile  as shown in \ninset of Fig.  1S and extracted the integrated intensities as a parameter  from the fit . The \ncalculated ratio of I(200) and I(400) with re spect to alloy composition ( x) is sho wn in Fi g. 1S. \nWe note that there is no significant change in the I(200)/I(400) ratio. This result indicates an \noverall good quality atomic site ordering in the broad range of samples used  in our study . \n \nFig. 1S. Variation of integrated intensity ratio I(200)/I(40 0) with x, obtained from XRD patterns. Inset \nshows the fit to the peaks with Lorentzian profile.  \n \nIV. Analysis of RHEED pattern  \n The growth quality of the CFMS thin films was experimentally investigated using in-situ \nRHEED technique. Figure 2S shows the RHEED images captured  along the MgO [100] \ndirection for all the samples. All the images contain main  thick  streak lines in between the thin \nstreak lines , which are marked by the white arrows, suggesting the formation o f ordered phases. \nThe presence of regularly -aligned streak lines confirms the epitaxial growth in all the films.     \n \nFig. 2S. In-situ RHEED images for all the Co 2FexMn 1-xSi films taken along the MgO [100] direction. \nWhite arrows mark the presence of thin streak lines originating from the L2 1 ordered phase.  \n \nV. Analysis of magnetic hysteresis loops  \nFigure 3SA represents the M-H loops measured at room temperature  using VSM for all the \nsamples. All the loops are square in nature, which indicates a very small saturation magnetic \nfield. We have estimated the values of saturation magnetization ( MS) and coercive field ( HC) \nfrom the M-H loops.  Figure 3SB represents MS as a function of x showing a nearly monotonic \nincreasing trend, which is consistent with the Slater -Pauling rule for Heusler alloys [4S], i.e. \nthe increment in MS due to the increase in the number of valence electrons. However, it deviates \nremarkably at x = 1.0. This deviation towards the Fe -rich region is probably due to the slight \ndegradation in the film quality. Figure 3SC shows that HC remains almost constant with \nvariation of x. \n   \nFig. 3S. (A) Variation of M with H for all the samples. (B) Variation of MS as a function of x. \nSymbols are experimentally obtained values and dashed line is a linear fit. (C) Variation of HC \nwith x.    \n \nVI. Anal ysis of frequency ( f) versus bias magnetic field ( Hb) from TRMOKE \nmeasurements  \nWe have experimentally investigated the precessional dynamics of all the samples using \nTRMOKE technique. By varying the external bias magnetic field ( Hb), various precessional \ndynamics have been measured. The post -processing of these data foll owed by fast Fourier \ntransform (FFT) provides the precessional frequency (f) and this is plotted against Hb as shown \nin Fig. 4S .  \nTo determine the value of in-plane magnetic anisotropy  constant , obtained f-Hb curves have \nbeen analysed  with Kittel formula  which is given below:  \n                                  \n2 1 2\nS\nS S S2K 2K 2K γ(4πM )( )2π M M Mbb f H H= + + + +\n                                      (1S) \n  where MS is saturation magnetization and \nγ  denote the gyromagnetic ratio given by\nBgμγ=\nwhile K1 and K2 represent the two -fold uniaxial and four -fold biaxial magnetic anisotropy \nconstant, respectively.  \n \nFig. 4S. Variation of f as a function of Hb. Circular filled symbols represent the experimental data and \nsolid lines are Kittel fit.  \n \nWe have found the values of several parameters from the fit including K1 and K2. K1 has a \nnegligible value while K2 has reasonably large value in our samples. The e xtracted values of \nthe parameters from the fit are tabulated as follows  in Table 1S : \nTable 1S: The extracted values of Lande g factor and the four -fold biaxial magnetic anisotropy \nconstant K 2 for different values of x. \nx g K2 (erg/cm3) \n0.00 2.20 3.1×104 \n0.40 2.20 2.6×104 \n0.50 2.20 3.0×104 \n0.60 2.20 2.5×104 \n0.75 2.20 2.6×104 \n 1.00 2.20 3.4×104 \n \nVII. Variation of r elaxation frequency  with alloy composition   \nWe have estimated the damping coefficient (α) and presented its variation with alloy \ncomposition ( x) in the main manuscript. According to the Slater -Pauling rule, M S increases \nwhen the valence electron number systematically increases. As in our case the valence electron \nnumber changes with x, one may expect a marginal effect of M S on the estimation of damping. \nThus, to rule out any such possibilit ies, we have calculated the  variation of  relaxation \nfrequency ,\nS GαγM= with x, which is represented in Fig. 5S. It can be clearly observed from \nFig. 5S that relaxation frequen cy exactly follows the trend of\nα . This rules out any possible  \nspurious contribution of M S in magnetic damping.   \n \nFig. 5S. Non-monotonic v ariation of G with x for all the samples.  \n \nReferences:  \n[1S] S. Pan, S. Mondal,  T. Seki,  K. Takanashi, , and A. Barman, Influence of the thickness -dependent \nstructural evolution on ultrafast magnetization dynamics in Co 2Fe0.4Mn 0.6Si Heusler alloy thin films.  \nPhys. Rev. B  94, 184417 (2016).  \n[2S] S. Pan, T. Seki, K. Takanashi, and A. Barman, Role of the Cr buffer layer in the thickness -\ndependent ultrafast magnetization dynamics of Co 2Fe0.4Mn 0.6Si Heusler alloy thin films.  Phys. Rev. \nAppl.  7, 064012 (2017).  \n[3S] S. Panda,  S. Mondal, J. Sinha,  S. Choudhury, and A. Barman, All-optical det ection of interfacial \nspin transparency from spin pumping in β -Ta/CoFeB thin films.  Science Adv.  5, eaav7200 (2019).  \n[4S] I. Galanakis,  P. H.  Dederichs, and  N. Papanikolaou, Slater -Pauling behavior and origin of half -\nmetallicity of the full Hesuler alloys.  Phys. Rev. B  66, 174429 (2002).  \n  "    },    {        "title": "1909.09085v1.Magnetization_dynamics_of_the_compensated_ferrimagnet__Mn__2_Ru__x_Ga_.pdf",        "content": "Magnetisation dynamics of the compensated ferrimagnet Mn 2RuxGa\nG. Bon\fglio\nRadboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands\nK. Rode, K. Siewerska, J. Besbas, G. Y. P. Atcheson, P. Stamenov, and J.M.D. Coey\nCRANN, AMBER and School of Physics, Trinity College Dublin, Ireland\nA.V. Kimel, Th. Rasing, and A. Kirilyuk\nRadboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands and\nFELIX Laboratory, Radboud University, Toernooiveld 7c, 6525 ED Nijmegen, The Netherlands\nHere we study both static and time-resolved dynamic magnetic properties of the compensated\nferrimagnet Mn 2RuxGa from room temperature down to 10 K, thus crossing the magnetic compen-\nsation temperature TM. The behaviour is analysed with a model of a simple collinear ferrimagnet\nwith uniaxial anisotropy and site-speci\fc gyromagnetic ratios. We \fnd a maximum zero-applied-\n\feld resonance frequency of \u0018160 GHz and a low intrinsic Gilbert damping \u000b\u00180:02, making it a\nvery attractive candidate for various spintronic applications.\nI. INTRODUCTION\nAntiferromagnets (AFM) and compensated ferrimag-\nnets (FiM) have attracted a lot of attention over the last\ndecade due to their potential use in spin electronics1,2.\nDue to their lack of a net magnetic moment, they are\ninsensitive to external \felds and create no demagnetis-\ning \felds of their own. In addition, their spin dynamics\nreach much higher frequencies than those of their ferro-\nmagnetic (FM) counterparts due to the contribution of\nthe exchange energy in the magnetic free energy3.\nDespite these clear advantages, AFMs are scarcely\nused apart from uni-directional exchange biasing rela-\ntively in spin electronic applications. This is because\nthe lack of net moment also implies that there is no\ndirect way to manipulate their magnetic state. Fur-\nthermore, detecting their magnetic state is also compli-\ncated and is usually possible only by neutron di\u000braction\nmeasurements4, or through interaction with an adjacent\nFM layer5.\nCompensated, metallic FiMs provide an interesting al-\nternative as they combine the high-speed advantages of\nAFMs with those of FMs, namely, the ease to manipu-\nlate their magnetic state. Furthermore, it has been shown\nthat such materials are good candidates for the emerging\n\feld of All-Optical Switching (AOS) in which the mag-\nnetic state is solely controlled by a fast laser pulse6{8.\nA compensated, half-metallic ferrimagnet was \frst en-\nvisaged by van Leuken and de Groot9. In their model\ntwo magnetic ions in crystallographically di\u000berent po-\nsitions couple antiferromagnetically and perfectly com-\npensate each-other, but only one of the two contributes\nto the states at the Fermi energy responsible for elec-\ntronic transport. The \frst experimental realisation of\nthis, Mn 2RuxGa (MRG), was provided by Kurt et al.10.\nMRG crystallises in the XAHeusler structure, space\ngroupF\u001643m, with Mn on the 4 aand 4csites11.\nSubstrate-induced bi-axial strain imposes a slight tetrag-\nonal distortion, which leads to perpendicular magneticanisotropy. Due to the di\u000berent local environment of\nthe two sublattices, the temperature dependence of their\nmagnetic moments di\u000ber, and perfect compensation is\ntherefore obtained at a speci\fc temperature TMthat\ndepends on the Ru concentration xand the degree of\nbiaxial strain. It was previously shown that MRG ex-\nhibits properties usually associated with FMs: a large\nanomalous Hall angle12, that depends only on the mag-\nnetisation of the 4 cmagnetic sublattice13; tunnel magne-\ntoresistance (TMR) of 40 %, a signature of its high spin\npolarisation14, was observed in magnetic tunnel junc-\ntions (MTJs) based on MRG15; and a clear magneto-\noptical Kerr e\u000bect and domain structure, even in the ab-\nsence of a net moment16,17. Strong exchange bias of a\nCoFeB layer by exchange coupling with MRG through\na Hf spacer layer18, as well as single-layer spin-orbit\ntorque19,20showed that MRG combined the qualities of\nFMs and AFMs in spin electronic devices.\nThe spin dynamics in materials where two distinct\nsublattices are subject to di\u000bering internal \felds (ex-\nchange, anisotropy, . . . ) is much richer than that of a\nsimple FM, as previously demonstrated by the obers-\nvation of single-pulse all-optical switching in amorphous\nGdFeCo21,22and very recently in MRG8. Given that the\nmagnetisation of MRG is small, escpecially close to the\ncompensation point, and the related frequency is high,\nnormal ferromagnetic resonance (FMR) spectroscopy is\nunsuited to study their properties. Therefore, we used\nthe all-optical pump-probe technique to characterize the\nresonance frequencies at di\u000berent temperatures in vicin-\nity of the magnetic compensation point. This, together\nwith the simulation of FMR, make it possible to deter-\nmine the e\u000bective g-factors, the anisotropy constants and\ntheir evolution across the compensation point. We found,\nin particular, that our ferrimagnetic half-metallic Heusler\nalloy has resonance frequency up to 160 GHz at zero-\feld\nand a relatively low Gilbert damping.arXiv:1909.09085v1  [cond-mat.mtrl-sci]  19 Sep 20192\nFIG. 1. Net moment measured by magnetometry and coercive\n\feld measured by static Faraday e\u000bect. The upturn of the net\nmoment below T\u001850 K is due to paramagnetic impurities\nin the MgO substrate. TMis indicated by the vertical dotted\nline. As expected the maximum available applied \feld \u00160H=\n7 T is insu\u000ecient to switch the magnetisation close to TM.\nII. EXPERIMENTAL DETAILS\nThin \flm samples of MRG were grown in a `Sham-\nrock' sputter deposition cluster with a base pressure of\n2\u000210\u00008Torr on MgO (001) substrates. Further infor-\nmation on sample deposition can be found elsewhere23.\nThe substrates were kept at 250\u000eC, and a protective\n\u00183 nm layer of aluminium oxide was added at room tem-\nperature. Here we focus on a 53 nm thick sample with\nx= 0:55, leading to TM\u001980 K as determined by SQUID\nmagnetometry using a Quantum Design 5 T MPMS sys-\ntem (see FIG. 1). We are able to study the magneto-\noptical properties both above and below TM.\nThe magnetisation dynamics was investigated using an\nall-optical two-colour pump-probe scheme in a Faraday\ngeometry inside a \u00160Hmax= 7 T superconducting coil-\ncryostat assembly. Both pump and probe were produced\nby a Ti:sapphire femtosecond pulsed laser with a cen-\ntral wavelength of 800 nm, a pulse width of 40 fs and\na repetition rate of 1 kHz. After splitting the beam in\ntwo, the high-intensity one was doubled in frequency by\na BBO crystal (giving \u0015= 400 nm) and then used as\nthe pump while the lower intensity 800 nm beam acted\nas the probe pulse. The time delay between the two was\nadjusted by a mechanical delay stage. The pump was\nthen modulated by a synchronised mechanical chopper\nat 500 Hz to improve the signal to noise ratio by lock-in\ndetection. Both pump and probe beams were linearly\npolarized, and with spot sizes on the sample of 150 µm\nand 70 µm, respectively. The pump pulse hit the sample\nat an incidence angle of \u001910\u000e. After interaction with\nthe sample, we split the probe beam in two orthogonally\npolarized parts using a Wollaston prism and detect the\nchanges in transmission and rotation by calculating the\nFIG. 2. Comparison of hysteresis loops obtained by Faraday,\nAHE, and magnetometry recorded at room temperature. The\ntwo former were recorded with the applied \feld perpendicular\nto the sample surface, while for the latter we show results for\nboth \feld applied parallel and perpendicular to the sample.\nsum and the di\u000berence in intensity of the two signals.\nThe external \feld was applied at 75\u000eto the easy axis of\nmagnetization thus tilting the magnetisation away from\nthe axis. Upon interaction with the pump beam the mag-\nnetisation is momentarily drastically changed24and we\nmonitor its return to the initial con\fguration via remag-\nnetisation and then precession through the time depen-\ndent Faraday e\u000bect on the probe pulse.\nThe static magneto-optical properties were examined\nin the same cryostat/magnet assembly.\nIII. RESULTS & DISCUSSION\nA. Static magnetic properties\nWe \frst focus on the static magnetic properties as\nobserved by the Faraday e\u000bect, and compare them to\nwhat is inferred from magnetometry and the anomalous\nHall e\u000bect. In FIG. 2 we present magnetic hysteresis\nloops as recorded using the three techniques. Due to the\nhalf metallic nature of the sample, the magnetotrans-\nport properties depend only on the 4 csublattice. As the\nmain contribution to the MRG dielectric tensor in the\nvisible and near infrared arises from the Drude tail16,\nboth AHE and Faraday e\u000bect probe essentially the same\nproperties (mainly the spin polarised conduction band of\nMRG), hence we observe overlapping loops for the two\ntechniques. Magnetometry, on the other hand, measures\nthe net moment, or to be precise the small di\u000berence\nbetween two large sublattice moments. The 4 asublat-\ntice, which is insigni\fcant for AHE and Faraday here\ncontributes on equal footing. FIG. 2 shows a clear di\u000ber-\nence in shape between the magnetometry loop and the3\nFIG. 3. Time resolved Faraday e\u000bect recorded at T= 290 K\nin applied \felds ranging from 1 T to 7 T. After the initial\ndemagnetisation seen as a sharp increase in the signal at t\u0018\n0 ps, magnetisation is recovered and followed by precession\naround the e\u000bective \feld until fully damped. The lines are\n\fts to the data. The inset shows the experimental geometry\nfurther detailed in the main text.\nAHE or Faraday loops. We highlight here that the ap-\nparent `soft' contribution that shows switching close to\nzero applied \feld, is not a secondary magnetic phase, but\na signature of the small di\u000berences in the \feld-behaviour\nof the two sublattices. We also note that this behaviour\nis a result of the non-collinear magnetic order of MRG.\nA complete analysis of the dynamic properties therefore\nrequires knowledge of the anisotropy constants on both\nsublattices as well as the (at least) three intra and in-\nter sublattice exchange constants. Such an analysis is\nbeyond the scope of this article, and we limit our anal-\nysis to the simplest model of a single, e\u000bective uniaxial\nanisotropy constant Kuin the exchange approximation\nof the ferrimagnet.\nB. Dynamic properties\nWe now turn to the time-resolved Faraday e\u000bect and\nspin dynamics. Time-resolved Faraday e\u000bect data were\nrecorded at \fve di\u000berent temperatures 10 K, 50 K, 100 K,\n200 K and 290 K, with applied \felds ranging from 1 T to\n7 T.\nFIG. 3 shows the \feld-dependence of the Faraday ef-\nfect as a function of the delay between the pump and\nthe probe pulses, recorded at T= 290 K. Negative de-\nlay indicates the probe is hitting the sample before the\npump. After the initial demagnetisation, the magneti-\nsation recovers and starts precessing around the e\u000bec-\ntive \feld which is determined by the anisotropy and the\napplied \feld. The solid lines in FIG. 3 are \fts to the\ndata to extract the period and the damping of the pre-cession in each case. The \ftting model was an expo-\nnentially damped sinusoid with a phase o\u000bset. We note\nthat the apparent evolution of the amplitude and phase\nwith changing applied magnetic \feld is due to the quasi-\nresonance of the spectrum of the precessional motion\nwith the low-frequency components of the convolution\nbetween the envelope of the probe pulse and the phys-\nical relaxation of the system. The latter include both\nelectron-electron and electron-lattice e\u000bects. A rudimen-\ntary model based on a classical oscillator successfully re-\nproduces the main features of the amplitude and phase\nobserved.\nIn two-sublattice FiMs, the gyromagnetic ratios of the\ntwo sublattices are not necessarily the same. This is par-\nticularly obvious in rare-earth/transition metal alloys,\nand is also the case for MRG despite the two sublat-\ntices being chemically similar; they are both Mn. Due\nto the di\u000berent local environment however, the degree\nof charge transfer for the two di\u000bers. This leads to two\ncharacteristic temperatures, a \frst TMwhere the mag-\nnetic moments compensate, and a second TAwhere the\nangular momenta compensate. It can be shown that for\nthe ferromagnetic mode, the e\u000bective gyromagnetic ratio\n\re\u000bcan then be written25\n\re\u000b=M4c(T)\u0000M4a(T)\nM4c(T)=\r4c\u0000M4a(T)=\r4a(1)\nsubscripti= 4a;4cdenotes sublattice i,Mi(T)\nthe temperature-dependent magnetisation, and \rithe\nsublattice-speci\fc gyromagnetic ratio. \re\u000bis related to\nthe e\u000bective g-factor\nge\u000b=\re\u000bh\n\u0016B(2)\nwherehis the Planck constant and \u0016Bthe Bohr magne-\nton.\nThe frequency of the precession is determined by the\ne\u000bective \feld, which can be inferred from the derivative\nof the magnetic free energy density with respect to M.\nFor an external \feld applied at a given \fxed angle with\nrespect to the easy axis this leads to the Smit-Beljers\nformula26\n!FMR =\re\u000bvuut1\nM2ssin2\u001e\"\n\u000e2E\n\u000e\u00122\u000e2E\n\u000e\u001e2\u0000\u0012\u000e2E\n\u000e\u0012\u000e\u001e\u00132#\n(3)\nwhere\u0012and\u001eare the polar and azimuthal angles of the\nmagnetisation vector, and Ethe magnetic free energy\ndensity\nE=\u0000\u00160H\u0001M+Kusin2\u0012+\u00160M2\nscos2\u0012=2 (4)\nwhere the terms correspond to the Zeeman, anisotropy\nand demagnetising energies, respectively, and Msis the\nnet saturation magnetisation. It should be mentioned\nthat the magnetic anisotropy constant Kuis related to\nM, which is being considered constant in magnitude, via\nKu=\f\u00160M2\ns=2,\fa dimensionless parameter.4\nFIG. 4. Observed precession frequency as a function of the\napplied \feld for various temperatures. The solid lines are \fts\nto the data as described in the main text.\nBased on Eqs. (1) through (4) we \ft our entire data set\nwith\re\u000bandKuas the only free parameters. The exper-\nimental data and the associated \fts are shown as points\nand solid lines in FIG. 4. At all temperatures our simple\nmodel with one e\u000bective gyromagnetic ratio \re\u000band a\nsingle uniaxial anisotropy parameter Kureproduces the\nexperimental data reasonably well. The model systemat-\nically underestimates the resonance frequency for inter-\nmediate \felds, with the point of maximum disagreement\nincreasing with decreasing temperature. We speculate\nthis is due to the use of a simple uniaxial anisotropy in\nthe free energy (see Eq. 4), while the real situation is\nmore likely to be better represented as a sperimagnet. In\nparticular, the non-collinear nature of MRG that leads\nto a deviation from 180\u000eof the angle between the two\nsublattice magnetisations, depending on the applied \feld\nand temperature.\nFrom the \fts in FIG. 4 we infer the values of ge\u000band\nthe anisotropy \feld \u00160Ha=2Ku=Ms. The result is shown\nin FIG. 5. The anisotropy \feld is monotonically increas-\ning with decreasing temperature as the magnetisation\nof the 4csublattice increases in the same temperature\nrange. We highlight here the advantage of determining\nthis \feld through time-resolved magneto-optics as op-\nposed to static magnetometry and optics. Indeed the\nanisotropy \feld as seen by static methods is sensitive to\nthe combination of anisotropy and the netmagnetic mo-\nment, as illustrated in FIG. 1, where the coercive \feld\ndiverges as T!TM. In statics one would expect a di-\nvergence of the anisotropy \feld at the same temperature.\nThe time-resolved methods however distinguish between\nthe net and the sublattice moments, hence better re\rect-\ning the evolution of the intrinsic material properties of\nthe ferrimagnet.\nThe temperature dependence of the anisotropy con-\nstants was a matter for discussion for many years27,28.\nFIG. 5. E\u000bective g-factor,ge\u000b, and the anisotropy \feld\nas determined by time-resolved Faraday e\u000bect. ge\u000b, orange\nsquares, increases from near the free electron value of 2 to 4\njust belowTM, while the anisotropy \feld, blue triangles, in-\ncreases near-linearly with decreasing temperature. A M3\ft,\nred dashes line, of the anisotropy behaviour shows the almost-\nmetallic origin of it, indicating the dominant character of the\n4c sublattice.\nWritten in spherical harmonics the 3 danisotropy can\nbe expressed as, k2Y0\n2(\u0012) +k4Y0\n4(\u0012)29wherek2/\nM(T)3andk4/M(T)10. The experimental measured\nanisotropy is then, K2(T) =ak2(T)+bk4(T), withaand\nbthe contributions of the respective spherical harmonics.\nFIG. 5 shows that a reasonable \ft of our data is ob-\ntained with M(T)3which means, \frst, that the contri-\nbution of the 4thorder harmonic can be neglected, and\nsecond, that the contribution of the TMand 2ndsublat-\ntice is negligible, indicating the dominant character of\nthe 4c sublattice.\nIn addition, we should note here that the high fre-\nquency exchange mode was never observed on our exper-\niments. While far from TMits frequency might be too\nhigh to be observable, in the vicinity of TM, in contrast,\nits frequency is expected to be in the detection range.\nMoreover, given the di\u000berent electronic structure of the\ntwo sublattices, it is expected that the laser pulse should\nselectively excite the sublattice 4c, and therefore lead to\nthe e\u000bective excitation of the exchange mode. We argue\nthat it is the non-collinearity of the sublattices (see sec-\ntion III A) that smears out the coherent precession at\nhigh frequencies.\nThe e\u000bective gyromagnetic ratio, ge\u000b, shows a non-\nmonotonic behaviour. It increases with decreasing Tto-\nwardsTM, reaching a maximum at about 50 K before\ndecreasing again at T= 10 K. We alluded above to\nthe di\u000berence between the magnetic and the angular mo-\nmenta compensation temperatures. We expect that ge\u000b\nreaches a maximum when T=TA30, here between the\nmeasurement at T= 50 K and the magnetic compensa-\ntion temperature TM\u001980 K.5\nFIG. 6. Intrinsic and anisotropic broadening in MRG across\ntheTM. The inset shows the evaluation process of the two\ndamping parameters. A linear \ft is used to evaluate intercept\n(anisotropic broadening) and slope (intrinsic damping) of the\nfrequencies versus the inverse of the decay time. The data\npoint are obtained from the \ft of time-resolved Faraday e\u000bect\nmeasurements (an example is shown in Fig.4).\nFrom XMCD data11, we could estimate spin and or-\nbital moment components of the magnetic moments of\nthe two sublattices, what allowed us to derive the ef-\nfective g-factors for the sublattices as g4a= 2:05 and\ng4c= 2:00. In this case we expect the angular momentum\ncompensation temperature TAto be below TM, opposite\nto what is observed for GdFeCo21. Given this small dif-\nference however, TAandTMare expected to be rather\nclose to each other, consistent with the limited increase\nofge\u000bacross the compensation points.\nWe turn \fnally to the damping of the precessional mo-\ntion of Maround the e\u000bective \feld \u00160He\u000b. Damping is\nusually described via the dimensionless parameter \u000bin\nthe Landau-Lifshiz-Gilbert equation, and it is a measure\nof the dissipation of magnetic energy in the system. In\nthis model, \u000bis a scalar constant and the observed broad-\nening in the time domain is therefore a linear function of\nthe frequency of precession31{33. We infer \u000b0, the total\ndamping, from our \fts of the time-resolved Faraday e\u000bect\nas\u000b0= (\u001cd)\u00001, where\u001cdis the decay time of the \fts. We\nthen, for each temperature, plot \u000b0as a function of the\nobserved frequency and regress the data using a straight\nline \ft. The intrinsic \u000bis the slope of this line, while the\nintercept represents the anisotropic broadening.\nFIG. 6 shows the intrinsic damping \u000band the\nanisotropic broadening as a function of temperature.\nAnisotropic broadening is usually attributed to a vari-\nation of the anisotropy \feld in the region probed by the\nprobe pulse34. For MRG this is due to slight lateral vari-\nations in the Ru content xin the thin \flm sample. Such a\nvariation leads to a variation in e\u000bective TMandTAand\ncan therefore have a large in\ruence on the broadening asa function of temperature. Despite this, the anisotropic\nbroadening is reasonably low in the entire temperature\nrange above TM, and a more likely explanation for its\nrapid increase below TMis that the applied magnetic\n\feld is insu\u000ecient to completely remagnetize the sam-\nple between two pump pulses. As observed in Fig.5, the\nanisotropy \feld reaches almost 4 T at low temperature,\ncomparable to our maximum applied \feld of 7 T. The\nintrinsic damping \u000bis less than 0.02 far from TM, but\nincreases sharply at T= 100 K. We tentatively attribute\nthis to an increasing portion of the available power be-\ning transferred into the high-energy exchange mode, al-\nthough we underline that we have not seen any direct\nevidence of such a mode in any of the experimental data.\nIV. CONCLUSION\nWe have shown that the time-resolved Faraday e\u000bect\nis a powerful tool to determine the spin dynamic proper-\nties in compensated, metallic ferrimagnets. The high spin\npolarisation of MRG enables meaningful Faraday data to\nbe recorded even near TMwhere the net magnetisation\nis vanishingly small, and the dependence of the dynamics\non the sublattice as opposed to the net magnetic prop-\nerties provides a more physical understanding of the ma-\nterial. Furthermore, we \fnd that the ferromagnetic-like\nmode of MRG reaches resonance frequencies as high as\n160 GHz in zero applied \feld, together with a small in-\ntrinsic damping. This value is remarkable if compared\nto well-known materials such as GdFeCo which, at zero\n\feld, resonates at tens of GHz21or [Co/Pt] nmultilay-\ners at 80 GHz35but with higher damping. We should\nhowever stress that, in the presence of strong anisotropy\n\felds, higher frequencies can be reached. Example of that\ncan be found for ferromagnetic Fe/Pt with \u0019280 GHz\n(Ha= 10T)36, and for Heusler-like ferrimagnet (Mn 3Ge\nand Mn 3Ga) with\u0019500 GHz (Ha= 20T)37,38. Never-\ntheless, the examples cited above show a considerably\nhigher intrinsic damping compared to MRG. In addi-\ntion, it was recently shown that MRG exhibits unusu-\nally strong intrinsic spin-orbit torque20. 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Yildirim,\net al. , Applied Physics Letters 109, 032403 (2016)."    },    {        "title": "1204.5342v1.Nonlocal_feedback_in_ferromagnetic_resonance.pdf",        "content": "Nonlocal feedback in ferromagnetic resonance\nThomas Bose and Steffen Trimper\nInstitute of Physics, Martin-Luther-University, D-06099 Halle, Germany\u0003\n(Dated: April 27, 2022)\nAbstract\nFerromagnetic resonance in thin films is analyzed under the influence of spatiotemporal feedback\neffects. The equation of motion for the magnetization dynamics is nonlocal in both space and time\nandincludesisotropic, anisotropicanddipolarenergycontributionsaswellastheconservedGilbert-\nand the non-conserved Bloch-damping. We derive an analytical expression for the peak-to-peak\nlinewidth. It consists of four separate parts originated by Gilbert damping, Bloch-damping, a mixed\nGilbert-Bloch component and a contribution arising from retardation. In an intermediate frequency\nregimetheresultsarecomparablewiththecommonlyusedLandau-Lifshitz-Gilberttheorycombined\nwith two-magnon processes. Retardation effects together with Gilbert damping lead to a linewidth\nthe frequency dependence of which becomes strongly nonlinear. The relevance and the applicability\nof our approach to ferromagnetic resonance experiments is discussed.\nPACS numbers: 76.50.+g; 76.60.Es; 75.70.Ak; 75.40.Gb\n\u0003thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de\n1arXiv:1204.5342v1  [cond-mat.mes-hall]  24 Apr 2012I. INTRODUCTION\nFerromagnetic resonance enables the investigation of spin wave damping in thin or ul-\ntrathin ferromagnetic films. The relevant information is contained in the linewidth of the\nresonance signal [1–3]. Whereas the intrinsic damping included in the Gilbert or Landau-\nLifshitz-Gilbert equation [4, 5], respectively, predicts a linear frequency dependence of the\nlinewidth [6], the extrinsic contributions associated with two-magnon scattering processes\nshow a nonlinear behavior. Theoretically two-magnon scattering was analyzed for the case\nthat the static external field lies in the film plane [7, 8]. The theory was quantitatively\nvalidated by experimental investigations with regard to the film thickness [9]. Later the\napproach was extended to the case of arbitrary angles between the external field and the\nfilm surface [10]. The angular dependence of the linewidth is often modeled by a sum of\ncontributions including angular spreads and internal field inhomogeneities [11]. Among oth-\ners, two-magnon mechanisms were used to explain the experimental observations [12–17]\nwhereas the influence of the size of the inhomogeneity was studied in [18]. As discussed in\n[3, 14] the two-magnon contribution to the linewidth disappears for tipping angles between\nmagnetization and film plane exceeding a critical one \bcrit\nM=\u0019=4. Recently, deviations from\nthis condition were observed comparing experimental data and numerical simulations [17].\nSpin pumping can also contribute to the linewidth as studied theoretically in [19]. How-\never, a superposition of both the Gilbert damping and the two-magnon contribution turned\nout to be in agreement very well with experimental data illustrating the dependence of the\nlinewidth on the frequency [16, 20–23]. Based on these findings it was put into question\nwhether the Landau-Lifshitz-Gilbert equation is an appropriate description for ferromag-\nnetic thin films. The pure Gilbert damping is not able to explain the nonlinear frequency\ndependence of the linewidth when two-magnon scattering processes are operative [3, 24].\nAssuming that damping mechanisms can also lead to a non-conserved spin length a way\nout might be the inclusion of the Bloch equations [25, 26] or the the Landau-Lifshitz-Bloch\nequation [27, 28] into the concept of ferromagnetic resonance.\nAnother aspect is the recent observation [29] that a periodic scattering potential can alter\nthe frequency dependence of the linewidth. The experimental results are not in agreement\nwith those based upon a combination of Gilbert damping and two-magnon scattering. It\nwas found that the linewidth as function of the frequency exhibits a non monotonous be-\n2havior. The authors [29] suggest to reconsider the approach with regard to spin relaxations.\nMoreover, it would be an advantage to derive an expression for the linewidth as a measure\nfor spin damping solely from the equation of motion for the magnetization.\nTaking all those arguments into account it is the aim of this paper to propose a gener-\nalized equation of motion for the magnetization dynamics including both Gilbert damping\nand Bloch terms. The dynamical model allows immediately to get the magnetic susceptibil-\nity as well as the ferromagnetic resonance linewidth which are appropriate for the analysis\nof experimental observations. A further generalization is the implementation of nonlocal\neffects in both space and time. This is achieved by introducing a retardation kernel which\ntakes into account temporal retardation within a characteristic time \u001cand a spatial one\nwith a characteristic scale \u0018. The last one simulates an additional mutual interaction of\nthe magnetic moments in different areas of the film within the retardation length \u0018. Re-\ncently such nonlocal effects were discussed in a complete different context [30]. Notice that\nretardation effects were already investigated for simpler models by means of the Landau-\nLifshitz-Gilbert equation. Here the existence of spin wave solutions were in the focus of the\nconsideration [31]. The expressions obtained for the frequency/damping parameters were\nconverted into linewidths according to the Gilbert contribution which is a linear function\nof the frequency [31, 32]. In the present approach we follow another line. The propagating\npart of the varying magnetization is supplemented by the two damping terms due to Gilbert\nand Bloch, compare Eq. (9). Based on this equation we derive analytical expressions for the\nmagnetic susceptibility, the resonance condition and the ferromagnetic resonance linewidth.\nDue to the superposition of damping and retardation effects the linewidth exhibits a non-\nlinear behavior as function of the frequency. The model is also extended by considering\nthe general case of arbitrary angles between the static external field and the film surface.\nMoreover the model includes several energy contributions as Zeeman and exchange energy\nas well as anisotropy and dipolar interaction. The consequences for ferromagnetic resonance\nexperiments are discussed.\nII. DERIVATION OF THE EQUATION OF MOTION\nIn order to define the geometry considered in the following we adopt the idea presented\nin [10], i.e. we employ two coordinate systems, the xyz-system referring to the film surface\n3ΘMey\nex,eX\nezMS\neZeY\nΘHH0\n/Bullet\n/Bullet\n/BulletξM(z1)\nM(z2)\nM(z3)hrf\nd\nlxlzFIG. 1. (Color online) The geometry referring to the film and the magnetization. Further descrip-\ntion in the text.\nand the XYZ-system which is canted by an angle \u0002Mwith respect to the film plane. The\nsituation for a film of thickness dis sketched in Fig. 1. The angle \u0002Mdescribing the direction\nof the saturation magnetization, aligned with the Z-axis, originates from the static external\nfieldH0which impinges upon the film surface under an angle \u0002H. Therefore, it is more\nconvenient to use the XYZ-system for the magnetization dynamics. As excitation source\nwe consider the radio-frequency (rf) magnetic field hrfpointing into the x= X-direction. It\nshould fulfill the condition hrf\u001cH0. To get the evolution equation of the magnetization\nM(r;t),r= (x;y;z )we have to define the energy of the system. This issue is well described\nin Ref. [10], so we just quote the most important results given there and refer to the cited\nliterature for details. Since we consider the thin film limit one can perform the average along\nthe direction perpendicular to the film, i.e.\nM(rk;t) =1\ndZd=2\n\u0000d=2dyM(r;t); (1)\nwhere rk= (x;0;z)lies in the film plane. In other words the spatial variation of the\nmagnetization across the film thickness dis neglected. The components of the magnetization\npoint into the directions of the XYZ-system and can be written as [33]\nM(rk;t) =MX(rk)eX+MY(rk)eY+\u0012\nMS\u0000M2\nX(rk) +M2\nY(rk)\n2MS\u0013\neZ:(2)\n4Typically the transverse components MX;Yare assumed to be much smaller than the satu-\nration magnetization MS. Remark that terms quadratic in MX;Yin the energy will lead to\nlinear terms in the equation of motion. The total energy of the system can now be expressed\nin terms of the averaged magnetization from Eq. (1) and reads\nH=Hz+Hex+Ha+Hd: (3)\nThe different contributions are the Zeeman energy\nHz=\u0000Z\nd3rH0sin (\u0002 H\u0000\u0002M)MY(rk)\n\u0000Z\nd3rH0cos (\u0002 H\u0000\u0002M)\u0012\nMS\u0000MX(rk)2+MY(rk)2\n2MS\u0013\n;(4)\nthe exchange energy\nHex=D\n2MSZ\nd3r\u0002\nrMX(rk)\u00032+\u0002\nrMY(rk)\u00032; (5)\nthe surface anisotropy energy\nHa=HSMSV\n2sin2(\u0002M) +HS\n2sin(2\u0002 M)Z\nd3rM Y(rk)\n+HS\n2MScos(2\u0002 M)Z\nd3rM Y(rk)2\u0000sin2(\u0002M)Z\nd3rM X(rk)2;(6)\nand the dipolar energy\nHd=2\u0019M2\nSVsin2(\u0002M) +\u0019Z\nd3r\u001a\n2MSsin(2\u0002 M)MY(rk)\n+\u0012dk2\nz\nkksin2(\u0002M)\u0000(dkk\u00002) cos2(\u0002M)\u00002 sin2(\u0002M)\u0013\nMY(rk)2\n+\u0012dk2\nx\nkk\u00002 sin2(\u0002M)\u0013\nMX(rk)2\u00002dkxkz\nkksin(\u0002 M)MX(rk)MY(rk)\u001b\n:(7)\nIn these expressions V=lxlzdis the volume of the film, Ddesignates the exchange stiffness\nandHS/d\u00001represents the uniaxial out-of-plane anisotropy field. If HS<0the easy axis\nis perpendicular to the film surface. The in-plane anisotropy contribution to the energy is\nneglected but it should be appropriate for polycrystalline samples [16]. Moreover kk=jkkj\nis introduced where kk=kxex+kzezis the wave vector of the spin waves parallel to the\nfilm surface. Eqs. (3)-(7) are valid in the thin film limit kkd\u001c1. In order to derive Hdin\nEq. (7) one defines a scalar magnetic potential and has to solve the corresponding boundary\n5value problem inside and outside of the film [34]. As result [10] one gets the expressions in\nEq. (7).\nIn general if the static magnetic field is applied under an arbitrary angle \u0002Hthe mag-\nnetization does not align in parallel, i.e. \u0002M6= \u0002 H. The angle \u0002Mcan be derived from\nthe equilibrium energy Heq=H(MX= 0;MY= 0). Defining the equilibrium free energy\ndensity asfeq(\u0002M) =Heq=Vaccording to Eqs. (3)-(7) one finds the well-known condition\nsin(\u0002 H\u0000\u0002M) =4\u0019M S+HS\n2H0sin(2 \u0002 M) (8)\nby minimizing feqwith respect to \u0002M. We further note that all terms linear in MYin\nEqs. (3)-(7) cancel mutually by applying Eq. (8) as already pointed out in Ref. [10].\nThe energy contributions in Eqs. (3) and the geometric aspects determine the dynamical\nequation for the magnetization. The following generalized form is proposed\n@\n@tM(rk;t) =ZZ\ndr0\nkdt0\u0000(rk\u0000r0\nk;t\u0000t0)(\n\r\u0002\nHeff(r0\nk;t0)\u0002M(r0\nk;t0)\u0003\n+\u000b\u0014\nM(r0\nk;t0)\u0002@\n@t0M(r0\nk;t0)\u0015\n\u00001\nT2M?(r0\nk;t0))\n;(9)\nwhere\r=g\u0016B=~is the absolute value of the gyromagnetic ratio, T2is the transverse\nrelaxation time of the components M?=MXeX+MYeYand\u000bdenotes the dimensionless\nGilbertdampingparameter. Thelatterisoftentransformedinto G=\u000b\rM Srepresentingthe\ncorresponding damping constant in unit s\u00001. The effective magnetic field Heffis related to\nthe energy in Eqs. (3)-(7) by means of variational principles [35], i.e. Heff=\u0000\u000eH=\u000eM+hrf.\nHere the external rf-field hrf(t)is added which drives the system out of equilibrium.\nRegarding the equation of motion presented in Eq. (9) we note that a similar type was\napplied in [12] for the evaluation of ferromagnetic resonance experiments. In this paper\nthe authors made use of a superposition of the Landau-Lifshitz equation and Bloch-like\nrelaxation. Here we have chosen the part which conserves the spin length in the Gilbert form\nand added the non-conserving Bloch term in the same manner. That the combination of\nthesetwodistinctdampingmechanismsissuitablefortheinvestigationofultrathinmagnetic\nfilms was also suggested in [24]. Since the projection of the magnetization onto the Z-axis is\nnot affected by T2this relaxation time characterizes the transfer of energy into the transverse\ncomponents of the magnetization. This damping type is supposed to account for spin-spin\nrelaxation processes such as magnon-magnon scattering [33, 36]. In our ansatz we introduce\n6another possible source of damping by means of the feedback kernel \u0000(rk\u0000r0\nk;t\u0000t0). The\nintroduction of this quantity reflects the assumption that the magnetization M(rk;t2)is\nnot independent of its previous value M(rk;t1)providedt2\u0000t1< \u001c. Here\u001cis a time\nscale where the temporal memory is relevant. In the same manner the spatial feedback\ncontrols the magnetization dynamics significantly on a characteristic length scale \u0018, called\nretardation length. Physically, it seems to be reasonable that the retardation length differs\nnoticeably from zero only in z-direction which is shown in Fig. 1. As illustrated in the figure\nM(x;z1;t)is affected by M(x;z2;t)while M(x;z3;t)is thought to have negligible influence\nonM(x;z1;t)sincejz3\u0000z1j>\u0018. Therefore we choose the following combination of a local\nand a nonlocal part as feedback kernel\n\u0000(rk\u0000r0\nk;t\u0000t0) =\u0000 0\u000e(rk\u0000r0\nk)\u000e(t\u0000t0)\n+\u00000\n4\u0018\u001c\u000e(x\u0000x0) exp\u0014\u0000jz\u0000z0j\n\u0018\u0015\nexp\u0014\u0000(t\u0000t0)\n\u001c\u0015\n; t>t0:(10)\nThe intensity of the spatiotemporal feedback is controlled by the dimensionless retardation\nstrength \u00000. The explicit form in Eq. (10) is chosen in such a manner that the Fourier-\ntransform \u0000(kk;!)!\u00000for\u0018!0and\u001c!0, and in case \u00000= 1the ordinary equation\nof motion for the magnetization is recovered. Further,R\ndrkdt\u0000(rk;t) = \u0000 0<1, i.e. the\nintegral remains finite.\nIII. SUSCEPTIBILITY AND FMR-LINEWIDTH\nIf the rf-driving field, likewise averaged over the film thickness, is applied in X-direction,\ni.e.hrf(rk;t) =hX(rk;t)eX, the Fourier transform of Eq. (9) is written as\n\u0014i!\n\r\u0000(kk;!)+1\n\rT2+H21(kk)\u0015\nMX(kk;!) =\u0000\u0014\nH1(kk) +i\u000b!\n\r\u0015\nMY(kk;!);\n\u0014i!\n\r\u0000(kk;!)+1\n\rT2+H12(kk)\u0015\nMY(kk;!) =\u0014\nH2(kk) +i\u000b!\n\r\u0015\nMX(kk;!)\u0000MShX(kk;!):\n(11)\n7The effective magnetic fields are expressed by\nH1(kk) =H0cos(\u0002 H\u0000\u0002M) + (4\u0019M S+HS) cos(2 \u0002 M)\n+ 2\u0019dkkMS \nk2\nz\nk2\nksin2(\u0002M)\u0000cos2(\u0002M)!\n+Dk2\nk\nH2(kk) =H0cos(\u0002 H\u0000\u0002M)\u0000(4\u0019M S+HS) sin2(\u0002M)\n+ 2\u0019dM Sk2\nx\nkk+Dk2\nk;(12)\nand\nH12(kk) = 2\u0019dM Skxkz\nkksin(\u0002 M) =\u0000H21(kk): (13)\nThe Fourier transform of the kernel yields\n\u0000(kk;!) =\u00000(1 + i!\u001c) + \u0000 1\n2 (1 + i!\u001c)(!2\u001c2\u001c1)'\u00000+ \u0000 1\n2\u0000i\n2\u00001!\u001c;\n\u00001=\u00000\n1 +\f2; \f =\u0018kz;(14)\nwhere the factor 1=2arises from the condition t > t0when performing the Fourier trans-\nformation from time into frequency domain. In Eq. (14) we discarded terms !2\u001c2\u001c1.\nThis condition is fulfilled in experimental realizations. So, it will be turned out later the\nretardation time \u001c\u001810 fs. Because the ferromagnetic resonance frequencies are of the order\n10:::100 GHz one finds!2\u001c2\u001810\u00008:::10\u00006. The retardation parameter \f=\u0018kz, introduced\nin Eq. (14), will be of importance in analyzing the linewidth of the resonance signal. With\nregard to the denominator in \u00001, compare Eq. (14), the parameter \fmay evolve ponderable\ninfluence on the spin wave damping if this quantity cannot be neglected compared to 1.\nAs known from two-magnon scattering the spin wave modes can be degenerated with the\nuniform resonance mode possessing wave vectors kk\u0018105cm\u00001. The retardation length \u0018\nmay be estimated by the size of inhomogeneities or the distance of defects on the film sur-\nface, respectively. Both length scales can be of the order \u001810:::1000 nm, see Refs. [18, 29].\nConsequently the retardation parameter \fcould reach or maybe even exceed the order of 1.\nLet us stress that in case \f= 0,\u001c= 0,\u00000= 1and neglecting the Gilbert damping,\ni.e.\u000b= 0, the spin wave dispersion relation is simply \rp\nH1(kk)H2(kk)\u0000H2\n12(kk). This\nexpression coincides with those ones given in Refs. [7] and [10].\nProceeding the analysis of Eq. (11) by defining the magnetic susceptibility \u001fas\nM\u000b(kk;!) =X\n\f\u001f\u000b\f(kk;!)h\f(kk;!);f\u000b;\fg=fX;Yg;(15)\n8whereh\fplays the role of a small perturbation and the susceptibility \u001f\u000b\fexhibits the\nresponse of the system. Eq. (15) reflects that there appears no dependence on the direction\nofkk.\nSince the rf-driving field is applied along the eX-direction it is sufficient to focus the\nfollowing discussion to the element \u001fXXof the susceptibility tensor. From Eq. (11) we\nconclude\n\u001fXX(kk;!) =MSh\nH1(kk;!) +i\u000b!\n\ri\nh\nH1(kk;!) +i\u000b!\n\rih\nH2(kk;!) +i\u000b!\n\ri\n+h\ni!\n\r\u0000(kk;!)+1\n\rT2i2:(16)\nBecause at ferromagnetic resonance a uniform mode is excited let us set kk= 0in Eqs. (12)-\n(13). Considering the resonance condition we can assume \f=\u0018kz= 0. For reasons men-\ntioned above we have to take \f=\u0018kz6= 0when the linewidth as a measure for spin damping\nis investigated. Physically we suppose that spin waves with non zero waves vectors are not\nexcited at the moment of the ferromagnetic resonance. However such excitations will evolve\nduring the relaxation process. In finding the resonance condition from Eq. (16) it seems to\nbe a reasonable approximation to disregard terms including the retardation time \u001c. Such\nterms give rise to higher order corrections. In the same manner all the contributions orig-\ninated from the damping, characterized by \u000bandT2, are negligible. Let us justify those\napproximation by quantitative estimations. The fields H1,H2and!=\rare supposed to\nrange in a comparable order of magnitude. On the other hand one finds \u000b\u001810\u00003:::10\u00002,\n!T2\u001810\u00002and!\u001c\u001810\u00004. Under these approximations the resonance condition reads\n\u0012!r\n\r\u00132\n= \u00002\n0H1(kk= 0)H2(kk= 0): (17)\nThisresultiswellknownforthecasewithoutretardationwith \u00000= 1. Althoughtheretarda-\ntion time\u001cand the retardation length \u0018are not incorporated in the resonance condition, the\nstrength of the feedback may be important as visible in Eq. (17). Now the consequences for\nthe experimental realization will be discussed. To address this issue the resonance condition\nEq. (17) is rewritten in terms of the resonance field Hr=H0(!=!r)leading to\nHr=1\n2 cos(\u0002 H\u0000\u0002M)8\n<\n:s\n(4\u0019M S+HS)2cos4(\u0002M) +\u00121\n\u000002!r\n\r\u00132\n\u0000(4\u0019M S+HS)(1\u00003 sin2(\u0002M))9\n=\n;:(18)\n9ΘM[deg]\nΘH[deg]Γ0= 0 .7\nΓ0= 1 .0\nΓ0= 1 .3FIG. 2. (Color online) Dependence of the magnetization angle \u0002Mon the angle \u0002Hunder which the\nstatic external field is applied for !r=(2\u0019) = 10 GHz . The parameters are taken from [16]: 4\u0019MS=\n16980 G,HS=\u00003400 G;\r= 0:019 GHz=G.\nThe result is arranged in the in the same manner as done in [16]. The difference is the\noccurrence of the parameter \u00000in the denominator. In [16] the gyromagnetic ratio \rand\nthe sum (4\u0019M S+HS)were obtained from \u0002H-dependent measurements and a fit of the\ndata according to Eq. (18) with \u00000= 1under the inclusion of Eq. (8). If the saturation\nmagnetization can be obtained from other experiments [16] the uniaxial anisotropy field HS\nresults. Thus, assuming \u000006= 1the angular dependence \u0002M(\u0002H)and the fitting parameters\nas well would change. In Fig. 2 we illustrate the angle \u0002M(\u0002H)for different values of \u00000and\na fixed resonance frequency. If \u00000<1the curve is shifted to larger \u0002Mand for \u00000>1to\nsmaller magnetization angles. To produce Fig. 2 we utilized quantitative results presented\nin [16]. They found for Co films grown on GaAs the parameters 4\u0019M S= 16980 G ,HS=\n\u00003400 Gand\r= 0:019 GHz=G. As next example we consider the influence of HSand denote\nH(0)\nS=\u00003400 Gthe anisotropy field for \u00000= 1andH(R)\nSthe anisotropy field for \u000006= 1. The\nabsolute value of their ratio jH(R)\nS=H(0)\nSj, derived from Hr(H(0)\nS;\u00000= 1) =Hr(H(R)\nS;\u000006= 1),\nisdepictedinFig.3forvariousfrequencies. Inthisgraphweassumedthatallotherquantities\nremain fixed. The effect of a varying retardation strength on the anisotropy field can clearly\nbeseen. Thechangeinthesignoftheslopeindicatesthattheanisotropyfield H(R)\nSmayeven\nchange its sign. From here we conclude that the directions of the easy axis and hard axis\nare interchanged. For the frequencies 4 GHzand10 GHzthis result is not observed in the\nrange chosen for \u00000. Moreover, the effects become more pronounced for higher frequencies.\n10/vextendsingle/vextendsingle/vextendsingleH(R)\nS/H(0)\nS/vextendsingle/vextendsingle/vextendsingle\nΓ04 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHzFIG. 3. (Color online) Effect of varying retardation strength on the uniaxial anisotropy field for\nvarious frequencies and \u0002M=\u0019=3.4\u0019MS= 16980 G ,HS=\u00003400 G;\r= 0:019 GHz=G, see [16].\nIn Fig. 3 we consider only a possible alteration of the anisotropy field. Other parameters like\nthe experimentally obtained gyromagnetic ration were unaffected. In general this parameter\nmay also experiences a quantitative change simultaneously with HS.\nLet us proceed by analyzing the susceptibility obtained in Eq. (16). Because the following\ndiscussion is referred to the energy absorption in the film, we investigate the imaginary part\nofthesusceptibility \u001f00\nXX. SinceexperimentallyoftenaLorentziancurvedescribessufficiently\nthe resonance signal we intend to arrange \u001f00\nXXin the form A0=(1 +u2), whereA0is the\nabsolute value of the amplitude and uis a small parameter around zero. The mapping to a\nLorentzian is possible under some assumptions. Because the discussion is concentrated on\nthe vicinity of the resonance we introduce \u000eH=H0\u0000Hr, whereHris the static external\nfield when resonance occurs. Consequently, the fields in Eq. (12) have to be replaced by\nH1;2!H(r)\n1;2+\u000eHcos(\u0002 H\u0000\u0002M). Additionally, we take into account only terms of the order\np\n\u000f\u0015in the final result for the linewidth where f\u000f;\u0015g/f!=\r[\u000b+!\u001c] + 1=(\rT2)g. After a\nlengthy but straightforward calculation we get for \u000eH=H(r)\n1;2\u001c1and using the resonance\ncondition in Eq. (17)\n\u001f00\nXX(!) =A0\n1 +h\nH0\u0000Hr\n\u0001Ti2; A0=MS\n(1 +\u0014) cos(\u0002 H\u0000\u0002M) \u0001T; \u0014=H(r)\n2\nH(r)\n1:(19)\nHere we have introduced the total half-width at half-maximum (HWHM) \u0001Twhich can be\n11brought in the form\n\u0001T=1\ncos(\u0002 H\u0000\u0002M)q\n\u00012\nG+ \u00012\nB+ \u00012\nGB+ \u00012\nR: (20)\nThe HWHM is a superposition of the Gilbert contribution \u0001G, the Bloch contribution \u0001B,\na joint contribution \u0001GBarising from the combination of the Gilbert and Bloch damping\nparts in the equation of motion and the contribution \u0001Rwhich has its origin purely in the\nfeedback mechanisms introduced into the system. The explicit expressions are\n\u0001G=!\n\rs\n\u000b\u0014\n\u000b\u000016p\u0014\n(1 +\u0014)\u00000\u00001!\u001c\n(\u00000+ \u0000 1)3\u0015\n; (21a)\n\u0001B=4 \u00000\n(\u00000+ \u0000 1)p\u0014\n(1 +\u0014)s\n1\n(\rT2)2\u00004 \u00001\n(\u00000+ \u0000 1)2!\n\r!\u001c\n\rT2; (21b)\n\u0001GB=s\n8\u00000\n(\u00000+ \u0000 1)p\u0014\n(1 +\u0014)\u000b!\n\r2T2; (21c)\n\u0001R=8p\u0014\n(1 +\u0014)!\n\r\u00000\u00001!\u001c\n(\u00000+ \u0000 1)3: (21d)\nThe parameter \u00001is defined in Eq. (14). If the expressions under the roots in Eqs. (21a)\nand (21b) are negative we assume that the corresponding process is deactivated and does\nnot contribute to the linewidth \u0001HT. Typically, experiments are evaluated in terms of the\npeak-to-peak linewidth of the derivative d\u001f00\nXX=dH0, denoted as \u0001H\u0011. One gets\n\u0001H\u0011=2p\n3\u0001\u0011; (22)\nwhere the index \u0011stands for G(Gilbert contribution), B(Bloch contribution), GB(joint\nGilbert-Bloch contribution), R(pure retardation contribution) or Tdesignating the total\nlinewidth according to Eq. (20) and Eqs. (21a)-(21d). Obviously these equations reveal a\nstrong nonlinear frequency dependence, which will be discussed in the subsequent section.\nIV. DISCUSSION\nAs indicated in Eqs. (20) - (22) the quantity \u0001H\u0011consists of well separated distinct\ncontributions. Thebehaviorof \u0001H\u0011isshowninFigs.4-6asfunctionofthethreeretardation\nparameters, the strength \u00000, the spatial range \fand the time scale \u001c. In all figures the\nfrequencyf=!=(2\u0019)is used. In Fig. 4 the dependence on the retardation strength \u00000is\n12∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nΓ0∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 4. (Color online) Influence of the retardation strength \u00000on the peak-to-peak linewidth \u0001HT\nfor various frequencies (top graph) and on the single contributions \u0001H\u0011forf= 70 GHz (bottom\ngraph). \u0001B= 0is this frequency region. The parameters are: \u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,\nT2= 5\u000210\u00008s;\u001c= 1:7\u000210\u000014s. The other parameters are 4\u0019MS= 16980 G ,HS=\u00003400 G;\r=\n0:019 GHz=G, compare [16].\nshown. As already observed in Figs. 2 and 3 a small change of \u00000may lead to remarkable\neffects. Hence we vary this parameter in a moderate range 0:5\u0014\u00000\u00142. The peak-to-peak\nlinewidth \u0001HTas function of \u00000remains nearly constant for f= 4 GHz andf= 10 GHz ,\nwhereas for f= 35 GHz a monotonous growth-up is observed. Increasing the frequency\nfurther tof= 50 GHz and70 GHzthe curves offers a pronounced kink. The subsequent\nenhancement is mainly due to the Gilbert damping. In the region of negative slope we\nset\u0001HG(\u00000) = 0, while in that one with a positive slope \u0001HG(\u00000)>0grows and tends\nto2\u000b!=(p\n3\r)for\u00000!1. The other significant contribution \u0001HR, arising from the\nretardation decay, offers likewise a monotonous increase for growing values of the retardation\nparameter \u00000. This behavior is depicted in Fig. 4 for f= 70 GHz . Now let us analyze the\ndependence on the dimensionless retardation length \f=\u0018kz. Because\fis only nonzero if\n13∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nβ∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 5. (Color online) Influence of the dimensionless retardation length \f=\u0018kzon the total\npeak-to-peak linewidth \u0001HTfor various frequencies (top graph) and on the single contributions\n\u0001H\u0011forf= 70 GHz (bottom graph); \u0001B= 0in this range. The parameters are: \u0002H= \u0002 M= 0,\n\u00000= 1:1,\u000b= 0:01,T2= 5\u000210\u00008s;\u001c= 1:7\u000210\u000014s. The other parameters: 4\u0019MS= 16980 G ,\nHS=\u00003400 Gand\r= 0:019 GHz=Gare taken from [16].\nkz6= 0this parameter \u0018accounts the influence of excitations with nonzero wave vector. We\nargue that both nonzero wave vector excitations, those arising from two-magnon scattering\nand those originated from feedback mechanisms, may coincide. Based on the estimation\nin the previous section we consider the relevant interval 10\u00002\u0014\f\u001410. The results are\nshown in Fig.5. Within the range of \fone recognizes that the total peak-to-peak linewidths\n\u0001HTforf= 4 GHz andf= 10 GHz offer no alteration when \fis changed. The plotted\nlinewidths are characterized by a minimum followed by an increase which occurs when \f\nexceeds approximately 1. This behavior is the more accentuated the larger the frequencies\nare. The shape of the curve can be explained by considering the single contributions as\nis visible in the lower part in Fig. 5. While both quantities \u0001HG(\f)and\u0001HR(\f)remain\nconstant for small \f,\u0001HG(\f)tends to a minimum and increases after that. The quantity\n14∆HT[G]4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHz∆Hη[G]\nτ[fs]∆HG\n∆HB\n∆HGB\n∆HR\n∆HTf= 70 GHzFIG. 6. (Color online) Influence of the retardation time \u001con the total peak-to-peak linewidth\n\u0001HTfor various frequencies (top graph) and on the single contributions \u0001H\u0011forf= 70 GHz\n(bottom graph). \u0001B= 0in this region. The parameters are \u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,\nT2= 5\u000210\u00008s;\u00000= 1:1; the other parameters are taken from [16]: 4\u0019MS= 16980 G ,HS=\n\u00003400 G;\r= 0:019 GHz=G.\n\u0001HR(\f)develops a maximum around \f\u00191. Thus, both contributions show nearly opposite\nbehavior. The impact of the characteristic feedback time \u001con the linewidth is illustrated\nin Fig. 6. In this figure a linear time scale is appropriate since there are no significant\neffects in the range 1 fs\u0015\u001c\u00150. The total linewidth \u0001HT(\u001c)is again nearly constant\nforf= 4 GHz andf= 10 GHz . In contrast \u0001HT(\u001c)reveals for higher frequencies two\nregions with differing behavior. The total linewidth decreases until \u0001HG(\u001c)becomes zero.\nAfter that one observes a positive linear slope which is due to the retardation part \u0001HR(\u001c).\nThis linear dependency is recognizable in Eq. (21d), too. Below we will present arguments\nwhy the feedback time \u001cis supposed to be in the interval 0< \u001c < 100 fs. Before let us\nstudy the frequency dependence of the linewidth in more detail. The general shape of the\ntotal linewidth \u0001HT(!)is depicted in Fig. 7. Here both the single contribution to the\n15∆Hη[G]\nf[GHz]∆HG\n∆HB\n∆HGB\n∆HR\n∆HTFIG. 7. (Color online) Frequency dependence of all contributions to the peak-to-peak linewidth for\n\u0002H= \u0002 M= 0,\f= 0:5,\u000b= 0:01,T2= 5\u000210\u00008s,\u001c= 1:7\u000210\u000014sand\u00000= 1:2. Parameters taken\nfrom Ref. [16]: 4\u0019MS= 16980 G ,HS=\u00003400 Gand\r= 0:019 GHz=G. The Bloch contribution\n\u0001HBis shown in the inset.\nlinewidth and the total linewidth are shown. Notice that the total linewidth is not simply\nthe sum of the individual contributions but has to be calculated according to Eq. (20). One\nrealizes that the Bloch contribution \u0001HBis only nonzero for frequencies f\u00146 GHzin the\nexamples shown. Accordingly \u0001HB= 0in Figs. 4-6 (lower parts) since these plots refer to\nf= 70 GHz . The behavior of the Gilbert contribution deviates strongly from the typically\napplied linear frequency dependence. Moreover, the Gilbert contribution will develop a\nmaximum value and eventually it disappears at a certain frequency where the discriminant\nin Eq. (21a) becomes negative. Nevertheless, the total linewidth is a nearly monotonous\nincreasing function of the frequency albeit, as mentioned before, for some combinations of\nthe model parameters there might exist a very small frequency region where \u0001HGreaches\nzero and the slope of \u0001HTbecomes slightly negative. The loss due to the declining Gilbert\npart is nearly compensated or overcompensated by the additional line broadening originated\nbytheretardationpartandthecombinedGilbert-Blochterm. Thelatteroneis \u0001HGB/pf\nand\u0001HR/f2, see Eqs. (21c)-(21d). In the frequency region where \u0001HG= 0only \u0001HGB\nand\u0001HRcontribute to the total linewidth, the shape of the linewidth is mainly dominated\nby\u0001HR. Thispredictionisanewresult. Thebehavior \u0001HR/f2, obtainedinourmodelfor\nhigh frequencies, is in contrast to conventional ferromagnetic resonance including only the\nsum of a Gilbert part linear in frequency and a two-magnon contribution which is saturated\n16at high frequencies. So far, experimentally the frequency ranges from 1 GHzto225 GHz,\nsee [21]. Let us point out that the results presented in Fig. 7 can be adjusted in such a\nmanner that the Gilbert contribution will be inoperative at much higher frequencies by the\nappropriate choice of the model parameters. Due to this fact we suggest an experimental\nverification in more extended frequency ranges. Another aspect is the observation that\nexcitations with a nonzero wave vector might represent one possible retardation mechanism.\nRegarding Eqs. (21a)-(21d) retardation can also influence the linewidth in case kz= 0\n(i.e.\f= 0and\u00001= \u0000 0). Only if\u001c= 0the retardation effects disappear. Therefore let us\nconsider the time domain of retardation and its relation to the Gilbert damping. The Gilbert\ndamping and the attenuation due to retardation can be considered as competing processes.\nSo temporal feedback can cause that the Gilbert contribution disappears. In the same\nsense the Bloch contribution is a further competing damping effect. In this regard temporal\nfeedback has the ability to reverse the dephasing process of spin waves based on Gilbert and\nBloch damping. On the other hand the retardation part \u0001Rin Eq. (21d) is always positive\nfor\u001c > 0. Thus, the retardation itself leads to linewidth broadening in ferromagnetic\nresonance and consequently to spin damping. Whether the magnitude of retardation is able\nto exceed the Gilbert damping depends strongly on the frequency. With other words, the\nfrequency of the magnetic excitation ’decides’ to which damping mechanisms the excitation\nenergyistransferred. Ourcalculationsuggeststhatforsufficienthighfrequenciesretardation\neffects dominate the intrinsic damping behavior. Thus the orientation and the value of the\nmagnetization within the retardation time \u001cplays a major role for the total damping.\nGenerally, experimental data should be fit according to the frequency dependence of the\nlinewidth in terms of Eqs. (20)-(22). To underline this statement we present Fig. 8. In this\ngraph we reproduce some results presented in [7] for the case \u0002H= \u0002 M= 0. To be more\nspecific, we have used Eq. (94) in [7] which accounts for the two-magnon scattering and\nthe parameters given there. As result we find a copy of Fig. 4 in [7] except of the factor\n2=p\n3. Further, we have summed up the conventional Gilbert linewidth /fwith the Gilbert\ndamping parameter \u000b1= 0:003. This superposition yields to the dotted line in Fig. 8. The\nresult is compared with the total linewidth resulting from our retardation model plotted as\nsolid line. To obtain the depicted shape we set the Gilbert damping parameter according\nto the retardation model \u000b2= 0:0075, i.e. to get a similar behavior in the same order of\nmagnitude of \u0001HTwithin both approaches we have to assume that \u000b2is more than twice\n17∆HT[G]\nf[GHz]retardation model\nGilbert+2-magnonFIG. 8. (Color online) Comparison with the two-magnon model. Frequency dependence of the total\npeak-to-peak linewidth \u0001HTfor\u0002H= \u0002 M= 0,\f= 0:5,\u000b1= 0:003,\u000b2= 0:0075,T2= 5\u000210\u00008s,\n\u001c= 1:22\u000210\u000014sand\u00000= 1:2. Parameters taken from [7]: 4\u0019MS= 21000 G ,HS=\u000015000 Gand\nfrom [37]:\r= 0:018 GHz=G(derived from g= 2:09for bulk Fe). The dotted line is a superposition\nof Fig. 4 in [7] reflecting the two-magnon contribution and the Gilbert contribution (denoted as\n\u000b1in the text) linear in the frequency.\nas large compared to \u000b1.\nFinally we discuss briefly the \u0002H-dependence of the linewidth which is shown in Fig. 9.\nIn the upper part of the figure one observes that \u0001HT(\u0002H)exhibits a maximum which is\nshifted towards lower field angles as well as less pronounced for increasing frequencies. The\nlower part of Fig. 9, referring to f= 10 GHz , displays that the main contribution to the total\nlinewidth arises from the Gilbert part \u0001HG. This result for f= 10 GHz is in accordance\nwith the results discussed previously, compare Fig. 7. For higher frequencies the retardation\ncontribution \u0001HRmay exceed the Gilbert part.\nV. CONCLUSIONS\nA detailed study of spatiotemporal feedback effects and intrinsic damping terms offers\nthat both mechanisms become relevant in ferromagnetic resonance. Due to the superposi-\ntion of both effects it results a nonlinear dependence of the total linewidth on the frequency\nwhich is in accordance with experiments. In getting the results the conventional model in-\ncluding Landau-Lifshitz-Gilbert damping is extended by considering additional spatial and\n18linewidth ∆ HT[G]\n4 GHz\n10 GHz\n35 GHz\n50 GHz\n70 GHzlinewidth ∆ Hη[G]\nΘH[deg]∆HB\n∆HR\n∆HGB\n∆HG\n∆HTf= 10 GHzFIG. 9. (Color online) Angular dependence of the total peak-to-peak linewidth \u0001HTfor various\nfrequencies (top graph) and all contributions \u0001H\u0011forf= 10 GHz (bottom graph) with \f= 0:5,\n\u000b= 0:01,T2= 5\u000210\u00008s,\u001c= 1:7\u000210\u000014sand\u00000= 1:1. The parameters are taken from\n[16]: 4\u0019MS= 16980 G ,HS=\u00003400 Gand\r= 0:019 GHz=G.\ntemporal retardation and non-conserved Bloch damping terms. Our analytical approach\nenables us to derive explicit expressions for the resonance condition and the peak-to-peak\nlinewidth. We were able to link our results to such ones well-known from the literature.\nThe resonance condition is affected by the feedback strength \u00000. The spin wave damping is\nlikewise influenced by \u00000but moreover by the characteristic memory time \u001cand the retar-\ndation length \u0018. As expected the retardation gives rise to an additional damping process.\nFurthermore, the complete linewidth offers a nonlinear dependence on the frequency which\nis also triggered by the Gilbert damping. From here we conclude that for sufficient high\nfrequencies the linewidth is dominated by retardation effects. Generally, the contribution of\nthedifferentdampingmechanismstothelinewidthiscomprisedofwellseparatedrateswhich\nare presented in Eqs. (20)-(22). Since each contribution to the linewidth is characterized\nby adjustable parameters it would be very useful to verify our predictions experimentally.\n19Notice that the contributions to the linewidth in Eqs. (20)-(22) depend on the shape of\nthe retardation kernel which is therefore reasonable not only for the theoretical approach\nbut for the experimental verification, too. One cannot exclude that other mechanisms as\nmore-magnon scattering effects, nonlinear interactions, spin-lattice coupling etc. are likewise\nrelevant. Otherwise, we hope that our work stimulates further experimental investigations\nin ferromagnetic resonance.\nWe benefit from valuable discussions about the experimental background with Dr. Khali\nZakeri from the Max-Planck-Institute of Microstructure Physics. 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Lett. 49, 658\n(2000)\n22"    },    {        "title": "1703.03198v3.Material_developments_and_domain_wall_based_nanosecond_scale_switching_process_in_perpendicularly_magnetized_STT_MRAM_cells.pdf",        "content": "Material developments and domain wall based nanosecond-scale switching process in\nperpendicularly magnetized STT-MRAM cells\nThibaut Devolder\u0003and Joo-V on Kim\nCentre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud,\nUniversit ´e Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France\nJ. Swerts, S. Couet, S. Rao, W. Kim, S. Mertens, and G. Kar\nIMEC, Kapeldreef 75, B-3001 Leuven, Belgium\nV . Nikitin\nSAMSUNG Electronics Corporation, 601 McCarthy Blvd Milpitas, CA 95035, USA\nWe investigate the Gilbert damping and the magnetization switching of perpendicularly magnetized FeCoB-\nbased free layers embedded in magnetic tunnel junctions adequate for spin-torque operated magnetic memories.\nWe first study the influence of the boron content in MgO / FeCoB /Ta systems alloys on their Gilbert damping pa-\nrameter after crystallization annealing. Increasing the boron content from 20 to 30% increases the crystallization\ntemperature, thereby postponing the onset of elemental diffusion within the free layer. This reduction of the in-\nterdiffusion of the Ta atoms helps maintaining the Gilbert damping at a low level of 0.009 without any penalty on\nthe anisotropy and the magneto-transport properties up to the 400\u000eC annealing required in CMOS back-end of\nline processing. In addition, we show that dual MgO free layers of composition MgO/FeCoB/Ta/FeCoB/MgO\nhave a substantially lower damping than their MgO/FeCoB/Ta counterparts, reaching damping parameters as\nlow as 0.0039 for a 3 ˚A thick Tantalum spacer. This confirms that the dominant channel of damping is the\npresence of Ta impurities within the FeCoB alloy. On optimized tunnel junctions, we then study the duration of\nthe switching events induced by spin-transfer-torque. We focus on the sub-threshold thermally activated switch-\ning in optimal applied field conditions. From the electrical signatures of the switching, we infer that once the\nnucleation has occurred, the reversal proceeds by a domain wall sweeping though the device at a few 10 m/s.\nThe smaller the device, the faster its switching. We present an analytical model to account for our findings. The\ndomain wall velocity is predicted to scale linearly with the current for devices much larger than the wall width.\nThe wall velocity depends on the Bloch domain wall width, such that the devices with the lowest exchange\nstiffness will be the ones that host the domain walls with the slowest mobilities.\nI. INTRODUCTION\nTunnel magnetoresistance (TMR) and spin transfer torque\n(STT) – the fact that spin-polarized currents manipu-\nlate the magnetization of nanoscale magnets and in par-\nticular magnetic tunnel junction (MTJ) nanopillars – are\nthe basic phenomena underpinning an emerging technol-\nogy called Spin-Transfer-Torque Magnetic Random Access\nMemory (STT-MRAM)1, which combines high endurance,\nlow power requirement2,3, CMOS back-end-of-line (BEOL)\ncompatibility4and potentially large capacity5.\nThe core of an STT-MRAM stack is a magnetic tunnel\njunction composed6of an FeCoB/MgO/FeCoB central block.\nOne of the FeCoB layer is pinned to a high anisotropy syn-\nthetic ferrimagnet to create a fixed reference layer (RL) sys-\ntem while the second FeCoB acts as a free layer (FL). Histor-\nically, the FL is capped with (or deposited on) an amorphous\nmetal such as Ta4,7and more recently capped with a second\nMgO layer to benefit from a second interface anisotropy7–9\nin the so-called ’dual MgO’ configuration. So far, it is un-\nclear whether this benefit of anisotropy can be obtained with-\nout sacrificing the other important properties of the free layer,\nin particular the Gilbert damping.\nIn this paper, we will first tailor the Boron content inside\nthe FeCoB alloy to improve the properties of Ta / FeCoB /\nMgO ’single MgO’ free layers and their resilience to thermal\nannealing. The idea is to postpone the FeCoB crystalliza-tion till the very last stage of the BEOL annealing. Indeed\nmaintaining the amorphous state of FeCoB allows to mini-\nmize the interdiffusion of materials –in our case: tantalum–\nwithin the stack. This interdiffusion is otherwise detrimental\nto the Gilbert damping.\nWe then turn to dual MgO systems comprising a Ta spacer\nlayer in the midst of the FL. This spacer is empirically needed\nto allow proper crystallization and to effectively get perpen-\ndicular magnetic anisotropy (PMA)8,10–14. Unfortunately, the\npresence of heavy elements inside the FeCoB free layer is ex-\npected to alter its damping and to induce some loss of mag-\nnetic moment usually referred as the formation of magneti-\ncally dead layers. We study to what extend the Ta spacer in\nthe dual MgO free layers affects the damping and how this\ndamping compares with the one that can be obtained with sin-\ngle MgO free layers. Once optimized, damping factors as low\nas 0.0039 can be obtained a dual MgO free layer.\nBesides the material issues, the success of STT-MRAM\nalso relies on the capacity to engineer devices in accordance\nwith industry roadmaps concerning speed and miniaturiza-\ntion. To achieve fast switching and design devices accordingly\noptimized, one needs to elucidate the physical mechanism by\nwhich the magnetization switches by STT. Several categories\nof switching modes – macrospin15, domain-wall based16,\nbased on sub-volume nucleation17or based on the spin-wave\namplification18– have been proposed, but single-shot time-\nresolved experimental characterization of the switching patharXiv:1703.03198v3  [cond-mat.mtrl-sci]  4 Sep 20172\nare still scarce19–21. Here we study the nanosecond-scale spin-\ntorque-induced switching in perpendicularly magnetized tun-\nnel junctions with sizes from 50 to 300 nm. Our time-resolved\nexperiments argue for a reversal that happens by the motion\nof a single domain wall, which sweeps through the sample\nat a velocity set by the applied voltage. As a result, the\nswitching duration is proportional to the device length. We\nmodel our finding assuming a single wall moving in a uni-\nform material as a result of spin torque. The wall moves with\na time-averaged velocity that scales with the product of the\nwall width and the ferromagnetic resonance linewidth, such\nthat the devices with the lowest nucleation current densities\nwill be the ones that host the domain walls with the lowest\nmobilities.\nThe paper is split in first a material science part, followed\nby a study of the magnetization reversal dynamics. After a de-\nscription of the samples and the caracterization methods, sec-\ntion II C describes how to choose the optimal Boron content\nin an FeCoB-based free layer for STT-MRAM applications.\nSection II D discusses the benefits of ’dual MgO’ free layers\nwhen compared to ’single MgO’ free layers. Moving to the\nmagnetization switching section, the part III A gathers the de-\nscription of the main properties of the samples and the experi-\nmental methods used to characterize the STT-induced switch-\ning speed. Section III B describes the electrical signatures of\nthe switching mechanism at the nanosecond scale. The latter\nis modeled in section III C in an analytical framework meant\nto clarify the factors that govern the switching speed when the\nreversal involves domain wall motion.\nII. ADVANCED FREE LAYER DESIGNS\nA. Model systems under investigation\nOur objective is to study advanced free layer designs in\nfull STT-MRAM stacks. The stacks were deposited by phys-\nical vapor deposition in a Canon-Anelva EC7800 300 mm\ncluster tool. The MgO tunnel barriers were deposited by\nRF-magnetron sputtering. In dual MgO systems, the top\nMgO layer was fabricated by oxidation of a thin metallic Mg\nfilm. All stacks were post-deposition annealed in a TEL-MSL\nMRT5000 batch furnace in a 1 T perpendicular magnetic field\nfor 30 minutes. Further annealing at 400\u000eC were done in a\nrapid thermal annealing furnace in a N 2atmosphere for a pe-\nriod of 10 minutes.\nWe will focus on several kinds of free layers embod-\nied in state-of-the art bottom-pinned Magnetic Tunnel Junc-\ntions (MTJ) with various reference systems comprising ei-\nther [Co/Ni] and [Co/Pt] based hard layers22,23. Although we\nshall focus here on FLs deposited on [Co/Ni] based synthetic\nantiferromagnet (SAF) reference layers, we have conducted\nthe free layer development also on [Co/Pt] based reference\nlayers. While specific reference layer optimization leads to\nslightly different baseline TMR properties, we have found that\nthe free layer performances were not impacted provided the\nSAF structure is stable with the concerned heat treatment (not\nshown).The first category of samples are the so-called ’single-\nMgO’ free layers. We shall focus on samples with a free\nlayer consists of a 1.4 nm thick Fe 60Co20B20or a 1.6 nm\nthick Fe 52:5Co17:5B30layer sandwiched between the MgO\ntunnel oxide and a Ta (2 nm) metal cap. Note that these\nso-called ”boron 20%” and ”boron 30%” samples have dif-\nferent boron contents but have the same number of Fe+Co\natoms. A sacrificial4Mg layer is deposed before the Ta cap\nto avoid Ta and FeCoB mixing during the deposition, and\navoid the otherwise resulting formation of a dead layer. The\nMg thickness is calibrated so that the Mg is fully sputtered\naway upon cap deposition. This advanced capping method has\nproven to provide improved TMR ratios and lower RA prod-\nucts thanks to an improved surface roughness and a higher\nmagnetic moment4.\nThe second category of free layers are the so-called ’dual\nMgO’ free layers in which the FeCoB layer is sandwiched\nby the MgO tunnel oxide and an MgO cap which concur to\nimprove the magnetic anisotropy. The exact free layer com-\npositions are MgO (1.0 nm) / Fe 60Co20B20(1.1 nm) / spacer\n/ Fe 60Co20B20(0.9 nm) / MgO (0.5 nm). We study shall two\nspacers: a Mg/Ta(3 ˚A) spacer and a Mg/Ta(4 ˚A) spacer, both\ncomprising a sacrificial Mg layer.\nB. Experimental methods used for material quality assessment\nWe studied our samples by current-in-plane tunneling\n(CIPT), vibrating sample magnetometry (VSM) and Vector\nNetwork Ferromagnetic resonance (VNA-FMR)24in out-of-\nplane applied fields. CIPT was performed to extract the tun-\nnel magneto-resistance (TMR) and the resistance-area product\n(RA) of the junction. VSM measurements of the free layer\nminor loops have been used to extract the areal moments. We\nthen use VNA-FMR to identify selectively the properties of\neach subsystem. Our experimental method is explained in\nFig. 1, which gathers some VNAFMR spectra recorded on\noptimized MTJs. The first panel records the permeability of\na single MgO MTJ in the ffield-frequencygparameter space.\nWe systematically investigated a sufficiently large parameter\nspace to detect 4 different modes whose spectral characters\ncan be used to index them22. Three of the modes belong to the\nreference system that comprises 3 magnetic blocks coupled\nby interlayer exchange coupling through Ru and Ta spacers\nas usually done22,23; the properties of these 3 modes are inde-\npendent from the nature of the free layer. While we are not\npresently interested in analyzing the modes of the fixed sys-\ntem – thorough analyses can be found in ref.22,23– we empha-\nsize that it is necessary to detect all modes to unambiguously\nidentify the one belonging to the free layer, in order to study\nit separately. The free layer modes are the ones having V-\nshaped frequency versus field curves [Fig. 1(a)], whose slope\nchanges at the free layer coercivity. in each sample, the free\nlayer modes showed an asymmetric Lorentzian dispersion for\nthe real part of the permeability and a symmetric Lorentzian\ndispersion for the imaginary part [see the examples Fig. 1(b,\nc)]. As we found no signature of the two-layer nature of the\ndual MgO free layers, we modeled each free layer as a sin-3\nSingle MgOfree layerModes of the reference layers\nDual MgOTa spacer\u0000f2f=0.006\u0000f2f=0.016Contrastx 10Permeabilitymap\n↵=12@\u0000f@f=0.0039\nFIG. 1. (Color online). Examples of MTJ dynamical properties to\nillustrate the method of analysis. (a) Microwave permeability versus\nincreasing out-of-plane field and frequency for an MTJ with a sin-\ngle MgO free layer after an annealing of 300\u000eC. Note that the scale\nof the permeability was increased by a factor of 10 above 58 GHz\nfor a better contrast. The apparent vertical bars are the eigenmode\nfrequency jumps at the different switching fields of the MTJ. (b)\nReal and imaginary parts of the experimental (symbols) and modeled\n(lines) permeability for an out-of-plane field of 1.54 T for the same\nMTJ. The model is for an effective linewidth \u0001f=(2f) = 0:016,\nwhich includes both the Gilbert damping and a contribution from the\nsample inhomogeneity. (c) Same but for a dual MgO free layer based\non a 3 ˚A Ta spacer, modeled with \u0001f=(2f) = 0:006. (d) Cross\nsymbols: FMR half frequency linewidth versus FMR frequency for\na dual MgO free layer based on a 3 ˚A Ta spacer. The line is a guide\nto the eye corresponding to a Gilbert damping of 0.0039.\nglemacrospin, disregarding whether it was a single MgO or a\ndual MgO free layer.\nFMR frequency versus field fits [see one example in\nfig. 2(c)] were used to get the effective anisotropy fields\nHk\u0000Msof all free layers25. The curve slopes are \r0, where\n\r0= 230 kHz.m/A is the gyromagnetic factor \rmultiplied\nby the vacuum permeability \u00160. It was consistent with a spec-\ntroscopic splitting Land ´e factor ofg\u00192:08. Damping analy-\nsis was conducted as follows: the free layer composition can\nyield noticeable differences in the FMR linewidths [see for in-\nstance Fig. 1(b) and (c)]. To understand these differences, we\nsystematically separated the Gilbert damping contribution to\nthe linewidth from the contribution of the sample’s inhomo-\ngeneity using standard VNA-FMR modeling25. This is doneby plotting the half FMR linewidth \u0001f=2versus FMR fre-\nquencyfFMR [see one example in Fig. 1(d)]. The Gilbert\ndamping is the curve slope and the line broadening arising\nfrom the inhomogeneity of the effective field within the free\nlayer is the zero frequency intercept1\n2\r0\u0001fjf=0) of the curve.\nC. Boron content and Gilbert damping upon annealing of\nsingle MgO free layers\nDesigning advanced free layer in STT-MRAM stacks re-\nquires to minimize the Gilbert damping of the used raw ma-\nterial. In Ta/FeCoB/MgO ’single MgO’ free layers made of\namorphous FeCoB alloys or made of FeCoB that has been\njust crystallized, a damping of 0.008 to 0.011 can be found\ntypically19,25. (Note that lower values can be obtained but for\nthicknesses and anisotropies that are not adequate for spin-\ntorque application26). The damping of Ta/FeCoB/MgO sys-\ntems generally degrades substantially when further annealing\nthe already crystallized state27. Let us emphasize than even in\nthe best cases26, the damping of FeCoB based free layers are\nstill very substantially above the values of 0.002 or slightly\nless than can be obtained on FeCo of Fe bcc perfect single\ncrystals28,29.\nThere are thus potentially opportunities to improve the\ndamping of free layers by material engineering. We illustrate\nthis in fig. 2 in which we show that a simple increase of the\nBoron content is efficient to maintain the damping unaffected,\neven upon annealing at 400\u000eC in a single MgO free layer. In-\ndeed starting from Ta/FeCoB/MgO ’single MgO’ free layers\nsharing the same damping of 0.009 after annealing at 300\u000eC\n(not shown), an additional 100\u000eC yields\u000b= 0:015for the\nfree layer with 20% of boron, while the boron 30% free lay-\ners keep a damping of \u000b= 0:009[see fig. 2(d)]. Meanwhile\nthe anisotropies of these two free layers remain perpendicu-\nlar [fig. 2(c)] with \u00160(Hk\u0000Ms)being 0.27 and 0.17 T, re-\nspectively, after annealing at 400\u000eC. Let us comment on this\ndifference of damping.\nTwo mechanisms can yield to extra damping: spin-\npumping30and spin-flip impurity scattering of the conduc-\ntion electrons by a spin-orbit process31. Tantalum is known\nto be a poor spin-sink material as this early transition metal\nhas practically no delectrons and therefore its spin-pumping\ncontribution to the damping of an adjacent magnetic layer is\nweak30. We expect a spin pumping contribution to the damp-\ning of Ta (2 nm) / FeCoB (1.4 nm) / MgO ’single MgO’\nfree layers that compares with for instance that measured by\nMizukami et al. on Ta (3 nm) / Fe 20Ni80(3 nm) which was\nundetectable32since below 0.0001; we therefore expect that\nthe spin-pumping contribution to the total free layer damping\nis too negligible to account for the differences observed be-\ntween a free layer and the corresponding perfect single crys-\ntals. The main remaining contribution to the damping is the\nmagnon scattering by the paramagnetic impurities within the\nFeCoB material33. Indeed the Ta atoms within an FeCoB layer\nare paramagnetic impurities that contribute to the damping ac-\ncording to their concentration like any paramagnetic dopant;\nhowever the effect with Ta is particularly large34as Fe and Co4\n/s50/s50 /s50/s52 /s50/s54/s48 /s49\n/s49/s48/s50/s48/s51/s48\n/s49/s48 /s50/s48 /s51/s48/s49/s48 /s50/s48 /s51/s48\n/s48/s46/s51/s48/s46/s54\n/s40/s100/s41/s66/s111/s114/s111/s110/s32/s51/s48/s37\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41/s40/s99/s41\n/s40/s98/s41/s40/s97/s41/s66/s111/s114/s111/s110/s32/s50/s48/s37/s84/s114/s97/s110/s115/s118/s101/s114/s115/s101/s32/s112/s101/s114/s109/s101/s97/s98/s105/s108/s105/s116/s121\n/s32/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s32/s66/s32/s40/s84/s41\n/s72/s97/s108/s102/s32/s108/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s71/s72/s122/s41\n/s32/s84\n/s97/s110/s110/s101/s97/s108/s32/s61/s32/s52/s48/s48/s176/s67\nFIG. 2. (Color online). Properties of single MgO free layers after\nannealing at 400\u000eC. (a) and (b): Real part (narrow lines) and imagi-\nnary part (bold lines) of the free layer permeability in a field of 0.7 T.\nThe lines are macrospin fits. (c) Ferromagnetic resonance frequency\nversus field curves. (d) Half linewidth versus FMR frequencies. The\nlines have slopes of \u000b= 0:009(red, B30%) and \u000b= 0:015(green,\nB20%)\natoms in direct contact with Ta atoms loose part of their mo-\nment and get an extra paramagnetic character, an effect usu-\nally referred as a ”magnetically dead layer”. Qualitatively, the\nTa atoms in the inner structure of the free layer degrade its\ndamping.\nAs the cap of Ta / FeCoB / MgO ’single MgO’ free lay-\ners contain many Ta atoms available for intermixing, a strong\ndegradation of the damping can be obtained in single MgO\nsystems when interdiffusion occurs. To prevent interdiffu-\nsion, we used the following strategy. Amorphous materials\n(including the glassy metals like FeCoB) are known to be ef-\nficient diffusion barriers, as they exhibit atom mobilities that\nare much smaller than their crystalline counterparts. To avoid\nthe diffusion of Ta atoms to the inner part of the FeCoB free\nlayer, a straightforward way is to maintain the FeCoB in an\namorphous state as long as possible during the annealing.\nIn metal-metalloid glasses, the crystallization temperature in-\ncreases with the metalloid content. In our FeCoB free lay-\ners, we find crystallization temperatures of 200, 300, 340 and\n375\u000eC for boron contents of respectively 10%, 20%, 25% and\n30%. Increasing the boron content in FeCo alloys is a way\nto conveniently increase the crystallization temperature and\nthus preserve a low damping. However since to obtain large\nTMR requires the FeCoB to be crystalline35,36, one should en-\ngineer the boron content such that the crystallization tempera-\nture matches with that used in the CMOS final BEOL anneal-\ning of 400\u000eC. In practice, we have found that this situation\nis better approached with a boron content of 30% than 0% to\n25%.D. Gilbert damping in single MgO and dual MgO free layers\nIn our search to further improve the free layers for STT-\nMRAM applications, we have compared the damping of op-\ntimized ’single MgO’ and optimized ’dual MgO’ free layers.\nFor a fair comparison, we first compare samples made from\nFeCoB with the same boron content of 20% and the same\n300\u000eC annealing treatement. From Fig. 1(b) and (c), there\nis a striking improvement of the FMR linewidths when pass-\ning from a single MgO to a dual MgO free layer. To discuss\nthis difference in linewidth, we have separated the Gilbert\ndamping contribution to the linewidth from the contribution\nof the sample’s inhomogeneity. We find that dual MgO sys-\ntems have systematically a substantially lower damping than\nsingle MgO free layers which confirms the trends indepen-\ndently observed by other authors9. Damping values as low\nas low as 0.0039\u00060:005were obtained in Ta 3 ˚A-spacer dual\nMgO stacks [Fig. 1(d)] after 300\u000eC annealing. Samples with\na thicker Ta spacer exhibit an increased damping (not shown).\nThis trend –lower damping in dual MgO systems –is main-\ntained after 400\u000eC annealing; for that annealing temperature,\nthe best damping are obtained for a slightly different internal\nconfiguration of the dual MgO free layer. Indeed a damping of\n0.0048 was obtained (not shown) in MgO / Fe 52:5Co17:5B30\n(1.4 nm) / Ta (0.2 nm) / Fe 52:5Co17:5B30(0.8 nm). This\nshould be compared with that the corresponding single MgO\nfree layer which had a damping of 0.009 for the same an-\nnealing condition [Fig. 2(d)]. This finding is consistent with\nthe results obtained on the single MgO free layer if we as-\nsume that the Ta impurities within an FeCoB layer contribute\nto the damping according to their concentration. Somehow,\nthe number of Tantalum atoms in the initial structure of the\nfree layer sets an upper bound for the maximum degradation\nof the damping upon its interdiffusion that can occur during\nthe annealing. Notably, the single MgO free layers contain\nmuch more Ta atoms (i.e. 2 nm compared to 0.2 to 0.4 nm)\navailable for intermixing: not only the initial number of Ta\nimpurities within the FeCoB layer directly after deposition is\nlarger in the case of single MgO free layer, but in addition a\nmuch stronger degradation of the damping can be obtained in\nsingle MgO systems when interdiffusion occurs, in line with\nour experimental findings. This interpretation – the dominant\nsource of damping is the Ta content – is further strengthened\nby the fact that the thickness of the Ta spacer strongly impacts\nthe damping in dual MgO free layers.\nLet us now study the spin-torque induced switching process\nin nanopillars processed from optimized MTJs.\nIII. SPIN-TORQUE INDUCED SWITCHING PROCESS\nA. Sample and methods for the switching experiments\nIn this section we use two kinds of perpendicularly magne-\ntized MTJ: a ’single MgO’ and a ’dual MgO’ free layer whose\nproperties are detailed respectively in ref.19and20. Note that\nthe devices are made from stacks that do not include all the\nlatest material improvement described in the previous sec-5\ntions and underwent only moderate annealing processes of\n300\u000eC. The ’single MgO’ free layer samples include a 1.4 nm\nFeCoB 20%free layer and a Co/Pt based reference synthetic\nantiferromagnet. Its most significant properties include19an\nareal moment of Mst\u00191:54mA, a damping of 0.01, an ef-\nfective anisotropy field of 0.38 T, a TMR of 150% . The ’dual\nMgO’ devices are made from tunnel junctions with a 2.2 nm\nthick FeCoB-based free layer and a hard reference system also\nbased on a well compensated [Co/Pt]-based synthetic antifer-\nromagnet. The perpendicular anisotropy of the (much thicker)\nfree layer is ensured by a dual MgO encapsulation and an iron-\nrich composition. After annealing, the free layer has an areal\nmoment of Mst\u00191:8mA and an effective perpendicular\nanisotropy field 0.33 T. Before pattering, standard ferromag-\nnetic resonance measurements indicated a Gilbert damping\nparameter of the free layer being \u000b= 0:008. Depending on\nthe size of the patterned device, the tunnel magnetoresistance\n(TMR) is 220 to 250%.\nBoth types of MTJs were etched into pillars of various size\nand shapes, including circles from sub-50 nm diameters to 250\nnm and elongated rectangles with aspect ratio of 2 and foot-\nprint up to 150\u0002300 nm. The MTJs are inserted in series\nbetween coplanar electrodes [Fig. 3(a)] using a device integra-\ntion scheme that minimizes the parasitic parallel capacitance\nso as to ensure an electrical bandwidth in the GHz range. The\njunction properties19,20are such that the quasi-static switching\nthresholds are typically 500 mV . Spin-wave spectroscopy ex-\nperiments similar to ref.37indicated that the main difference\nbetween the two sample series lies in the FL intralayer ex-\nchange stiffness. It is A= 8\u00009pJ/m in the 2.2 nm thick\ndual MgO free layers of the samples of ref.20and more usual\n(\u001920pJ/m) in the 1.4 nm thick ’single MgO’ free layers of\nthe samples of ref.19.\nFor switching experiments, the sample were characterized\nin a set-up whose essential features are described in Fig. 3(a):\na slow triangular voltage ramp is applied to the sample in se-\nries with a 50 \n oscilloscope. As the device impedance is\nmuch larger than the input impedance of the oscilloscope, we\ncan consider that the switching happens at an applied voltage\nthat is constant during the switching. We capture the elec-\ntrical signature of magnetization switching by measuring the\ncurrent delivered to the input of the oscilloscope [Fig. 3(b)].\nWhen averaging several switching events [as conducted in\nFig. 3(b)], the stochasticity of the switching voltage induces\nsome rounding of the electrical signature of the transition.\nHowever, the single shot switching events can also be cap-\ntured (Fig. 4-5). In that case, we define the time origins in\nthe switching as the time at which a perceivable change of the\nresistance suddenly happens (see the convention in Fig. 5).\nThis will be referred hereafter as the ”nucleation” instant.\nThis measurement procedure – slow voltage ramp and time-\nresolved current – entails that the studied reversal regime is\nthe sub-threshold thermally activated reversal switching. This\nsub-threshold thermally-activated switching regime is not di-\nrectly relevant to understand the switching dynamics in mem-\nory devices in which the switching will be forced by short\npulses of substantially higher voltage21. However elucidat-\ning the sub-threshold switching dynamics is of direct inter-\n50 ΩMTJVoltagebias\nOscilloscope50 Ω(a)(b)FIG. 3. (Color online). (a) Sketch of the experimental set-up. Mea-\nsurement procedure: the device is biased with a triangular kHz-rate\nvoltage (green) and the current (red) is monitored by a fast oscillo-\nscope connected in series. (b) The switching transitions are seen as\nabrupt changes of the current (red) followed by a change of the cur-\nrent slope. The resistance (blue) can be computed from the voltage-\nto-current ratio when the current is sufficiently non-zero. In this fig-\nure, the displayed currents and resistances are the averages over 1000\nevents for a 250 nm device with a dual MgO free layer of thickness\n2.2 nm and a weak exchange stiffness.\nest for the quantitative understanding of read disturb errors\nthat may happen at applied voltages much below the writing\npulses. Note finally that sending directly the current to the os-\ncilloscope has a drawback: the current decreases as the MTJ\narea such that the signal-to-noise ratio of our measurement\ndegrades substantially for small device areas (Fig. 5). As a\nresult, the comfortable signal-to-noise ratio allows for a very\nprecise determination of the onset of the reversal in large de-\nvices, but the precision degrades substantially to circa 500 ps\nfor the smallest (40 nm) investigated devices.\nB. Switching results\nIn samples whose (i) reference layers are sufficiently fixed\nto ensure the absence of back-hopping19and (ii) in which the\nstray field from the reference layer is rather uniform20, opti-\nmized compensation of the stray field of the reference layers\nleads to a STT-induced switching with a simple and abrupt\nelectrical signature [Fig. 4(a)]. If examined with a better time\nresolution, the switching event [Fig. 4(b)] appears to induce\na monotonic ramp-like evolution of the device conductance.\nFor a given MTJ stack, the switching voltage is practically in-\ndependent from the device size and shape in our interval of\ninvestigated sizes (not shown). This finding is consistent with\nthe consensual conclusion that the switching energy barrier\nis almost independent from the device area38,39for device ar-\neas above 50 nm. In spite of this quasi-independence of the\nswitching voltage and the device size, the switching duration\nwas found to strongly depend on device size (Fig. 5); we have\nfound that smaller devices switch faster, and the trend is that\nthe switching duration correlates linearly with the longest di-\nmension of the device. This is shown in Fig. 5: 40 nm devices\nswitch in typically 2 to 3 ns whereas devices that are 6 times6\n33PAP\nFIG. 4. (Color online). Single-shot time-resolved absolute value\nof the current during a spin-torque induced switching for parallel to\nantiparallel switching for a circular device of diameter 250 nm made\nwith a weak exchange stiffness, dual MgO 2.2 nm thick free layer.\n(a) Two microsecond long time trace, illustrating that the switching\nis complete, free of back-hopping phenomena, and occurs between\ntwo microwave quiet states. (b) 30 ns long time trace illustrating\nthe regular monotonic change of the device conductance during the\nswitching.\nlarger switch in 10 to 15 ns.\nSuch a reversal path can be interpreted this way: once a do-\nmain is nucleated at one edge of the device, the domain wall\nsweeps irreversibly through the system at a velocity set by\nthe applied voltage [sketch in Fig. 4(b)]. The average domain\nwall speed is then about 20 nm/ns for the low-exchange-free-\nlayers of ref.20. The other devices (not shown but described in\nref.19) based on a ’single MgO’ free layer with a more bulk-\nlike exchange switch with a substantially higher apparent do-\nmain wall velocity, reaching 40 m/s.\nC. Switching Model: domain wall-based dynamics\nTo model the switching, we assume that there is a domain\nwall (DW) which lies at a position qand moves along the\nlongest axis xof the device. The domain wall is assumed\nto be straight along the ydirection, as sketched in Fig. 4(b).\nWe describe the wall in the so-called 1D model40: the wall is\nassumed to be a rigid object of fixed width \u0019\u000epresenting a\ntilt\u001eof its magnetization in the device plane; by convention\n\u001e= 0is for a wall magnetization along x, i.e. a N ´eel wall.\n-10-5051015202501002000510m\nodelduration (ns)switchingd\nevice diam. (nm) \n90 nm \n60 nm 150 nm4\n0 nm80 nmCurrent (norm.)T\nime after nucleation (ns)250 nm  FIG. 5. (Color online). Single-shot time-resolved conductance traces\nfor parallel to antiparallel switching events occurring at at -0.5 V\nfor circular devices of various diameters. The curves are for the de-\nvices whose dual MgO free layer has a thickness of 2.2 nm and has a\nweak exchange stiffness. The curves have been vertically offset and\nvertically normalized to ease the comparison. The time origins and\nswitching durations are chosen at the perceivable onset and end of\nthe conductance change: they are defined by fitting the experimental\nconductance traces by 3 segments (see the sketch labelled ”model”).\nInset: duration of the switching events versus free layer diameter\n(symbols) and linear fit thereof with an inverse slope of 20 m/s.\nThe local current density at the domain wall position is\nwrittenj. The wall is subjected to an out-of-plane field Hz\nassumed to vary slowly in space at the scale of the DW width.\njis assumed to transfer p\u00191spin per electron to the DW by\na pure Slonczewski-like STT. We define\n\u001b=~\n2e\r0\n\u00160MSt(1)\nas the spin-transfer efficiency in unit such that \u001bjis a fre-\nquency. With typical FeCoB parameters, i.e. magnetization\nMs2[1:1;1:4]MA/m and free layer thickness t2[1:4;2:2]\nnm, we have \u001b2[0:018;0:036] Hz / (A/m2) where the low-\nest value corresponds to the largest areal moment Mst. With\nswitching current density of the order of 4\u00021010A/m2, this\nyields\u001bjdcbetween 0.72 and 1.4 GHz.\nFollowing ref.41, the wall position qand and wall tilt \u001eare7\nlinked by the two differential equations:\n_\u001e+\u000b\n\u000e_q=\r0Hz; (2)\n_q\n\u000e\u0000\u000b_\u001e=\u001bjdc+\r0HDW\n2sin(2\u001e) (3)\nin which\u0019\u000eis the width of a Bloch domain wall in an ultra-\nthin film, with \u000e2= 2A=(\u00160MsHeff\nk)whereAis the exchange\nstiffness. A wall parameter \u000e= 12 nm will be assumed for\nthe normal exchange 1.4 nm free layer from various estimates\nincluding ref.37for the exchange stiffness and ref.22for the\nanisotropy of the free layer. The domain wall stiffness field42\nHDWis the in-plane field that one would need to apply to have\nthe wall transformed from a Bloch wall to a N ´eel wall. As it\nexpresses the in-plane demagnetization field within the wall,\nit depends on the wall width \u0019\u000eand on the wall length when\nthe finite size of the device constrains the wall dimensions.\nUsing42, the domain wall stiffness field can be estimated\nto be at the most 20 mT in our devices. In circular devices,\nthe domain wall has to elongate upon its propagation38such\nthat the domain wall stiffness field HDWdepends in princi-\nple on the DW position. It should be maximal when the wall\nis along the diameter of the free layer. However we will see\nthatHDWis not the main determinant of the dynamics. In-\ndeed in the absence of stray field and current, the Walker field\nHWalker is proportional to the domain wall stiffness field times\nthe damping parameter, i.e. HWalker =\u000bH DW=2. As the sam-\nples required for STT switching are typically made of low\ndamped materials with \u000b < 0:01, the Walker field is very\nsmall and likely to be smaller than the stray fields emanating\nfrom either the reference layers or the applied field. This very\nsmall Walker field has implications: in practice as soon as\nthere is some field of some applied current, any domain wall\nin the free layer is bound to move in the Walker regime and\nto make the back-and-forth oscillatory movements that are in-\nherent to this regime. The DW oscillates at a generally fast\n(GHz) frequency43such that only the time-averaged velocity\nmatters to define how much it effectively advances.\nTo see quantitatively the effect of a constant current on the\ndomain wall dynamics, we assume that the sample is invari-\nant along the domain wall propagation direction (x) (like in\nan hypothetical stripe-shaped sample). Solving numerically\nEq. 2 and 3, we find that the Walker regime is maintained for\njdc6= 0(not shown). Two points are worth noticing:\nThe time-averaged domain wall velocity h_qivaries linearly\nwith the applied current density. When in the Walker regime,\nthe current effect can be understood from Eq. 3. Indeed the\nsin(2\u001e)term essentially averages out in a time integration as\n\u001eis periodic, and the term \u000b_\u001eis neligible, such that the time-\naveraged wall velocity reduces to:\nh_qi\u0019\u000e\u001bj dc (4)\nFor\u000e= 12 nm and\u001bjin the range of 1.4 GHz at the switching\nvoltage for the bulk-like exchange stiffness sample with free\nlayer thcikness 1.4 nm, the previous equation would predict atime-averaged domain wall velocity of 17 m/s (or nm/ns) dur-\ning the switching. More compact domain walls are expected\nfor the samples with a weaker exchange stiffness; the twice\nlower\u001bj\u00190:72GHz related to the larger thickness would\nreinforce this trend to a much a lower domain wall velocity (9\nm/s for our material parameters estimates). This expectation\ncompares qualitatively well with our experimental findings of\nslower walls in weakly exchanged materials (Fig. 5).\nWe wish to emphasize that Eq. 4 can be misleading regard-\ning the role of damping. Indeed a too quick look at Eq. 4 could\nlet people wrongly conclude the domain wall velocity is es-\nsentially set by the areal moment Mstand that the wall veloc-\nity under STT from a current perpendicular to the plane (CPP\ncurrent) is independent from the damping factor (see Eq. 1).\nHowever this is not the case as the switching current jdcis\na sweep-rate-dependent and temperature-determined fraction\n\u00112[1\n2;1]of the zero temperature instability current jc0of a\nmacrospin in the parallel state, which reads15,44:\njc0=\u000b4e\n~1 +p2\np\u00160MstHeff\nk\n2(5)\nwherep\u00191is an effective spin polarization.\nUsing Eq. 1, 4 and 5, the time-averaged wall velocity at the\npractical switching voltage is:\nh_qi\u0019\u000b\u000e\r 0Heff\nk\u0011 (6)\nThis expression indicates that the samples performing best\nin term of switching current (minimal damping and easy nu-\ncleation thanks to a small exchange) will host domain walls\nthat are inherently slow when pushed by the CPP current in\nthe Walker regime. The domain wall speed scales with the\ndomain wall width, which may be the reason why the low\nexchange stiffness samples host domain walls that are experi-\nmentally slower.\nTo summarize, once nucleated at the instability of the uni-\nformly magnetized state at jdc=\u0011jc0, the domain wall flows\nin a Walker regime through the device. The switching dura-\ntion varies thus simply with the inverse current:\n\u001cswitch =L\n\u000e\u001bj dc\u0019L\n\u000e\u00021\n\u000b\r0Heff\nk\u00021\n\u0011(7)\nLet us comment on this equation which is the main con-\nclusion of this section. The underlying simplifications are:\n(i) a rigid wall (ii) that does not sense the sample’s edges\n(iii) that moves at a speed equal to its average velocity in the\nWalker regime (iv) at a switching voltage that is independent\nfrom the sample geometry. Under these assumptions, the du-\nration of the switching scales with the length Lof the sam-\nple, as observed experimentally. It also scales with the inverse\nof the zero-field ferromagnetic resonance linewith 2\u000b\r0Heff\nk.\nThe practical switching voltage is below the zero temperature\nmacrospin switching voltage by a factor \u0011, which gathers the\neffect of the thermal activation and of the sweeping rate of the\napplied voltage45.\u0011\u00191=2for quasi-static experiments like\nreported here and \u0011!1for experiments in which the voltage\nrise timeVmax=_Vis short enough compared to the switching\nduration (Eq. 7).8\nIV . SUMMARY AND CONCLUSION\nIn summary, we have investigated the Gilbert damping of\nadvanced free layer designs: they comprise FeCoB alloys with\nvariable B contents from 20 to 30% and are organized in the\nsingle MgO or dual MgO free layer configuration fully em-\nbedded in functional STT-MRAM magnetic tunnel junctions.\nIncreasing the boron content increases the cristallization tem-\nperature, thereby postponing the onset of elemental diffusion\nwithin the free layer. This reduction of the interdiffusion of\nthe Ta atoms helps maintaining the Gilbert damping at a low\nlevel without any penalty on the anisotropy and the transport\nproperties. Thereby, increasing the Boron content to at least\n30% is beneficial for the thermal robustness of the MTJ up\nto the 400\u000erequired in CMOS back-end of line processing.\nIn addition, we have shown that dual MgO free layers have a\nsubstantially lower damping than their single MgO counter-\nparts, and that the damping increases as the thickness of the\nTa spacer within dual MgO free layers. This indicates that\nthe dominant source of extra damping is the presence of Ta\nimpurities within the FeCoB alloy. Using optimized MTJs,\nwe have studied the duration of the switching events as in-\nduced by spin-transfer-torque. Our experimental procedure –\ntime-resolving the switching with a high bandwidth but dur-\ning slow voltage sweep – ensures that we are investigating\nonly sub-threshold thermally activated switching events. In\noptimal conditions, the switching induces a ramp-like mono-\ntonic evolution of the device conductance that we interpret\nas the sweeping of a domain wall through the device. The\nswitching duration is roughly proportional to the device size:\nthe smaller the device, the faster it switches. We studied twoMTJ stacks and found domain wall velocities from 20 to 40\nm/s. A simple analytical model using a rigid wall approxima-\ntion can account for our main experimental findings. The do-\nmain wall velocity is predicted to scale linearly with the cur-\nrent for device sizes much larger than the domain wall widths.\nThe domain wall velocity depends on the material parame-\nters, such that the samples with the thinnest domain walls will\nbe the ones that host the domain walls with the lowest mo-\nbilities. Schematically, material optimization for low current\nSTT-induced switching (i.e. in practice: fast nucleation be-\ncause of low exchange stiffness Aand low damping \u000b) will\ncome together with slow STT-induced domain wall motion at\nleast in the range of device sizes in which the STT-induced re-\nversal proceeds through domain wall motion. 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Available:\nhttp://link.aps.org/doi/10.1103/PhysRevLett.92.088302"    },    {        "title": "1102.4551v1.Ab_initio_calculation_of_the_Gilbert_damping_parameter_via_linear_response_formalism.pdf",        "content": "arXiv:1102.4551v1  [cond-mat.mtrl-sci]  22 Feb 2011APS/123-QED\nAb-initio calculation of the Gilbert damping parameter via linear response formalism\nH. Ebert, S. Mankovsky, and D. K¨ odderitzsch\nUniversity of Munich, Department of Chemistry,\nButenandtstrasse 5-13, D-81377 Munich, Germany\nP. J. Kelly\nFaculty of Science and Technology and MESA+ Institute for Na notechnology,\nUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands\n(Dated: October 14, 2018)\nA Kubo-Greenwood-like equation for the Gilbert damping par ameterαis presented that is based\non the linear response formalism. Its implementation using the fully relativistic Korringa-Kohn-\nRostoker (KKR) band structure method in combination with Co herent Potential Approximation\n(CPA) alloy theory allows it to be applied to a wide range of si tuations. This is demonstrated with\nresults obtained for the bcc alloy system Fe xCo1−xas well as for a series of alloys of permalloy with\n5d transition metals. To account for the thermal displaceme nts of atoms as a scattering mechanism,\nan alloy-analogy model is introduced. The corresponding ca lculations for Ni correctly describe the\nrapid change of αwhen small amounts of substitutional Cu are introduced.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nThe magnetization dynamics that is relevant for the\nperformance of any type of magnetic device is in gen-\neral governed by damping. In most cases the magneti-\nzation dynamics can be modeled successfully by means\nof the Landau-Lifshitz-Gilbert (LLG) equation [1] that\naccounts for damping in a phenomenological way. The\npossibility to calculate the corresponding damping pa-\nrameter from first principles would open the perspective\nof optimizing materials for devices and has therefore mo-\ntivated extensive theoretical work in the past. This led\namong others to Kambersky’s breathing Fermi surface\n(BFS) [2] and torque-correlation model (TCM) [3], that\nin principle provide a firm basis for numerical investi-\ngations based on electronic structure calculations [4, 5].\nThe spin-orbit coupling that is seen as a key factor in\ntransferring energy from the magnetization to the elec-\ntronic degrees of freedom is explicitly included in these\nmodels. Most ab-initioresults havebeen obtained for the\nBFS model though the torque-correlation model makes\nfewer approximations [4, 6]. In particular, it in principle\ndescribes the physical processes responsible for Gilbert\ndamping over a wide range of temperatures as well as\nchemical(alloy)disorder. However,inpractice,likemany\nother models it depends on a relaxation time parame-\nterτthat describes the rate of transfer due to the vari-\nous types of possible scattering mechanisms. This weak\npoint could be removed recently by Brataas et al. [7]\nwho described the Gilbert damping by means of scatter-\ning theory. This development supplied the formal basis\nfor the first parameter-free investigations on disordered\nalloys for which the dominant scattering mechanism is\npotential scattering caused by chemical disorder [8].\nAs pointed out by Brataas et al. [7], their approach is\ncompletelyequivalenttoaformulationintermsofthelin-\nearresponseorKuboformalism. Thelatterrouteistakenin this communication that presents a Kubo-Greenwood-\nlike expression for the Gilbert damping parameter. Ap-\nplication of the scheme to disordered alloys demonstrates\nthat this approach is indeed fully equivalent to the scat-\ntering theory formulation of Brataas et al. [7]. In addi-\ntion a scheme is introduced to deal with the temperature\ndependence of the Gilbert damping parameter.\nFollowing Brataas et al. [7], the starting point of our\nscheme is the Landau-Lifshitz-Gilbert (LLG) equation\nfor the time derivative of the magnetization /vectorM:\n1\nγd/vectorM\ndτ=−/vectorM×/vectorHeff+/vectorM×/bracketleftBigg˜G(/vectorM)\nγ2M2sd/vectorM\ndτ/bracketrightBigg\n,(1)\nwhereMsis the saturation magnetization, γthe gyro-\nmagnetic ratio and ˜Gthe Gilbert damping tensor. Ac-\ncordingly, the time derivative of the magnetic energy is\ngiven by:\n˙Emag=/vectorHeff·d/vectorM\ndτ=1\nγ2˙/vector m[˜G(/vector m)˙/vector m] (2)\nin terms of the normalized magnetization /vector m=/vectorM/Ms.\nOntheotherhandtheenergydissipationoftheelectronic\nsystem ˙Edis=/angbracketleftBig\ndˆH\ndτ/angbracketrightBig\nis determined by the underlying\nHamiltonian ˆH(τ). Expanding the normalized magne-\ntization/vector m(τ), that determines the time dependence of\nˆH(τ) about its equilibrium value, /vector m(τ) =/vector m0+/vector u(τ), one\nhas:\nˆH=ˆH0(/vector m0)+/summationdisplay\nµ/vector uµ∂\n∂/vector uµˆH(/vector m0). (3)\nUsing the linear response formalism, ˙Ediscan be written2\nas [7]:\n˙Edis=−π/planckover2pi1/summationdisplay\nii′/summationdisplay\nµν˙uµ˙uν/angbracketleftBigg\nψi|∂ˆH\n∂uµ|ψi′/angbracketrightBigg/angbracketleftBigg\nψi′|∂ˆH\n∂uν|ψi/angbracketrightBigg\n×δ(EF−Ei)δ(EF−Ei′),(4)\nwhereEFis the Fermi energy and the sums run over all\neigenstates αof the system. Identifying ˙Emag=˙Edis,\none gets an explicit expression for the Gilbert damping\ntensor˜Gor equivalently for the damping parameter α=\n˜G/(γMs):\nαµν=−π/planckover2pi1γ\nMs/summationdisplay\nii′/angbracketleftBigg\nψi|∂ˆH\n∂uµ|ψi′/angbracketrightBigg/angbracketleftBigg\nψi′|∂ˆH\n∂uν|ψi/angbracketrightBigg\n×δ(EF−Ei)δ(EF−Ei′).(5)\nAn efficient way to deal with Eq. (5) is achieved by ex-\npressing the sum over the eigenstates by means of the\nretarded single-particle Green’s function Im G+(EF) =\n−π/summationtext\nα|ψα/angbracketright/angbracketleftψα|δ(EF−Eα). This leads for the parame-\nterαto a Kubo-Greenwood-like equation:\nαµν=−/planckover2pi1γ\nπMsTrace/angbracketleftBigg\n∂ˆH\n∂uµImG+(EF)∂ˆH\n∂uνImG+(EF)/angbracketrightBigg\nc(6)\nwith/angbracketleft.../angbracketrightcindicating a configurational average in case of\na disordered system (see below). Identifying ∂ˆH/∂uµ\nwith the magnetic torque Tµthis expression obviously\ngives the parameter αin terms of a torque-torque corre-\nlation function. However, in contrast to the conventional\nTCM the electronic structure is not represented in terms\nof Bloch states but using the retarded electronic Green\nfunction giving the present approach much more flexibil-\nity. As it corresponds one-to-one to the standard Kubo-\nGreenwood equation for the electrical conductivity, the\ntechniques developed to calculate conductivities can be\nstraightforwardly adopted to evaluate Eq. (6).\nThe most reliable way to account for spin-orbit cou-\npling as the source of Gilbert damping is to evaluate\nEq. (6) using a fully relativistic Hamiltonian within the\nframework of local spin density formalism (LSDA) [9]:\nˆH=c/vector α/vector p+βmc2+V(/vector r)+β/vector σ/vector mB(/vector r).(7)\nHereαiandβare the standard Dirac matrices and /vector pis\nthe relativistic momentum operator [10]. The functions\nVandBare the spin-averagedand spin-dependent parts\nrespectively of the LSDA potential. Eq. (7) implies for\nthe magnetic torque Tµoccurring in Eq. (6) the expres-\nsion:\nTµ=∂\n∂uµˆH=βBσµ. (8)\nThe Green’s function G+in Eq. (5) can be obtained in a\nvery efficient way by using the spin-polarized relativisticversion of multiple scattering theory [9] that allows us to\ntreat magnetic solids:\nG+(/vector rn,/vector rm′,E) =/summationdisplay\nΛΛ′Zn\nΛ(/vector rn,E)τnm\nΛΛ′(E)Zm×\nΛ′(/vector rm′,E)\n−/summationdisplay\nΛZn\nΛ(/vector r<,E)Jn×\nΛ′(/vector r>,E)δnm.(9)\nHere coordinates /vector rnreferring to the center of cell n\nhave been used with |/vector r<|=min(|/vector rn|,|/vector rn′|) and|/vector r>|=\nmax(|/vector rn|,|/vector rn′|). The four component wave functions\nZn\nΛ(/vector r,E) (Jn\nΛ(/vector r,E)) are regular (irregular) solutions to\nthe single-site Dirac equation for site nandτnm\nΛΛ′(E) is\nthe so-called scattering path operator that transfers an\nelectronic wave coming in at site minto a wave going\nout from site nwith all possible intermediate scattering\nevents accounted for coherently.\nUsing matrix notation, this leads to the following ex-\npression for the damping parameter:\nαµµ=g\nπµtot/summationdisplay\nnTrace/angbracketleftbig\nT0µ˜τ0nTnµ˜τn0/angbracketrightbig\nc(10)\nwith the g-factor 2(1 + µorb/µspin) in terms of the spin\nand orbital moments, µspinandµorb, respectively, the\ntotal magnetic moment µtot=µspin+µorb, and ˜τ0n\nΛΛ′=\n1\n2i(τ0n\nΛΛ′−τ0n\nΛ′Λ) and the energy argument EFomitted.\nThematrix elementsofthe torqueoperator Tnµareiden-\ntical to those occurring in the context of exchange cou-\npling [11] and can be expressed in terms of the spin-\ndependent part Bof the electronic potential with matrix\nelements:\nTnµ\nΛ′Λ=/integraldisplay\nd3rZn×\nΛ′(/vector r) [βσµBxc(/vector r)]Zn\nΛ(/vector r).(11)\nAs indicated above, the expressions in Eqs. (6) – (11)\ncan be applied straightforwardly to disordered alloys.\nIn this case the brackets /angbracketleft.../angbracketrightcindicate the necessary\nconfigurational average. This can be done by describ-\ning in a first step the underlying electronic structure\n(forT= 0 K) on the basis of the Coherent Potential\nApproximation (CPA) alloy theory. In the next step\nthe configurational average in Eq. (6) is taken follow-\ning the scheme worked out by Butler [12] when dealing\nwith the electrical conducting at T= 0 K or residual\nresistivity respectively, of disordered alloys. This im-\nplies in particular that so-called vertex corrections of the\ntype/angbracketleftTµImG+TνImG+/angbracketrightc− /angbracketleftTµImG+/angbracketrightc/angbracketleftTνImG+/angbracketrightcthat\naccount for scattering-in processes in the language of the\nBoltzmann transport formalism are properly accounted\nfor.\nThermal vibrations as a source of electron scattering\ncan in principle be accounted for by a generalization of\nEqs. (6) – (11) to finite temperatures and by includ-\ning the electron-phonon self-energy Σ el−phwhen calcu-\nlating the Greens function G+. Here we restrict our-\nselves to elastic scattering processes by using a quasi-\nstatic representation of the thermal displacements of the3\n0 0.1 0.2 0.3 0.4 0.50.6 0.7\nconcentration xCo0123456α(x) x10-3Expt\nTheory (CPA), bcc\nTheory (NL CPA)Fe-Co\nn(EF)\nn(EF) (sts./Ry)\n102030405060\n0\nFIG. 1: Gilbert damping parameter for bcc Fe xCo1−xas a\nfunction of Co concentration: full circles - the present res ults\nwithin CPA, empty circles - within non-local CPA (NL CPA),\nand full diamonds - experimental data by Oogane [14].\natoms from their equilibrium positions. We introduce\nan alloy-analogy model to average over a discrete set\nof displacements that is chosen to reproduce the ther-\nmal root mean square average displacement/radicalbig\n/angbracketleftu2/angbracketrightTfor\na given temperature T. This was chosen according to\n/angbracketleftu2/angbracketrightT=1\n43h2\nπ2mkΘD[Φ(ΘD/T)\nΘD/T+1\n4] with Φ(Θ D/T) the De-\nbye function, hthe Planck constant, kthe Boltzmann\nconstant and Θ Dthe Debye temperature [13]. Ignoring\nthe zero temperature term 1 /4 and assuming a frozen\npotential for the atoms, the situation can be dealt with\nin full analogy to the treatment of disordered alloys de-\nscribed above.\nThe approach described above has been applied to\nthe ferromagnetic 3d-transition metal alloy systems bcc\nFexCo1−x, fcc Fe xNi1−xand fcc Co xNi1−x. Fig. 1 shows\nas an example results for bcc Fe xCo1−xforx≤0.7. The\ncalculated damping parameter α(x) forT= 0 K is found\nin very good agreement with the results based on the\nscatteringtheoryapproach[8]demonstratingnumerically\nthe equivalence of the two approaches. An indispensable\nrequirement to achieve this agreement is to include the\nvertex corrections mentioned above. In fact, ignoring\nthem leads in some cases to completely unphysical re-\nsults. To check the reliability of the standard CPA, that\nimplies a single-site approximation when performing the\nconfigurationalaverage,weperformedcalculationsonthe\nbasis of the non-local CPA [15]. In this case four atom\ncluster have been used leading - apart from the very di-\nlute case - practically to the same results as the CPA. As\nfound before for fcc Fe xNi1−x[8] the theoretical results\nforαreproduce the concentration dependence of the ex-\nperimental data quite well but are found too low (see\nbelow). As suggested by Eq. (10) the variation of α(x)\nwith concentration xmay reflect to some extent the vari-\nation of the average magnetic moment µtotof alloy. As\nthe moments as well as the spin-orbit coupling strength\nof Fe and Co don’t differ too much, the variation of α(x)\nshould be determined in the concentrated regime primar-\nily by the electronic structure at the Fermi energy EF.As Fig. 1 shows, there is indeed a close correlation of the\ndensity of states n(EF) that may be seen as a measure\nfor the available relaxation channels.\nWhile the scattering and linear response approach are\ncompletely equivalent when dealing with bulk alloys the\nlatter allows us to perform the necessary configuration\naveragingin a much more efficient way. This allows us to\nstudy with moderate effort the influence of varying the\nalloy composition on the damping parameter α. Corre-\nsponding work has been done in particular using permal-\nloy as a starting material and adding transition metals\n(TM) [16] orrareearthmetals [17]. Fig. 2(top) showsre-\nsultsobtainedbysubstitutingFeandNiatomsinpermal-\nloy by 5d TMs. As found by experiment [16] αincreases\n00.05 0.1 0.15x01234 α x10-2\nTa W Re OsIr PtAu02468α x10-2\nTa W Re OsIr PtAu-0.300.30.60.9mspin5d (µB)WOs\nIr\nPt\nAuRe\nTa\nn5d(EF)\n5d spin moment\n061218\nn5d(EF) (sts./Ry)\nFIG. 2: Top: Change of the Gilbert damping parameter ∆ α\nw.r.t. permalloy(Py)forvariousPy/5dTM systemsasafunc-\ntion of 5d TM concentration; Middle: Gilbert damping pa-\nrameter αfor Py/5d TM systems with 10 % 5d TM content\nin comparison with experiment [16]; Bottom: spin magnetic\nmoment m5d\nspinand density of states n(EF) at the Fermi en-\nergy of the 5 dcomponent in Py/5d TM systems with 10 %\n5d TM content.\nin all cases nearly linearly with the 5d TM content. The\ntotal damping for 10 % 5d TM content shown in the\nmiddle panel of Fig. 2 varies roughly parabolically over\nthe 5d TM series. In contrast to the Fe xCo1−xalloys\nconsidered above, there is now an S-like variation of the\nmomentsµ5d\nspinover the series (Fig. 2, bottom), char-\nacteristic of 5d impurities in the pure hosts Fe and Ni\n[18, 19]. In spite of this behaviour of µ5d\nspinthe variation4\n00.050.10.15α(T)expt:    pure Ni\ntheory: pure Ni\n00.050.10.15α(T)expt:    Ni + 0.17 wt.%Cu\ntheory: Ni + 0.2 at.%Cu\n0 100 200 300 400 500\nTemperature (K)00.050.10.15α(T)expt: Ni + 5 wt.%Cu\ntheory: Ni + 5 at.%Cu\nFIG. 3: Temperature variation of Gilbert damping of pure\nNi and Ni with Cu impurities: present theoretical results vs\nexperiment [20]\nofα(x) seems again to be correlated with the density of\nstatesn5d(EF) (Fig. 2 bottom). Again the trend of the\nexperimental data is well reproduced by the theoretical\nones that are however somewhat too low.\nOne of the possible reasons for the discrepancy of the\ntheoretical and experimental results shown in Figs. 1 and\n2 might be the neglect of the influence of finite temper-\natures. This can be incorporated as indicated above by\naccounting for the thermal displacement of the atoms in\na quasi-static way and performing a configurational av-\nerage over the displacements using the CPA. This leads\neven for pure systems to a scattering mechanism and this\nwaytoafinite valuefor α. Correspondingresultsforpure\nNi are given in Fig. 3 that show in full accordance with\nexperiment a rapid decrease of αwith increasing tem-\nperature until a regime with a weak variation of αwith\nTis reached. This behavior is commonly interpreted as\na transition from conductivity-like to resistivity-like be-\nhaviour reflecting the dominance of intra- and inter-band\ntransition, respectively [4], that is related to the increase\nof the broadening of electron energy bands caused by the\nincrease of scattering events with temperature. Adding\nonly less than 1 at. % Cu to Ni, the conductivity-like\nbehavior at low temperatures is strongly reduced whilethe high temperature behavior is hardly changed. A fur-\nther increase of the Cu content leads to the impurity-\nscattering processes responsible for the band broaden-\ning dominating α. This effect completely suppresses the\nconductivity-likebehavior in the low-temperatureregime\nbecause of the increase of scattering events due to chem-\nical disorder. Again this is fully in line with the experi-\nmental data, providing a straightforward explanation for\ntheir peculiar variation with temperature and composi-\ntion.\nFromtheresultsobtainedforNionemayconcludethat\nthermal lattice displacements are only partly responsible\nfor the finding that the damping parameters obtained\nfor Py doped with the 5 dTM series, and Fe xCo1−xare\nsomewhatlowcomparedwith experiment. This indicates\nthat additional relaxation mechanisms like magnon scat-\ntering contribute. Again, these can be included at least\nin a quasi-static way by adopting the point of view of a\ndisordered local moment picture. This implies scatter-\ning due to random temperature-dependent fluctuations\nof the spin moments that can also be dealt with using\nthe CPA.\nInsummary, aformulationforthe Gilbert dampingpa-\nrameterαin terms of a torque-torque-correlation func-\ntion was derived that led to a Kubo-Greenwood-like\nequation. The scheme was implemented using the fully\nrelativistic KKR band structure method in combination\nwith the CPA alloy theory. This allows us to account for\nvarious types of scattering mechanisms in a parameter-\nfree way. Corresponding applications to disordered tran-\nsition metal alloys led to very good agreement with re-\nsults based on the scattering theory approach of Brataas\net al. demonstrating the equivalence of both approaches.\nThe flexibility and numerical efficiency of the present\nscheme was demonstrated by a study on a series of\npermalloy-5dTMsystems. Toinvestigatetheinfluenceof\nfinite temperatures on α, a so-called alloy-analogymodel\nwas introduced that deals with the thermal displacement\nof atoms in a quasi-static manner. Applications to pure\nNi gave results in very good agreement with experiment\nand in particular reproduced the dramatic change of α\nwhen Cu is added to Ni.\nAcknowledgments\nThe authors would like to thank the DFG for finan-\ncial support within the SFB 689 “Spinph¨ anomene in re-\nduzierten Dimensionen” and within project Eb154/23for\nfinancialsupport. PJKacknowledgessupportbyEUFP7\nICT Grant No. 251759 MACALO.\n[1] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004).[2] V. Kambersky, Can. J. Phys. 48, 2906 (1970).\n[3] V. Kambersky, Czech. J. Phys. 26, 1366 (1976), URL5\nhttp://dx.doi.org/10.1007/BF01587621 .\n[4] K. Gilmore, Y. U. Idzerda, and M. D. Stiles,\nPhys. Rev. Lett. 99, 027204 (2007), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.99.0272 04.\n[5] M. F¨ ahnle and D. Steiauf, Phys. Rev. B 73, 184427\n(2006).\n[6] V. Kambersky, Phys. Rev. B 76, 134416 (2007).\n[7] A. Brataas, Y. Tserkovnyak, and G. E. W.\nBauer, Phys. Rev. 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Rev.B 10, 179 (1974)."    },    {        "title": "2303.01343v2.Spin_Pumping_into_Carbon_Nanotubes.pdf",        "content": "Spin Pumping into Carbon Nanotubes\nK. Fukuzawa1, T. Kato1, M. Matsuo2,3,4,5, T. Jonckheere6, J. Rech6, and T. Martin6\n1Institute for Solid State Physics, The University of Tokyo, Kashiwa, 277-8581, Japan\n2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China\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\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, Japan\n6Aix Marseille Univ, Universit´ e de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, France\n(Dated: October 31, 2023)\nWe theoretically study spin pumping from a ferromagnetic insulator (FI) into a carbon nanotube\n(CNT) . By employing the bosonization method, we formulate the Gilbert damping induced by the\nFI/CNT junction, which can be measured by ferromagnetic resonance. We show that the increase\nin the Gilbert damping has a temperature dependence characteristic of a Luttinger liquid and is\nhighly sensitive to the Luttinger parameter of the spin sector for a clean interface. We also discuss\nthe experimental relevance of our findings based on numerical estimates, using realistic parameters.\nI. INTRODUCTION\nSpin pumping induced by ferromagnetic resonance\n(FMR) [1, 2] is a fundamental technique in spintronics\nfor generating spin current from a ferromagnet to an ad-\njacent material [3, 4]. While spin pumping has been used\nfor injecting spin into various materials, it can also be uti-\nlized for detecting spin excitations in various systems [5–\n17]. Compared with bulk measurement techniques, such\nas nuclear magnetic resonance (NMR) and neutron scat-\ntering experiments, spin pumping has an advantage in\nsensitivity for nanostructured systems such as surfaces,\nthin films and atomic-layer compounds [5].\nThe study of exotic spin excitations which emerge in\nspecific materials is one of the forefront topics of con-\ndensed matter physics. A typical example is spin exci-\ntation in quasi-one-dimensional interacting electron sys-\ntems, whose low-energy excitation can be described by\nthe Tomonaga-Luttinger liquid [18–20]. Spin excitations\ninherent to the Tomonaga-Luttinger liquid have been\nstudied in carbon nanotubes (CNTs) by using NMR [21–\n23]. While NMR can detect the local spin susceptibility\nin CNTs, the use of spin pumping to detect spin excita-\ntions is expected to provide useful information reflecting\nthe exotic character of the Luttinger liquid, which can-\nnot be captured by NMR. It is thus important to clarify\nwhat kind of information about the Luttinger liquid can\nbe obtained from a spin pumping experiment.\nIn this work, we theoretically formulate the increase\nin the Gilbert damping due to spin pumping in a setup\nin which spin is injected into CNTs. We consider a\nmagnetic junction composed of a ferromagnetic insula-\ntor (FI) and a single-wall CNT (see Fig. 1) and take in-\nterfacial randomness into account with a simple model.\nWe derive an analytic expression for the increase in the\nGilbert damping by utilizing the bosonization method\nand second-order perturbation with respect to the inter-\nfacial exchange coupling.\nWe will focus on the two limiting cases, i.e., a clean in-\nterface and a dirty interface. We show that for both cases\nFIG. 1. Magnetic junction composed of a ferromagnetic in-\nsulator (FI) and a single-wall carbon nanotube (CNT). The\ndimension of the FI is W×W′×d′.\nthe temperature dependence of the increase of the Gilbert\ndamping shows a power-law behavior, with an exponent\nreflecting the Luttinger parameters. For a clean inter-\nface, the exponent includes information on the Luttinger\nparameters in the spin sector and is shown to be sensitive\nto small deviations from unity (which is the value of the\nSU(2) symmetric model in the spin sector). For a dirty\ninterface, the exponent depends on the Luttinger param-\neters of both the spin and charge sectors as in an NMR\nmeasurement. We estimate the increase of the Gilbert\ndamping using realistic parameters and discuss the ex-\nperimental feasibility.\nOur paper is organized as follows. We introduce the\nmicroscopic model of the FI/CNT magnetic junction in\nSec. II. We analytically calculate the increase in the\nGilbert damping in Sec. III and subsequently estimate\nit with realistic parameters in Sec. IV. Finally, we briefly\ndiscuss the experimental relevance of our findings in\nSec. V and summarize our results in Sec. VI. A detailedarXiv:2303.01343v2  [cond-mat.mes-hall]  28 Oct 20232\nderivation of the analytic expressions is given in the two\nAppendices.\nII. MODEL\nLet us consider a junction composed of a CNT and\nFI, whose Hamiltonian is given by H=HCNT+HFI+\nHint. Here, HCNTandHFIdescribe electrons in the CNT\nand FI, respectively, and Hintrepresents the interfacial\nexchange interaction between the CNT and FI. We will\ngive their explicit forms in the subsections that follow.\nA. Carbon nanotube\nThe low-energy Hamiltonian of electrons in CNTs is\ngiven by\nHCNT=HK+HC, (1)\nwhere HKandHCrepresent the kinetic energy and\nthe forward scattering potential due to the screened\nCoulomb interaction, respectively. Using standard con-\nventions [24], the Hamiltonians describing these energies\nof electrons in CNTs are given by\nHK=−ivFZ\ndxX\nrασrψ+\nrασ(x)∂xψrασ(x), (2)\nHC=1\n2Z\ndx dy ρ (x)V(x−y)ρ(y), (3)\nwhere ψrασ(x) is the slowly varying part of the\nfield operator of electrons, vFis the Fermi velocity,\nV(x) is the screened Coulomb potential, and ρ(x) =P\nrασψ†\nrασ(x)ψrασ(x) is the electron density operator.\nThe subscripts, r(=±),α(=±), and σ(=±), repre-\nsent the direction of propagation, the nanotube branch\n(the valley), and the spin orientation, respectively. Using\nthe bosonization method [19, 24], the annihilation oper-\nator describing fermions in the CNT can be expressed in\nterms of bosonic fields, θασ(x) and ϕασ(x), as\nψrασ(x) =ηrασ√\n2πaei(−rθασ(x)+ϕασ(x)), (4)\nwhere ηrασis the Klein factor, and ais a short-length\ncutoff which can be identified with the lattice constant of\nthe CNT. To diagonalize the Hamiltonian, we introduce\nnew bosonic fields for the charge and spin sectors, θjδ(x)\nandϕjδ(x) as\nθασ(x) =1\n2X\njδhjδ(α, σ)θjδ(x), (5)\nϕασ(x) =1\n2X\njδhjδ(α, σ)ϕjδ(x), (6)\nwhere δ(=±) represents symmetric/antisymmetric\nmodes, j(=c, s) indicates the charge/spin mode, hc+=1,hc−=α,hs+=σ, and hs−=ασ. The Hamiltonian\nof the CNTs can be written as\nHCNT=X\nj,δvjδ\n2πZ\ndx[K−1\njδ(∂xθjδ)2+Kjδ(∂xϕjδ)2],(7)\nwhere Kjδis the Luttinger parameter and vjδ=vF/Kjδ.\nB. Ferromagnetic insulator\nWe consider a bulk FI described by the quantum\nHeisenberg model and employ the spin-wave approxi-\nmation assuming that the temperature is much lower\nthan the magnetic transition temperature and the mag-\nnitude of the localized spin, S0, is much larger than\none [8, 9, 11, 15–17, 25]. In this situation, the Hamil-\ntonian for the FI is approximately written as a superpo-\nsition of magnon modes:\nHFI=X\nkℏωkb†\nkbk, (8)\nwhere bkis the annihilation operator of magnons, ℏωk=\nDk2+ℏγghdcis the magnon dispersion, Dis spin stiffness,\nγgis the gyromagnetic ratio, and hdcis the static mag-\nnetic field. We will only focus on uniform spin precession\ninduced by external microwaves. For this purpose, it is\nsufficient to consider the magnon mode of k=0with the\nsimplified Hamiltonian\nHFI=ℏω0b†\n0b0. (9)\nMicrowave absorption in FMR can be related to the\nimaginary part of the retarded spin correlation function,\nwhich is defined as\nGR(ω) =−i\nℏZ∞\n0dt ei(ω+iδ)t⟨[S+\n0(t), S−\n0]⟩, (10)\nwhere S+\n0=√2S0b0and S−\n0=√2S0b†\n0are spin\nladder operators of the FI for k=0andS+\n0(t) =\neiHt/ℏS+\n0e−iHt/ℏ. For an isolated bulk FI, the spin sus-\nceptibility is calculated as:\nGR\n0(ω) =2S0/ℏ\nω−ω0+iδ. (11)\nIn real experiments, the FMR linewidth is finite due to\nthe the Gilbert damping. To represent this finite spin\nrelaxation in the bulk FI, we introduce a phenomeno-\nlogical dimensionless parameter αGand express the spin\ncorrelation function as\nGR\n0(ω) =2S0/ℏ\nω−ω0+iαGω. (12)3\nC. Interfacial exchange interaction\nNow let us consider the interfacial exchange interaction\nbetween the FI and the CNT with the Hamiltonian,\nHint=S+\n0s−+S−\n0s+, (13)\nwhere s±is the spin ladder operator of the CNT, defined\nas\ns−=r\n1\nNFIX\nr,r′X\nα,α′ZW\n0dx J(x)\n×e−i(α−α′)kFx−i(r−r′)qFxψ†\nrα−(x)ψr′α′+(x),(14)\nands+= (s−)†. Here, Wis the length of the interface,\nJ(x) is the interfacial exchange coupling, NFIis the num-\nber of unit cells in the FI, kFis the Fermi wavenumber,\nandqF(≪kF) is the momentum mismatch associated\nwith the two modes. Because the interfacial exchange\ncoupling J(x), which is induced by quantum mechanical\nmixing between CNT and FI, is sensitive to distances of\natoms across the junction, we assumed that it depends\non the position xdue to random atomic configuration\nnear the interface. A simplified model for randomness in\nJ(x) will be accounted for in the next section.\nIII. FORMULATION\nA. Gilbert damping\nUsing second-order perturbation with respect to the\ninterfacial exchange coupling, the spin susceptibility is\ncalculated as\nG(iωn) =1\nG0(iωn)−1−Σ(iωn)(15)\nΣ(iωn) =−1\nℏZℏβ\n0dτeiωnτ⟨Tτs+(τ)s−(0)⟩ (16)\nwhere s±(τ) = eHCNTτ/ℏs±e−HCNTτ/ℏ. The retarded\nspin correlation function is obtained by analytic contin-\nuation iωn→ω+iδas\nGR(ω) =2S0/ℏ\nω−(ω0+δω0) +i(αG+δαG)ω0,(17)\nδω0\nω0≃2S0\nℏω0Re ΣR(ω0), (18)\nδαG≃ −2S0\nℏω0Im ΣR(ω0), (19)\nwhere ΣR(ω) is the retarded self-energy defined by\nΣR(ω) =Z\ndt eiωtΣR(t), (20)\nΣR(t) =−iθ(t)\nℏ⟨[s+(t), s−(0)]⟩, (21)θ(t) is the step function, and αG+δαG≪1 has been\nassumed. In our work, we focus on the increase in the\nGilbert damping due to the junction, δαG, which is writ-\nten in terms of the dynamic spin susceptibility of CNTs.\nB. Self-energy of electrons in CNTs\nBy substituting Eq. (14) into Eq. (21), we obtain\nΣR(t) =−i\nℏθ(t)2S0\nNFIX\nr,r′X\nα,α′ZW\n0dxZW\n0dy⟨J(x)J(y)⟩imp\n×e−i(kF(α−α′)+qF(r−r′))(x−y)Crαr′α′(x, y, t ),(22)\nCrαr′α���(x, y, t ) =⟨[ψ†\nrα,+(x, t)ψr′α′,−(x, t),\nψ†\nr′α′,−(y,0)ψrα,+(y,0)]⟩0. (23)\nHere, ⟨···⟩ impindicates a random average for the in-\nterfacial exchange coupling. For simplicity, we assume\nthat the exchange coupling follows a Gaussian distribu-\ntion whose average and variance are given by\n⟨J(x)⟩imp=J1, (24)\n⟨δJ(x)δJ(y)⟩imp=J2\n2aδ(x−y), (25)\nwhere δJ(x) =J(x)− ⟨J(x)⟩imp. Here, J1andJ2repre-\nsent respectively the average and the standard deviation\nof the distribution. The ratio J2/J1reflects the random-\nness of the interfacial exchange coupling. In particular,\nthe case of J2/J1= 0 corresponds to a clean interface\nwithout randomness.\nAccordingly, the self-energy is calculated as\nΣR(t) = ΣR\n1(t) + ΣR\n2(t), (26)\nΣR\n1(t) =−iθ(t)2S0J2\n1\nℏNFIX\nr,r′,α,α′ZW\n0dxZW\n0dy\n×e−i(kF(α−α′)+qF(r−r′))(x−y)Crαr′α′(x, y, t ),(27)\nΣR\n2(t) =−iθ(t)2S0J2\n2a\nℏNFIX\nr,r′,α,α′ZW\n0dx C rαr′α′(x, x, t ).\n(28)\nSince the integrand of ΣR\n1(t) includes a rapidly oscillating\npart as a function of ( x−y), the integral is negligibly\nsmall except for the case of α=α′andr=r′. There,\nwe obtain\nΣR\n1(t) =−iθ(t)2S0J2\n1\nℏNFIX\nr,αZW\n0dxZW\n0dy C rαrα(x, y, t ).\n(29)\nWe should note that ΣR\n1(t) corresponds to the process\nof electron creation and annihilation in the same branch\nand represents momentum-conserving spin relaxation for\na clean junction. In contrast, ΣR\n2(t) represents spin re-\nlaxation for a “dirty” junction that is independent of the4\nelectron momentum. Here, the word “dirty” means that\nduring spin exchange process the momentum of electrons\nin the CNT is not conserved and transitions between dif-\nferent branches of valleys and propagation directions are\nallowed. The following discussion will consider two lim-\niting cases for the interface. For the clean interface limit\n(J1≫J2), the magnon self-energy is represented with\nΣR\n1(t), while in the dirty interface limit ( J1≪J2), it is\nrepresented with ΣR\n2(t).\nC. Clean interface\nSince the correlation function Crαr′α′(x, y, t ) can be\ncalculated using the bosonization method (see Ap-\npendix A), the self-energy ΣR\n1(t) can be obtained ana-\nlytically. Therefore, the corresponding increase in the\nGilbert damping is obtained as\nδαG,1=−2S0\nℏω0Im ΣR\n1(ω0)\n=−4S0J2\n1\nℏ2ω0(2πa)2NFIZW\n0dxZW\n0dyZ∞\n0dtsinω0t\n×Im\"\u0012sinh(iπa/β ℏvF)\nsinh(π(ia−(x−y)−vFt)/βℏvF)\u0013γ−1\n×\u0012sinh(iπa/β ℏvF)\nsinh(π(ia+ (x−y)−vFt)/βℏvF)\u0013γ+1#\n,(30)\nγ≡Ks+\n4+Ks−\n4+1\n4Ks++1\n4Ks−. (31)\nAfter analytic integration with respect to t(see Ap-\npendix B for details), we obtain\nδαG,1=2\nπΓ(γ)2\nΓ(2γ)S0J2\n1Wa\nℏ2v2\nFNFI\u00122πa\nβℏvF\u00132γ−3\n×I(πW/β ℏvF, γ), (32)\nI(w, γ) =1\nwZw\n0dz′Zz′\n0dz e−2(γ−1)z\n×F(γ−1, γ,2γ; 1−e−4z), (33)\nwhere F(a, b, c ;x) is the hypergeometric function.\nD. Dirty interface\nThe self-energy ΣR\n2(t) can be obtained in a similar\nway as above. The corresponding increase in the Gilbertdamping is given by\nδαG,2=−2S0\nℏω0Im ΣR\n2(ω0)\n=−S0J2\n2aW\nℏ2ω0(πa)2NFIX\nr,r′,α,α′Z∞\n0dtsinω0t\n×Im\"\u0012sinh(iπa/β ℏvF)\nsinh(π(ia−vFt)/βℏvF)\u00132γrαr ′α′#\n,(34)\nγrαr′α′=γ1δr,r′δα,α′+γ2δr,−r′δα,α′\n+γ3δr,r′δα,−α′+γ4δr,−r′δα,−α′,(35)\nγ1= (Ks++Ks−+ 1/Ks++ 1/Ks−)/4, (36)\nγ2= (Kc++Kc−+ 1/Ks++ 1/Ks−)/4, (37)\nγ3= (Ks++Kc−+ 1/Kc++ 1/Ks−)/4, (38)\nγ4= (Kc++Ks−+ 1/Ks++ 1/Kc−)/4. (39)\nWe should note that δαG,2is proportional to W, since the\nspin relaxation rate is determined through spatially-local\nspin exchange in the dirty interface and is proportional\nto the number of spin-exchange channels. After analytic\nintegration with respect to t(see Appendix B for details),\nwe obtain\nδαG,2=1\n2πS0J2\n2aW\nℏ2v2\nFNFI\n×X\nr,r′,α,α′Γ(γrαr′α′)2\nΓ(2γrαr′α′)\u00122πa\nβℏvF\u00132γrαr ′α′−2\n.(40)\nIV. NUMERICAL ESTIMATE\nNext, we evaluate numerically the increase in the\nGilbert damping by using realistic experimental parame-\nters. While the increase was formulated for a single CNT\nin the previous section, to increase the signal, it would be\nmore useful if we considered a junction with a bundle of\nCNTs. Thus, in the following, we will consider a junction\ncomposed of a FI and a bundle of CNTs with an area of\nW×W′(see Fig. 1) and multiply δαG,1andδαG,2by\nthe number of CNTs in the junction, NCNT=W′/d(d:\nthe diameter of CNTs).\nThe parameters are given in Table I. The Fermi ve-\nlocity vF, lattice constant a, diameter d, Luttinger pa-\nrameters of CNTs, Kc+,Kc−, and Ks−, are taken from\nRefs. [20, 24, 26]. The value of Ks+is an experiment\nresult [22] under a magnetic field of 3 .6 T[27]. The spin\namplitude S0and the lattice constant a′are determined\nby assuming that the FI is made from yttrium iron gar-\nnet (YIG). The interfacial exchange coupling ( J1orJ2)\nis roughly estimated to be 2 K [28]. The number of unit\ncells is estimated as NFI=WW′d′/a′3, where d′is the\nthickness of the FI.5\nTABLE I. Parameters used for the numerical estimate.\nMicrowave frequency ω0 1 GHz\nFermi velocity of CNT vF 106m/s\nLattice constant of CNT a 2.46˚A\nDiameter of CNT d 1.5 nm\nAmplitude of spins of FI S0 10\nLattice constant of FI a′12.376˚A\nThickness of FI d′10 nm\nInterfacial exchange couplings J1,J2\nclean interface J1= 2 K, J2= 0\ndirty interface J1= 0 K, J2= 1,2,3 K\nLuttinger parameters Kc+ 0.20\nKs+ 1.07\nKc−,Ks−1\n100 1010-2\n10-3\n10-4\n3 300\nFIG. 2. Temperature dependence of increase in the Gilbert\ndamping, δαG,1, for a clean interface ( J1≫J2).\nA. Clean interface\nThe estimated increase in the Gilbert damping for a\nclean interface ( J1= 2 K ≫J2) is shown in Fig. 2 as\na function of temperature. While δαG,1is proportional\nto 1/Tat high temperatures, it is almost constant at\nlow temperatures. The crossover temperature for a fixed\nlength Wis given by T∗=g(γ)ℏvF/(kBW) (kB: Boltz-\nmann constant), which is proportional to 1 /W. The fac-\ntorg(γ), which depends only on γ, is explicitly shown\nlater. The increase in the Gilbert damping is shown as a\nfunction of the junction length Win Fig. 3. While δαG,1\nis proportional to Wfor a short junction, it is almost\nconstant for a long junction. The crossover length for a\nfixed temperature Tis given by W∗=g(γ)ℏvF/(kBT).\nIn the present estimate, the condition Lth≪vF/ω0\nalways holds, where Lth=ℏvF/kBTis a thermal length.\nUnder this condition, the increase in the Gilbert damping\nbecomes independent of ω0and is approximately given\n3K\n10K\n30K\n100K\n300K\n10-310-410-510-610-2\n10-510-3\n10-4\nFIG. 3. Junction-length dependence of increase in the Gilbert\ndamping, δαG,1, for a clean interface ( J1≫J2).\nby\nδαG,1=Γ(γ)2\nΓ(2γ)S0J2\n1a′3a\n(ℏvF)2dd′\u00122πa\nLth\u00132γ−3\nf(γ, πW/L th),\n(41)\nf(γ, w) =(\nw/π, (w/π≪g(γ)),\ng(γ),(w/π≫g(γ)),(42)\ng(γ) =2\nπZ∞\n0dz e−2(γ−1)zF(γ−1, γ,2γ; 1−e−4z).\n(43)\nFrom this analytic expression, we obtain\nδαG,1∝(\nT2γ−2W, (W≪g(γ)Lth),\nT2γ−3g(γ),(W≫g(γ)Lth).(44)\nThe exponent γ= (Ks++Ks−+K−1\ns++K−1\ns−)/4 cor-\nresponds to unity when Ks+=Ks−= 1. Even in the\npresent estimate employing Ks+= 1.07, the exponent is\nalmost unity ( γ= 1.00114). By setting γ= 1, we can\nreproduce the power in the temperature and junction-\nlength dependence of δαG,1shown in Figs. 2 and 3.\nFinally, let us discuss the factor g(γ). If γis slightly\nlarger than 1 as in the present estimate, the geometric\nfunction is approximated as F(γ−1, γ,2γ;x)≃1. Then,\nthe factor g(γ) is approximately given as\ng(γ) =1\nπ(γ−1). (45)\nThis expression indicates that the increase in the Gilbert\ndamping in the high-temperature limit ( T≫T∗) or the\nlong-junction limit ( W≫W∗) is highly sensitive to the\ndeviation of γfrom unity. The crossover temperature\nT∗and the crossover length W∗also include the factor6\n100 1010-5\n10-810-6\n10-7\n3 300\nFIG. 4. Temperature dependence of increase in the Gilbert\ndamping, δαG,2, for a dirty interface ( J2≫J1). The three\nlines correspond to J2= 1, 2, and 3 K, respectively.\ng(γ)∝(γ−1)−1. Thus, the increase in the Gilbert damp-\ning can be used to investigate small deviations of γfrom\nunity. Then, the Luttinger parameter Ks,+in the spin\nsector can also be determined from Eq. (31) if we know\nwhether it is greater or less than unity. We note that in\nthe NMR measurement [22] Ks,+decreases as the mag-\nnetic field increases. Using this experimental tendency,\nwe expect that Ks,+can be determined uniquely.\nB. Dirty interface\nNext, we consider a dirty interface ( J2≫J1). Figure 4\nshows the increase of the Gilbert damping, δαG,2, as a\nfunction of the temperature for J2= 1, 2, and 3 K. In\nthis case, δαG,2is proportional to T−0.43in the whole\ntemperature range and shows a nontrivial exponent in-\nherent to the Tomonaga-Luttinger liquid.\nThe condition Lth≪vF/ω0also holds for a dirty in-\nterface. Therefore, δαG,2can be approximated as\nδαG,2=1\n2πS0J2\n2aa′3\n(ℏvF)2dd′\n×X\nr,r′,α,α′Γ(γrαr′α′)2\nΓ(2γrαr′α′)\u00122πa\nLth\u00132γrαr ′α′−2\n.(46)\nNoting that a≪Lth, the factor (2 πa/L th)2γrαr ′α′in\nEq. (46) is largely reduced as γrαr′α′increases. There-\nfore, in the sum of Eq. (46), it is sufficient to keep the\nterms in which γrαr′α′takes a minimum value. In the\npresent estimate, γrαr′α′is given by Eq. (35) with\n(γ1, γ2, γ3, γ4) = (1 .001,0.784,1.001.0.784). (47)\nUpon setting the minimum exponent to be γmin= 0.784,\nwe obtain δαG,2∝T2γmin−2=T−0.432, which is consis-tent with the numerical results shown in Fig. 4. There-\nfore, the nontrivial exponent inherent to the Tomonaga-\nLuttinger liquid appears in spin pumping through a dirty\njunction. Note that the approximate expression is inde-\npendent of the junction length Wfor a fixed thickness,\nsince the W-linear factor in NFI=WW′d/a′3cancels\nout the factor of Win Eq. (34).\nThe equation for the increase in the Gilbert damping\nfor the dirty interface has almost the same form as that\nfor 1/T1Tin NMR experiments where T1is the longitu-\ndinal relaxation time of nuclear spins[21–23]. Therefore,\nthe power law of the temperature dependence for the\ndirty interface is the same as in NMR experiments. This\nis because the spin transfer occurs at a spatially localized\npoint due to the impurity average at the dirty interface,\nleading to the same situation as the NMR experiment in\nwhich 1 /T1Tis related to the local dynamic spin suscep-\ntibility.\nV. EXPERIMENTAL RELEVANCE\nWe estimated the increase in the Gilbert damping δαG\nin two limiting situations, i.e., clean and dirty interfaces.\nIf we choose YIG as the ferromagnet, δαGshould be\nroughly in the range 10−5–10−2, because it should be\ncomparable to the Gilbert damping of bulk YIG, αG,\nwhich is of order of 10−5–10−3. For a clean interface,\nδαGis large enough to be measured in FMR experiments\n(see Figs. 2 and 3). Note that δαGcan be reduced by in-\ncreasing the thickness of YIG (denoted by d′). On the\nother hand, for a dirty interface, δαGis too small for it to\nbe observable by spin pumping (see Fig. 4). However, we\nwill moderate judgement on the possibility of observing\nδαGfor a dirty interface, because detailed information\non the interfacial exchange coupling is still lacking. We\nshould note that in the present modeling of randomness,\nthe increase in the Gilbert damping is given by a sum of\nthese two contributions, i.e., δαG=δαG,1+δαG,2, for an\narbitrary strength of interfacial randomness.\nOur calculation can be applied straightforwardly to\nother one-dimensional electron systems such as quasi-\none-dimensional magnets, whose low-energy states are\nalso described by the Tomonaga-Luttinger liquid model.\nIn particular, the low-energy states of spin systems with\nin-plane anisotropy are characterized by a Luttinger pa-\nrameter Kssmaller than 1. If Ksis sufficiently smaller\nthan 1, δαGshould show nontrivial power-law behavior\nwith respect to the temperature even for a clean inter-\nface.\nVI. SUMMARY\nWe theoretically studied spin pumping from a ferro-\nmagnetic insulator into carbon nanotubes. First, we for-\nmulated the increase in the Gilbert damping in terms\nof the spin susceptibility and described the interfacial7\nexchange coupling with a simple model, in which two\ntypes of spin-flip process, i.e., momentum-conserving and\nmomentum-nonconserving processes, coexist. Then, we\nanalytically calculated the increase in the Gilbert damp-\ning by treating electrons in carbon nanotubes in the\nframework of the Luttinger liquid. For a clean interface,\nthe increase in damping is proportional to the inverse of\nthe temperature at high temperatures while it is almost\nconstant at low temperatures. The crossover tempera-\nture includes information on the Fermi velocity in carbon\nnanotubes. We also found that the increase in damping is\nhighly sensitive to the deviation of the Luttinger param-\neter in the spin sector from unity. For a dirty interface,\nthe increase in damping shows a power-law dependence\non the temperature with a nontrivial exponent reflecting\nthe nature of the Tomonaga-Luttinger liquid. We also\nestimated the increase of the Gilbert damping using re-\nalistic parameters. Our results indicate a possible appli-\ncation of spin pumping for detecting power-law behavior\nof spin excitation in low-dimensional systems. Detection\nof other types of spin excitation in exotic many-body\nstates will be left as a future study.\nACKNOWLEDGMENTS\nThis French-Japanese collaboration is supported by\nthe CNRS International Research Project “Excitations\nin Correlated Electron Systems driven in the Giga-\nHertz range” (ESEC). This work received support from\nthe French government under the France 2030 invest-\nment plan, as part of the Initiative d’Excellence d’Aix-\nMarseille Universit´ e - A*MIDEX. We acknowledge sup-\nport from the institutes IPhU (AMX-19-IET-008) and\nAMUtech (AMX-19-IET-01X). T. K. acknowledges sup-\nport from the Japan Society for the Promotion of Sci-\nence (JSPS KAKENHI Grants No. 20K03831). M.\nM. acknowledges support by a Grant-in-Aid for Sci-\nentific Research B (23H01839 and 21H01800) and A\n(21H04565) from MEXT, Japan, and by the Priority Pro-\ngram of Chinese Academy of Sciences, under Grant No.\nXDB28000000.\nAppendix A: Correlation functions\nHere, we briefly summarize the calculation of the corre-\nlation function Crαr′α′(x, y, t ) defined in Eq. (23). Using\nthe bosonic fields, the correlation function is written as\nCrαr′α′(x, y, t )\n=1\n(2πa)2\u0002\n⟨eAeBeCeD⟩0− ⟨eCeDeAeB⟩0\u0003\n, (A1)\nA=−i(−rθα+(x, t) +ϕα+(x, t)), (A2)\nB=i(−r′θα′−(x, t) +ϕα′−(x, t)), (A3)\nC=−i(−r′θα′−(y,0) +ϕα′−(y,0)), (A4)\nD=i(−rθα+(y,0) +ϕα+(y,0)), (A5)where we set r= +1 ( r=−1) for the left-going (right-\ngoing) branch. Using the formula,\n⟨eA1eA2···eAN⟩= exp\n1\n2X\ni⟨A2\ni⟩+X\ni<j⟨AiAj⟩\n,\n(A6)\nwhich holds when [ Ai, Aj] is a c-number, we obtain\n⟨eAeBeCeD⟩0≡eFrαr ′α′(x−y,t)\n= exph1\n2⟨(A2+B2+C2+D2)⟩+⟨AB⟩+⟨CD⟩\n+⟨AC⟩+⟨AD⟩+⟨BC⟩+⟨BD⟩i\n, (A7)\n⟨eCeDeAeB⟩0=eFr′α′rα(y−x,−t). (A8)\nThe correlation functions of the bosonic fields, which are\ndefined as GXY\njδ=⟨X(x, t)Y(y,0)⟩, are calculated as [19]\nGθθ\njδ(x, t) =Kjδ\n4(I(x, t) +I(−x, t)), (A9)\nGϕϕ\njδ(x, t) =1\n4Kjδ(I(x, t) +I(−x, t)), (A10)\nGθϕ\njδ(x, t) =Gϕθ\njδ(x, t) =1\n4(I(x, t)−I(−x, t)),(A11)\nI(x, t) =−log\u00142iβℏvF\nLsinh\u0012π(ia−x−vFt)\nβℏvF\u0013\u0015\n.\n(A12)\nUsing these correlation functions, we obtain\nFrαr′α′(x, t) =F1δr,r′δα,α′+F2δr,−r′δα,α′\n+F3δr,r′δα,−α′+F4δr,−r′δα,−α′,(A13)\nF1(x, t) =˜Gθθ\ns++˜Gθθ\ns−+˜Gϕϕ\ns++˜Gϕϕ\ns−\n−r(˜Gθϕ\ns++˜Gθϕ\ns−+˜Gϕθ\ns++˜Gϕθ\ns−),(A14)\nF2(x, t) =˜Gθθ\nc++˜Gθθ\nc−+˜Gϕϕ\ns++˜Gϕϕ\ns−, (A15)\nF3(x, t) =˜Gθθ\ns++˜Gθθ\nc−+˜Gϕϕ\ns++˜Gϕϕ\nc−\n−r(˜Gθϕ\ns++˜Gθϕ\nc−+˜Gϕθ\ns++˜Gϕθ\nc−),(A16)\nF4(x, t) =˜Gθθ\nc++˜Gθθ\ns−+˜Gϕϕ\ns++˜Gϕϕ\nc−, (A17)\nwhere ˜GXY\njδ(x, t)≡GXY\njδ(x, t)−GXY\njδ(0,0). Com-\nbining these results enables the correlation function\nCrαr′α′(x, y, t ) to be obtained analytically.8\nAppendix B: Analytic expressions of integrals\nFor a clean interface, the increase in damping is given\nas\nδαG,1=−4S0J2\n1\nℏ2ω0(2πa)2NFIIγ, (B1)\nIγ=v2\nF\u0012βℏ\nπ\u00133Zw\n0dx′Zw\n0dy′Z∞\n0dusin(˜ω0u)\n×Im(\u0014sinh(iα)\nsinh(iα+x−y−u)\u0015γ+1\n×\u0014sinh(iα)\nsinh(iα−x+y−u)\u0015γ−1)\n, (B2)\nwhere ˜ ω0=βℏω0/π,u=πt/β ℏ,w=πW/β ℏvF,x′=\nπx/β ℏvF,y′=πy/β ℏvF, and α=πa/β ℏvF. Changing\nvariables from x′andy′with Z= (x+y)/2 and z=x−y,\nthe integral is modified as\nIγ=v2\nF\u0012βℏ\nπ\u00133Z∞\n0dusin(˜ω0u)\n×\"Zw/2\n0dZZ2Z\n−2Zdz+Zw\nw/2dZZ2(w−Z)\n−2(w−Z)dz#\n×Im(\u0014sinh(iα)\nsinh(iα+z−u)\u0015γ+1\n×\u0014sinh(iα)\nsinh(iα−z−u)\u0015γ−1)\n=−v2\nF\n4\u0012βℏ\nπ\u00133Zw/2\n0dZZ−2z\n2zdzZ∞\n−∞du\n×(ei˜ω0u−e−i˜ω0u)\n×\u0014sinh(iα)\nsinh(iα+z−u)\u0015γ+1\u0014sinh(iα)\nsinh(iα−z−u)\u0015γ−1\n.\n(B3)\nIn the last equation, we have used the relation\n\u0012sinh(iα)\nsinh(iα±z−u)\u0013∗\n=sinh(iα)\nsinh(iα∓z+u)(B4)\nand the symmetry of the integrand with respect to Z↔\nw−Zandz↔ −z. At this stage, it is useful to introduce\nBγ(ζ, z) =Z∞\n−∞du eiζu\u0014sinh(iα)\nsinh(iα+z−u)\u0015γ+1\n×\u0014sinh(iα)\nsinh(iα−z−u)\u0015γ−1\n, (B5)\nso that\nIγ=−v2\nF\n2\u0012βℏ\nπ\u00133Zw/2\n0dZZ2Z\n−2Zdz\n×[Bγ(˜ω0, z)− Bγ(−˜ω0, z)]. (B6)Setting v= 2uand rearranging the hyperbolic sine func-\ntion, we obtain\nBγ(ζ, z) =1\n2(1−e−2iα)2γe−2zZ∞\n−∞dv\n×e−v(γ−iζ/2)\n[e−v+e−2z+i(π−2α)]γ+1[e−v+e−2z+i(π−2α)]γ−1,\n(B7)\nwhich can be computed analytically, invoking the formula\n3.315.1 in Ref. [29] as\nBγ(ζ, z)\n=1\n2(1−e−2iα)2γe−i(π−2α)(γ+iζ/2)e−2z(γ−1−iζ/2)\n×|Γ(γ+iζ/2)|2\nΓ(2γ)F(γ−1, γ,2γ; 1−e−4z), (B8)\nwhere F(a, b, c ;x) is the Gauss hypergeometric function.\nTo leading order in the small parameter α(≪1), this\nreduces to\nBγ(ζ, z) =1\n2(2α)2γeπζ/2+iζze−2z(γ−1)\n×|Γ(γ+iζ/2)|2\nΓ(2γ)F(γ−1, γ,2γ; 1−e−4z).(B9)\nIn practice, we are mostly interested in the regime where\n˜ω0=βℏω0≪1 so we can focus on values of ζsuch\nthat|ζ| ≪1. This allows us to expand Bγ(ζ, z) for small\nvalues of ζ, which yields, after substituting back into the\nexpression for Iγ,\nIγ=−π\n2v2\nFω0\u0012βℏ\nπ\u00134\n(2α)2γΓ(γ)2\nΓ(2γ)Zw/2\n0dZ\n×Z2Z\n0dze−2z(γ−1)F(γ−1, γ,2γ; 1−e−4z),(B10)\nwhere we have used the symmetry of the integrand with\nrespect to z↔ −z. Combining this result with Eq. (B1),\nEqs. (32) and (33) can be derived, after rewriting the\nintegral variable as z′= 2Z.\nIn the limiting case, the integral I(w, γ) given in\nEq. (33) can be approximated into a simple form. For the\nshort-junction limit ( w/π=W/β ℏvF=W/L th≪g(γ)),\nwe obtain\nI(w, γ)≃1\nwZw\n0dz′Zz′\n0dz,e−2z(γ−1)=w\n2.(B11)\nFor the long-junction limit ( w/π≫g(γ)),\nI(w, γ)\n≃1\nwZw\n0dz′Z∞\n0dz,e−2z(γ−1)F(γ−1, γ,2γ; 1−e−4z)\n=π\n2g(γ), (B12)9\nwhere g(γ) is defined by Eq. (43). These analytical ex-\npressions lead to Eqs. (41) and (42) in the main text.\nFor a dirty interface, the increase in damping is ex-\npressed as\nδαG,2=−S0J2\n2aW\nℏ2ω0(πa)2NFIX\nr,r′,α,α′I′\nγrαr ′α′, (B13)\nI′\nγ=βℏ\nπZ∞\n0dusin(˜ω0u)Im(\u0014sinh(iα)\nsinh(iα−u)\u00152γ)\n,\n(B14)\nwith the same dimensionless variables as for a clean in-\nterface. By a similar way as the clean case, the integral\nI′\nγis modified as\nI′\nγ=−βℏ\n4π[Aγ(−˜ω0)−Aγ(−˜ω0)], (B15)\nAγ(ζ) =Z∞\n−∞du e−iζu\u0014sinh(iα)\nsinh(iα−u)\u00152γ\n. (B16)\nSetting v= 2uand rearranging the hyperbolic sine func-tion, we obtain\nAγ(ζ) =1\n2(1−e−2iα)2γZ∞\n−∞dve−(γ+iζ/2)v\n(e−v+ei(π−2α))2γ.\n(B17)\nInvoking the formula 3.314 in Ref. [29], this can be com-\nputed as\nAγ(ζ) =1\n2(2 sin α)2γeαζe−πζ/2|Γ(γ+iζ/2)|2\nΓ(2γ),(B18)\nwhich then yields, to leading order in α(≪1),\nI′\nγ=−βℏ\n4π(2α)2γ|Γ(γ+iζ/2)|2\nΓ(2γ)sinh(π˜ω0/2).(B19)\nAssuming ˜ ω0=βℏω0/π≪1, this is further simplified as\nI′\nγ=−β2ℏ2\n8πω0\u00122πa\nβℏvF\u00132γΓ(γ)2\nΓ(2γ), (B20)\nwhich finally leads to Eq. 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However, unlike the charge density waves, the theoretical analysis of the sliding\nmotion of SkX and HL remains unclear, especially regarding the similarities and differences in sliding\ndynamics between these two spin density waves. In this work, we systematically explore the sliding\ndynamics of SkX and HL in chiral magnets in the limit of large current density. We demonstrate\nthat the sliding dynamics of both SkX and HL can be unified within the same theoretical framework\nas density waves, despite their distinct microscopic orders. Furthermore, we highlight the significant\nrole of gyrotropic sliding induced by impurity effects in the SkX state, underscoring the impact of\nnontrivial topology on the sliding motion of density waves. Our theoretical analysis shows that\nthe effect of impurity pinning is much stronger in HL compared with SkX, i.e., χSkX/χHL∼α2\n(χSkX,χHL: susceptibility to the impurity potential, α(≪1) is the Gilbert damping). Moreover,\nthe velocity correction is mostly in the transverse direction to the current in SkX. These results are\nfurther substantiated by realistic Landau-Lifshitz-Gilbert simulations.\nIntroduction.— Density waves in solids represent a\nprevalent phenomenon, particularly in low-dimensional\nsystems [1, 2]. They break the translational symmetry\nof the crystal, leading to the emergence of Goldstone\nbosons, i.e., phasons, which remain gapless when the pe-\nriod of density waves is incommensurate with the crystal\nperiodicity. The sliding motion of density waves under an\nelectric field Ehas been extensively studied. In this con-\ntext, the impurity pinning of phasons results in a finite\nthreshold field [1, 2]. In general, exploring the dynam-\nics of pinning and depinning offers valuable insights into\nunderstanding the behavior of density waves.\nThe skyrmion crystal (SkX) and helix (HL) phases\nin chiral magnets can be recognized as periodic density\nwaves of spins, as depicted in Figs. 1 (a) and (b). The\nHL phase is stabilized in chiral magnet at small mag-\nnetic field regions, with spins of neighboring magnetic\nmoments arranging themselves in a helical pattern. SkX\nis a superposition of three phase-locked HL and comprises\narrays of magnetic skyrmions, nanoscale vortex-like spin\ntextures characterized by a non-zero skyrmion number\nNsk=1\n4πR R\nd2rs·(∂xs×∂ys) (sbeing the unit vec-\ntor of spin). Theoretically proposed magnetic skyrmions\n[3–5] were initially observed in the chiral magnet MnSi\nunder magnetic fields [6–8], wherein the skyrmion lattice\nstructure produces a six-fold neutron scattering pattern.\nSince then, the chiral magnetic states encompassing SkX\nand HL states have been the focus of extensive research\n[9–13].\nThe dynamics of SkX in a random environment, specif-\nically the pinning effects from impurities, are manifested\nthrough the topological Hall effect. The current depen-\ndence of topological Hall resistivity ρxywas initially ex-\nplored theoretically by Zang et al [14] and experimentally\n∗nagaosa@riken.jpby Schulz et al. [15]. To illustrate, a schematic plot is\npresented in Fig. 1(c). Typically, there are three distinct\nregions characterizing the dynamics of SkX: the pinned,\ncreep, and flow regions. The topological Hall resistiv-\nity decreases when SkX is depinned because the motion\nof SkX induces temporal changes in the emerging mag-\nnetic fields Be, subsequently generating emergent elec-\ntric fields Eeand an opposing Hall contribution. Theo-\nretically, the pinning problem of both SkX and HL was\ninvestigated in terms of replica symmetry breaking [16],\nrevealing a distinct difference in glassy states between\nSkX and HL. The key factor lies in the nontrivial topol-\nogy of SkX, contrasting with the trivial topology in HL\nand most density wave states. However, this difference\nhas not been theoretically explored in the context of slid-\ning/moving density wave states for chiral magnets.\nIn this work, we systemically study the current-driven\nsliding dynamics of the SkX and HL in chiral magnets.\nWe employ the methodology proposed by Sneddon et al.\n[17] in their investigation of charge density waves and\napply it to magnetic materials. This method allows us\nto investigate the current-driven dynamics of SkX and\nHL, considering both deformation and impurity pinning\neffects. Through this method, we reveal that the drift ve-\nlocity correction ∆ vddue to the impurity pinning effects\nversus the current density jsin the flow region, follows\n∆vd∝(vd0)d−2\n2(−e∥+G\nαDe⊥) for the SkX phase, while\n∆vd∝ −(vd0)d−2\n2e∥for the HL phase with the spatial di-\nmension denoted as d. Here, e∥represents the direction\nof the intrinsic drift velocity vd0(the magnitude of vd0is\nproportional to the current density jsdue to the univer-\nsal linear current-velocity relation [18]), G= 4πNsk,Dis\na form factor at the order of unity, α≪1 is the Gilbert\ndamping parameter so that G/αD ≫1. Although the\nscaling relation ( vd0)d−2\n2applies to both SkX and HL, we\ncan see that the gyrodynamics of the SkX state inducedarXiv:2312.07116v2  [cond-mat.mes-hall]  13 Dec 20232\n(a)\n(c)\nvd\nvs(b)\n(d)\nHL SkX\nρxy\nvdCreep Flow pinned\njs\nFIG. 1. (a), (b) The current-driven motion of the SkX and\nHL, respectively. (c) Schematic of the Hall resistivity ρxy\nand drift velocity vdversus current density jswith pinned\n(yellow), creeping (green), and flowing (purple) highlighted.\n(d) The collective flow motion of the SkX, where the center\nof each skyrmion (red dots) and the impurities (black crosses)\nare highlighted.\nby its nontrivial topology results in its sliding dynamics\nmore robust than in HL and mostly in the transverse di-\nrection. Finally, we explicitly conduct the micromagnetic\nsimulations on both the SkX and HL systems, aligning\nwell with our theoretical expectations.\nOur work demonstrates the unification of sliding dy-\nnamics between spin density waves and charge density\nwaves within the same theoretical framework. Our re-\nsults also vividly illuminate both the similarities and dif-\nferences in the sliding dynamics between SkX and HL\nphases. This insight significantly enhances our under-\nstanding of the sliding dynamics associated with topo-\nlogical density wave phenomena, which possesses pos-\nsible applications in areas, such as skyrmion-based de-\nvices [19–21], depinning dynamics [22–27], Hall responses\n[14, 15, 28], and current-driven motion of Wigner crystals\nunder out-of-plane magnetic fields [29–31].\nSliding dynamics for skyrmion crystals.— The current-\ndriven motion of SkX is described by the Thiele equation\nassuming that its shape does not change [18, 32, 33]:\nG×(vs−vd) +D(βvs−αvd) +F= 0. (1)\nHere, the first term on the left represents the Magnus\nforce, the second term is the dissipative force, and the\nlast term arises from deformation and impurity-pinning\neffects. Here, vsis the velocity of conduction electrons,\nαis the damping constant of the magnetic system, and β\ndescribes the non-adiabatic effects of the spin-polarized\ncurrent. The gyromagnetic coupling vector is denoted as\nG= (0,0,4πNsk), and the dissipation matrix Dij=δijD\nwhere i, j∈ {x, y}. It is noteworthy that the Thieleequation respects out-of-plane rational symmetry [Sup-\nplementary Material (SM) Sec. IA [34]].\nTo obtain the equation of motion of SkX, the displace-\nment vector field of skyrmions is defined as u(r, t) so\nthat the drift velocity vd=∂u(r,t)\n∂t, where ris the posi-\ntion vector, tis the time. The force Fcan be expressed\nwithu(r, t) asF(r, t) =Fimp+Fde, where the impurity\npinning force Fimp=−P\ni∇U(r+u(r, t)−ri)ρ(r) =\nfimp(r+u(r, t))ρ(r) and the deformation force Fde=R\ndr′D(r−r′)u(r′, t′). Here, U(r−ri) is the impu-\nrity potential around site ri,D(r−r′) characterizes the\nrestoration strength after deformation, and ρ(r) is the\nskyrmion density. Based on these definitions, the Thiele\nequation can be expressed as an equation of motion:\n∂u(r, t)\n∂t=ˆM0vs+ˆM1Z\ndr′D(r−r′)u(r′, t)\n+ˆM1fimp(r+u(r, t))ρ(r). (2)\nwhere ˆM0=1\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\nand\nˆM1=1\nG2+α2D2\u0012\nαD G\n−G αD\u0013\n. Note that each skyrmion is\nnow considered as a center-of-mass particle, and these\nskyrmions form a triangular lattice and move collec-\ntively with scatterings from impurities, as illustrated in\nFig. 1(d).\nThe displacement vector can be expanded around the\nuniform motion,\nu(r, t) =vdt+˜u(r, t). (3)\nHere,vdis the dominant uniform skyrmion motion veloc-\nity,˜u(r, t) characterizes a small non-uniform part. Using\nthe Green’s function approach to solve the differential\nequation Eq. (2), ˜u(r, t) can be obtained as [17, 29, 34]\n˜u(r, t) =Z\ndr′Z\ndt′G(r−r′, t−t′){vd0−vd\n+M1fimp(r′+vdt′+˜u(r′, t′))ρ(r′)},(4)\nwhere the intrinsic drift velocity vd0=ˆM0vs, the Fourier\ncomponent of the Green’s function Gis given by\nG���1(k, ω) =−iω−ˆM1D(k). (5)\nHere,D(k) arises from the Fourier transformation of de-\nformation D(k) =R\ndd(r)D(r)e−ik·r(the spatial dimen-\nsion is denoted as d).\nIn the flow region, ˜u(r, t) in Eq. (4) can be solved\nperturbatively. Up to the second order, ˜u(r, t)≈\n˜u0(r, t) +˜u1(r, t) +˜u2(r, t), which, respectively, are ob-\ntained by replacing the terms in the brackets of Eq. (4) as\nM0vs−vd, M1fimp(r′+vdt′)ρ(r′), M1∇fimp(r′+vdt′)·\n˜u1(r, t)ρ(r′). Based on this approximation and making\nuse of ⟨˜u(r,t)\n∂t⟩= 0, the self-consistent equation for the3\nvelocity reads (for details see SM Sec. IB [34])\nvd=vd0+X\ngZddq\n(2π)d|ρ(g)|2Λ(q)ˆM1×\n\u0012q2\nxqxqy\nqxqyq2\ny\u0013\nIm [G(q−g,−q·vd)]ˆM1\u0012\nqx\nqy\u0013\n.(6)\nwhere ρ(g) is the Fourier component of ρ(r) with gas the\nreciprocal skyrmion lattice vectors, and Λ( q) arises from\nthe impurity average U(q1)U(q2) = (2 π)dΛ(q2)δ(q1+\nq2). Note that the crucial aspects for the above method\nto be valid are (i) the impurity strength is weak, (ii) the\ndrift velocity is large compared to the impurity effects\nand the SkX remains elastic, (iii) the deformation within\neach skyrmion is negligible so that each skyrmion can be\nregareded as a point object.\nTo proceed further, we adopt the following approxi-\nmations. The current-driven distortion is expected to be\nweak so that D(k) would be dominant by the long-wave\nlimit. In this case, D(k) can be expanded as Kxk2\nx+Kyk2\ny\nfor the 2D case and as Kxk2\nx+Kyk2\ny+Kzk2\nzfor the 3D\ncase. On the other hand, the characterized frequency\nthat enters into the Green’s function is q·vd∼vd/a.\nUsing a reasonable parameter vd= 10 m/s, the skyrmion\nlattice constant a= 25 nm, we estimate vd/a∼0.4 GHz.\nThis frequency is much smaller compared with the one of\nKj, which is roughly the scale of exchange energy J∼1\nmeV∼240 GHz [14, 18]. As a result, the dominant\ncontribution to the integral is given by the elastic modes\nvdgj≈ωk≈ D(k)/√\nG2+α2D2withk=q−g→0,\naround which the imaginary part of Green’s function is\nthe largest.\nWith the above approximations, we perform the in-\ntegral in Eq. (6) and sum over the smallest gvectors:\ngj=√\n3κ0(sin(j−1)π\n3,cos(j−1)π\n3) with jas integers from\n1 to 6 and κ0=4π\n3a. Since the Thiele equation exhibits\nout-of-plane rotational symmetry, without loss of gen-\nerality, we set vd0along x-direction here. After some\nsimplifications (for details see SM. Sec IB), we find the\ncorrection (∆ vd=vd−vd0) on the drift velocity due to\nthe impurity and deformation are given by\n∆vd≈χSkX\nd(vd0)d−2\n2(−e∥+G\nαDe⊥), (7)\nwhere the susceptibility to the impurity potential\nχSkX\nd =9κ3\n0|ρ1|2Λ0αD\n4√\nKxKy(G2+α2D2)ford= 2, while χSkX\nd =\n9√\n3κ7/2\n0Γ(G2+α2D2\n4α2D2)|ρ1|2Λ0(αD)3/2\nπ2√\nKxKyKz(G2+α2D2)ford= 3 (the function\nΓ(a) =R+∞\n0dxx6\n(x4−a)2+x4). Note that we have replaced\nρ(gj) =ρ1,Λ(gj) = Λ 0given the six-fold rotational sym-\nmetry of the skyrmion lattice.\nThe first important aspect in Eq. (7) is that the cor-\nrection ∆ vdis insensitive to vd0in 2D limit but follows\na square-root scaling: ( vd0)1/2in 3D limit. Similar to\nmany scaling phenomena, the dimension plays a critical\n0 0.005 0.01-1.5-1-0.50\n0 0.005 0.0100.510 0.005 0.01-1.5-1-0.50\n0 0.005 0.0100.511.5\n0 0.005 0.01-1.5-1-0.50\nSkyrmion 3DSkyrmion 2D, G < 0\ni=i=||Helix 2D\nSkyrmion 2D, G > 0(a) (b)\n(c) (d)\n(e) (f)-10 -8 -6 -4-2.5-2-1.5-1-0.50\nLog(vd0/K)Log(|∆vd|) Slope ≈ 0.5\n0°30°60°90°\n120°\n150°\n180°\n210°\n240°\n270°300°330°00.20.40.60.81\ni=i=||\ni=i=||\nHelix 3DFIG. 2. (a) and (b) The correction ∆ vdversus vd0(in units\nofK) for the 2D and 3D HL state, respectively. (c) and (d)\nThe correction ∆ vdversus vd0(in units of K) for the 2D\nSkX with G= 4πandG=−4π, respectively, where the\nlongitudinal (transverse) component is in blue (red). The 3D\nSkX case is plotted in (f) with G= 4π. (e) The angular-\ndependence of |∆vd(θ)|, where θis the angle of vd0. All the\nof ∆vdin these plots has been normalized. The parameters\nD= 5.577π,α= 0.04.\nrole here. The second important aspect is that the cor-\nrection along the transverse direction directly reflects the\nskyrmion topological number Gwith the ratio compared\nto the longitudinal one as G/αD . These interesting as-\npects embedded in the Eq. (7) will be further highlighted\nlater.\nHelix case.— It is straightforward to generalize the\nabove treatment to the helical spin order. The Thiele\nequation is reduced to one dimension:\nD(βvs−αvd) +F= 0. (8)\nThe essential difference here is the absence of gyrotropic\ncoupling ( G= 0). Following the same procedure [SM\nSec. II], the self-consistent equation for the drift velocity\nis given by\nvd=vd0+Zddq\n(2π)dX\ng|ρ(g)|2\nα2D2Λ(q)q3\nxIm[G(q−g,−qxvd)].\n(9)\nHere, vd0=β\nαvs, the flow direction of the HL is defined\nasx-direction. After adopting the approximation in the4\nprevious section, the analytical expression of the correc-\ntion ∆ vdof the helical magnetic state is\n∆vd≈ −χHL\nd(vd0)d−2\n2 (10)\nwhere χHL\nd =(KxKy)−1/2|ρ1|2Λ0g3\n0\n4αD,ford= 2\n(KxKyKz)−1/2|ρ1|2Λ0g7/2\n0\n2√\n2π(αD)1/2 ,ford= 3 with g0=π\na. Despite\ndifferent magnetic state nature, the ∆ vdas a function\nvd0in Eq. (10) for the HL displays a consistent scaling\nbehavior as the one of SkX shown in Eq. (7).\nNumerical evaluation.— To further justify our analyt-\nical results, we calculate the ∆ vdnumerically according\nto Eqs. (6) and (9). For simplicity, we set the elastic\ncoefficient Kjas isotropic with K≡Kj. Figs. 2(a) and\n(b) display the correction ∆ vdas a function of vd0of\nHL. Note that the zero drify velocity limit should be ig-\nnored since the impurity pinning effect would be dom-\ninant in practice. When the vdis beyond this limit so\nthat the pinning effect can be treated as a perturbation,\nwhich is true for the flow region, the plots clearly indi-\ncate ∆ vd∝(vd0)d−2\n2. The square root behavior in 3D\n(d= 3) is explicitly checked with the log-log plot (inset\nof Fig. 2(b)).\nFigs. 2(c) and (d) show the longitudinal component\n(blue) and transverse component (red) of the correction\nfor the SkX case with positive Gand negative G, respec-\ntively. It is consistent with Eq. (7) that the transverse\ncomponent is odd with respect to Gand is much larger\nthan the longitudinal component as G/αD ≫1. This\ngyrotropic type correction is inherited from the Magnus\nforces in the Thiele equation, and this correction also\nimplies that there exists a net change on the skyrmion\nHall angle due to the impurities. Moreover, the angu-\nlar dependence of the total correction |∆vd|is shown in\nFigs. 2(e), where the anisotropy is very small. The ∆ vd\nas a function vd0also displays distinct scaling behavior\nbetween 2D [Figs. 2(c),(d)] and 3D case [Fig. 2(f)].\nOverall, the scaling behavior of skyrmion similar to\nthat of the HL in Fig. 2, as expected from our theoretical\nanalysis. Moreover, the intrinsic drift velocity vd0is lin-\nearly proportional to the current density jsfor both SkX\nand HL ( vd0∝js) at large js. As a result, we can replace\nvd0with jsin the scaling relation, i.e., ∆ vd∝(js)d−2\n2. It\nis worth noting that the charge density wave also respects\nthis scaling relation [17], despite its distinct microscopic\nnature.\nPhysical interpretation.— Now we provide a physical\ninterpretation of the observed scaling behavior: ∆ vd∝\n(vd0)d−2\n2. As we mentioned earlier, the dominant con-\ntribution to the drift velocity correction arises from the\nexcitation of elastic modes. Hence, we expect the correc-\ntion to be proportional to the number of excited elastic\nmodes at a fixed vd0. For the SkX case, these modes\nfollow the dispersion: vd0|gj|=D(k)/√\nG2+α2D2,\nwhich can be rewritten as vd0=kd/2m′with m′=\n2π√\nG2+α2D2/(√\n3aK). Next, the problem is mapped\nto evaluate the density of states of free fermions with\n0.050.10.150.20.251234\n0.10.20.30.40.52D HL2D SkX0102030051015202530vd,x clean  \njs (1010 A/m2)vd (m/s)2D SkX, weak impuritiesvd,x weak impurities  vd,y defect  vd,y weak impurities  \n0102030051015202530cleanweak impurities2D HL, weak impuritiesvd (m/s)\njs (1010 A/m2)(a)(b)\n(d)∆vd (m/s)α(c)0102030012345|Δvd/<Δv∥>||Δv∥/<Δv∥>||Δv⊥/<Δv∥>|G/αD⃗vd⃗y⃗xΔ⃗v⊥Δ⃗v∥⃗vd0finalColumn 6Column 5|Δvd|FIG. 3. Simulation results of LLG equation. (a) and (c)\nThe drift velocity vdversus current density js(in units of\n1010A/m2) of the 2D SkX and HL at the clean and impurity\ncase. (b) The magnitdue of longitutinal (∆ d,∥) and trans-\nverse (∆ d,⊥) drift velocity correction versus the current den-\nsity (normalized the avegare value of ∆ d,∥), where the G/αD\nratio is highlighted (red dashed line). The coordinate relation\nbetween different vectors are shown in the inset. (d) the drift\nvelocity correction as a function of the damping parameter α\natjs= 2×1011A/m2(only vd,xis used for the SkX). For (a)\nto (c), α= 0.2 and β= 0.5αare employed in the simulations.\nan energy vd0. Recall that the density of states of\nfree fermion N(E)∝Ed−2\n2at energy E. Hence, it is\nexpected that the correction follows the same scaling:\n∆vd∝(vd0)d−2\n2according to this argument. We em-\nphasize that the microscopic nature of the density waves\nin this argument are not essential, which mainly stems\nfrom the long-wave characteristic of elastic modes. This\nexplains why the HL and charge density wave also follow\nthe same scaling behavior.\nMicromagnetic simulation.— We now further validate\nour theory through solving the Landau–Lifshitz–Gilbert\n(LLG) equation with the spin transfer torque effect [35–\n39] (for details see SM). The calculated drift velocity vd\nversus current density jscurves are shown in Fig. 3 for\nboth the SkX and HL. For simplicity, we mainly focus\non 2D SkX and HL with weak impurities here, where\nour analytical expressions from perturbation theory are\napplicable.\nFigs. 3(a) and (c) show vdin the clean and disordered\ncase with α= 0.2. The correction between these two\ncases at both SkX and HL is indeed insensitive to the\ncurrent density within the flow limit. It is noteworthy\nthat due to the gyrodynamcs, the SkX exhibits a much\nsmaller depinning critical current density. Fig. 3(b) is to\nshow that the correction along the transverse direction is\nobviously larger than the longitudinal one with the ratio5\n∼G/αD , being consistent with Eq. (7). Interestingly,\nthe longitudinal correction versus the damping parame-\nterαof 2D SkX and HL show a positive and negative\ncorrelation, respectively [Fig. 3(d)], which is also consis-\ntent with our analytical expressions [see χSkX\ndandχHL\nd\nin Eqs. (7) and (10)]. It can also be seen that the im-\npurity correction along longitudinal direction is typically\nmuch stronger in HL than in SkX as χSkX\nd/χHL\nd∼α2.\nThese distinct features between SkX and HL highlight\nthe importance of the nontrivial topology in the sliding\ndynamics of density waves.\nDiscussion.— We have provided a thorough analy-\nsis of the sliding dynamics exhibited by the SkX and\nHL phases, highlighting both their similarities and dif-\nferences in terms of density waves sliding with distinct\ntopologies. Our theory could have broader applica-\ntions. For instance, one can explore the relationship be-\ntween the topological Hall effect and the current den-\nsity in the flow region. In the clean limit, the uni-\nversal linear current-velocity relation vd0∝jsimplies\nthat the topological Hall resistivity ρxy, proportional to\n|(vs−vd0)×Be|/|vs|[15], is expected to exhibit a plateau\nin the flow region, as illustrated in Fig. 1(c). In thepresence of impurities, the topological Hall resistivity is\nmodified to ρxy∝ |(vs−vd0−∆vd)×Be|/|vs|. Consid-\nering that ∆ vd∝(vd0)(d−2)/2, we anticipate a modified\nrelation ρxy=a+bj−2+d/2\ns , where aandbremain inde-\npendent of the current magnitude js. The second term,\nbj−2+d/2\ns , represents the correction from impurities. 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Nagaosa, Sci Rep 8, 6328 (2018).1\nSupplementary Material for “ Sliding Dynamics of Current-Driven Skyrmion Crystal\nand Helix in Chiral Magnets ”\nYing-Ming Xie,1Yizhou Liu,1Nato Nagaosa,1\n1RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\nI. THE SLIDING DYNAMICS OF SKYRMION CRYSTAL WITH PINING AND DEFORMATION\nEFFECTS\nA. The skyrmion dynamics and Thiele equation\nFrom the Landau-Lifshitz-Gilbert equation, it was obtained that the current-driven skyrmion dynamics are captured\nby the Thiele equation:\nG×(vs−vd) +D(βvs−αvd) +F= 0. (S1)\nOne can rewrite the equation as\n−G(vsy−vdy) +D(βvsx−αvdx) +Fx= 0, (S2)\nG(vsx−vdx) +D(βvsy−αvdy) +Fy= 0. (S3)\nIn the matrix form:\n\u0012\nG αD\nαD−G\u0013\u0012\nvdx\nvdy\u0013\n=\u0012\nG βD\nβD−G\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nFy\nFx\u0013\n. (S4)\nThen,\n\u0012\nvdx\nvdy\u0013\n=\u0012\nG αD\nαD−G\u0013−1\u0012\nG βD\nβD−G\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nG αD\nαD−G\u0013−1\u0012\nFy\nFx\u0013\n. (S5)\n\u0012\nvdx\nvdy\u0013\n=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nG αD\nαD−G\u0013\u0012\n0 1\n1 0\u0013\u0012\nFx\nFy\u0013\u0015\n,\n=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nαD G\n−G αD\u0013\u0012\nFx\nFy\u0013\u0015\n(S6)\nWithout loss of generality, we can choose the current direction to be x-direction: vs= (vs,0). When the pinning\nforce is set to be F= 0, one can solve\nvd∥,0=G2+αβD2\nG2+α2D2vs,vd⊥,0=(α−β)GD\nG2+α2D2ˆz×vs. (S7)\nTherefore, the longitudinal drift velocity vdxis proportional to the electric current when the force Fis neglectable.\nIn a general direction, we can write the intrinsic drift velocity as\nvd0=1\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n(S8)\n=s\n1 +β2γ2\n1 +α2γ2\u0012\ncosθSkH−sinθSkH\nsinθSkH cosθSkH\u0013\u0012\nvscosθs\nvssinθs\u0013\n(S9)\n=vd0\u0012\ncos(θs+θSkH)\nsin(θs+θSkH)\u0013\n. (S10)\nHere, the angle θsis to characterize the applied current direction, the skyrmion Hall angle θSkH= atanγ(α−β)\n1+αβγ2with\nγ=D\nG, and the magnitude of drift velocity vd0=|vd0|=vsq\n1+β2γ2\n1+α2γ2.2\nNow we show that the Thiele equation respects rotational symmetry with the principal axis along z-direction from\nEq. (S6). The rotational operator is defined as Rz=\u0012\ncosϕ−sin(ϕ)\nsin(ϕ) cos( ϕ)\u0013\nwith ϕas the rotational angle. Under this\nrotational operation, Eq. (S6) becomes\nRz\u0012\nvdx\nvdy\u0013\n=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nG αD\nαD−G\u0013\u0012\n0 1\n1 0\u0013\u0012\nFx\nFy\u0013\u0015\n,\n=1\nG2+α2D2\u0014\nRz\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\nR−1\nzRz\u0012\nvsx\nvsy\u0013\n+Rz\u0012\nαD G\n−G αD\u0013\nR−1\nzRz\u0012\nFx\nFy\u0013\u0015\n(S11)\nIt is easy to show\nRz\u0012\nA B\n−B A\u0013\nR−1\nz=\u0012\nA B\n−B A\u0013\n(S12)\nwith AandBas constant. The Eq. (S13) is simplified as\nRz\u0012\nvdx\nvdy\u0013\n=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nG αD\nαD−G\u0013\u0012\n0 1\n1 0\u0013\u0012\nFx\nFy\u0013\u0015\n,\n=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\nRz\u0012\nvsx\nvsy\u0013\n+\u0012\nαD G\n−G αD\u0013\nRz\u0012\nFx\nFy\u0013\u0015\n(S13)\nHence, we have shown that the Thiele equation respects out-of-plane rotational symmetry.\nB. The correction on the drifted velocity due to the pining and deformation effects\nLet us define the displacement of skyrmion lattice as u(r, t) so that the drift velocity vd(r, t) =∂u(r,t)\n∂t. The force\nis given by\nF(r, t) = Fimp+Fde (S14)\nFimp =−X\ni∇U(r+u(r, t)−ri)ρ(r) =fimp(r+u(r, t))ρ(r) (S15)\nFde=Z\ndr′D(r−r′)u(r′, t′), (S16)\nwhere Fimdescribes the pining effect from impurities and Fdearises from the deformation of skyrmion lattice, U(r−ri)\nis the impurity potential around the site ri.ρ(r) is the skyrmion densities.\n∂u(r, t)\n∂t=1\nG2+α2D2\u0014\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\u0012\nαD G\n−G αD\u0013Z\ndr′D(r−r′)\u0012\nux(r′, t)\nuy(r′, t)\u0013\u0015\n+1\nG2+α2D2\u0012\nαD G\n−G αD\u0013\nfimp(r+u(r, t))ρ(r). (S17)\nThe displacement vector can be expanded around the uniform motion,\nu(r, t) =vdt+˜u(r, t). (S18)\nHere, vdis the dominant uniform skyrmion motion velocity, ˜u(r, t) characterizes a small non-uniform motion. Then\nthe equation of motion is written as\n\u0014∂\n∂t−1\nG2+α2D2\u0012\nαD G\n−G αD\u0013Z\ndr′D(r−r′)\u0015\u0012\n˜ux(r′, t)\n˜uy(r′, t)\u0013\n=1\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\n+1\nG2+α2D2\u0012\nαD G\n−G αD\u0013\nfimp(r+vdt+˜u(r, t))ρ(r).\n(S19)3\nHere, we have used D(q) = 0 in the long wave limit ( q→0) so thatR\ndr′D(r−r′) = 0.\nLet us try to solve the Green’s function of the operator at the left-hand side, which is given by\n\u0014∂\n∂t−1\nG2+α2D2\u0012\nαD G\n−G αD\u0013Z\ndr′D(r−r′)\u0015\nG(r′, t) =δ(t)δ(r)\u0012\n1 0\n0 1\u0013\n(S20)\nIt is more economical to work in the momentum space with\nG(r, t) =Zdω\n2πZddk\n(2π)de−iωt+ik·rG(k, ω). (S21)\nLet us define D(r) =Rddq\n(2π)deiq·rD(q), and then\nZ\ndr′Z\nD(r−r′)G(r′, t) =Z\ndr′Zddq\n(2π)deiq·(r−r′)D(q)Zddk\n(2π)dZdω\n2πG(k, ω)ei(k·r′−ωt)\n=Zddk\n(2π)dZdω\n(2π)G(k, ω)D(k)ei(k·r−ωt). (S22)\nIn the momentum space, we find\nG−1(k, ω) =−iω−1\nG2+ (Dα)2\u0012\nαD G\n−G αD\u0013\nD(k) (S23)\nTherefore, Eq. (S19) can be rewritten as\n˜u(r, t) =Z\ndr′Z\ndt′G(r−r′, t−t′){1\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd (S24)\n+1\nG2+α2D2\u0012\nαD G\n−G αD\u0013\nfimp(r′+vdt′+˜u(r′, t′))ρ(r′)}.\nIn the flow limit, the perturbation from the deformation and impurity can be regarded as small in comparison with\nthe leading order term. As a result, the displacement vector can be expanded as\n˜u0(r, t) =Z\ndr′Z\ndt′G(r−r′, t−t′)\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n, (S25)\n˜u1(r, t) =1\nG2+α2D2Z\ndr′Z\ndt′G(r−r′, t−t′)\u0012\nαD G\n−G αD\u0013\nfimp(r′+vdt′)ρ(r′), (S26)\n˜u2(r, t) =1\nG2+α2D2Z\ndr′Z\ndt′G(r−r′, t−t′)\u0012\nαD G\n−G αD\u0013\n∇fimp(r′+vdt′)·˜u1(r′, t′)ρ(r′).(S27)\nHere, u0,u1, and u2are the leading, first, and second order terms, respectively. Next, let us evaluate the volume-\naverage velocity\n\u001c∂˜u(r, t)\n∂t\u001d\n=\u001c∂˜u0(r, t)\n∂t\u001d\n+\u001c∂˜u2(r, t)\n∂t\u001d\n(S28)\nNote the fact that under the impurity average fimp(r′+vt′) = 0 has been used so that u1(r, t) would not contribute\ndirectly. Since non-uniform motion must vanish over the volume average, we can obtain a self-consistent equation for\nthe velocity vd. Next, let us work out the self-consistent equation for vd.\nThe leading order\n∂u0(r, t)\n∂t=Z\ndr′Z\ndt′G(r−r′, t−t′)\n∂t\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n=Z\ndr′Z\ndt′Zddk\n(2π)dZdω\n2πeik·(r−r′)−iω(t−t′)(−iω)G(k, ω)\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n= lim\nω→0−iωG(k= 0, ω)\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n=\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n(S29)4\nAs mentioned theD\n∂˜u1(r,t)\n∂tE\nwould not contribute, now let us show it explicitly. Recall that\n˜u1(r, t) =1\nG2+α2D2Z\ndr′Z\ndt′G(r−r′, t−t′)\u0012\nαD G\n−G αD\u0013\nfimp(r′+vdt′)ρ(r′) (S30)\nThen,\n∂˜u1(r, t)\n∂t=Z\ndr′Z\ndt′ 1\nG2+α2D2G(r−r′, t−t′)\u0012\nαD G\n−G αD\u0013\nfimp(r′+vdt′)ρ(r′) (S31)\n=−1\nG2+α2D2Z\ndr′Z\ndt′Zddk\n(2π)dZdω\n2πeik·(r−r′)−iω(t−t′)G(k, ω)\u0012\nαD G\n−G αD\u0013\n× (S32)\nZddq\n(2π)d\u0012\niqx\niqy\u0013\nUqeiq·(r′+vdt′)ρ(r′) (S33)\n= 0, (S34)\nbecause after averaging over the impurity configurations, Uk= 0.\nNow let us look at the second-order term\n∂˜u2(r, t)\n∂t=1\nG2+α2D2Z\ndr′Z\ndt′∂G(r−r′, t−t′)\n∂t\u0012\nαD G\n−G αD\u0013\n∇fimp(r′+vdt′)·˜u1(r′, t′)ρ(r′).(S35)\nNote that\n∇fimp(r′+vdt′)·˜u1(r′, t′) =−Zddq\n(2π)dU(q)eiq·(r′+vdt′)\u0012q2\nxqxqy\nqxqyq2\ny\u0013\u0012\n˜u1x(r′, t′)\n˜u1y(r′, t′)\u0013\n(S36)\nSubstitute the form of ˜u1(r′, t),\n∂˜u2(r, t)\n∂t=−1\n(G2+α2D2)2Z\ndr′Z\ndt′∂G(r−r′, t−t′)\n∂t\u0012\nαD G\n−G αD\u0013\nρ(r′)×\nZddq1\n(2π)dU(q1)eiq1(·r′+vdt′)\u0012q2\n1xq1xq1y\nq1xq1yq2\n1y\u0013\n×\nZ\ndt′′Z\ndr′′G(r′−r′′, t′−t′′)\u0012\nαD G\n−G αD\u0013\nρ(r′′)Zddq2\n(2π)d\u0012\niq2x\niq2y\u0013\nU(q2)eiq2·(r′′+vdt′′)(S37)\nThen write the terms at the right-hand side of the equation with their Fourier components,\n∂˜u2(r, t)\n∂t=−1\n(G2+α2D2)2Z\ndt′Z\ndr′Zddk\n(2π)dZdω\n2πG(k, ω)(−iω)eik·(r−r′)−iω(t−t′)\u0012\nαD G\n−G αD\u0013\n×\nZddq1\n(2π)d\u0012q2\n1xq1xq1y\nq1xq1yq2\n1y\u0013\nU(q1)eiq1·(r′+vdt′)X\ng1ρ(g1)eig1·r′×\nZ\ndt′′Z\ndr′′Zddk′\n(2π)dZdω′\n2πG(k′, ω′)eik′·(r′−r′′)−iω′(t′−t′′)\u0012\nαD G\n−G αD\u0013X\ng2ρ(g2)eig2·r′′×\nZddq2\n(2π)d\u0012\niq2x\niq2y\u0013\nU(q2)eiq2·(r′′+vdt′′)(S38)\nWe can take integrals with respect to the space and time, and take the average over disorders, several delta functions5\nwould appear on the right-hand side:\nU(q1)U(q2) = (2 π)dΛ(q2)δ(q1+q2), (S39)Z\ndt′eiωt′eiq1·vdt′e−iω′t′= 2πδ(ω−ω′+q1·vd), (S40)\nZ\ndr′e−ik·r′eiq1·r′eig1·r′eik′·r′= (2π)dδ(k′−k+q1+g1), (S41)\nZ\ndt′′eiω′t′′eiq2·vdt′′= 2πδ(ω′+q2·vd), (S42)\nZ\ndr′′e−ik′·r′′eig2·r′′eiq2·r′′= (2π)dδ(g2+q2−k′). (S43)\nTake the volume average, and consider constraints from delta functions: q2=−q1=q,g1=−g2=g,ω′=−q·vd,\nwe find\n\u001c∂˜u2(r, t)\n∂t\u001d\n=1\n(G2+α2D2)2X\ngZddq\n(2π)d|ρ(g)|2Λ(q)\u0012\nαD G\n−G αD\u0013\u0012q2\nxqxqy\nqxqyq2\ny\u0013\n×\nIm [G(q−g,−q·vd)]\u0012\nαD G\n−G αD\u0013\u0012\nqx\nqy\u0013\n. (S44)\nTherefore,\n\u001c∂˜u(r, t)\n∂t\u001d\n=\u00141\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n−vd\u0015\n+\n1\n(G2+α2D2)2X\ngZddq\n(2π)d|ρ(g)|2Λ(q)\u0012\nG αD\nαD−G\u0013\u0012q2\nxqxqy\nqxqyq2\ny\u0013\n×\nIm [G(q−g,−q·vd)]\u0012\nG αD\nαD−G\u0013\u0012\nqx\nqy\u0013\n(S45)\nSetD\n∂˜u(r,t)\n∂tE\n= 0, the self-consistent equation for the velocity is\nvd=1\nG2+α2D2\u0012\nG2+αβD2GD(β−α)\nGD(α−β)G2+αβD2\u0013\u0012\nvsx\nvsy\u0013\n+\n1\n(G2+α2D2)2X\ngZddq\n(2π)d|ρ(g)|2Λ(q)\u0012\nαD G\n−G αD\u0013\u0012q2\nxqxqy\nqxqyq2\ny\u0013\n×\nIm [G(q−g,−q·vd)]\u0012\nαD G\n−G αD\u0013\u0012\nqx\nqy\u0013\n. (S46)\nAs argued in the main text, the largest imaginary part is contributed by k=q−gin long wave limit ( kis small).\nTo further proceed, let us evaluate Im[ G(k, ω)].\nG(k, ω) =1\n−iω−1\nG2+(Dα)2\u0012\nαD G\n−G αD\u0013\nD(k)=−G2+α2D2\nλ(k)iω−D(k)\nλ(k)\u0012\nαD−G\nG αD\u0013\n, (S47)\nwhere\nλ(k) = [D(k) +ω(iαD+G)][D(k) +ω(iαD−G)] =D(k)2−(G2+α2D2)ω2+ 2iαDωD(k). (S48)\nThe imaginary part of Green’s function is given by\nIm[G(k, ω)] = −(G2+α2D2)[D2(k)−(G2+α2D2)ω2]ω\n[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)\n+2αDωD2(k)\n[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)\u0012\nαD−G\nG αD\u0013\n. (S49)6\nThe largest imaginary part is given by the real mode ωk=D(k)/√\nG2+α2D2. As a result, the first term in Im[ G(k, ω)]\ncan be negligible. In 2D, we can expand\nD(k) =Kxk2\nx+Kyk2\ny. (S50)\nNow we can show that\nZd2k\n(2π)22αDωD2(k)\n[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)\n= (KxKy)−1/2Z+∞\n0dk′2πk′\n(2π)22αDωk′4\n(k′4−(G2+α2D2)ω2)2+ 4α2D2ω2k′4\n=(KxKy)−1/2sgn(ω)\n4. (S51)\nwhere the integralR+∞\n0dtt2\n(t2−√\nG2+α2D2\n2αD)2+t2=π\n2is used with t=k′2\n2αD|ω|. Note that αD̸= 0 is taken. The\nmultiplications between matrices give\n\u0012\nαD G\n−G αD\u0013\u0012q2\nxqxqy\nqxqyq2\ny\u0013\u0012\nαD−G\nG αD\u0013\u0012\nαD G\n−G αD\u0013\u0012\nqx\nqy\u0013\n= (G2+α2D2)(q2\nx+q2\ny)\u0012\nGqy+αDq x\n−Gqx+αDq y\u0013\n.(S52)\nFinally, we obtain\nδvd=(KxKy)−1/2\n4(G2+α2D2)X\ng|ρ(g)|2Λ(g)sgn(−g·vd0)|g|2\u0012\nGgy+αDg x\n−Ggx+αDg y\u0013\n(S53)\nWe have shown that δvdrespects out-of-plane rotational symmetry in the main text, which is inherited from the\nTheiele equation. Without loss of generality, let us set vd0to be along x-direction. In this case, after summing over\nthe six smallest gvectors: gj=√\n3κ0(sin(j−1)π\n3,cos(j−1)π\n3) with κ0=4π\n3a,jare integers from 1 to 6, we find\nδvd=9κ3\n0|ρ1|2Λ0αD\n4p\nKxKy(G2+α2D2)\u0012−1\nG\nαD\u0013\n(S54)\nwhere ρ1=ρ(gj),Λ0= Λ(gj). In the 3D case,\nD(k) =Kxk2\nx+Kyk2\ny+Kzk2\nz. (S55)\nThen,\nZd3k\n(2π)32αDωD2(k)\n[D2(k)−(G2+α2D2)ω2]2+ 4α2D2ω2D2(k)\n= (KxKyKz)−1/2Z+∞\n04πk′2dk′\n(2π)32αDωk′4\n(k′4−(G2+α2D2)ω2)2+ 4α2D2ω2k′4\n=(KxKyKz)−1/2Γ(G2+α2D2\n4α2D2)sgn( ω)p\n2αD|ω|\n2π2. (S56)\nwhere Γ( a) =R+∞\n0dxx6\n(x4−a)2+x4. Similarly, we can obtain\nδvd=(2αD)1/2(KxKyKz)−1/2Γ(G2+α2D2\n4α2D2)\n2π2(G2+α2D2)X\ng|ρ(g)|2Λ(g)sgn(−g·vd0)p\n|g·vd0||g|2\u0012\nGgy+αDg x\n−Ggx+αDg y\u0013\n.(S57)\nAfter summing over the six smallest gvectors, we find\nδvd=9√\n3κ7/2\n0Γ(G2+α2D2\n4α2D2)|ρ1|2Λ0αD√αDv d0\nπ2p\nKxKyKz(G2+α2D2)\u0012−1\nG\nαD\u0013\n. (S58)7\nII. HELICAL SPIN ORDER CASE\nIn this section, we consider the helical spin order case. Without loss of generality, we denote the helical spin order\nhas a variation along x-direction. The Thiele equation for the helical spin order would be\nD(βvs−αvd) +F= 0. (S59)\nFor simplicity, we have omitted the index xin the following. The equation of motion becomes\n∂u(r, t)\n∂t=β\nαvs+1\nαDZ\ndr′D(r−r′)u(r′, t′) +1\nαDfimp(r+u(r, t))ρ(r). (S60)\nSimilarly, by defining u(r, r) =vdt+ ˜u(r, t), the equation of motion can be rewritten as\n[∂\n∂t−1\nαDZ\ndr′D(r−r′)]˜u(r′, t) =β\nαvs+1\nαDfimp(r+u(r, t))ρ(r). (S61)\nLet us define the Green’s function G(r, t) so that\n[∂\n∂t−1\nαDZ\ndr′D(r−r′)]˜u(r′, t)G(r′, t) =δ(r)δ(t). (S62)\nIt is easy to obtain\nG−1(k, ω) =−iω−D(k)\nαD. (S63)\nThe displacement ˜ u(r, t) is given by\n˜u(r, t) =Z\ndr′Z\ndt′G(r−r′, t−t′)[β\nαvs−vd+1\nαDfimp(r′+u(r′, t′))ρ(r′)]. (S64)\nThen up to the second order,\n˜u0(r, t) =Z\ndr′dt′G(r−r′, t−t′)[β\nαvs−vd], (S65)\n˜u1(r, t) =1\nαDZ\ndr′Z\ndt′G(r−r′, t−t′)fimp(r′+vdt′)ρ(r′), (S66)\n˜u2(r, t) =1\nαDZ\ndr′Z\ndt′G(r−r′, t−t′)∂xfimp(r′+vdt′)˜u1(r′, t′)ρ(r′). (S67)\nFollowing a similar procedure in Sec. II, using ⟨∂u(r,t)\n∂t⟩= 0, we obtain\nvd=vd0+1\nα2D2X\ng|ρ(g)|2Zddq\n(2π)dΛ(q)q3\nxIm[G(q−g,−qxvd)]. (S68)\nwhere the intrinsic drift velocity vd0=β\nαvs. Consider the Im[ G(k, ω)] is dominant in the long wave limit ( k→0) and\nexpand D(k) =Kxk2\nx+Kyk2\nyind= 2, we find\nvd≈vd0−(KxKy)−1/2\n8αDX\ng|ρ(g)|2Λ(g)|gx|3(S69)\nand similarly in d= 3 case, D(k) =Kxk2\nx+Kyk2\ny+Kzk2\nz, we obtain\nvd≈vd0−(KxKyKz)−1/2\n4√\n2π(αD)1/2X\ng|ρ(g)|2Λ(g)|gx|3p\n|gxvd0|. (S70)\nAfter summing over the smallest reciprocal lattice vectors for the helix: g= (±1,0)g0with g0=π\na, the correction\n∆vd=β\nαvs−χHL\nd(vd0)d−2\n2 (S71)\nwhere\nχHL\nd=\n\n(KxKy)−1/2|ρ1|2Λ0g3\n0\n4αD, ford= 2\n(KxKyKz)−1/2|ρ1|2Λ0g7/2\n0\n2√\n2π(αD)1/2 ,ford= 3(S72)\nHere, we have set ρ(g0) =ρ1and Λ( g0) = Λ 0in this case.8\nIII. NUMERICAL METHOD DETAILS\nA. Details for the main text Fig.2\nThe main text Fig.2 is obtained from the main text Eq. (6) and (9) by numerically integrating qand summing over\nthe smallest reciprocal lattice gvectors. For the SkX, the 4000 ×4000 in-plane momentum grids of qare taken with\na hexagonal boundary (the boundary length is 4 κ0); while for the HL, the 1000 ×1000 in-plane momentum grids are\ntaken with a square boundary (the boundary length is 4 g0). In the 3D case, the 1000 out-of-plane momentum points\nofqwithin [ −2g0,2g0] are used for both SkX and HL in evaluating the integral. Also, we set the elastic coefficients\nKj= 10000, the lattice constant aas a natural unit of one, the damping parameter α= 0.04, the dissipative coefficient\nD= 5.577π, the additional parameter G=±4πfor the SkX.\nB. Micromagnetic simulations\nThe micromagnetic simulations were performed using MuMax3 [35]. The Landau-Lifshitz-Gilbert (LLG) equation\nis numerically solved\n˙s=−|γ0|s×Heff+αs×˙s+pa3\n2eS(js· ∇)s−pa3β\n2eM2ss×(js· ∇)s, (S73)\nwhere sis the unit vector of spin, γ0is the gyromagnetic constant, αis the Gilbert damping constant, Msis the\nsaturation magnetization, and Heff=−1\nµ0MsδH\nδSis the effective field. The spin transfer torque effect of the current\nis described by the last two terms [36–38]. βdescribes the non-adiabaticity of the spin transfer torque effect. The\ncurrent is applied along the x-direction in the simulations.\nA typical chiral magnet can be described by the following Hamiltonian density\nH=A(∇S)2−DS·(∇ ×S)−µ0MsB·S (S74)\nThe corresponding parameters and their values employed in the simulations are: the saturation magnetization\nMs= 111 kA/m, the exchange stiffness A= 3.645 pJ/m, and the Dzyaloshinskii-Moriya interaction strength\nD= 0.567 mJ/m2. An external magnetic field B= 0.3 T (with its direction perpendicular to the skyrmion plane) is\nused in the simulations to stabilize the skyrmions. The simulations for helical state are performed at zero-field. The\ncell size is 1 nm ×1 nm×1 nm.\nWe consider magnetic impurities with uniaxial magnetic anisotropy Himp=−KimpS2\nz, where the easy-axis is\nperpendicular to the skyrmion 2D plane. For the weak impurity case, an impurity concentration x= 0.1% and\nimpurity strength Kimp= 0.2A/l2(lis the cell size) were used. For the strong impurity case, an impurity concentration\nx= 0.5% and impurity strength Kimp= 0.6A/l2were used. The simulation results are averaged over 100 impurity\ndistributions. The skyrmion velocity is extracted by using the emergent electric field method [39]. For each current\ndensity, the emergent electric field is also averaged over 100 time steps in order to get the skyrmion velocity. For the\ntransverse correction, the damping value α= 0.2 is employed in the main text Figs. 3(a) and (b) for computational\nefficiency, as using smaller damping values results in significantly longer simulation times to obtain a reasonable\ncorrection along the transverse direction."    },    {        "title": "1003.6092v1.Magnonic_Crystal_with_Two_Dimensional_Periodicity_as_a_Waveguide_for_Spin_Waves.pdf",        "content": "arXiv:1003.6092v1  [cond-mat.mes-hall]  31 Mar 2010APS/123-QED\nMagnonic Crystal with Two-Dimensional Periodicity as a\nWaveguide for Spin Waves\nRakesh P. Tiwari and D. Stroud\nDepartment of Physics,\nOhio State University,\nColumbus, OH 43210\n(Dated: October 29, 2021)\nAbstract\nWe describe a simple method of including dissipation in the s pin wave band structure of a\nperiodic ferromagnetic composite, by solving the Landau-L ifshitz equation for the magnetization\nwith the Gilbert damping term. We use this approach to calcul ate the band structure of square\nand triangular arrays of Ni nanocylinders embedded in an Fe h ost. The results show that there are\ncertain bands and special directions in the Brillouin zone w here the spin wave lifetime is increased\nby more than an order of magnitude above its average value. Th us, it may be possible to generate\nspin waves in such composites decay especially slowly, and p ropagate especially large distances, for\ncertain frequencies and directions in k-space.\nPACS numbers:\n1Theexistenceofaperiodicsuperlatticestronglyaffectsmanytype sofexcitationsinsolids.\nFor example, the electronic band structure of a conventional sem iconductor or semimetal[1],\nandthedispersionrelationsofelectromagneticwaves[2], elasticwav es[3–6], andspinwaves[7–\n11] are all greatly influenced by a periodic superlattice potential. In many cases, such\npotentials can give rise to new, and even complete, electronic, phot onic, elastic, or magnonic\nband gaps which may have important implications for the properties o f these materials.\nThese excitations have, by now, been extensively studied numerica lly and analytically, using\na variety of methods, and have been probed in many experiments[12 –15].\nInthepresent paper, we consider aparticular classofsuch excita tions, namely, spinwaves\nin periodic magnetic materials. Such magnetic superlattices are ofte n called magnonic crys-\ntals. We go beyond previous work by calculating the spin wave lifetimes in such materials.\nOur most striking finding is that the “figure of merit” (FOM) of these spin waves (product\nof spin wave frequency and lifetime) is strongly dependent on the Blo ch wave vector k, even\nthough, in our model, the same spin waves would have a k-independen t FOM in a homoge-\nneous magnetic material. This strong k-dependence suggests tha t magnetization in periodic\nmagnetic materials may be transported most efficiently by spin waves propagating along\nspecial directions in k-space. Possibly this k-dependence could be t ested by experiments in\nwhich spin waves are launched in particular directions corresponding to the largest FOMs.\nThis spin wave generation could be accomplished using real magnetic fi elds, or (via the spin\ntorque effect[16]) using spin currents. Measurements of spin wave lifetimes might be carried\nout, e. g., by neutron spin-echo techniques/citebayrakci.\nOur calculations are carried out for an array of infinitely long circular cylinders made\nof a ferromagnetic material Aembedded in another infinite ferromagnetic material B. All\nthe cylinders are taken to be parallel to the ˆ zaxis and their intersection with the xyplane\nforms a two-dimensional periodic lattice. We consider two arrangem ents of such cylinders: a\ntriangular and a square superlattice. An external static magnetic fieldH0is applied parallel\nto the axis of the cylinders, and both ferromagnets are assumed t o be magnetized parallel\ntoH0.\nTheequationofmotionforthisperiodiccompositeisgivenbytheLand au-Lifshitz-Gilbert\n(LLG) equation[18]:\n∂\n∂tM(r,t) =γµ0M(r,t)×Heff(r,t)+α\nMs(r)/parenleftbigg\nM(r,t)×∂\n∂tM(r,t)/parenrightbigg\n.(1)\n2Hereγis the gyromagnetic ratio, which is assumed to be the same in both fer romagnets,\nHeffis the effective field acting on the magnetization M(r,t),ris the position vector, αis\nthe Gilbert damping parameter and Msis the spontaneous magnetization. For this inho-\nmogenous composite Heffcan be written as\nHeff(r,t) =H0ˆz+h(r,t)+2\nµ0Ms/parenleftbigg\n∇·A\nMs∇/parenrightbigg\nM(r,t), (2)\nwhereh(r,t) is the dynamic dipolar field and Adenotes the exchange constant. The\nlast term on the right-hand side of eq. (2) denotes the exchange fi eld. For the two-\ncomponent composite we consider, the exchange constant, the s pontaneous magnetiza-\ntion and the Gilbert damping parameter take the forms A(r) =AB+ Θ(r)(AA−AB),\nMs(r) =Ms,B+ Θ(r)(Ms,A−Ms,B), andα(r) =αB+ Θ(r)(αA−αB), where the step\nfunction Θ( r) = 1 ifris inside ferromagnet A, and Θ( r) = 0 otherwise.\nWe separate the static and time-dependent parts of the magnetiz ation by writing\nM(r,t) =Msˆz+m(r,t), where m(r,t) =m(r)e−iωtis the time-dependent part of the\nmagnetization. The time-dependent dipolar field h(r)e−iωt, whereh(r) =−∇Ψ(r) and Ψ(r)\nis the magnetostatic potential. Since ∇ ·(h(r) +m(r)) = 0, the magnetostatic potential\nΨ(r) obeys∇2Ψ(r)−∇·m(r) = 0.\nWithinthelinear-magnonapproximation[19], thesmalltermsofsecon dorderin m(r)and\nh(r) are neglected in the equation of motion. This is equivalent to setting m(r)·ˆ z= 0[20].\nSubstituting the above equations into eqs. (1), we obtain\niΩmx(r)+∇·[Q∇my(r)]−my(r)−Ms\nH0∂Ψ\n∂y+iΩαmy(r) = 0,\niΩmy(r)−∇·[Q∇mx(r)]+mx(r)+Ms\nH0∂Ψ\n∂x−iΩαmx(r) = 0, (3)\nwhere Ω = ω/(|γ|µ0H0) andQ= 2A/(Msµ0H0).\nNext, usingtheperiodicityof Q,Msandαinthexyplane, wecanexpandthesequantities\nin Fourier series as Q(x)≡Q(x,y) =/summationtext\nGQ(G)eiG·x, with analogous expressions for Ms(x)\nandα(x). Here xandGare two-dimensional position and reciprocal lattice vectors in\nthexyplane. The vector r= (x,z), but none of the above quantities will have any z\ndependence for the composite we consider. The inverse Fourier tr ansforms are of the form\nQ(G) =1\nS/integraltext /integraltext\nd2xQ(x)e−iG·x, whereSis the area of the unit cell; similar expressions hold\nforMs(G) andα(G).\n3To calculate the band structure for spin waves propagating in the xyplane, we con-\nsider the two-dimensional Bloch vector, kand use Bloch’s theorem to write mx(x) =\neik·x/summationtext\nGmx,K(G)eiG·x,my(x) =eik·x/summationtext\nGmy,K(G)eiG·x, and Ψ(x) =eik·x/summationtext\nGΨK(G)eiG·x.\nAfter some straightforward algebra, the equations of motion red uce to\niΩ/summationdisplay\nG′˜A(G,G′)\nmx,K(G)\nmy,K(G)\n=/summationdisplay\nG′˜M(G,G′)\nmx,K(G′)\nmy,K(G′)\n; (4)\nthe 2×2 matrix\n˜A(G,G′) =\nδGG′α(G−G′)\n−α(G−G′)δGG′,\n (5)\nwhereδGG′is the Kronecker delta and the four components of the 2 ×2 matrix ˜M(G,G′)\nare given by\n˜M(G,G′)xx=Ms(G−G′)\nH0(Kx+G′\nx)(Ky+G′\ny)\n(K+G′)2\n˜M(G,G′)xy=δGG′+Q(G−G′)(K+G)·(K+G′)+Ms(G−G′)\nH0(Ky+G′\ny)2\n(K+G′)2\n˜M(G,G′)yx=−δGG′−Q(G−G′)(K+G)·(K+G′)−Ms(G−G′)\nH0(Kx+G′\nx)2\n(K+G′)2\n˜M(G,G′)yy=−Ms(G−G′)\nH0(Kx+G′\nx)(Ky+G′\ny)\n(K+G′)2. (6)\nOn left-multiplying eq. (4) by the inverse of the matrix ˜A, we reduce the band structure\nproblem, including Gilbert damping, to that of finding the (complex) eig envalues of ˜A−1˜M.\nA similar plane wave expansion has been previously used to calculate th e magnonic band\nstructure, for the case of zero damping, by several others (se e, e. g., Refs. [7] and [18]).\nWe have used this formalism to calculate band structures for both a triangular Bravais\nlattice, with basis vectors a1=aˆx,a2=a/parenleftBig\n1\n2ˆx+√\n3\n2ˆy/parenrightBig\n, and a square Bravais lattice, with\na1=aˆx,a2=aˆy, whereais the edge of the magnonic crystal unit cell. Since Fourier\ntransformsareavailableanalyticallyforcylinders ofcircular crosss ection, thebandstructure\nis easily calculated in this plane wave representation.\nIn order to solve Eq. (4), we restrict the sum over G′to the first 625 reciprocal lattice\nvectors, which requires the diagonalization of a 1250 ×1250 complex matrix. The resulting\neigenvalues of the matrix ˜B(G,G′) are all complex. For a given k, the imaginary part of the\neigenvalue for gives the spin wave frequency, while the real part re presents the inverse spin\n4wave lifetime. We have found that both the frequencies and lifetimes are well converged to\nwithin 0.1 % for this number of plane waves.\nFor each eigenvalue, the figure of merit (FOM) mentioned above is th e ratio of the\nimaginary part to the real part of the eigenvalue. If the Gilbert dam ping parameters αA=\nαB, the FOM would be same for all k’s and all bands. By contrast, when αA/negationslash=αBwe find\nthat the FOM varies from band to band and depends strongly on k. In particular, the FOM\nis particularly large in certain high symmetry directions. As a result, s pin waves will have\na longer lifetime when they are launched at special kvalues and with special frequencies.\nWe first consider the case of zero damping. In the left panel of Fig. 1, we plot the band\nstructure of a composite of Fe cylinders arranged on a triangular la ttice and embedded in\na Ni host, as calculated at an applied ��eld µ0H0= 0.1T. The lattice constant a= 10 nm\nand the Fe filling fraction f= 0.5 (f is the area fraction occupied by the cylinders). The\ncenter-hand panel shows a similar composite, but for Fe cylinders a rranged on a square\nlattice, again with f= 0.5. The right-hand panel shows the Brillouin zones of the square\nand triangular lattices with symmetry points indicated. In calculating the band structure,\nwe use an exchange constant and spontaneous magnetization at r oom temperature of 8.3\npJ/m and 1.71092 ×106A/m for Fe, and 3.4 pJ/m and 0.485423 ×106A/m for Ni[21].\nWe have not found band structures for exactly these materials in t he literature, but when\nwe carry out analogous calculations for Co cylinders in a Permalloy mat rix (not shown),\nusing the plane wave method, we obtain nearly identical results to th ose found by Vasseur\net al[18], who also used a plane wave expansion.\nIn Fig. 2, we show analogous calculations including damping for a squar e lattice. We use\nthe same parameters, magnetic field, and value of fas in Fig. 1, except that the Gilbert\ndamping parameters are αFe= 0.0019 and αNi=0.064, following Ref. [22]. In the left panel,\nthe width of each cross-hatched region is proportional to the figu re of merit (FOM) for\nthe given band and kvalue. The right panel shows the FOM for the fourth lowest spin\nwave band, as a function of magnonic crystal wave vector k, along specified directions in\nthe superlattice (or magnonic crystal) Brillouin zone (SBZ), and at t hree different filling\nfractions f. The inset again shows the SBZ and symmetry points. We plot the firs t nine\nbands. The scales for the FOM and the real frequencies are differe nt, as indicated.\nIn Fig. 3, we show the corresponding quantities for a triangular mag nonic crystal, again\nusing a superlattice constant 10 nm and f= 0.5. The other parameters are the same as\n5in Fig. 2, except that now the right hand panel shows the FOM for th e third lowest spin\nwave band. In Fig. 4, we show how the FOM for the optimal special sy mmetry points of\nFigs. 2 and 3 and bands depends on the superlattice filling fraction f. Note, in particular,\nthat the FOM increases strongly near the close-packing values of ffor both the square and\ntriangular lattices.\nThe most striking feature of these plots is the strong dependence of the FOM on both k\nand band index. For example, in the square superlattice, the FOM is la rgest in the fourth\nband at the symmetry point M, and in the triangular superlattice, it is largest for the\nthird band at K. The physics behind these strong maxima in the FOM is that, in both\nsuperlattices, the spin waves at these k-points propagate mainly through the Fe host, which\nisthelow-damping component. Thisresult suggests somepossible wa ys toincrease theFOM\neven further at these points: if we can arrange that a spin wave pr opagates entirely through\nthe low-dissipation material, this should give an FOM close to the theor etical maximum,\nwhich is that of this material in its homogeneous form. Thus, a judicio us exploration of\ndifferent periodic composites made of Fe and Ni, or other materials, c ould well lead to an\neven stronger dependence of spin lifetime on kvalue.\nWe should add a few words of caution regarding the “spin waveguiding effect.” In prin-\nciple, a measure of distance traveled by a propagating spin waves is g iven by the coherence\nlength (or spin wave mean free path) lc[23]. This coherence length, for a given band nat\nwave vector k, is defined as lc(k,n) =|Vg\nkn|/γkn, whereVg\nknrepresents the groupvelocity and\nγknrepresents the imaginary part of the eigenfrequency, i. e., the inv erse lifetime. Since the\ngroup velocity may itself depend strongly on n and k, the behavior of lc(k,n) may be quite\ndifferent from that of the lifetime. Nevertheless, we expect that lc(k,n), likeτ(k,n) and the\nFOMγkn, will depend strongly on both kand n, with sharp extrema near special symmetry\npoints. Hence the waveguiding effect is likely to remain when one consid erslc(k,n) rather\nthanγkn. A full answer to this question would require a calculation of Vg\nknfor different k\nandn.\nSince single crystal Fe and Ni already have some intrinsic anisotropy , one might expect\nthat this anisotropy could be exploited to obtain a strongly n and k-dependent FOM even\nin single crystals. However, in practice, most magnetic studies of Fe and Ni are carried out\non polycrystalline samples, which no longer have this anisotropy. The present work provides\na possible way of recovering this anisotropy, and even more, by use of a periodic lattice of\n6inclusions.\nThe present work can be generalized in various other ways. For exa mple, if a homoge-\nneous magnetic layer is perturbed by a periodic array of spin torque oscillators, this would\ngenerate an artificial magnetic superlattice, because the spin tor que would provide another\ncontribution to Heff. Another possibility is to extend the present work to magnonic crys -\ntalswith three-dimensional periodicity, thoughthismight beanexperimental challenge. The\npresent work could conceivably have applications, e. g., in magnonic c ircuits which exploit\nthe strong anisotropy in magnon lifetimes found in the present work .\nIn summary, we have calculated the spin wave spectrum of a magnet ic superlattice with\ntwo-dimensional periodicity, including for the first time the effects o f dissipation. We find a\nstriking anisotropy ofthespin wave figureofmerit, which fortypica l materials ismuch larger\nin certain bands near particular points of symmetry in the Brillouin zon e. This anisotropy\nimplies that propagating spin waves will have much longer lifetimes at ce rtain frequencies\nand in certain directions in k-space , which could be interpreted as a w aveguiding effect for\nthese excitations. We suggest that this anisotropy might be furth er increased with suitable\ntuning of the array parameters.\nFunding for this research was provided by the Center for Emergen t Materials at the Ohio\nState University, an NSF MRSEC (Award Number DMR-0820414).\n7[1] R. Tsu, Superlattice to Nanoelectronics (Elsevier, Oxford, 2005).\n[2] E. Yablonovitch, J. Opt. Soc. Am. B 10, 283 (1993).\n[3] M. M. Sigalas and E. N. Economou, J. Sound Vib. 158,377 (1992).\n[4] M. M. Sigalas and E. N. Economou, Solid State Commun. 86, 141 (1993).\n[5] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari- Rouhani, Phys. Rev. Lett. 71, 2022\n(1993).\n[6] M. S. Kushwaha, P. Halevi, G. Martinez, L. Dobrzynski, an d B. Djafari-Rouhani, Phys. Rev.\nB49, 2313 (1994).\n[7] M. Krawczyk and H. Puszkarski, Phys. Rev. B 77, 054437 (2008).\n[8] H. Puszkarski and M. Krawczyk, Solid State Phenom. 94, 125 (2003).\n[9] V. V. Kruglyak and R. J. Hicken, J. Magn. Magn. Mater. 306, 191 (2006).\n[10] S. A. Nikitov, P. Tailhades, and C. S. Tai, J. Magn. Magn. Mater.236, 320 (2001).\n[11] V. V. Kruglyak and A. N. Kuchko, Physica B 339, 130 (2003).\n[12] J. Cheon, J.-I. Park, J.-S. Choi, Y.-W. Jun, S. Kim, M. G. Kim, Y.-M. Kim, and Y. J. Kim,\nProc. Natl. Acad. Sci. U.S.A. 103, 3023 (2006).\n[13] S. L. Vysotskii, S. A. Nikitov, and Yu. A. Filimonov, JET P101, 547 (2005).\n[14] Z. K. Wang, V. L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok, S. Ja in, and A. O. Adeyeye,\nAppl. Phys. Lett. 94, 083112 (2009).\n[15] N. I. Polushkin, Phys. Rev. B 77, 180401(R) (2008).\n[16] See, e. g., S. I. Kisilev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A.\nBuhrman, and D. C. Ralph, Nature 425, 380 (2003); S. Kaka. M. R. Pufall, W. H. Rippard,\nT. J. Silva, S. E. Russak, and J. A. Katine, Nature 137, 389 (2005).\n[17] S. P. Bayrakci, T. Keller, K. Habicht, and B. Keimer, Sci ence312, 1926 (2006).\n[18] J. O. Vasseur, L. Dobrzynski, B. Djafari-Rouhani, and H . Puszkarski, Phys. Rev. B 541043\n(1996).\n[19] M. G. Cottam and O. J. Lockwood, Light Scattering in Magnetic Solids (Wiley, New York,\n1987).\n[20] M. Vohl, J. Barnas and P. Gr¨ unberg, Phys. Rev. B 39, 12003 (1989).\n[21] R. Skomski and D. J. Sellmyer, Handbook of Advanced Magnetic Materials, Nanostructural\n8Effects, Vol. 1, edited by Yi Liu, D. J. Sellmyer, and Daisuke Shindo ( Springer, New York,\n2006), p. 20.\n[22] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando and T. Miyazaki, Jpn. J. Appl. Phys.\n45(2006), 3889.\n[23] M. P. Kostylev and A. A. Stashkevich, Phys. Rev. B 81, 054418 (2010).\n9050100150200Re[iΩ]=Re[iω/γµ0H0]\nΓ MK Γ050100150Re[iΩ]=Re[iω/γµ0H0]\nΓ XM ΓXM\nΓ\nΓΜK\nFIG. 1: (Color online) Left panel: band structure for a trian gular lattice of Fe cylinders in Ni, with\nlattice constant a= 10 nm, Fe filling fraction f= 0.5, and no Gilbert damping. Other parameters\nare given in the text. Center panel: same as left panel but for a square lattice. Right panel:\nBrillouin zone for square and triangular lattices with symm etry points indicated.\n10050100150Re[iΩ]=Re[iω/γµ0H0]\n0100200300400 FOMf=0.5\nf=0.77\nf=0.1\nΓ XM Γ ΓMX Γ\nFIG. 2: (Color Online) Left panel: same as center panel of Fig . 1, but with Gilbert damping pa-\nrameters αFe= 0.0019 and αNi= 0.064. The widths of the cross-hatched regions are proportion al\nto the figure of merit (FOM) for the given band, as defined in the text. Right panel: FOM for the\nfourth lowest spin wave band, as a function of superlattice w ave vector k, along specified directions\nin the superlattice Brillouin zone (SBZ), and at three differe nt filling fractions f.\n11050100150200Re[iΩ]=Re[iω/γµ0H0]\n050100150200250300350400 FOMf=0.5\nf=0.9\nf=0.1\nΓ MK Γ Γ MK Γ\nFIG. 3: (Color online.) Same as Fig. 2 but for a triangular lat tice of Fe cylinders in Ni, with lattice\nconstant a= 10 and f= 0.5 (left panel) and f= 0.1, 0.5, and 0.9 (right panel).\n120100200300400 FOM\n0 0.2 0.4 0.6 0.8\nfilling fraction, fSquare superlattice\nTriangular superlattice\nFIG. 4: (Color Online) Same as Figs. 2 and 3, but showing the FO M as a function of filling fraction\nfforµ0H0= 0.1T,a= 10 nm.\n13"    },    {        "title": "2209.00558v1.Growth_parameters_of_Bi0_1Y2_9Fe5O12_thin_films_for_high_frequency_applications.pdf",        "content": "1 \n Growth parameters of Bi 0.1Y2.9Fe5O12 thin films for high frequency \napplications  \n \nGanesh  Gurjar1,4, Vinay Sharma2, S. Patnaik1,*, Bijoy K. Kuanr3 \n1School of Physical S ciences, Jawaharlal Neh ru University, New Delhi, INDIA 110067  \n2Department of Physics, Morgan State University, Baltimore, MD, USA 21251  \n3Special C entre for Nanosciences, Jawaharlal Nehru University, New Delhi , INDIA 110067  \n4Shaheed Rajguru College of Applied Sciences for Women, University of Delhi, INDIA 110096  \n \n \nAbst ract \n \nThe growth and characterization of Bismuth  (Bi) substituted YIG ( Bi-YIG, Bi0.1Y2.9Fe5O12) thin \nfilms are reported.  Pulsed laser deposited (PLD) films with thicknesses ranging from 20 to 150 nm \nwere grown o n Gadolinium Gallium Garnet substrates . Two substrate orientations of (100) and \n(111)  were considered . The enhanced distribution of Bi3+ ions at dodecahedral site along (111) is \nobserved to lead to  an increment in lattice constant from 12.379 Å in (1 00) to 12.415 Å in (1 11) \norient ed films.  Atomic force microscopy images show ed decreasing roughness with increasing \nfilm thickness.  Compared to  (100) grown films, (111) orient ed films  showed an increase in \nferromagnetic resonance  linewid th and consequent increase in Gilbert dampin g. The lowest  \nGilbert damping values are found  to be  (1.06±0. 12) × 10-4 for (100)  and (2.30±0. 36) × 10-4 for (111)  \noriented  films  with thickness  of ≈150 nm . The observed value s of extrinsic linewidth,  effective \nmagnetization , and anisotropic field  are related  to thickness  of the films  and substrate orientation. \nIn addition, the in-plane angular variation establishe d four-fold symmetry for the (100) deposited \nfilms unlike the case of (111) deposited films.  This study prescribes growth condition s for PLD \ngrow n single-crystalline Bi -YIG films towards desired high frequency  and magneto -optic al device \napplications.  \n \nKeyword s: Bi-Yttrium iron oxide; Thin film; Lattice mismatch; Pulsed Laser Deposition; \nFerromagnetic resonance; Gilbert damping; Inhomogeneous br oadening . \nCorresponding authors:  spatnaik@mail.jnu.ac.in  2 \n 1.1 Introduction  \n  \n One of the most important magnetic materials for studying high frequency magnetization \ndynamics is the Yttrium Iron Garnet (YIG, Y 3Fe5O12). Thin film  form  of YIG have attracted  a \nhuge attention in the field of spintronic devices due to its  large spin -wave propagation length , high \nCurie temperature T c ≈ 560 K  [1], lowest Gilbert damping and strong magneto -crystalline \nanisotropy  [2-7]. Due to these merits of YIG, it finds  several ap plications such as in magneto -\noptical (MO) devices, spin-caloritronics  [8,9] , and microwave resonators  and filters  [10-14].  \n The crystal structure of YIG is body centered cubic  under Ia3̅d space group . In Wyckoff \nnotation, t he yttrium (Y) ions are located at the dodecahedral 24c sites, whereas the Fe ions are \nlocated at two  distinct sites ; octahedral 16a and tetrahedral 24d .  The  oxygen ions are located in \nthe 96h sites  [7]. The ferrimagnetism of YIG is induced via a super -exchange interaction at the ‘d’ \nand ‘a’ site between the non-equivalent Fe3+ ions. It has already been observed that substituting \nBi/Ce for Y in YIG improves magneto -optical responsiv ity [13,15 -21]. In addition, Bi substitution \nin YIG (Bi -YIG) is known to generate growth -induced anisotropy, therefore, perpendicular \nmagnetic anisotropy (PMA) can be achieved in Bi doped YIG, which is beneficial in applications \nlike magnetic memory and logic devices  [7,22,23] . Due to its u sage in magnon -spintronics and \nrelated  disciplines such as caloritronics, the study of fundamental characteristics of Bi -YIG \nmaterials is of major current interest due to their high uniaxial anisotropy and F araday rotation  \n[17, 24-27]. Variations in the concentration of Bi3+ in YIG, as well as substrate orientation and \nfilm thickness, can improve strain tuned structural properties and magneto -optic characteristics . \nAs a result, selecting the appropriate substrate orientation and film thickness is important for \nidentifying the  growth of Bi-YIG thin films.  3 \n  The structural and magnetic characteristics of Bi -YIG [Bi 0.1Y2.9Fe5O12] thin film have been \nstudied in the current study. Gadolinium Gallium Garnet (GGG) substrates with orientations of \n(100) and (111)  were used to grow thin films . The Bi-YIG films of  four different thickness  (≈20 \nnm, 50 nm, 100 nm and 150 nm ) were deposited in -situ by pulsed laser deposit ion (PLD) method  \n[19,2 8] over single -crystalline GGG  substrates . Along with structural characterization  of PLD  \ngrown films , magnetic properties were ascertained by using  vibrating sample magnetometer \n(VSM) in conjunction with ferromagnetic resonance (FMR) techniques.  FMR is a highly effective \ntool for studying magnetization dynamics. The FMR response not only provides information about \nthe magnetization dynamic s of the material such as Gilbert damping and anisotropic field, but also \nabout the static magnetic properties such as saturation magnetization and anisotropy field.  \n \n1.2 Experiment  \nPolycrystalline YIG an d Bi -YIG targets were synthesized via the solid -state reaction \nmethod. Briefly, yttrium oxide (Y 2O3) and iron oxide (Fe 2O3) powders from Sigma -Aldrich were \ngrounded for ≈14 hours before calcination at 1100 ℃. The calcined powders were pressed into \npellets of one inch and sintered at 1300 ℃.  Using these polycrystalline YIG and Bi -YIG targets, \nthin films of four thicknesses ( ≈20 nm, 50 nm, 100 nm, and 150 nm) were synthesized in -situ on \n(100) - and (111) -oriented GGG substrates using the PLD method.  The samples are labelled in the \ntext as 20 nm (100), 20 nm (111) , 50 nm (100) , 50 nm (111),  100 nm (100) , 100 nm (111) , 150 nm \n(100),  and 150 nm (111) . Before deposition,  GGG substrates were cleaned in an ultrasonic bath \nwith acetone and isopropanol for 30  minute s.  The deposition chamber was cleaned and evacuated  \nto 5.3×10-7 mbar. For PLD growth, a 248 nm KrF excimer laser  (Laser fluence (2.3 J cm-2) with \n10 Hz pulse rate was used to ablate the material from the target . Oxygen pressure, target -to-4 \n substrate distance, and substrate temperature were maintained at 0.15 mbar, 5.0 cm, and 825 oC, \nrespectively.  Growth  rate of deposited films were  6 nm/min . The as -grown films were annealed \nin-situ for 2 hours at 825 oC in the presence of oxygen (0.15 mbar).  The structural characterization \nof thin films were ascertained  using X -ray diffraction (XRD)  with Cu-Kα radiation  (1.5406  Å). We \nhave performed the XRD me asurement at room temperature in -2 geometry and incidence angle \nare 20 degrees.   The film's surface morphology and thickness were estimated using atomic force \nmicroscopy  (AFM)  (WITec GmbH , Germany ). The magnetic properties were studied using a  \nvibrating sample magnetomet ry (VSM) in  Cryogenic 14 Tesla Physical Property Measurement \nSystem (PPMS).  FMR measurements were done on a coplanar waveguide (CPW) in a flip -chip \narrangement with a dc magnetic field applied perpendicular to the high -frequency  magnetic field \n(hRF). A Keysight Vector Network Analyzer was used  for this purpose. The CPW was rotated in \nthe film plane from 0º to 360º  for in -plane () measurement s and from 0º to 18 0º for out of plane \n(θ) measurement.  \nIn this study, the thickness of Bi -YIG was determined by employing methods such as laser \nlithography and AFM. We have calibrated the thickness of thin films with PLD laser shots. \nPhotoresist by spin coating is applied to a silicon substrate, and then straight-line patterns were \ndrawn on the photoresist coated substrates using laser photolithography. The PLD technique was \nused to deposit thin films of the required material onto a pattern -drawn substrate. It is then \nnecessary to wet etch the PLD grown thin fi lm in order to remove the photoresist coating. Then, \nAFM tip is scanned over the line pattern region in order to estimate the thickness of the grown \nsamples from the AFM profile image.  \n \n 5 \n 1.3 Results and Discussion  \n \n1.3.1  Structural properties  \n \nFigure  1 (a)-(d) show the XRD pattern of  (100)- and (111)-oriented Bi-YIG grown thin \nfilms  with thickness ≈20-150 nm  (Insets depict the zoomed image of XRD patterns) . XRD data \nindicate single -crystalline growth of Bi -YIG thin films . Figures 1 (e) and 1 (f) show the l attice \nconstant and lattice mismatch  (with respect to substrate) determined from XRD data, respectively . \nThe cubic lattice constant 𝒂 is calculated using the formula , \n𝒂=𝜆√ℎ2+𝑘2+𝑙2\n2sin𝜃                                                        (1) \n where the wavelength of Cu -Kα radiation  is represented by 𝜆, diffraction angle  by 𝜃, and the Miller \nindices of the corresponding XRD peak  by [h, k, l] . Further, the l attice  mismatch  parameter (𝛥𝑎\n𝑎) \nis calculated  using  the equation , \n   𝛥𝑎\n𝑎=(𝑎𝑓𝑖𝑙𝑚 − 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 )\n𝑎𝑓𝑖𝑙𝑚 100               (2) \nHere lattice constant of film and substrate are represented by  𝑎𝑓𝑖𝑙𝑚 and 𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 , respectively . \nThe reported lattice constant values are consistent with prior findings  [15,17,21].  Lattice constant \nslightly increases with the increase in thickness of the film in the case of (111) as compared to \n(100) . Since the distribution of Bi3+ in the dodecahedral sit e is dependent on the substrate \norientation  [7,23,2 9], the (111) oriented films show an increase in the lattice constant . In Bi -YIG \nfilms, this slight increase in the lattice constant (in the 111 direction) leads to a \ncompar atively  larger  lattice mismatch  as seen in Fig. 1 (f). For 50 nm (111) Bi -YIG film, we \nachieved a lattice mismatch  of ~0.47 , which is close to what has been reported earlier  [30,31]. 6 \n Smaller value of lattice mismatch can reduc e the damping constant of the film  [31]. We want to  \nunderline the importance of lattice plane dependen t growth in conjunction with film thickness in \nindicating structural and magnetic property changes.  \n \n1.3.2  Surface morphology  \n  \nFigure 2 (a) -(h) shows room temperature AFM images with root mean square (RMS) \nroughness. Roughness is essential from an application standpoint because the roughness directly \nimpacts the inhomogeneous linewidth broadening which leads to increase in the Gil bert damping. \nWe have observed RMS roughness  around  0.5 nm  or less  for all grown Bi -YIG films which are \ncomparable to previous reported YIG films  [32,33]. We have observed that  RMS  roughness  \ndecreases with increase  in thickness of the film. With (100) and (111) orientations, there is no \ndiscernible difference in roughness. Furthermore, roughness would be more affected by changes \nin growth factors and by substrate orientation  [7,33,34]. \n \n1.3.3  Static magnetization study  \n \n The room temperature ( ≈296 K ) VSM magnetization measurements were carried out with \napplied  magnetic field parallel to the film plane  (in-plane) . The paramagnetic contribution s from \nthe GGG substrate were carefully subtracted. F igure 3 (a)-(h) show s the magnetization plot s of Bi-\nYIG thin films of thickness ≈20-150 nm . Inset of Fig. 3 (i) shows the measured saturation \nmagnetization ( µ0MS) data of as-grown (100) and (111) -oriented Bi -YIG films which are \nconsistent with the previous reports  [6,17,22,3 5,36]. Figure 3 ( i) shows plot of µ0Ms × t Vs. t, \nwhere ‘t’ is film thickness . This is done t o determine thickness of dead layer via linear 7 \n extrapolati on plot to the x -axis. The obtaine d magnetic dead -layer for (100) and (111) -oriented \nGGG substrates are 2.88 nm and 5.41 nm , which are  comparable to previous reports  [37-39]. The \nsaturation magnetization of Bi -YIG films increases as the thickness of the films increases . The \nincrease in saturation magnetization with increase in thickness can be understood by the following \nways .  Firstly, ferromagnetic thin films are generally deposited with a thin magnetically dead layer \nover the interface with the substrate. This magnetic dead layer effect is larger in thinner films  that \nleads to the decrease in net magnetization with the decrease in thickness  [40,41]. Figure 3 (i) shows \nthe effect of magnetic dead laye r region near to the substrate.  Secondly, t hicker films exhibit the \nbulk effect of YIG which, in turn, results in increas ed magnetization.  \n \n1.3.4 Ferromagnetic r esonance study  \n \n Figure 4 (a) -(d) shows  the FMR absorption spectra  of (100) and (111)  -oriented films that  \nare labeled with open circle ( Ο) and open triangle ( Δ) respectively . FMR experiment s were carried \nout at room temperature. In -plane dc magnetic field was a pplied  parallel to film surface . To find \nthe effective magnetization and Gilbert damping, the  FMR linewidth (∆H) and resonance magnetic \nfield (H r) are calculated using a Lorentzian fit of the FMR absorption spectra measured at 𝑓 = 1 \nGHz to 12 GHz. Effective magnetization field ( 0𝑀𝑒𝑓𝑓) were obtained from the fitting of Kittel's \nin-plane equation (Eq. 3) [42].  \n𝑓=𝛾\n2𝜋0√(𝐻𝑟)(𝐻𝑟+𝑀𝑒𝑓𝑓)     (3), \nHere, 0𝑀𝑒𝑓𝑓=0(𝑀𝑠−𝐻𝑎𝑛𝑖), anisotropy field 𝐻𝑎𝑛𝑖=2𝐾1\n0𝑀𝑠, and   𝛾 being the gyromagnetic \nratio. Further, the dependence of FMR linewidth on microwave frequency shows a linear variation \n(Eq. 4) [42]  from which the Gilbert damping parameter (α) and FMR linewidth broad ening ( 𝛥𝐻 0) \nwere obtained:  \nµ0𝛥𝐻=µ0𝛥𝐻 0+4𝜋𝛼\n𝛾𝑓                                     (4) 8 \n where, 𝛥𝐻 0 is the inhomogeneous broadening linewidth and α is the Gilbert damping. Figures 4 \n(e) and 4 (f) show Kittel and linewidth fitted graphs, respectively . Figure 5 (a) -(d) shows the \nderived parameters acquired from the FMR  study. The estimated Gilbert damping is con sistent \nwith data reported for sp in-wave propagation  [3,22] . The value of α decreases as the thickness of \nthe film increases ( Fig. 5 (c)). Howe ver, in the instance of Bi -YIG with (111) orientation, there is \na substantial increase. This might be attributed qualitatively to the presence of Bi3+ ions, which \ncause strong spin-orbit coupling  [43-45] as well as electron scattering inside the lattice when the \nlattice mismatch (or strain) increases  [46]. Our earlier study  [7] revealed a clear  distribution  of \nBi3+ ions along (11 1) planes, as well as slightly  larger lattice mismatch  in Bi -YIG (111). These \nresults  explain the larger values of Gilbe rt damping,  0𝑀𝑒𝑓𝑓, and ΔH 0 values in Bi -YIG (111)  (Fig. \n5). The change in 0𝑀𝑒𝑓𝑓 is due to u niaxial in -plane magne tic anisotropy and it  is observed from \nmagnetization measurements  using  0𝑀𝑒𝑓𝑓=0(𝑀𝑠−𝐻𝑎𝑛𝑖) [36,47,48]. The enhanced \nanisotropy field in the lower thic kness of Bi -YIG ( Fig. 5 (d) ) signifies the effect of dead magnetic \nlayer at the interface.  The lattice mismatch between films and GGG substrates induces uniaxial in -\nplane magnetic anisotropy  [36,47]. ΔH 0 has a magnitude that is similar to previously published \nvalues for the same substrate orientation  [7,47]. In conclusion, Bi -YIG with (100) orientation \nproduces  the lowest Gilbert damping facto r and inhomogeneous broadening linewidth . These are \nthe required  optimal parameters for  spintronics  based devices.  \n Figure 6 (a) shows the variation of resonance field with polar angle ( ) for the grown 20 \nnm-150 nm films , H is the angle measured between applied magnetic field and surface of film \n(shown in inset of Fig. 4 (a)).  The FMR linewidth (ΔH) were extracted fr om fitting of FMR spectra \nwith L orentzian absorption functions.  From Fig. 6 (a), we observe change in H r value for 50 nm \nBi-YIG film as  0.22 T and 0.27 T for (100) and  (111)  orientation respectively. Similarly, 0.21 T \nand 0.31 T  change is observed in (100) and (111) orientation respectively  for 100 nm Bi -YIG film .  \nWe see that H r increases slightly in case o f (111) oriented film by changing  the di rection of H from \n0º to 90º  with regard  to sample surface (inset of Fig. 4 (a)).  The change in H r decreases with \nincrease in film thickness in cas e of (100) while it is reversed  in case of (111 ). Figure 6 (b) shows 9 \n the variation of FMR linewidth with polar angle  for 150 nm Bi -YIG film . Maximum FMR  \nlinewidth is observed at 90º  and it is slightly more as compared with (100) orientation. The \nenhanced variation of FMR linewidth in (111) oriented samples is generated due to the higher \ncontribution of two -magnon scatte ring in perpendicular geometry  [49]. This can be understood \ndue to the higher anisotropy field in (111) oriented samples ( Fig. 5 (d)). \n Figure 6 (c) & (e) shows the azimuthal angle ( ) variation of H r. Frequency of 5 GHz  is \nused in the measurement . From  variation data  (by changing  the direction of H  from 0 º to 360  \nwith regard  to sample  surface  (inset of Fig. 4 (a)). We can see clearly  in-plane anisotropy of  four-\nfold in Bi-YIG (100)  (Fig. 6 (c)) unlike in  Bi-YIG (111)  (Fig. 6 (e)). According to crystalline \nsurface symmetry there would be six -fold in -plane anisotropy in case of (111) orientation but we \nhave not observe d it, based  on previous reports, it can be superseded by a mis cut-induced uniaxial \nanisotropy  [33,50]. This reinforces our grown films' single -crystalline nature . The observed change \nin H r (H=0 to 45) is 6 .6 mT in 50 nm (100 ), 0.17 mT for 50 nm (111) , 6.2 mT in 100 nm (100) , \n0.17 mT for 100 nm (111) ) and 5.1 mT in 150 nm (100 ). As a result, during in -plane rotation, the  \nhigher FMR field change observed  along the (100) orientation.  The  dependent FMR field data \nshown in figure 6 (c) were fitted using the following Kittel relation [50]  \n𝑓=𝛾\n2𝜋0√([𝐻𝑟cos(𝐻−𝑀)+𝐻𝑐cos4(𝑀−𝐶)+𝐻𝑢cos2(𝑀−𝑢)])×\n(𝐻𝑟cos(𝐻−𝑀)+𝑀𝑒𝑓𝑓+1\n4𝐻𝑐(3+cos4(𝑀−𝐶))+𝐻𝑢𝑐𝑜𝑠2(𝑀−𝑢))        (5) \nWith respect to the [100] direction of the GGG substrate, in -plane directions of the magnetic field, \nmagnetization, uniaxial, and cubic anisotropies are given by H, M, u and c, respectively. \n𝐻𝑢=2𝐾𝑢\nµ0𝑀𝑠  and 𝐻𝑐=2𝐾𝑐\nµ0𝑀𝑠 correspond to the uniaxial and cubic anisotropy fields, respectively, \nwith 𝐾𝑢 and 𝐾𝑐 being the uniaxial and cubic magnetic anisotropy constants, respectively.  10 \n Figure 6 (d) shows t he obtained uniaxial anisotropy field, cubic anisotropy field and saturation \nmagnetization field for (100) orientation. The obtained saturation magnetization field follows the \nsame pattern as we have obtained from the VSM measurements. The cubic anisotropy  field \nincreases and then saturates with the thickness of the film. A large drop in the uniaxial anisotropy \nfield is observed with the thickness of the grown films. We have not got the in -plane angular \nvariation data for the 20 nm thick Bi -YIG sample and m ay be due to the low thickness of the Bi -\nYIG, it is not detected by our FMR setup.  \n \n1.4 Conclusion  \n In conclusion, we compare  the properties of high-quality Bi -YIG thin films of four distinct \nthicknesses (20 nm, 50 nm, 100 nm, and 150 nm) grown on GGG substrates with orientations of \n(100) and (111). Pulsed laser deposition was used to synthesize the se films. AFM and XRD \ncharacterizations reveal th at the deposited thin films have  smooth surfaces and are phase pure. \nAccording to FMR data, t he Gilbert  damping value  decreases with increase in film  thickness . This \nis explained i n the context of a dead m agnetic layer . The (100) orientation has a lower va lue of \nGilbert damping, indicating that it is the preferable substrate for doped YIG thin films  for high \nfrequency application . Bi-YIG on (111) orientation , on the other hand, exhibits anisotropic \ndominance, which is necessary for magneto -optic devices. Th e spin -orbit coupled Bi3+ ions are \nresponsible  for the enhanced Gilbert damping in (111). We have also correlated  ∆H 0, anisotropic \nfield, and effective magnetization  to the variations in film thickness and substrate ori entation . In \n(100) oriented films, there is unambiguous observation of four-fold in -plane anisotropy. In  \nparticular, Bi-YIG grown on (111) GGG substrates  yields best result for optim al magnetization \ndynamics. This is linked to an enhanced  magnetic anisotropy.  Therefore,  proper substrate 11 \n orientation and thickness are found to be important parameters for growth of Bi-YIG thin film \ntowards  high frequency applications.  \n \nAcknowledgments  \nThis work is supported by the MHRD -IMPRINT grant, DST (SERB, AMT , and PURSE -\nII) gran t of Govt. of India. Ganesh Gurjar acknowledges CSIR, New Delhi for financial support . \nWe acknowledge AIRF, JNU for access of PPMS facility.  \n  12 \n References  \n[1] V. Cherepanov, I. Kolokolov, V. L’vov, The saga of YIG: Spectra, thermodynamics, \ninteraction and relaxation of magnons in a complex magnet, Phys. Rep. 229 (1993) 81 –\n144. https://doi.org/10.1016/0370 -1573(93)90107 -O \n[2] S.A. Manuilov, C.H. Du, R. Adur, H.L. Wang, V.P. Bhallamudi, F.Y. Yang, P.C. Hammel, \nSpin pumping from spinwaves in th in film YIG, Appl. Phys. Lett. 107 (2015) 42405.  \nhttps://doi.org/10.1063/1.4927451  \n[3] C. Hauser, T. Richter, N. Homonnay, C. Eisenschmidt, M. Qaid, H. Deniz, D. Hesse, M. \nSawicki, S.G. 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Svedlindh, \nThickness -dependent enhancement of damping in Co 2FeAl/β -Ta thin films , Physical \nReview B 97.13 (2018): 134421. https://doi.org/10 .1103/PhysRevB.97.134421  \n \n \n \n \n \n \n  20 \n List of f igure caption s \n \nFigure 1:  (a)-(d) X -ray diffraction (XRD)  patterns of 20 nm -150 nm Bi -substituted YIG films in \n(100) and (111) orientations. Insets in (a) -(d) depict the zoomed image of XRD patterns. Variation \nof lattice constant (e) and (f) lattice mismatch with thickness  are shown . \n \nFigure 2:  (a)-(h) A tomic force microscopy  images of 20 nm -150 nm Bi -YIG film in (100) and \n(111) orientations  are shown . \n \nFigure 3:  (a)-(h) Static magnetization graph of 2 0 nm -150 nm Bi-substituted YIG  (Bi-YIG) films \nin (100) and (111) orientations. ( i) Graph to determine the magnetic dead -layer thickness of Bi -\nYIG films on (100) and (111) -oriented GGG substrates is depicted (inset shows the variation of  \nsaturation magnetiz ation value with the film thickness).  \n \nFigure 4:  (a)-(d) Ferromagnetic resonance ( FMR ) absorption spectra of 20 nm -150 nm Bi-\nsubstituted YIG  films with (100) and (111) orientations. Inset in (a) shows the geometry of an \napplied field angle measured from the sample surface.  (e) shows frequency -dependent FMR \nmagnetic field data fitted with Kittel Eq. 3 . (f) shows frequency -dependent FMR linewidth  data \nfitted with Eq. 4 . \n \nFigure 5:  Variation s of (a) extrinsic linewidth, (b) effective magnetization, (c) Gilbert damping, \nand (d) magnetic anisotropy with thickness for (100) and (111) oriented Bi-substituted YIG  films  \nare depicted .  21 \n Figure 6:  (a) Angular variation of Ferromagnetic resonance (FMR) magnetic field for 20 nm -150 \nnm Bi -substituted YIG (Bi -YIG) film with (100) and (111) orientations is shown. (b) Angular \nvariation of FMR linewidth of 150 nm thick Bi -YIG film with (100) and (111) orientation is \nshown. Variations of FMR magnetic field as a  function of azimuthal angle ( ) for (c) 50 nm, 100 \nnm and 150 nm Bi -YIG film with (100) orientation is depicted (d) obtained uniaxial anisotropy \nfield, cubic anisotropy field and saturation magnetization field for (100) orientation. (e)  \ndependent FMR fi eld data for 50 nm and 100 nm Bi -YIG film with (111) orientation is depicted.  \n \n \n  22 \n Figure 1  \n \n \n \n \n \n \n23 \n  \nFigure 2  \n \n \n24 \n  \nFigure 3  \n  \n25 \n Figure  4 \n \n26 \n  \nFigure 5  \n \n \n \n \n \n \n \n \n \n \n27 \n  \nFigure 6  \n \n \n"    },    {        "title": "1612.07020v2.Spin_Pumping__Dissipation__and_Direct_and_Alternating_Inverse_Spin_Hall_Effects_in_Magnetic_Insulator_Normal_Metal_Bilayers.pdf",        "content": "Spin Pumping, Dissipation, and Direct and Alternating Inverse Spin Hall E\u000bects in\nMagnetic Insulator-Normal Metal Bilayers\nAndr\u0013 e Kapelrud and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nWe theoretically consider the spin-wave mode- and wavelength-dependent enhancement of the\nGilbert damping in magnetic insulatornormal metal bilayers due to spin pumping as well as the\nenhancement's relation to direct and alternating inverse spin Hall voltages in the normal metal. In\nthe long-wavelength limit, including long-range dipole interactions, the ratio of the enhancement\nfor transverse volume modes to that of the macrospin mode is equal to two. With an out-of-\nplane magnetization, this ratio decreases with both an increasing surface anisotropic energy and\nmode number. If the surface anisotropy induces a surface state, the enhancement can be an order of\nmagnitude larger than for to the macrospin. With an in-plane magnetization, the induced dissipation\nenhancement can be understood by mapping the anisotropy parameter to the out-of-plane case\nwith anisotropy. For shorter wavelengths, we compute the enhancement numerically and \fnd good\nagreement with the analytical results in the applicable limits. We also compute the induced direct-\nand alternating-current inverse spin Hall voltages and relate these to the magnetic energy stored\nin the ferromagnet. Because the magnitude of the direct spin Hall voltage is a measure of spin\ndissipation, it is directly proportional to the enhancement of Gilbert damping. The alternating spin\nHall voltage exhibits a similar in-plane wave-number dependence, and we demonstrate that it is\ngreatest for surface-localized modes.\nPACS numbers: 76.50.+g, 75.30.Ds, 75.70.-i, 75.76.+j, 75.78.-n\nI. INTRODUCTION\nIn magnonics, one goal is to utilize spin-based sys-\ntems for interconnects and logic circuits1. In previous\ndecades, the focus was to gain control over these systems\nby exploiting long-range dipole interactions in combina-\ntion with geometrical shaping. However, the complex\nnature of the nonlinear magnetization dynamics persis-\ntently represents a challenge in using geometrical shaping\nalone to realize a variety of desired properties1.\nIn magnonic systems, a unique class of materials con-\nsists of magnetic insulators. Magnetic insulators are elec-\ntrically insulating, but localized magnetic moments cou-\nple to form a long-range order. The prime example is\nYttrium Iron Garnet (YIG). YIG is a complex crystal2\nin the Garnet family, where the Fe2+and Fe3+ions at\ndi\u000berent sites in the unit cell contribute to an overall fer-\nrimagnetic ordering. What di\u000berentiates YIG from other\nferromagnetic (ferrimagnetic) systems is its extremely\nlow intrinsic damping. The Gilbert damping parame-\nter measured in YIG crystals is typically two orders of\nmagnitude smaller than that measured in conventional\nmetallic ferromagnets (Fe, Co, Ni, and alloys thereof).\nThe recent discovery that the spin waves in mag-\nnetic insulators strongly couple to spin currents in ad-\njacent normal metals has re-invigorated the \feld of\nmagnonics3{12. Although there are no mobile charge car-\nriers in magnetic insulators, spin currents \row via spin\nwaves and can be transferred to itinerant spin currents in\nnormal metals via spin transfer and spin pumping13,14.\nThese interfacial e\u000bects open new doors with respect to\nlocal excitation and detection of spin waves in magnonic\nstructures. Another key element is that we can transfer\nknowledge from conventional spintronics to magnonics,opening possibilities for novel physics and technologies.\nTraditionally, spin-wave excitation schemes have focused\non the phenomenon of resonance or the use of \u001frsted\n\felds from microstrip antennas.\nA cornerstone for utilizing these systems is to estab-\nlish a good understanding of how the itinerant elec-\ntrons in normal metals couple across interfaces with\nspin-wave dynamics in magnetic insulators. Good mod-\nels for adressing uniform (macrospin) magnetization\nthat agrees well with experiments have been previously\ndeveloped13{15. We recently demonstrated that for long-\nwavelength magnons the enhanced Gilbert damping for\nthe transverse volume modes is twice that of the uniform\nmode, and for surface modes, the enhancement can be\nmore than ten times stronger. These results are con-\nsistent with the theory of current-induced excitations\nof the magnetization dynamics16because spin pump-\ning and spin transfer are related by Onsager reciprocity\nrelations17. Moreover, mode- and wave-vector-dependent\nspin pumping and spin Hall voltages have been clearly\nobserved experimentally4.\nIn this paper, we extend our previous \fndings18in the\nfollowing four aspects. i) We compute the in\ruence of\nthe spin back\row on the enhanced spin dissipation. ii)\nWe also compute the induced direct and alternating in-\nverse spin Hall voltages. We then relate these voltages to\nthe enhanced Gilbert damping and the relevant energies\nfor the magnetization dynamics. The induced voltages\ngive additional information about the spin-pumping pro-\ncess, which can also be directly measured. iii) We also\nprovide additional information on the e\u000bects of interfa-\ncial pinning of di\u000berent types in various \feld geometries.\niv) Finally, we explain in more detail how the numerical\nanalysis is conducted for a greater number of in-planearXiv:1612.07020v2  [cond-mat.mes-hall]  6 Apr 20172\nwave numbers.\nIt was discovered19{23and later quantitatively\nexplained13,15,24,25that if a dynamic ferromagnetic mate-\nrial is put in contact with a normal metal, the magnetiza-\ntion dynamics will exert a torque on the spins of electrons\nin the immediate vicinity of the magnet. This e\u000bect is\nknown as spin pumping (SP)13,15,25. As the electrons are\ncarried away from the ferromagnet-normal metal inter-\nface, the electrons spin with respect to each other, caus-\ning an overall loss of angular momentum. The inverse\ne\u000bect, in which a spin-polarized current can a\u000bect the\nmagnetization of a ferromagnet, is called spin-transfer\ntorque (STT)26{28.\nThe discovery that a precessing magnetization in mag-\nnetic insulators3, such as YIG, also pumps spins into an\nadjacent metal layer was made possible by the fact that\nthe mixing conductance in YIG-normal metal systems is\nof such a size that the extra dissipation of the magneti-\nzation due to the spin pumping is of the same order of\nmagnitude as the intrinsic Gilbert damping. A conse-\nquence of this e\u000bect is that the dissipation of the magne-\ntization dynamics is enhanced relative to that of a system\nin which the normal metal contact is removed.\nThis paper is organized in the following manner. Sec-\ntion II presents the equation of motion for the magne-\ntization dynamics and the currents in the normal metal\nand the appropriate boundary conditions, both for gen-\neral nonlinear excitations and in the fully linear response\nregime. In Section III, we derive approximate solutions\nto the linearized problem, demonstrating how the mag-\nnetization dissipation is enhanced by the presence of an\nadjacent metal layer. Section IV presents our numerical\nmethod and results. Finally, we summarize our \fndings\nin Section V.\nII. EQUATIONS OF MOTION\nThe equation of motion for the magnetization is given\nby the Landau-Lifshitz-Gilbert equation29(presented\nhere in CGS units)\n@M\n@t=\u0000\rM\u0002He\u000b+\u000b\nMsM\u0002@M\n@t; (1)\nwhere\r=jg\u0016B=~jis the magnitude of the gyromagnetic\nratio;g\u00192 is the Land\u0013 e g-factor for the localized elec-\ntrons in the ferromagnetic insulator (FI); and \u000bis the di-\nmensionless Gilbert damping parameter. In equilibrium,\nthe magnitude of the magnetization is assumed to be\nclose to the saturation magnetization Ms. The magneti-\nzation is directed along the z-axis in equilibrium. Out of\nequilibrium, we assume that we have a small transverse\ndynamic magnetization component, such that\nM=M(r;t) =Ms+m(r;t) =Ms^z+m(r;t);(2)\nwherejmj\u001cMsandm\u0001^z= 0. Furthermore, we assume\nthat the dynamic magnetization can be described by a\nx\nhzx\nyz\nfq(a)\nd\nL2\n-L2NM\nFI\nSUBx (b)\nFIG. 1. a) The coordinate system. ^\u0018is the \flm normal\nand ^\u0010is the spin-wave propagation direction. \u0018\u0011\u0010form a\nright-handed coordinate system. The ^zaxis is the direc-\ntion of the magnetization in equilibrium, such that xyis the\nmagnetization-precession plane. b) The \flm stack is in the\nnormal direction.\nplane wave traveling along the in-plane \u0010-axis. In the\n(\u0018;\u0011;\u0010 ) coordinate system (see Figure 1), we have\nm(r;t) =m(\u0018;\u0010;t ) =mQ(\u0018)ei(!t\u0000Q\u0010); (3)\nwhere!is the harmonic angular frequency, Qis the in-\nplane wave number, and mQ(\u0018) =XQ(\u0018)^x+YQ(\u0018)^y,\nwhereXQandYQare complex functions. Note that m\nis independent of the \u0011coordinate due to translational\ninvariance.\nHe\u000bis the e\u000bective \feld, given as the functional deriva-\ntive of the free energy29,30\nHe\u000b(r;t) =\u0000\u000eU[M(r;t)]\n\u000eM(r;t)=Hi+2A\nM2sr2M(r;t)+\n+ 4\u0019ZL\n2\n\u0000L\n2d\u00180bGxy(\u0018\u0000\u00180)m(\u00180;\u0010;t);(4)\nwhere Hiis the internal \feld, which is composed of\nthe applied external \feld and the static demagnetization\n\feld. The direction of Hide\fnes the z-axis (see Fig-\nure 1). The second term of Eq. (4) is the \feld, Hex,\ninduced by to the exchange interaction (assuming cu-\nbic symmetry), where Ais the exchange sti\u000bness pa-\nrameter. The last term is the dynamic \feld, hd(r;t),\ninduced by dipole-dipole interactions, where bGxyis the\nupper 2\u00022 part of the dipole{dipole tensorial Green's\nfunctionbG\u0018\u0011\u0010in the magnetostatic approximation31ro-\ntated to the xyzcoordinate system (see Appendix A for\ncoordinate-transformation matrices).32\nThe e\u000bect of the dipolar interaction on the spin-wave\nspectrum depends on the orientation of the internal \feld\nwith respect to both the interface normals of the thin\n\flm, ^\u0018, and the in-plane spin-wave propagation direc-\ntion, ^\u0010. Traditionally, the three main con\fgurations are\nthe out-of-plane con\fguration ( \u0012= 0), in the forward\nvolume magnetostatic wave (FVMSW) geometry (see\nFig. 2a); the in-plane and parallel-to- ^\u0010con\fguration, in3\nthebackward volume magnetostatic wave (BVMSW) ge-\nometry (see Fig. 2b); and the in-plane and perpendicular-\nto-^\u0010con\fguration, in the magnetostatic surface wave\n(MSSW) geometry (see Fig. 2c).1,32{36Here, the term\n\\forward volume modes\" denotes modes that have posi-\ntive group velocities for all values of QL, whereas back-\nward volume modes can have negative group velocities\nin the range of QL, where both exchange and dipolar\ninteractions are signi\fcant. Volume modes are modes in\nwhich mQ(\u0018) is distributed across the thickness of the\nentire \flm, whereas the surface modes are localized more\nclosely near an interface.\nA. Spin-Pumping Torque\nWe consider a ferromagnetic insulator (FI) in contact\nwith a normal metal (NM) (see Figure 1). If the magneti-\nzation in the FI close to the interface is precessing around\nthe e\u000bective \feld, electron spins in the NM re\rected at\nthe interface will start to precess due to the local ex-\nchange coupling to the magnetization in the FI. The re-\n\rected electrons carry the angular momentum away from\nthe interface, where the spin information can get lost\nthrough dephasing of the spins within a typical spin di\u000bu-\nsion length lsf. This loss of angular momentum manifests\nitself as an increased local damping of the magnetization\ndynamics in the FI. The magnetization dissipation due\nto the spin-pumping e\u000bect can be taken into account by\nadding the local dissipation torque15\n\u001csp=\r~2g?\n2e2M2s\u000e(\u0018\u0000L\n2)M(r;t)\u0002@M(r;t)\n@t;(5)\nto the right-hand side (rhs) of Eq. (1). Here, g?is the\nreal part of the spin-mixing conductance per area, and e\nis the electron charge. We neglect the contribution from\nthe imaginary part of the mixing conductance, because\nthis has been shown to be signi\fcantly smaller than that\nof the real part, in addition to a\u000becting only the gyro-\nmagnetic ratio.15The spin-current density pumped from\nthe magnetization layer is thus given by\nj(s)\nsp=\u0000~2g?\n2e2M2s\u0014\nM(r;t)\u0002@M(r;t)\n@t\u0015\n\u0018=L=2;(6)\nin units of erg. Next, we will see how the spin pumping\na\u000bects the boundary conditions.\nB. Spin-Pumping Boundary Conditions\nFollowing the procedure of Rado and Weertman37, we\nintegrate Eq.(1) with the linear expansion of Eq. (2) over\na small pill-box volume straddling one of the interfaces\nof the FI. Upon letting the pill box thickness tend to\nzero, only the surface torques of the equation survive.\nAccounting for the direction of the outward normal ofthe lid on the di\u000berent top and bottom interfaces, we\narrive at the exchange-pumping boundary condition\n\u00142A\nM2sM\u0002@M\n@\u0018+~2\n2e2M2sg?M\u0002@m\n@t\u0015\n\u0018=\u0006L=2= 0:(7)\nThere is no spin current pumped at the interface to the\ninsulating substrate; thus, a similar derivation results in\na boundary condition that gives an unpinned magnetiza-\ntion,\n@M(r;t)\n@\u0018\f\f\f\f\n\u0018=\u0000L=2= 0: (8)\nIn the next section, we will generalize the bound-\nary conditions of Eqs. (7) by also considering possible\nsurface-anisotropy energies.\nIncluding surface anisotropy:\nIn the presence of surface anisotropy at an interface\nwith an easy-axis (EA) pointing along the direction ^n,\nthe surface free energy is\nUs[M(r;t)] =Z\ndV Ks\"\n1\u0000\u0012M(r;t)\u0001^n\nMs\u00132#\n\u000e(\u0018\u0000\u0018i);\n(9)\nwhereKsis the surface-anisotropy energy density at the\ninterface, which is assumed to be constant; ^nis the direc-\ntion of the anisotropy easy axis; and\u0018iis the transverse\ncoordinate of the interface. The contribution from the\nEA surface-anisotropy energy to the e\u000bective \feld is de-\ntermined by\nHs=\u0000\u000eUs[M(r;t)]\n\u000eM(r;t)=2Ks\nM2s(M\u0001^n)\u000e(\u0018\u0000\u0018j)^n:\nHowever, if we have an easy-plane (EP) surface\nanisotropy with, ^nbeing the direction of the hard axis,\nthe e\u000bective \feld is the same as that for the EA case,\nexcept for a change of sign of Ks. We unify both cases\nby de\fning Ks>0 to imply that we have an EA surface\nanisotropy with its easy axis along ^n, whereasKs<0\nimplies that we have an EP surface anisotropy with its\nhard axis along ^n.\nFollowing the approach from Section II B, the total\nboundary condition, including exchange, pumping and\nsurface anisotropy, becomes\n\u0014\n\u00062A\nM2sM\u0002@M\n@\u0018\u00002Ks\nM2s(M\u0001^n) (M\u0002^n) +\n+~2\n2e2M2sg?M\u0002@M\n@t\u0015\n\u0018=\u0006L=2= 0;(10)\nwhere the positive (negative) sign in front of the exchange\nterm indicates that the bulk FI is located below (above)\nthe interface coordinate.4\nx,z\nz\n(a)\nx\nz,zq (b)\nx\nz\nzfq (c)\nFIG. 2. Laboratory \feld con\fgurations, i.e., directions of ^z(green arrow) in relation to \flm normal ^\u0018and the spin-wave\npropagation direction ^\u0010, resulting in the di\u000berent geometries: a) FVMSW geometry; b) BVMSW geometry; c) MSSW geometry.\nC. Linearization\nWe linearize the equation of motion using Eq. (2) with\nrespect to the dynamic magnetization m. The linearized\nequation of motion for the bulk magnetization Eq. (1)\nbecomes32\n\u001a\ni!\n!M\u0012\n\u000b\u00001\n1\u000b\u0013\n+11\u0014!H\n!M+ 8\u0019\r2A\n!2\nM\u0012\nQ2\u0000d2\nd\u00182\u0013\u0015\u001b\n\u0001\n\u0001mQ(\u0018) =ZL\n2\n\u0000L\n2d\u00180bGxy(\u0018\u0000\u00180)mQ(\u00180);(11)\nwhere!H\u0011\rHi,!M\u00114\u0019\rMs, and 11 =\u00001 0\n0 1\u0001\n.\nNext, we linearize the boundary conditions of Eq. (10).\nWe choose the anisotropy axis to be perpendicular to the\n\flm plane, ^n=^\u0018, which in the xyzcoordinate system\nis given by ^\u0018xyz= (sin\u0012;0;cos\u0012), where\u0012is the angle\nbetween the z-axis and the \flm normal (see Fig. 1). The\n\fnite surface anisotropy forces the magnetization to be\neither perpendicular or coplanar with the \flm surface so\nthat\u0012= 0;\u0019=2;\u0019. Linearizing to 1st order in the dy-\nnamic magnetization, we arrive at the linearized bound-\nary conditions for the top interface\n\u0012\nL@\n@\u0018+i!\n!M\u001a+dcos(2\u0012)\u0013\nmQ;x(\u0018)\f\f\f\f\n\u0018=L\n2= 0;(12a)\n\u0012\nL@\n@\u0018+i!\n!M\u001a+dcos2(\u0012)\u0013\nmQ;y(\u0018)\f\f\f\f\n\u0018=L\n2= 0;(12b)\nwhered\u0011LKs=Ais the dimensionless surface-\npinning parameter that relates the exchange to the\nsurface anisotropy and the \flm thickness and \u001a\u0011\n!ML~2g?=4Ae2is a dimensionless constant relating the\nexchange sti\u000bness and the spin-mixing conductance.\nD. Spin Accumulation in NM and Spin Back\row\nThe pumped spin current induces a spin accumulation,\n\u0016(s)=\u0016(s)^s, in the normal metal. Here, ^sis the spin-\npolarization axis, and \u0016(s)= (\u0016\"\u0000\u0016#)=2 is half of thedi\u000berence between chemical potentials for spin-up and\nspin-down electrons in the NM.\nAs the spin accumulation is a direct consequence of\nthe spin dynamics in the FI (see Eq. (6)), the spin ac-\ncumulation cannot change faster than the magnetization\ndynamics at the interface. Thus, assuming that spin-\rip\nprocesses in the NM are must faster than the typical pre-\ncession frequency of the magnetization in the FI25, we can\nneglect precession of the spin accumulation around the\napplied \feld and any decay in the NM. With this assump-\ntion, the spin-di\u000busion equation@\u0016(s)\n@t=Dr2\u0016(s)\u0000\u0016(s)\n\u001csf,\nwhereDis the spin-di\u000busion constant, and \u001csfis the\nmaterial-speci\fc average spin-\rip relaxation time, be-\ncomes\n\u0016(s)\u0019l2\nsfr2\u0016(s); (13)\nwherelsf\u0011p\u001csfDis the average spin-\rip relaxation\nlength.\nThe spin accumulation results in a back\rowing spin-\ncurrent density, given by\nj(s)\nbf(L=2) =~g?\ne2M2sh\nM(r;t)\u0002\u0010\nM(r;t)\u0002\u0016(s)(r;t)\u0011i\n\u0018=L=2;\n(14)\nwhere the positive sign indicates \row from the NM into\nthe FI. This spin current creates an additional spin-\ntransfer torque on the magnetization at the interface\n\u001cbf=\u0000\r~g?\ne2M2s\u000e\u0010\n\u0018\u0000L\n2\u0011\nM(r;t)\u0002\u0010\nM(r;t)\u0002\u0016(s)\u0011\n:\n(15)\nBecause the spin accumulation is a direct result of the\npumped spin current, it must have the same orientation\nas the M(r;t)\u0002@tM(r;t) term in Eq. (5). That term is\ncomprised of two orthogonal components: the 1st-order\ntermMs^z\u0002_min thexyplane, and the 2nd-order term\nm\u0002_moriented along ^z. Because the magnetization is\na real quantity, care must be taken when evaluating the\n2nd-order term. Using Eq. (3), the 2nd-order pumped5\nspin current is proportional to\nRefmg\u0002@tRefmg\f\f\f\n\u0018=L=2=e\u00002Imf!gtRef!g\u0002\n\u0002^zh\nImXQReYQ\u0000ReXQImYQi\n;(16)\nwhich is a decaying direct-current (DC) term. This is in\ncontrast to the 1st-order term, which is an alternating-\ncurrent (AC) term. Thus, we write the spin accumula-\ntion as\n\u0016(s)=\u0016(s)\nAC(^z\u0002^mt) +\u0016(s)\nDC^z; (17)\nwhere we have used the shorthand notation mt=_m(\u0018=\nL=2), such that ^mt=mt=jmtj, which in general is not\nparallel to mbut guaranteed to lie in the xyplane. In-\nserting Eq. (17) into Eq. (13) gives one equation each for\nthe AC and DC components of the spin accumulation,\n@2\u0016(s)\nj\n@\u00182=l\u00002\nsf;j\u0016(s)\nj; (18)\nwherejdenotes either the AC or DC case and lsf,DC =lsf\nwhilelsf,AC =lsf(1+l2\nsfQ2)\u00001=2because mt/exp(i(!t\u0000\nQ\u0010)). Eq. (18) can be solved by demanding spin-current\nconservation at the NM boundaries: at the free surface of\nthe NM, there can be no crossing spin current; thus, the \u0018\ncomponent of the spin-current density must vanish there,\n@\u0018\u0016(s)\njj\u0018=L=2+d= 0. Similarly, by applying conservation\nof angular momentum at the FI-NM interface, the net\nspin-current density crossing the interface, due to spin\npumping and back\row, must equal the spin current in\nthe NM layer, giving\n\u0014\n\u0000~2g?\n2e2M2sM\u0002@M\n@t+~g?\ne2M2sM\u0002\u0010\nM\u0002\u0016(s)\u0011\u0015\n\u0018=L=2\n=\u0000~\u001b\n2e2@\u0018\u0016(s)j\u0018=L=2;(19)\nwhere\u001bis the conductivity of the NM. Using these\nboundary conditions, we recover the solutions (see,\ne.g.,25,38)\n\u0016(s)\nj=\u0016(s)\nj;0sinh\u0010\nl\u00001\nsf;j\u0002\n\u0018\u0000(L=2 +d)\u0003\u0011\nsinh\u0010\n\u0000d\nlsf;j\u0011; (20)\nwhere\u0016(s)\nj;0is time dependent, and depends on the \u0010co-\nordinate only in the AC case. We \fnd that the AC and\nDC spin accumulations \u0016(s)\nj;0are given by\n\u0016(s)\nAC;0=\u0000~\n2mt\nMs\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00001\n;\n(21)\n\u0016(s)\nDC;0=\u0000lsf~\n\u001bM2s~g?tanh\u0012d\nlsf\u0013\n^z\u0001[m\u0002_m]\u0018=L=2;\n(22)TABLE I. Typical values for the parameters used in the\ncalculations.6,7,11,39{41\nParameter Value Unit\nA 3:66\u000110\u00007erg cm\u00001\n\u000b 3\u000110\u00004{\nKs 0:05 erg cm\u00002\ng? 8:18\u00011022cm\u00001s\u00001\n\r 1:76\u0001107G\u00001s\u00001\n4\u0019M s 1750 G\n\u001b 8:45\u00011016s\u00001\nd 50 nm\nlsf 7.7 nm\n\u0002 0.1 {\nwhere ~g?is a renormalized mixing conductance, which is\ngiven by\n~g?=g?(\n1\u0000\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00001)\n:\n(23)\nThis scaling of g?occurring in the DC spin accumulation\noriginates from the second-order spin back\row due to the\nAC spin accumulation that is generated in the normal\nmetal.\nAdding both the spin-pumping and the back\row\ntorques to Eq. (1) and repeating the linearization pro-\ncedure from Sec. II C, we \fnd that the AC spin accumu-\nlation renormalizes the pure spin-mixing conductance.\nThus, the addition of the back\row torque can be ac-\ncounted for by replacing g?with ~g?in the boundary con-\nditions of Eqs. (12), making the boundary conditions Q-\ndependent in the process. Using the values from Table I,\nwhich are based on typical values for a YIG-Pt bilayer\nsystem, we obtain ~ g?=g?\u00180:4 forQL\u001c1, whereas\n~g?=g?!1 for large values of QL. Thus, AC back\row is\nsigni\fcant for long-wavelength modes and should be con-\nsidered when estimating g?from the linewidth broaden-\ning in ferromagnetic resonance (FMR) experiments.11\nInverse Spin Hall E\u000bect\nThe inverse spin Hall e\u000bect (ISHE) converts a spin\ncurrent in the NM to an electric potential through the\nspin-orbit coupling in the NM. For a spin current in\nthe^\u0018direction, the ISHE electric \feld in the NM layer\nisEISHE =\u0000e\u00001\u0002h(@\u0018\u0016(s))\u0002^\u0018i\u0018, where \u0002 is the di-\nmensionless spin-Hall angle, and h\u0001i\u0018is a spatial average\nacross the NM layer, i.e., for \u00182(L=2;L=2 +d). Using\nthe previously calculated spin accumulation, we \fnd that6\nthe AC electric \feld is\nEAC\nISHE =\u0000\u0002~\n2deMs\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00001\n\u0002\n\u0002\u0002\n\u0000^\u0011(\u0000mt;ycos\u0012cos\u001e+mt;xsin\u001e)+\n+^\u0010(\u0000mt;xcos\u001e\u0000mt;ycos\u0012sin\u001e)\u0003\n;(24)\nwhere\nmt;i=\u0000[Im!Remi+Re!Immi]\u0018=L=2; (25)\nandi=x;y. For BVMSW ( \u0012=\u0019=2;\u001e= 0) modes,\nthe AC \feld points along ^\u0010, whereas for MSSW ( \u0012=\n\u001e=\u0019=2) modes, it points along ^\u0011(i.e., in plane, but\ntransverse to \u0010; see Fig. 1). Notice that for both BVMSW\nand MSSW mode geometries, only the xcomponent of\nmtcontributes to the \feld. In contrast, for FVMSW\n(\u0012= 0) modes, the \feld points somewhere in the \u0011\u0010\nplane, depending on the ratio of mt;xtomt;y.\nSimilarly to the AC \feld, the DC ISHE electric \feld is\ngiven by\nEDC\nISHE = \u0002\u0016(s)\nDC;0\ndesin\u0012(^\u0011cos\u001e\u0000^\u0010sin\u001e);(26)\nwhich is perpendicular to the AC electric \feld and zero\nfor the FVMSW mode geometry.\nThe total time-averaged energy in the ferromagnet\nEtotal(see Morgenthaler42) is given by\nhEtotaliT=Z\nferriteRe\u0014\n\u0000i\u0019!\u0003\n!M(m\u0002m\u0003)^z\u0015\ndV; (27)\nwhere the integral is taken over the volume of the ferro-\nmagnet.\nBecause the DC ISHE \feld is in-plane, the voltage\nmeasured per unit distance along the \feld direction,\n^\u0003=^\u0011cos\u001e\u0000^\u0010sin\u001e, can be used to construct an esti-\nmate of the mode e\u000eciency. Taking the one-period time\naverage of Eq. (26) using Eq. (22) and normalizing it by\nEq. (27) divided by the in-plane surface area, A, we \fnd\nan amplitude-independent measure of the DC ISHE:\n\u000fDC=he^\u0003\u0001EDC\nISHEiT\nhEtotaliT=A=\u00002\r\u0002lsf~\nd\u001bMs~g?tanh\u0012d\nlsf\u0013\nsin\u0012\u0002\n\u0002Reh\n\u0000i!\u0003\n!M(m\u0002m\u0003)^zi\n\u0018=L=2RL=2\n\u0000L=2Reh\n\u0000i!\u0003\n!M(m\u0002m\u0003)^zi\nd\u0018;(28)\ngiven in units of cm, and where f\u0001g\u0003denotes complex\nconjugation.\nSimilarly, the AC ISHE electric \feld, being time-\nvarying, will contribute a power density that, when nor-\nmalized by the power density in the ferromagnet, be-comes\n\u000fAC=h\u001b\u0000\nEAC\nISHE\u00012iT\nRef!g\n2\u0019ALhEtotaliT=\u0019\u001b\nRef!g\u0012\u0002~\n2deMs\u00132\n\u0002\n\u0002\u0014\n1 +\u001b\n2g?lsf;ACcoth\u0012d\nlsf;AC\u0013\u0015\u00002\n\u0002\n\u0002jmt;xj2+ cos2\u0012jmt;yj2\n1\nLRL=2\n\u0000L=2Reh\n\u0000i!\u0003\n!M(m\u0002m\u0003)^zi\nd\u0018:(29)\nTo be able to calculate explicit realizations of the mode-\ndependent equations Eqs. (28) and (29), one will need to\n\frst calculate the dispersion relation and mode pro\fles\nin the ferromagnet.\nIII. SPIN-PUMPING THEORY FOR\nTRAVELLING SPIN WAVES\nBecause, the linearized boundary conditions (see\nEqs. (12)) explicitly depend on the eigenfrequency !, we\ncannot apply the method of expansion in the set of pure\nexchange spin waves, as was performed by Kalinikos and\nSlavin32. Instead, we analyze and solve the system di-\nrectly for small values of QL, whereas the dipole-dipole\nregime ofQL\u00181 is explored using numerical computa-\ntions in Sec. IV.\nA. Long-Wavelength Magnetostatic Modes\nWhenQL\u001c1 Eq. (11) is simpli\fed to\n( \nsin2\u00120\n0 0!\n+i!\n!M \n\u000b\u00001\n1\u000b!\n+\n+11\u0014!H\n!M\u00008\u0019\r2A\n!2\nMd2\nd\u00182\u0015\u001b\n\u0001mQ(\u0018) = 0;(30)\nwhere the 1st-order matrix term describe the dipole-\ninduced shape anisotropy and stems from bGxy(see32).\nWe make the ansatz that the magnetization vector in\nEq. (3) is composed of plane waves, e.g., mQ(\u0018)/eik\u0018.\nInserting this ansatz into Eq. (30) produces the disper-\nsion relation\n\u0010!\n!M\u00112\n=\u0010!H\n!M+\u00152\nexk2+i\u000b!\n!M\u0011\n\u0002\n\u0002\u0010!H\n!M+\u00152\nexk2+ sin2\u0012+i\u000b!\n!M\u0011\n;(31)\nwhere\u0015ex\u0011p\n8\u0019\r2A=!2\nMis the exchange length . Keep-\ning only terms to \frst order in the small parameter \u000b,\nwe arrive at\n!(k)\n!M=\u0006r\u0010!H\n!M+\u00152exk2\u0011\u0010!H\n!M+\u00152exk2+ sin2\u0012\u0011\n+\n+i\u000b\u0010!H\n!M+\u00152\nexk2+sin2\u0012\n2\u0011\n: (32)7\nThe boundary conditions in Eq. (12) depend explicitly on\n!andkand give another equation k=k(!) to be solved\nsimultaneously with Eq. (32). However, in the absence\nof spin pumping, i.e., when the spin-mixing conductance\nvanishesg?!0, it is su\u000ecient to insert the constant k\nsolutions from the boundary conditions into Eq. (32) to\n\fnd the eigenfrequencies.\nDi\u000berent wave vectors can give the same eigenfre-\nquency. It turns out that this is possible when !(k) =\n!(i\u0014), which has a non-trivial solution relating \u0014tok:\n\u00152\nex\u00142= sin2\u0012+\u00152\nexk2+ 2!H\n!M\u0006i2\u000b!(k)=!M:(33)\nWith these \fndings, a general form of the magnetiza-\ntion is\nmQ(\u0018) = \n1\nr(k)!h\nC1cos\u0000\nk(\u0018+L\n2)\u0001\n+C2sin\u0000\nk(\u0018+L\n2)\u0001i\n+\n+ \n1\nr(i\u0014)!h\nC3cosh\u0000\n\u0014(\u0018+L\n2)\u0001\n+C4sinh\u0000\n\u0014(\u0018+L\n2)\u0001i\n;\n(34)\nwherefCigare complex coe\u000ecients to be determined\nfrom the boundary conditions, and where \u0014=\u0014(k) is\ngiven by Eq. (33). The ratio between the transverse com-\nponents of the magnetization, r(k) =YQ=XQ, is deter-\nmined from the bulk equation of motion (see Eq. (30))\nand is in linearized form\nr(k) =\u0000\u000bsin2\u0012\u00062ir\u0010\n!H\n!M+\u00152exk2\u0011\u0010\n!H\n!M+\u00152exk2+ sin2\u0012\u0011\n2\u0010\n!H\n!M+\u00152exk2\u0011 ;\n(35)\nimplying elliptical polarization of mQwhen\u00126= 0.\nInserting Eq. (34) into Eq. (8) only leads to a solution\nwhenk= 0, such that C2=C4= 0 in the general case.\nBy solving Eq. (12b) for C3, we \fnd\nC3\nC1=\u0000!H\n!M+\u00152\nexk2+ sin2\u0012+i\u000b!\n!M\n!H\n!M\u0000\u00152ex\u00142+ sin2\u0012+i\u000b!\n!M\u0002\n\u0002(i!\n!M~\u001a+dcos2\u0012) cos(kL)\u0000kLsin(kL)\n(i!\n!M~\u001a+dcos2\u0012) cosh(\u0014L) +\u0014Lsinh(\u0014L);(36)\nwhere ~\u001a\u0011\u001ajg?!~g?is the pumping parameter altered by\nthe AC spin back\row from the NM (see Section II D). C1\nis chosen to be the free parameter that parameterizes the\ndynamic magnetization amplitude, which can be deter-\nmined given a particular excitation scheme. Lineariza-\ntion of Eq. (36) with respect to \u000bis straightforward, but\nthe expression is lengthy; we will therefore not show it\nhere.\nInserting the ansatz with C2=C4= 0 andC3given\nby Eq. (36) into Eq. (12a) gives the second equation for\nkand!(the \frst is Eq. (32)). In the general case, thenumber of terms in this equation is very large; thus, we\ndescribe it as\nf(k;!;\u000b; ~\u001a) = 0; (37)\ni.e., an equation that depends on the wave vector k, fre-\nquency!, Gilbert damping constant \u000band spin-pumping\nparameter ~\u001a.\nBecause both the bulk and interface-induced dissipa-\ntion are weak, \u000b\u001c1, ~\u001a\u001c1, the wavevector is only\nslightly perturbed with respect to a system without dis-\nsipation, i.e., k!k+\u000ekwhere\u0015ex\u000ek\u001c1. It is therefore\nsu\u000ecient to expand fup to 1storder in these small quan-\ntities:\nf(k;!; 0;0) + (~\u001a)@f\n@~\u001a\f\f\f\f\n0+\u000b@f\n@\u000b\f\f\f\f\n0+\n+ (\u0015ex\u000ek)@f\n@(\u0015ex\u000ek)\f\f\f\f\n0\u00190;(38)\nwhere the sub-index 0 means evaluation in a system with-\nout dissipation, i.e., when ( \u000b;~\u001a;\u000ek) = (0;0;0). By solv-\ning the system of equations in the absence of dissipation,\nf(k;!; 0;0) = 0, the dissipation-induced change in the\nwave vector \u000ekis given by\n\u000ek\u0019\u0000~\u001a@f\n@~\u001a\f\f\f\n0+\u000b@f\n@\u000b\f\f\f\n0\n\u0015ex@f\n@(\u0015ex\u000ek)\f\f\f\n0: (39)\nIn turn, this change in the wave vector should be in-\nserted into the dispersion relation of Eq. (31) to \fnd\nthe dissipation. Inspecting Eq. (31), we note that \u000ek-\ninduced additional terms proportional to !are of the\nform (k+\u000ek)2\u0000k2\u00192k\u000ekwhich renormalize the Gilbert-\ndamping term i\u000b!\n!M. Thus, in Eq. (39), there are terms\nproportional to the frequency in both terms in the numer-\nator. We extract these terms /i!\n!Mby di\u000berentiating\nwith respect to !and de\fne the renormalization of the\nGilbert damping, i.e., \u000b!\u000b+ \u0001\u000b, from spin pumping\nas\n\u0001\u000b=i2\u0015exk!M@!\u0000\n\u0015ex\u000ekj\u000b=0\u0001\ni2\u0015exk!M@!\u0000\n\u0015ex\u000ekj~\u001a=0\u0001\n\u00001; (40)\nwhere@!represents the derivative with respect to !and\nkis the solution to the 0th-order equation. Note that in\nperforming a further local analysis around some point k0\nin thek-space of Eq. (37), a series expansion of faround\nk0must be performed before evaluating Eqs. (39) and\n(40).\nEq. (40) is generally valid, except when d= 0 and\nkL!0, which we discuss below. In the following sec-\ntion, we will determine explicit solutions of the 0th-order\nequation for some key cases, and mapping out the spin-\nwave dispersion relations and dissipation in the process.8\nB. No Surface Anisotropy ( d= 0)\nLet us \frst investigate the case of a vanishing sur-\nface anisotropy. In this case, the 0th-order expansion\nof Eq. (37) has a simple form and is independent of the\nmagnetization angle \u0012. The equation to determine kis\ngiven by\nkLtan(kL) = 0; (41)\nwith solutions k=n\u0019=L , wheren2Z. Similarly, the\nexpression for \u000ekis greatly simpli\fed, \u000ekn=i!\n!M~\u001a\nn\u0019\u0015ex\nL,\nn6= 0, such that the mode-dependent Gilbert damping\nis\n\u0001\u000bn= 2~\u001a\u0012\u0015ex\nL\u00132\n; n6= 0: (42)\nFor the macrospin mode, when n= 0, the linear ex-\npansion in \u000ekbecomes insu\u000ecient. This is because\nkLtan(kL)\u0018(kL)2forkL!0; thus, we must expand\nthe function fto second order in the deviation \u000ekaround\nkL= 0. Ford= 0, we \fnd that the boundary condi-\ntion becomes \u000ek2L2=i!\n!M~\u001a\u00152\nex, and when inserted into\nEq. (31), it immediately gives\n\u0001\u000b0= ~\u001a\u0012\u0015ex\nL\u00132\n=1\n2\u0001\u000bn; (43)\nwhich is the macrospin renormalization factor found in\nRef. 15. Using a di\u000berent approach, our results in this\nsection reproduce our previous result that the renormal-\nization of the Gilbert damping for standing waves is\ntwice the renormalization of the Gilbert damping of the\nmacrospin.18Next, we will obtain analytical results be-\nyond the description in Ref. 18 for the enhancement of\nthe Gilbert damping in the presence of surface anisotropy.\nC. Including Surface Anisotropy ( d6= 0)\nIn the presence of surface anisotropy, the out-of-plane\nand in-plane \feld con\fgurations must be treated sepa-\nrately. This distinction is because the boundary condi-\ntion Eq. (37) has di\u000berent forms for the two con\fgura-\ntions in this scenario.\n1. Out-of-plane Magnetization\nWhen the magnetization is out of plane, i.e., \u0012= 0, the\nspin-wave excitations are circular and have a high degree\nof symmetry. A simpli\fcation in this geometry is that\nthe coe\u000ecient C3= 0. In the absence of dissipation,\nthe boundary condition Eq. (37) determining the wave\nvectors becomes\nkLtan(kL) =d: (44)\nLet us consider the e\u000bects of the two di\u000berent\nanisotropies in this geometry.\n0510152025300.00.51.01.52.02.5\nLKsADaEA,nDa0\nn=0n=5FIG. 3. The ratio of enhanced Gilbert damping \u0001 \u000bEA,n=\u0001\u000b0\nin a system with easy-axis surface anisotropy versus the en-\nhanced Gilbert damping of macrospin modes in systems with\nno surface anisotropy as a function of surface-anisotropy en-\nergy.nrefers to the mode number, where n= 0 is the\nuniform-like mode. The dashed line represents the ratio\n\u0001\u000bn=\u0001\u000b0in the case of no surface anisotropy (see Eq. (42)).\na. Easy-Axis Surface Anisotropy ( d > 0):When\nd\u00181 or larger, the solutions of Eq. (44) are displaced\nfrom the zeroes of tan( kL), i.e., the solutions we found in\nthe case of no surface anisotropy, and towards the upper\npoles located at kuL= (2n+1)\u0019=2, wheren= 0;1;2;:::.\nWe therefore expand fin Eq. (37) (and thus also in\nEq. (44)) into a Laurent series around the poles from\nthe \frst negative order up to the \frst positive order in\nkLto solve the boundary condition for kL, giving\nkL\u0019\u0015ex\nL3(1 +d) + 2(kuL)2\u0000p\n12(kuL)2+ 9(1 +d)2\n2kuL:\n(45)\nUsing this result and the Laurent-series expansion for\nfin Eq. (39) and Eq. (40), we \fnd the Gilbert-damping\nrenormalization term ( \u000b!\u000b+ \u0001\u000b(oop)\nEA,n) and the ratio\nbetween the modes\n\u0001\u000b(oop)\nEA,n\n\u0001\u000b0\u00193\u0000\n3(1 +d) + 2(kuL)2\u0000p\n12(kuL)2+ 9(1 +d)2\u0001\n\u0002\n\u0002\u0000p\n4(kuL)2+ 3(1 +d)2\u0000p\n3(1 +d)\u0001\n2(kuL)2p\n4(kuL)2+ 3(1 +d)2:\n(46)\nThis ratio is plotted in Figure 3 for n\u00145. We see that\nthe ratio vanishes for large values of d. For small values\nof the anisotropy energy d, the approximate ratio exceeds\nthe exact result of the ratio we found in the limiting case\nof no surface anisotropy (see Eq. (42)). For moderate\nvalues ofd\u00185, the expansion around the upper poles\nis su\u000ecient, but only for the \frst few modes. This im-\nplies that moderate-strength easy-axis surface anisotropy\nquenches spin pumping for the lowest excited modes but\ndoes not a\u000bect modes with higher transverse exchange\nenergy.9\n05101520253001234\nLÈKsÈADaEP,nDa0\nn=1n=5\nFIG. 4. Plot of \u0001 \u000b(oop)\nEP,n=\u0001\u000b0. The dashed line represents\nthe ratio \u0001 \u000bn=\u0001\u000b0in the case of no surface anisotropy (see\nEq. (42)).\nb. Easy-Plane Surface Anisotropy ( d < 0):Easy-\nplane surface anisotropy is represented by a negative sur-\nface anisotropy din Eq. (44). In this case, the boundary\ncondition must be treated separately for the uniform-\nlike (n= 0) mode and the higher excitations. When\njdj>1, we can obtain a solution by expanding along the\nimaginary axis of kL. This corresponds to expressing the\nboundary condition in the form \u0000ikLtanh(ikL) =\u0000jdj,\nwith the asymptotic behavior kL\u0019 \u0000ijdj. Using the\nasymptotic form of the boundary condition in Eqs. (39)\nand calculating the renormalization of the Gilbert damp-\ning using Eq. (40), we \fnd that the renormalization is\n\u000b!\u000b+ \u0001\u000b(oop)\nEP,0, where\n\u0001\u000b(oop)\nEP,0\n\u0001\u000b0= 2jdj: (47)\nThus, the Gilbert damping of the lowest mode is much\nenhanced by increasing surface anisotropy. The surface-\nanisotropy mode is localized at the surface because it\ndecays from the spin-active interface and into the \flm.\nBecause the e\u000bective volume of the mode is reduced,\nspin pumping more strongly causes dissipation out of the\nmode and into the normal metal.\nFor the higher modes ( n > 0), the negative term on\nthe rhs of Eq. (44) forces the kLsolutions closer to thenegative, lower poles of tan( kL), located at k(l)\nnL= (2n\u0000\n1)\u0019=2, wheren= 1;2;3;:::. We repeat the procedure\nused for the EA case by expanding finto a Laurent series\naround these lower poles, arriving at\nkL\u00193(1\u0000jdj) + 2(k(l)\nnL)2+q\n12(k(l)\nnL)2+ 9(1\u0000jdj)2\n2k(l)\nnL:\n(48)\nUsing this relation and the new lower-pole Laurent ex-\npansion for f, Eqs. (39) and (40) give us the renormal-\nization of the Gilbert damping ( \u000b!\u000b+ \u0001\u000b(oop)\nEP,n) and\nthe ratio\n\u0001\u000b(oop)\nEP,n\n\u0001\u000b0\u00193\u0000\n3(1\u0000jdj) + 2(kuL)2+p\n12(kuL)2+ 9(1\u0000jdj)2\u0001\n\u0002\n\u0002\u0000p\n4(kuL)2+ 3(1\u0000jdj)2+p\n3(1\u0000jdj)\u0001\n2(kuL)2p\n4(kuL)2+ 3(1\u0000jdj)2:\n(49)\nThis ratio is plotted in Figure 4 from n= 1 up ton= 5.\nWe see that the ratio vanishes for large values of jdj.\nSimilar to the case of EA surface anisotropy, the approx-\nimation breaks down for large nand/or small values of\njdj.\nWhereas the n= 0 mode exhibits a strong spin-\npumping enhanced dissipation in this \feld con\fguration,\nthe DC ISHE \feld vanishes when \u0012= 0 (see Eq. (26)).\nThis is one of the reasons why this con\fguration is sel-\ndom used in experiments. However, this con\fguration\ncan lead to a signi\fcant AC ISHE, and a similar AC sig-\nnal was recently detected12. Because of the strong dissi-\npation enhancement, the EP surface anisotropy induced\nlocalized mode in perpendicular magnetization geometry\ncould be important in future experimental work.\n2. In-plane Magnetization\nWe will now complete the discussion of the spin-\npumping enhanced Gilbert damping by treating the case\nin which the magnetization is in plane ( \u0012=\u0019=2). For\nsuch systems, the coe\u000ecient C36= 0, and the 0th-order\nexpansion of Eq. (37) becomes\nkLtankL=\u0000d\u0000\n(\u0015exk)2+!H\n!M\u0001q\n1 + (\u0015exk)2+ 2!H\n!Mq\n1 + (\u0015exk)2+ 2!H\n!M\u0000\n1 + 2(\u0015exk)2+ 2!H\n!M\u0001\n\u0000d\u0015ex\nL\u0000\n1 + (\u0015exk)2+!H\n!M\u0001\ncoth\u0010\nL\n\u0015exq\n1 + (\u0015exk)2+ 2!H\n!M\u0011:\n(50)\nFor typical \flm thicknesses, of some hundred nanome-\nters, we have L=\u0015 ex\u001d1 and (\u0015exk)2\u001c1 for the lowest\neigenmodes. Thus, we take the asymptotic coth \u00181 and\nneglect the ( \u0015exk)2terms, ridding the rhs of Eq. (50) ofanykdependence. Eq (50) now becomes similar to the\nout-of-plane case\nkLtan(kL) =de\u000b; (51)10\nwhere\nde\u000b=\u0000d!H\n!Mq\n1 + 2!H\n!M\n\u0000\n1 + 2!H\n!M\u00013=2\u0000d\u0015ex\nL\u0000\n1 +!H\n!M\u0001: (52)\nde\u000bis positive if d<0 and negative for d>0 up to a crit-\nical valued\u0015ex=L=\u0015exKs=A=\u0000\n1 + 2!H\n!M\u00013=2=\u0000\n1 +!H\n!M\u0001\n,\nwhere the denominator becomes zero. For negative d,\njde\u000bj<jdj, whereas for positive d,jde\u000bjis initially smaller\nthan that ofjdjbut quickly approaches the critical value.\nWith the value Ksfrom Tab. I, we have jde\u000bj<jdj, in-\ndependent of the sign of d.\nWith this relation, we can calculate an approximate\nGilbert damping renormalization in both the EA and\nEP cases using the EP and EA relations, respectively,\nobtained in the out-of-plane con\fguration. Thus,\n\u0001\u000bip\nEA,0\u0019\u0001\u000boop\nEP,0jd!deff= 2jde\u000bj; (53)\n\u0001\u000bip\nEA,n\u0019\u0001\u000boop\nEP,njd!deff; (54)\n\u0001\u000bip\nEP,n\u0019\u0001\u000boop\nEA,njd!deff: (55)\nTo summarize this section regarding the enhancement\nof Gilbert damping, we see that the enhancement can be\nvery strong for the surface modes because their e\u000bective\nsizes are smaller than the thickness of the \flm. For all\nother modes, the enhancement decreases with increasing\nmagnitude of the surface-anisotropy energy.\nIV. NUMERICAL CALCULATIONS\nThe \frst step in the numerical method is to approxi-\nmate the equation of motion of Eq. (11) into by \fnite-size\nmatrix eigenvalue problem. We discretize the transverse\ncoordinate\u0018on the interval [\u0000L=2;L=2] intoNpoints la-\nbeled byj= 1;2;:::;N , and characterize the transverse\ndiscrete solutions of the dynamic magnetization vectors\nmQby (mx;j;my;j) of size 2N.\nWe approximate the 2nd-order derivative arising from\nthe exchange interaction using a nth-order central dif-\nference method. For the n\u00002 discretized points next to\nthe boundaries, we also use nth-order methods, using for-\nward (backward) di\u000berence schemes for the lower (upper)\n\flm boundary. This strategy avoids the introduction of\n\\ghost\" points outside the interval [ \u0000L=2;L=2] to satisfy\nthe boundary conditions.\nThus, the total operator acting on the magnetiza-\ntion on the left-hand side of Eq. (11) becomes a sparse\n2N\u00022Nmatrix operator. On the right-hand side of\nEq. (11), we also represent the convolution integral as\na 2N\u00022Ndense matrix operator, where each row is\nweighted according to the extended integration formu-\nlas for closed integrals to nth order43. The four N\u0002N\nsub-blocks of this integration operator correspond to the\nfour tensor elements of bGxy. In the \fnal discrete form,\nwe obtained a 2 N\u00022N !-dependent matrix.Next, the 4 boundary conditions (at the left and right\nboundaries for the two components, mxandmy) are used\nto reduce the number of equations to 2 N\u00004. This is per-\nformed by algebraically solving the discretized boundary\nconditions with respect to the boundary points, i.e., by\ndetermining miwherei2f1;N;N + 1;2Ngin terms of\nthe magnetizations at the interior points.\nFinally, each (2 N\u00004)\u0002(2N\u00004) matrix is separated\ninto two parts: a term independent of the frequency !\nand a term proportional to !. The dipole interaction\ncauses the eigenvalue problem to be non-Hermitian and\ntherefore computationally more demanding than a gen-\neralized eigenvalue problem. We \fnd the dispersion rela-\ntion and magnetization vectors by solving this eigenvalue\nproblem. The resulting eigenvectors are used to \fnd the\nmagnetization at the boundary by back-substitution into\nthe equations for the boundary conditions.\nWe are interested in \fnding the mode and wave-vector\ndependence of the spin-pumping enhanced Gilbert damp-\ning. To obtain this information numerically, we perform\ntwo independent calculations of the (complex) eigenval-\nues. First, we calculate the complex eigenvalues !dwhen\nthere is no spin pumping, but dissipation occurs via the\nconventional bulk Gilbert damping. Second, we calcu-\nlate the complex eigenvalues !spwhen spin pumping is\nactive at the FI-NM interface but there is no bulk Gilbert\ndamping. A mode- and wave-vector-dependent measure\nof the e\u000bective enhanced Gilbert damping enhancement\nis then given by\n\u0001\u000b=\u000bIm!sp\nIm!d: (56)\nTo ensure that we treat the same modes in the two in-\ndependent calculations, we check the convergence of the\nrelative di\u000berence in the real part of the eigenvalues. Ta-\nble I lists the values for the di\u000berent system parameters\nthat are used throughout this section.\nLet us \frst discuss the renormalization of the Gilbert\ndamping when there is no surface anisotropy. We will\npresent the numerical results for the three main geome-\ntries described in Sec. I and compare the results to the\nanalytical results of Sec. III A.\nA. FVMSW ( \u0012= 0)\nFigure 5 shows the wave-vector dependent renormal-\nization of the Gilbert damping \u0001 \u000bdue to spin pumping\nat the FI-NM interface in the FVMSW geometry. In\nthis geometry, waves travelling along \u0006^\u0010have the same\nsymmetry; thus, each line is doubly degenerate and cor-\nresponds to two waves of \u0006!. The \\spikes\" in the \fgure\nare due to degeneracies, i.e., mode crossings, and upon\ninspection, these spikes can be observed in the dispersion\nrelation.11\n0.010.1110100QL0.050.100.150.200.250.30DaH10-2L\n0.11100.60.70.80.91.0Re8wwM<-L2xL2m®\nHxL¤\nFIG. 5. \u0001\u000bversus wave vector for the FVMSW geometry\nof the four smallest eigenvalues. Top inset: Magnitudes of\neigenvectors (in arbitrary units) across the \flm at QL= 10.\nBottom inset: dispersion relation in the dipole-dipole active\nregime.\n1. Easy-Axis Surface Anisotropy ( ^\u0018easy axis)\n0.010.1110100QL0.20.40.60.81.01.2DaH10-3L\n0.11100.60.70.80.91.0Re8wwM<\n-L2xL2m®\nHxL¤\nFIG. 6. \u0001\u000bEAversus wave vector for the FVMSW geometry\nshowing the four smallest eigenvalues. The horizontal dashed\nlines indicate solutions of Eq. (46). Left inset: Magnitudes of\neigenvectors (in arbitrary units) across the \flm at QL= 5.\nRight inset: Dispersion relation in the dipole-dipole active\nregime.\nFigure 6 shows \u0001 \u000bEAfor the FVMSW geometry with\nan EA surface anisotropy at the spin-active interface. As\npredicted in Sec. III C 1 a, all modes exhibit a decreased\n\u0001\u000bcompared with those in Eqs. (43) and (42). For small\nQLand the chosen value of Ks(see Tab. I), the 1stfour\nmodes match the analytical result of Eq. (46), which is\nconsistent with the plot in Figure 3. For even higher ex-\ncited modes, the e\u000bect of the EA surface anisotropy be-\ncomes weaker due to the increase in transverse exchange\nenergy. These modes (not shown in the \fgure) approach\nthe value of \u0001 \u000bn.2. Easy-Plane Surface Anisotropy ( ^\u0018hard axis)\naLDaH10-3L45\n0.010.11100.00.10.20.3\nQL0.11100.50.60.70.80.91.0Re8wwM<bL\n-L2xL2m®\nHxL¤cL\nFIG. 7. a) \u0001 \u000bEPversus wave vector for the FVMSW geom-\netry, showing the four smallest eigenvalues. The dashed lines\nrepresent the analytic solutions from Sec. III C 1 b. b) Disper-\nsion relation in the dipole-dipole active regime. c) Magnitude\nof eigenvectors (in arbitrary units) across the \flm at QL= 5.\nFigure 7 shows \u0001 \u000bEPfor the FVMSW geometry with\nan EP surface anisotropy. We see that the mode corre-\nsponding to n= 0 has been promoted to a surface mode\nwith a large \u0001 \u000b, which for small values of QLmatches\nEq. (47). For the higher excited modes, we observe a\ndecrease in \u0001 \u000bcompared to the case with no surface\nanisotropy.\nB. BVMSW ( \u0012=\u0019=2and\u001e= 0)\n0.010.1110100QL0.20.40.60.81.0DaH10-3L\n0.11100.70.80.91.01.11.2Re8wwM<\n-L2xL2m®\nHxL¤\nFIG. 8. \u0001\u000bversus wave vector for the BVMSW geometry\n(\u0012=\u0019=2 and\u001e= 0) withKs= 0, plotted for the four small-\nest eigenvalues. Left inset: magnitudes of normalized eigen-\nvectors across the \flm at QL= 5. Right inset: dispersion\nrelation in the dipole-dipole active regime.\nFigure 8 shows the QL-dependent renormalization of\nthe Gilbert damping due to spin pumping at the FI-\nNM interface in the BVMSW geometry. We see that\nthe enhancement \u0001 \u000bagrees with the analytic limits in12\nEqs. (43) and (42) for small values of QL. For large val-\nues ofQL, we are in the strong exchange regime, in which\nthe in-plane exchange energy becomes large compared to\nall other energy contributions. This in-plane exchange\nsti\u000bness e\u000bectively quenches the coupling to the normal\nmetal layer, causing \u0001 \u000b!0 for large values of QL.\nAlthough Figure 8 only appears to show the three \frst\neigenvalues and eigenvectors, it actually contains dou-\nble this amount. Because ^zis parallel to the wave-\npropagation direction ^\u0010in this geometry, there is no\nchange in dipolar energies, regardless of whether the wave\ntravels in the + ^\u0010direction or in the \u0000^\u0010direction; thus,\nthe Gilbert damping is enhanced equally in both wave\ndirections. A slight o\u000bset from this con\fguration, taking\neither\u0012 < \u0019= 2 or\u001e6= 0, would result in a splitting of\neach line in Figure 8 into two distinct lines.\nIncluding Surface Anisotropy\nFigure 9 shows both the EA and the EP surface-\nanisotropy calculations in the BVMSW geometry. In the\ncase of an EA surface anisotropy, the mode correspond-\ning ton= 0 gets promoted to a surface mode, similarly\nto the case in which there is EP surface anisotropy in the\nFVMSW geometry. The increase in \u0001 \u000bis much smaller\nfor the same magnitude of Ks, as explained in detail in\nSec. III C. The higher modes, corresponding to n > 0,\nexhibit increased quenching of the Gilbert damping en-\nhancement. In the case of EP surface anisotropy, all\nmodes exhibit quenched Gilbert damping enhancement.\nC. MSSW ( \u0012=\u001e=\u0019=2)\nFigure 10 shows the QL-dependent renormalization of\nthe Gilbert damping due to spin pumping at the FI-NM\ninterface in the MSSW geometry. The computed eigen-\nvalues agree with Eqs. (43) and (42) for small values of\nQL. We see in the inset of Figure 10 that in this geom-\netry, the macrospin-like mode behaves as predicted by\nDamon and Eshbach3433, cutting through the dispersion\nrelations of the higher excited modes for increasing val-\nues ofQLin the dipole-dipole regime. A prominent fea-\nture of this geometry is the manner in which the modes\nwith di\u000berent signs of Ref!gbehave di\u000berently due to\nthe dipole-dipole interaction. This is because the inter-\nnal \feld direction ( ^z) is not parallel to the direction of\ntravel ( ^\u0010) of the spin wave. Hence, changing the sign of\n!is equivalent to inverting the externally applied \feld,\nchanging the xyzcoordinate system in Figure 1 from a\nright-handed coordinate system to a left-handed system.\nIn the middle of the dipole regime, the lack of symme-\ntry with respect to propagation direction has di\u000berent\ne\u000bects on the eigenvectors; e.g., in the dipole-dipole ac-\ntive region the modes with positive or negative Ref!g\nexperience an increased or decreased magnitude of the\ndynamic magnetization, depending on the value of QL,as shown in Figure 10e & f. This magnitude di\u000berence\ncreates di\u000berent renormalizations of the Gilbert damp-\ning, as the plot of \u0001 \u000b(\u0006)in Figure 10b & c shows.\nIncluding Surface Anisotropy\nFigure 11 shows \u0001 \u000bcomputed for modes in the MSSW\ngeometry with EA and EP surface anisotropies. We can\nclearly see that for small QLan exponentially localized\nmode exists in the EA case, and as predicted in Sec. III C,\nall the lowest-energy modes have spin pumping quenched\nby EP surface anisotropy. This is similar to the corre-\nsponding case in the BVMSW geometry.\nD. AC and DC ISHE\nFigure 12 shows the DC and AC ISHE measures for\nthe BVMSW geometry corresponding to the data repre-\nsented in Figure 8. In this geometry, the angular term,\nsin\u0012, in Eq. (28) is to equal one, ensuring that the DC\nmeasure is nonzero. This is not the case for all geometries\nbecause the DC electric \feld vanishes in the FVMSW ge-\nometry. The mode-dependent DC ISHE measure exhibits\nthe sameQL-dependence as the spectrum of the Gilbert\ndamping enhancement in all geometries where sin \u00126= 0.\nWe have already presented the renormalization of the\nGilbert damping in the most general cases above. There-\nfore, we restrict ourselves to presenting the simple case of\nthe BVMSW geometry with no surface anisotropy here.\nThe AC ISHE measure plotted in Figure 12 exhibits\na similarQLdependence to the Gilbert damping renor-\nmalization (and hence the DC ISHE measure), but with a\nslight variation in the spectrum towards higher values of\nQL. Note that because Eq. (24) is non-zero for all values\nof\u0012, the AC e\u000bect should be detectable in the FVMSW\ngeometry. By comparing the computed renormalization\nof the Gilbert damping for the di\u000berent geometries in\nthe previous subsections, we see that the strong renor-\nmalization of the n= 0 induced surface mode that oc-\ncurs in the FVMSW geometry with easy-plane surface\nanisotropy (see Sec. IV A 2 and Fig. 7) can have a pro-\nportionally strong AC ISHE signal in the normal metal.\nV. CONCLUSION\nIn conclusion, we have presented analytical and numer-\nical results for the spin-pumping-induced Gilbert damp-\ning and direct- and alternating terms of the inverse spin-\nHall e\u000bect. In addition to the measures of the magnitudes\nof the DC and AC ISHE, the e\u000bective Gilbert damp-\ning constants strongly depend on the modes through the\nwave numbers of the excited eigenvectors.\nIn the long-wavelength limit with no substantial sur-\nface anisotropy, the spectrum is comprised of standing-\nwave volume modes and a uniform-like (macrospin)13\n0.010.1110100QL0.51.01.52.0aLDaH10-3L\n0.11100.70.80.91.01.11.2Re8wwM<\n-L2xL2m®\nHxL¤\n0.0010.010.1110100QL0.20.40.60.81.0bLDaH10-3L\n0.11100.70.80.91.01.11.2Re8wwM<\n-L2xL2m®\nHxL¤\nFIG. 9. a) Dispersion relation versus wave vector for the BVMSW geometry ( \u0012=\u0019=2,\u001e= 0) for the four lowest eigenvalues in\nthe case of EA surface anisotropy. b) Dispersion relation in the case of EP surface anisotropy. In both \fgures, the horizontal\ndashed lines mark the value of \u0001 \u000bnin the case of no surface anisotropy.\n0.010.11101000.00.20.40.60.8Da+H10-3LaL\n0.010.11101000.00.20.40.60.81.0\nQLDa-H10-3LbL\n0.010.11101000.91.1.1ÈReHwwMLÈcL\nQL\n-L2xL20.0.51.1.5m®\nHxL¤dL\n-L2xL20.0.51.m®\nHxL¤eL\nFIG. 10. Gilbert damping renormalization in the MSSW geometry. Subplots a) and b) show Gilbert damping renormalization\n\u0001\u000bfor modes with positive (negative) Ref!g. The horizontal dashed lines represent the analytical values \u0001 \u000b0and \u0001\u000bnfor\nsmallQL. c) Dispersion relation versus wave vector for the MSSW geometry ( \u0012=\u001e=\u0019=2) for the four smallest eigenvalues,\ncolored pairwise in \u0006!. Subplot d (e) shows the magnitude of normalized eigenvectors (in arbitrary units) at QL= 3 across\nthe \flm modes with positive (negative) Ref!g.\n0.010.11101000.0.51.1.5Da+H10-3LaL\n0.010.11101000.0.51.1.5\nQLDa-H10-3LbL\n0.010.11101000.00.20.40.60.8Da+H10-3LcL\n0.010.11101000.00.20.40.60.8\nQLDa-H10-3LdL\nFIG. 11. a) and b) Gilbert damping renormalization from spin pumping in the MSSW geometry ( \u0012=\u001e=\u0019=2) for modes with\npositive (negative) Ref!gin the case of EA surface anisotropy. The four smallest eigenvalues are colored pairwise in \u0006!across\nthe plots. c) and d) show the Gilbert damping renormalization in the case of EP surface anisotropy.\nmode. These results are consistent with our previous\n\fndings18: in the long-wavelength limit, the ratio be-\ntween the enhanced Gilbert damping for the higher vol-ume modes and that of the macrospin mode is equal\nto two. When there is signi\fcant surface anisotropy,\nthe uniform mode can be altered to become a pure lo-14\n0.010.11101000.00.10.20.30.40.50.6\nQLeACH10-4LaL\n0.010.11101000.00.51.01.5\nQLeDCH10-9cmLbL\nFIG. 12. ISHE as a function of in-plane wave vector in the\nBVMSW geometry with Ks= 0. a) AC ISHE measure of\nEq. (28); b) DC ISHE measure of Eq. (28).\ncalized surface mode (in the out-of-plane geometry and\nwith EP surface anisotropy), a blend between a uniform\nmode and a localized mode (in-plane geometries and EA\nsurface anisotropy), or quenched uniform modes (out-of-\nplane \feld con\fguration and EA surface anisotropy, or\nin-plane \feld con\fguration and EP surface anisotropy).\nThe e\u000bective Gilbert damping is strongly enhanced for\nthe surface modes but decreases with increasing surface-\nanisotropy energies for all the other modes.\nThe presented measures for both the AC and DC in-\nverse spin-Hall e\u000bects are strongly correlated with the\nspin-pumping renormalization of the Gilbert damping,\nwith the DC e\u000bect exhibiting the same QLdependency,\nwhereas the AC e\u000bect exhibits a slighthly di\u000berent vari-\nation for higher values of QL. Because the AC e\u000bect\nis nonzero in both in-plane and out-of-plane geometries\nand because both EP and EA surface anisotropies in-\nduce surface-localized waves at the spin-active interface,\nthe AC ISHE can be potentially large for these modes.\nACKNOWLEDGMENTS\nWe acknowledge support from EU-FET grant no.\n612759 (\\InSpin\"), ERC AdG grant no. 669442 (\\In-sulatronics\"), and the Research Council of Norway grant\nno. 239926.\nAppendix A: Coordinate transforms\nThe transformation for vectors from \u0018\u0011\u0010toxyzcoor-\ndinates (see Fig. 1) is given by an a\u000ene transformation\nmatrixT, so that\nf(xyz)=T\u0001f(\u0018\u0011\u0010);\nfor some arbitrary vector f. Tensor{vector products are\ntransformed by inserting a unity tensor I=T\u00001Tbe-\ntween the tensor and vector and by left multiplication by\nthe tensor T, such that the tensor transforms as TbGT\u00001\nfor some tensor bGwritten in the \u0018\u0011\u0010basis.\nTis given by the concatenated rotation matrices T=\nR2\u0001R1, whereR1is a rotation \u001earound the \u0018-axis,\nandR2is a rotation \u0012\u0000\u0019\n2around the new \u0011-axis/y-axis.\nHence,\nR1=0\nB@1 0 0\n0 cos\u001e\u0000sin\u001e\n0 sin\u001ecos\u001e1\nCA; (A1)\nR2=0\nB@sin\u00120\u0000cos\u0012\n0 1 0\ncos\u00120 sin\u00121\nCA; (A2)\nsuch that\nT=0\nB@sin\u0012\u0000cos\u0012sin\u001e\u0000cos\u0012cos\u001e\n0 cos\u001e\u0000sin\u001e\ncos\u0012sin\u0012sin\u001e sin\u0012cos\u001e1\nCA: (A3)\nThis transformation matrix consists of orthogonal trans-\nformations; thus, the inverse transformation, which\ntransforms xyz!\u0018\u0011\u0010, is just the transpose, T\u00001=TT.\n1A. Serga, A. Chumak, and B. 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(Cambridge University Press,\n2007)."    },    {        "title": "1502.02699v1.Large_amplitude_oscillation_of_magnetization_in_spin_torque_oscillator_stabilized_by_field_like_torque.pdf",        "content": "arXiv:1502.02699v1  [cond-mat.mes-hall]  9 Feb 2015Large amplitude oscillation of magnetization in spin-torq ue oscillator stabilized by\nfield-like torque\nTomohiro Taniguchi1, Sumito Tsunegi2, Hitoshi Kubota1, and Hiroshi Imamura1\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba 305-8568, Japan,\n2Unit´ e Mixte de Physique CNRS/Thales and Universit´ e Paris Sud 11, 1 av. A. Fresnel, Palaiseau, France.\n(Dated: July 8, 2021)\nOscillation frequency of spin torque oscillator with a perp endicularly magnetized free layer and\nan in-plane magnetized pinned layer is theoretically inves tigated by taking into account the field-like\ntorque. It is shown that the field-like torque plays an import ant role in finding the balance between\nthe energy supplied by the spin torque and the dissipation du e to the damping, which results in\na steady precession. The validity of the developed theory is confirmed by performing numerical\nsimulations based on the Landau-Lifshitz-Gilbert equatio n.\nSpin torque oscillator (STO) has attracted much at-\ntention as a future nanocommunication device because\nit can produce a large emission power ( >1µW), a high\nquality factor ( >103), a high oscillation frequency ( >1\nGHz), a wide frequency tunability ( >3 GHz), and a nar-\nrowlinewidth ( <102kHz) [1–9]. In particular,STOwith\na perpendicularly magnetized free layer and an in-plane\nmagnetizedpinnedlayerhasbeendevelopedafterthedis-\ncovery of an enhancement of perpendicular anisotropy of\nCoFeB free layer by attaching MgO capping layer [10–\n12]. In the following, we focus on this type of STO. We\nhave investigated the oscillation properties of this STO\nboth experimentally [6, 13] and theoretically [14, 15]. An\nimportant conclusion derived in these studies was that\nfield-like torque is necessary to excite the self-oscillation\nin the absence of an external field, nevertheless the field-\nlike torque is typically one to two orders of magnitude\nsmaller than the spin torque [16–18]. We showed this\nconclusion by performing numerical simulations based on\nthe Landau-Lifshitz-Gilbert (LLG) equation [15].\nThis paper theoretically proves the reason why the\nfield-like torque is necessary to excite the oscillation by\nusing the energy balance equation [19–27]. An effective\nenergy including the effect of the field-like torque is in-\ntroduced. It is shown that introducing field-like torque\nis crucial in finding the energy balance between the spin\ntorque and the damping, and as a result to stabilize a\nsteady precession. A good agreement with the LLG sim-\nulation on the current dependence of the oscillation fre-\nquency shows the validity of the presented theory.\nThesystemunderconsiderationisschematicallyshown\nin Fig. 1 (a). The unit vectorspointing in the magnetiza-\ntion directions of the free and pinned layers are denoted\nasmandp, respectively. The z-axis is normal to the\nfilm-plane, whereas the x-axis is parallel to the pinned\nlayer magnetization. The current Iis positive when elec-\ntrons flow from the free layer to the pinned layer. The\nLLG equation of the free layer magnetization mis\ndm\ndt=−γm×H−γHsm×(p×m)\n−γβHsm×p+αm×dm\ndt,(1)pxz+\n-\nm(a)\n(b)\nmxmy\n1 -1 001\n-1 \nFIG. 1: (a) Schematic view of the system. (b) Schematic\nviews of the contour plot of the effective energy map (dotted) ,\nEq. (2), and precession trajectory in a steady state with I=\n1.6 mA (solid).\nwhereγis the gyromagnetic ratio. Since the external\nfield is assumed to be zero throughout this paper, the\nmagnetic field H= (HK−4πM)mzezconsists of the per-\npendicular anisotropy field only, where HKand 4πMare\nthe crystalline and shape anisotropy fields, respectively.\nSinceweareinterestedintheperpendicularlymagnetized\nfree layer, HKshould be larger than 4 πM. The second\nand third terms on the right-hand-side of Eq. (1) are the\nspin torque and field-like torque, respectively. The spin\ntorque strength, Hs=/planckover2pi1ηI/[2e(1+λm·p)MV], includes\nthe saturation magnetization Mand volume Vof the\nfree layer. The spin polarization of the current and the2\ndependence of the spin torque strength on the relative\nangle of the magnetizations are characterized in respec-\ntive byηandλ[14]. According to Ref. [15], βshould\nbe negative to stabilize the self-oscillation. The values\nof the parameters used in the following calculations are\nM= 1448 emu/c.c., HK= 20.0 kOe,V=π×60×60×2\nnm3,η= 0.54,λ=η2,β=−0.2,γ= 1.732×107\nrad/(Oe·s), andα= 0.005, respectively [6, 15]. The crit-\nical current of the magnetization dynamics for β= 0 is\nIc= [4αeMV/(/planckover2pi1ηλ)](HK−4πM)≃1.2 mA, where Ref.\n[15] shows that the effect of βon the critical current is\nnegligible. Whenthecurrentmagnitudeisbelowthecrit-\nical current, the magnetization is stabilized at mz= 1.\nIn the oscillation state, the energy supplied by the spin\ntorquebalancesthedissipationdue tothedamping. Usu-\nally, the energy is the magnetic energy density defined as\nE=−M/integraltextdm·H[28], which includes the perpendic-\nular anisotropy energy only, −M(HK−4πM)m2\nz/2, in\nthe present model. The first term on the right-hand-side\nof Eq. (1) can be expressed as −γm×[−∂E/∂(Mm)].\nHowever, Eq. (1) indicates that an effective energy den-\nsity,\nEeff=−M(HK−4πM)\n2m2\nz−β/planckover2pi1ηI\n2eλVlog(1+λm·p),\n(2)\nshould be introduced because the first and third terms\non the right-hand-side of Eq. (1) can be summarized as\n−γm×[−∂Eeff/∂(Mm)]. Here, we introduce aneffective\nmagnetic field H=−∂Eeff/∂(Mm) = (β/planckover2pi1ηI/[2e(1 +\nλmx)MV],0,(HK−4πM)mz). Dotted line in Fig. 1 (b)\nschematically shows the contour plot of the effective en-\nergy density Eeffprojected to the xy-plane, where the\nconstant energy curves slightly shift along the x-axis be-\ncause the second term in Eq. (2) breaks the axial sym-\nmetry of E. Solid line in Fig. 1 (b) shows the preces-\nsion trajectory of the magnetization in a steady state\nwithI= 1.6 mA obtained from the LLG equation. As\nshown, the magnetization steadily precesses practically\non a constant energy curve of Eeff. Under a given cur-\nrentI, the effective energy density Eeffdetermining the\nconstant energycurve of the stable precessionis obtained\nby the energy balance equation [27]\nαMα(Eeff)−Ms(Eeff) = 0. (3)\nIn this equation, MαandMs, which are proportional to\nthe dissipation due to the damping and energy supplied\nby the spin torque during a precession on the constant\nenergy curve, are defined as [14, 25–27]\nMα=γ2/contintegraldisplay\ndt/bracketleftBig\nH2−(m·H)2/bracketrightBig\n, (4)\nMs=γ2/contintegraldisplay\ndtHs[p·H−(m·p)(m·H)−αp·(m×H)].\n(5)\nThe oscillation frequency on the constant energy curve(a)\n00.010.02\n-0.01\n-0.020 0.2 0.4 0.6 0.8 1.0\nmzMs, -αM α, Ms-αM αMs\n-αM αMs-αM α\n(b)\n00.010.02\n-0.01\n-0.030 0.2 0.4 0.6 0.8 1.0\nmzMs, -αM α, Ms-αM αMs\n-αM αMs-αM α\n-0.02β=0\nβ=-0.2\nFIG. 2: Dependences of Ms,−αMα, and their difference\nMs−Mαnormalized by γ(HK−4πM) onmz(0≤mz<1)\nfor (a)β= 0, and (b) β=−0.2, where I= 1.6 mA.\ndetermined by Eq. (3) is given by\nf= 1/slashbig/contintegraldisplay\ndt. (6)\nSince we are interested in zero-field oscillation, and from\nthe fact that the cross section of STO in experiment [6]\nis circle, we neglect external field Hextor with in-plane\nanisotropy field Hin−plane\nKmxex. However, the above\nformula can be expanded to system with such effects\nby adding these fields to Hand terms −MHext·m−\nMHin−plane\nKm2\nx/2 to the effective energy.\nIn the absence of the field-like torque ( β= 0), i.e.,\nEeff=E, thereisone-to-onecorrespondencebetween the\nenergy density Eandmz. Because an experimentally\nmeasurable quantity is the magnetoresistance propor-\ntional to ( RAP−RP)max[m·p]∝max[mx] =/radicalbig\n1−m2z,\nit is suitable to calculate Eq. (3) as a function of mz, in-\nstead ofE, whereRP(AP)is the resistance of STO in the\n(anti)parallel alignment of the magnetizations. Figure 2\n(a) shows dependences of Ms,−αMα, and their differ-\nenceMs−αMαonmz(0≤mz<1)forβ= 0, where Ms\nandMαare normalized by γ(HK−4πM). The current is\nset asI= 1.6 mA (> Ic). We also show Ms,−αMα, and\ntheir difference Ms−αMαforβ=−0.2 in Fig. 2 (b),\nwheremxis set as mx=−/radicalbig\n1−m2z. Because −αMα\nis proportional to the dissipation due to the damping,\n−αMαis always −αMα≤0. The implications of Figs.\n2 (a) and (b) are as follows. In Fig. 2 (a), Ms−αMαis\nalways positive. This means that the energy supplied by3\ncurrent (mA)frequency (GHz) \n1.2 1.4 1.6 1.8 2.012\n0345\n: Eq. (6): Eq. (1)\nFIG. 3: Current dependences of peak frequency of |mx(f)|\nobtained from Eq. (1) (red circle), and the oscillation fre-\nquency estimated by using (6) (solid line).\nthe spin torque is always larger than the dissipation due\nto the damping, and thus, the net energy absorbed in\nthe free layer is positive. Then, starting from the initial\nequilibrium state ( mz= 1), the free layer magnetization\nmoves to the in-plane mz= 0, as shown in Ref. [14]. On\nthe other hand, in Fig. 2 (b), Ms−αMαis positive from\nmz= 1to acertain m′\nz, whereasit is negativefrom m′\nzto\nmz= 0 (m′\nz≃0.4 in the case of Fig. 2 (b)). This means\nthat, starting from mz= 1, the magnetization can move\nto a point m′\nzbecause the net energy absorbed by the\nfree layer is positive, which drives the magnetization dy-\nnamics. However, the magnetization cannot move to the\nfilm plane ( mz= 0) because the dissipation overcomes\nthe energy supplied by the spin torque from mz=m′\nztomz= 0. Then, a stable and large amplitude precession\nis realized on a constant energy curve.\nWe confirm the accuracy of the above formula by com-\nparing the oscillation frequency estimated by Eq. (6)\nwiththenumericalsolutionoftheLLGequation, Eq. (1).\nIn Fig. 3, we summarize the peak frequency of |mx(f)|\nforI= 1.2−2.0 mA (solid line), where mx(f) is the\nFourier transformation of mx(t). We also show the oscil-\nlation frequency estimated from Eq. (6) by the dots. A\nquantitatively good agreement is obtained, guaranteeing\nthe validity of Eq. (6).\nIn conclusion, we developed a theoretical formula to\nevaluate the zero-field oscillation frequency of STO in\nthe presence of the field-like torque. Our approach was\nbasedon the energybalance equationbetween the energy\nsuppliedbythe spintorqueandthe dissipationdue tothe\ndamping. An effective energy density was introduced to\ntake into account the effect of the field-like torque. We\ndiscussed that introducing field-like torque is necessary\nto find the energy balance between the spin torque and\nthe damping, which as a result stabilizes a steady preces-\nsion. The validity of the developed theory was confirmed\nby performing the numerical simulation, showing a good\nagreement with the present theory.\nThe authors would like to acknowledge T. Yorozu,\nH. Maehara, H. Tomita, T. Nozaki, K. Yakushiji, A.\nFukushima, K. Ando, and S. Yuasa. This work was sup-\nported by JSPS KAKENHI Number 23226001.\n[1] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature425, 380 (2003).\n[2] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and\nT. J. Silva, Phys. Rev. Lett. 92, 027201 (2004).\n[3] D. Houssameddine, U. Ebels, B. Dela¨ et, B. Rodmacq,\nI. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P.\nMichel, L. Prejbeanu-Buda, et al., Nat. Mater. 6, 447\n(2007).\n[4] S.Bonetti, P.Muduli, F.Mancoff, andJ. Akerman, Appl.\nPhys. Lett. 94, 102507 (2009).\n[5] Z. Zeng, G. Finocchio, B. Zhang, P. K. Amiri, J. A. Ka-\ntine, I. N. Krivorotov, Y. Huai, J. Langer, B. Azzerboni,\nK. L. 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Pitaevskii, Statistical Physics\n(part 2), course of theoretical physics volume 9\n(Butterworth-Heinemann, Oxford, 1980), chap. 7, 1st ed."    },    {        "title": "1604.04688v1.A_broadband_Ferromagnetic_Resonance_dipper_probe_for_magnetic_damping_measurements_from_4_2_K_to_300_K.pdf",        "content": "arXiv:1604.04688v1  [cond-mat.mtrl-sci]  16 Apr 2016A broadband Ferromagnetic Resonance dipper probe for magne tic damping\nmeasurements from 4.2 K to 300 K\nShikun Hea)and Christos Panagopoulosb)\nDivision of Physics and Applied Physics, School of Physical\nand Mathematical Sciences, Nanyang Technological Univers ity,\nSingapore 637371\nAdipper probefor broadband FerromagneticResonance (FMR)op erating from4.2K\nto room temperature is described. The apparatus is based on a 2-p ort transmitted\nmicrowave signal measurement with a grounded coplanar waveguide . The waveguide\ngenerates a microwave field and records the sample response. A 3- stagedipper design\nis adopted for fast and stable temperature control. The tempera ture variation due\nto FMR is in the milli-Kelvin range at liquid helium temperature. We also desig ned\na novel FMR probe head with a spring-loaded sample holder. Improve d signal-to-\nnoise ratio and stability compared to a common FMR head are achieved . Using a\nsuperconducting vector magnet we demonstrate Gilbert damping m easurements on\ntwo thin film samples using a vector network analyzer with frequency up to 26GHz:\n1) A Permalloy film of 5 nm thickness and 2) a CoFeB film of 1.5nm thicknes s. Ex-\nperiments were performed with the applied magnetic field parallel and perpendicular\nto the film plane.\na)Electronic mail: skhe@ntu.edu.sg\nb)Electronic mail: christos@ntu.edu.sg\n1I. INTRODUCTION\nIn recent years, the switching of a nanomagnet by spin transfer t orque (STT) using a spin\npolarized current has been realized and intensively studied.1–3This provides avenues to new\ntypes of magnetic memory and devices, reviving the interest on mag netization dynamics in\nultrathinfilms.4,5Highfrequencytechniques playanimportantroleinthisresearchdir ection.\nAmongthem, FerromagneticResonance(FMR)isapowerfultool. M ostFMRmeasurements\nhave been performed using commercially available systems, such as e lectron paramagnetic\nresonance (EPR) or electron spin resonance (ESR).6These techniques take advantage of\nthe high Q-factor of a microwave cavity, where the field modulation a pproach allows for the\nutilization of a lock-in amplifier.7The high signal-to-noise ratio enables the measurement of\nevensub-nanometerthickmagneticfilms. However, theoperating frequencyofametalcavity\nis defined by its geometry and thus is fixed. To determine the damping of magnetization\nprecession, whichisinprincipleanisotropic, several cavitiesarereq uiredtostudytherelation\nbetween the linewidth and microwave frequency at a given magnetiza tion direction.8–10The\napparent disadvantage isthat changing cavities canbetedious and prolongthe measurement\ntime.\nRecently, an alternative FMR spectrometer has attracted consid erable attention.11–17\nThe technique is based on a state of the art vector network analyz er (VNA) and a coplanar\nwaveguide (CPW). Both VNA and CPW can operate in a wide frequency range hence this\ntechnique is also referred to as broadband FMR or VNA-FMR. The br oadband FMR tech-\nnique offers several advantages. First, it is rather straightforw ard to measure FMR over a\nwide frequency range. Second, one may fix the applied magnetic field and acquire spectra\nwith sweeping frequency in a matter of minutes.17Furthermore, a CPW fabricated on a chip\nusing standard photolithography enables FMR measurements on pa tterned films as well as\non a single device.18In brief, it is a versatile tool suitable for the characterization of ma g-\nnetic anisotropy, investigation of magnetization dynamics and the s tudy of high frequency\nresponse of materials requiring a fixed field essential to avoid any ph ase changes caused by\nsweeping the applied field.\nAlthough homebuilt VNA-FMRs are designed mainly for room temperat ure measure-\nments, a setup with variable temperature capability is of great inter est both for fundamental\nstudies and applications. Denysenkov et al. designed a probe with va riable sample temper-\nature, namely, 4-420K,19however, the spectrometer only operates in reflection mode. In\na more recent effort, Harward et al. developed a system operating at frequencies up to\n70GHz.12However the lower bound temperature of the apparatus is limited to 27K. Here\nwe present a 2-port broadband FMR apparatus based on a superc onducting magnet. A\n3-stage dipper probe has been developed which allows us to work in th e temperature range\n4.2- 300K. Taking advantage of a superconducting vector magnet , measurements can be\nperformed with the magnetic moment saturated either parallel or p erpendicular to the film\nplane. Wealsodesigned aspring-loadedsample holderforfastandre liablesample mounting,\nquicktemperatureresponseandimprovedstability. Thissetupallow sforswiftchangesofthe\nFMR probe heads and requires little effort for the measurement of d evices. To demonstrate\nthe capability of this FMR apparatus we measured the temperature dependence of magne-\n2FIG. 1. View of the FMR dipper probe. Top panel: The schematic of the entire design with a\nstraight type FMR head. All RF connectors are 2.4mm. The vacu um cap mounted on the 4K\nstage, using In seal, and the radiation shield mounted on the second stage are not shown for clarity.\nBottom panel: photograph of the components inside the vacuu m cap.\ntization dynamics of thin film samples of Permalloy (Py) and CoFeB in diffe rent applied\nmagnetic field configurations.\nII. APPARATUS\nA. Cryostat and superconducting magnet system\nOur customized cryogenic system was developed by Janis Research Company Inc. and\nincludes a superconducting vector magnet manufactured by Cryo magnetics Inc. A vertical\nfield up to 9T is generated by a superconducting solenoid. The field ho mogeneity is ±0.1%\nover a 10mm region. A horizontal split pair superconducting magnet provides a field up to\n4T with uniformity ±0.5% over a 10mm region. The vector magnet is controlled by a\nModel 4G-Dual power supply. Although the power supply gives field r eadings according to\nthe initial calibration, to avoid the influence of remnant field we employ an additional Hall\nsensor. The cryostat has a 50mm vertical bore to accommodate v ariable temperature\ninserts and dipper probes. Our dipper probe described below is confi gured for this\ncryostat, however, the principle can be applied also to other comme rcially available\nsuperconducting magnets and cryostats.\n3B. Dipper probe\nFig. 1shows a schematic of our dipper probe assembly and a photograph o f the com-\nponents inside the vacuum cap. The dipper probe is 1.2m long and is mou nted to the\ncryogenic system via a KF50 flange. The sliding seal allows a slow insert ion of the dipper\nprobe directly into the liquid helium space. Supporting arms (not show n) lock the probe\nand minimize vibration, with the sample aligned to the field center. The c onnector box on\ntop has vacuum tight Lemo and Amphenol connectors for 18 DC sign al feedthrough. Two\n2.4mm RF connection ports allow for frequencies up to 50GHz . A vacu um pump port can\nbe shut by a Swagelok valve. We adopted a three stage design as sho wn in the photograph\nofFig. 1. The 4K stage and the vacuum cap immersed in the He bath provide co oling power\nfor the probe. The intermediate second stage acts as an isolator o f heat flow and as thermal\nsink for the RF cables, providing improved temperature control. Fu rthermore, it allows one\nto change probe heads conveniently as we discuss later. A separat e temperature sensor on\nthe second stage is used for monitoring purpose. The third stage, namely, the FMR probe\nhead with the spring loaded sample holder, is attached to the lower en d of the intermediate\nstage using stainless steel rods.\nApairof0.086”stainlesssteelSemi-RigidRFcablesrunfromthetopo ftheconnectorbox\nto the non-magnetic bulkhead connector (KEYCOM Corp.) mounted on the second stage.\nBeCu non-magnetic Semi-Rigid cables (GGB Industries, Inc.) are use d for the connection\nbetween the second stage and the probe head. The cables are car efully bent to minimize\nlosses. The rods connecting the stages are locked by set screws. Loosening the set screw\nallows the rod length to be adjusted to match the length of the RF ca bles. Reflection\ncoefficient (S 11) and transmission coefficient (S 21) can be recorded simultaneously with this\n2-port design. The leads for the temperature sensors, heater, Hall sensor and for optional\ntransport measurements are wrapped around Cu heat-sinks at t he 4K stage before being\nsoldered to the connection pins.\nC. Probe head with spring-loaded sample holder\nThe key part of the dipper probe, namely, the FMR probe head is sch ematically shown in\nFig. 2. Theassemblyisplacedinaradiationshieldtubewithaninnerdiametero f32mm. To\nmaximize thermal conduction between parts, homebuilt component s are machined from Au\nplated Cu. The 1” long customized grounded coplanar waveguide (GC PW) has a nominal\nimpedanceof50Ohm. Thestraight-lineshapeGCPWwasmadeonduro idR/circlecopyrtR6010(Rogers)\nboard, with a thickness of 254 µm and dielectric constant 10.2. The width of the center\nconductor is 117 µm and the gap between the latter and the ground planes is 76 µm. For\nthe connection, first the GCPW is soldered to the probe head, and s ubsequently the center\npin of the flange connector (Southwest Microwave) is soldered to t he center conductor of the\nGCPW. The response of the dipper with the straight-line shape GCPW installed is shown\ninFig. 3. The relatively large insertion loss (-16.9dB at 26GHZ) is due to a tota l cable\nlength of more than 3m and multiple connectors. The high frequency current flowing in the\nCPW generates a magnetic field of the same frequency. This RF field d rives the precession\n4FIG. 2. Schematic of the spring-loaded FMR probe head with st raight shape grounded coplanar\nwaveguide (GCPW). 1 Au plated Cu housing; 2 straight shape GC PW; 3 flange connector; 4 strain\ngauge thin film heater; 5 CernoxTMtemperature sensor; 6 Hall-sensor housing; 7 housing for 4- pin\nDip socket or pingo pin; 8 sample; 9 sample holder; 10 Cu sprin g; 11 spring housing; 12 sample\nholder handle nut.\nof the magnetic material placed on top of the signal line, and gives ris e to a change in the\nsystem’s impedance, which in turn alters the transmitted and reflec ted signals.\nA spring-loaded sample holder depicted in Fig. 2by items 9 to 12 is designed to mount\nthe sample. The procedure for loading a sample is as follows: 1) Pull up the handle nut\nand apply a thin layer of grease (Apiezon N type) to the sample holder ; 2) Place the sample\nat the center of the sample holder; 3) Mount the spring-loaded sam ple holder to the FMR\nhead; 4) Release the handle nut gradually so that the spring pushes the sample towards\nthe waveguide. The mounting-hole of the spring-housing is slightly lar ger than the outer\ndiameterofthespring. Thisallowsthesampleholdertomatchthesur faceoftheGCPWself-\nadaptively. With the spring-loaded FMR head design, the sample moun ting is simple and\nleaves no residue from the commonly used tapes. It maximizes the sig nal by minimizing the\ngap between waveguide and sample, and enhances the stability. Fur thermore, it is suitable\nfor variable temperature measurements due to the enhanced the rmal coupling between the\nsample, cold head and sensors ( items 9 to 12 in Fig. 2.).\nThe temperature sensor is mounted at the backside of the probe h ead. Due to limited\nspace, the heater consists of three parallel connected strain ga uges with a resistance of 120\nOhm. TheHallsensor canbemountedaccordingtotherequiredmea surement configuration.\nThe position of the Hall sensor shown in Fig. 2is an example for measurements in the\npresence of a magnetic field applied parallel to the sample surface.\nD. Probe-head using end-launch connector\nAlthough the probe head with straight-line CPW works well in our expe riment, the\nnecessary replacement of CPW due to unavoidable performance fa tigue over time, or for\ntesting new CPW designs can be time consuming. In response, end-la unch connectors\n(ELC) utilizing a clamping mechanism allow for a smooth transition from R F cables to\nCPW. Soldering the launch pin to the center conductor of CPW is optio nal and reduces\nthe effort for modifications. In Fig. 4, we show our design of a FMR probe-head using ELC\n50 5 10 15 20 25-30-20-100S (dB)\nf (GHz) S11\n S21\nFIG. 3. The reflection (S 11) and transmission (S 21) coefficients of the dipper probe with the\nstraight-line shape GCPW mounted. The measurement was perf ormed at room temperature.\nform Southwest Microwave, Inc. and a homebuilt U-shape GCPW. Sim ilar to the design of\nFig. 2, a Au plated Cu housing is used to mount the GCPW, ELC and the tempe rature and\nHall sensors. There are two locations for sample mounting. In posit ion A, the vertical field\nis used for measurements with the magnetic field applied parallel to th e surface of the thin\nfilm sample whereas the horizontal field is used for measurements wit h field perpendicular\nto the sample surface. On the other hand, measurements for bot h configurations can be\naccomplishedonlybyusingthehorizontalfieldifthesampleisplacedinp ositionB.Asshown\ninFig. 4(b) and (c), to change between configurations simply requires rot ating the dipper\nprobeby90degrees. Nevertheless, weprefertoplacethesample inpositionAfortheparallel\nconfigurationsincethesolenoidfieldismoreuniform. However, weno tethatthesamedesign\nwith the sample placed in position B is suitable also for an electromagnet . Furthermore,\nadding a rotary stage to the probe enables angular dependent FMR measurements.\nIII. EXPERIMENTAL TEST\nIn this section, we present data to assess the performance of th e FMR probe head and\ndiscuss two sets of magnetic damping measurements, demonstrat ing the capabilities and\nperformance of the appratus.\nA. Spring-loaded sample holder\nWe tested our setup using a Keysight PNA N5222A vector network a nalyzer with maxi-\nmum frequency 26.5GHz. The output power of the VNA is always 0dBm in our test. Note\nthat with 2.4mm connectors and customized GCPW, our design can in p rinciple operate\nup to 50GHz. The performance of the spring-loaded sample holder is first studied at room\ntemperature with a 2nm thick Co 40Fe40B20film. For direct comparison, the FMR spectra\nare recorded with two sample loading methods: One with a spring-load ed sample holder\n6FIG. 4. FMR probe-head with u-shape GCPW and end-launch conn ector. (a) Photograph of the\nprobe-head using end-launch connector and U-shape GCPW. Se nsors are mounted at the backside\nand at the bottom of the Cu housing. Simplified sketch of the co nfiguration for measuring with an\nexternal field generated by the split coils (b) parallel and ( c) perpendicular to the sample plane.\nRotating the dipper probe in the horizontal plane changes fr om one configuration to the other.\n(Fig. 2) and the other using the common method12which only requires Kapton tape. The\nmagnetic field is applied parallel to the plane of the thin film sample. Six se ts of data were\nobtained by reloading the sample for each measurement. In Fig. 5, we show the amplitude\nof the power transmission coefficient from Port 1 to Port 2 (S 21) at a frequency of 10GHz\nand a temperature of 300K. The open circles represent a spectru m for a spring-loaded sam-\nple mounting whereas the open squares is the spectrum showing larg est signal for the six\nflip-sample loadings. The averaged spectra for all six spectra are s hown by solid line and\ndotted line, for spring and flip-sample loading, respectively. Two obs ervations are evident:\nFirst, the best signal we obtained using the flip sample method is appr oximately 20 percent\nlower compared to the spring-loaded method. Thus the spring-load ed method gives a better\nsignal to noise ratio and sensitivity. Second, for the spring-loaded method, the difference\nbetween the averaged spectrum and single spectra is negligibly small. On the other hand the\nvariation between measurements for the flip-sample method can be as large as 20 percent.\nHence the spring-loaded method has better stability and is reprodu cible.\nB. Temperature response\nAs detailed in the previous section, the probe head is made of Au plate d Cu blocks with\nhighinternalthermalconductionandgoodthermalcontact. Con sequently, theresponsetime\nof the temperature control will be small as the characteristic the rmal relaxation time of a\nsystem is C/k, whereCis the heat capacity and kis the overall thermal conduction. Also,\nthe temperature difference between sample and sensor is minimized e ven with the heater\nturned on. Shown in Fig. 6are the FMR spectra and temperature variation for a CoFeB\nfilm of 3nm thickness measured at 4.4K. The external field was swept at a rate of about -\n10Oe/s. Forfields close to which FMRpeaks areobserved, we detec ted a temperaturerise of\n7FIG. 5. Comparison between S 21signals obtained using spring-loaded sample holder mounti ng and\nflip-sample mounting at 300K. The sample has a stack of MgO(3n m)—CoFeB(2nm)—MgO(3nm)\ndeposited on silicon substrate. (Numbers in parenthesis of the sample composition represent the\nthickness of the respective layer.) The frequency is 10 GHz a nd the FMR center field is at 1520\nOe.\na few mK. In fact, the field values corresponding to maximum temper atures are about 20Oe\nlower than the fields satisfying FMR condition, showing that the char acteristic relaxation\ntime between the sample and cold head is approximately 2 seconds. Th e temperature rise\nof the probe head due to FMR indicates that the magnetic system ab sorbs energy from\nthe microwave and dissipates into the thermal bath. Specifically, at the field satisfying\nthe FMR condition, the damping torque is balanced by the torque gen erated by the RF\nfield. However, the dissipation power of such process is propotiona l to the thickness of\nthe magnetic film hence is very small. The successful detection of a t emperature rise adds\ncredence to the high thermal conduction within the probe head and relative low thermal\nconduction between different stages. This demonstration shows t hat the probe head is\ncapable of measuring samples with phase transitions in a narrow temp erature range, such\nas a superconducting/ferromagnetic bilayer system.20\nC. Magnetic damping measurements\nAlthough the FMR probe can be used to determine the energy anisot ropy of magnetic\nmaterials, our primary purpose is to study magnetic damping parame ter. In the following,\ntwo examples of such measurements are briefly described. Shown in Fig. 7is FMR response\nof a Py film of 5nm thickness deposited on a silicon substrate, measur ed at 4.4K. The\nsweeping external magnetic field is parallel to the sample surface. R eal and Imaginary parts\nof the spectra obtained at selected frequencies are plotted with o pen circles in Fig. 7(a)\nand (b), respectively. In FMR measurements, the change in the tr ansmittance, S21, is a\ndirect measure of the field-dependent susceptibility of the magnet ic layer. According to the\n8FIG. 6. Sample temperature variation due to FMR at selected f requencies. (upper panel) Ampli-\ntude of S 21and (lower panel) temperature variation of MgO(3nm)—CoFeB (3nm)—MgO(3nm) at\n4.4K measured with external field parallel to the film plane.\nLandauLifshitzGilbert formalism, the dynamic susceptibility of the ma gnetic material in the\nconfiguration where the field is applied parallel to the plane of the thin film be described\nas:21\nχIP=4πMs(H0+Huni+4πMeff+i∆H/2)\n(H0+Huni)(H0+Huni+4πMeff)−H2\nf+i(∆H/2)·[2(H0+Huni)+4πMeff](1)\nHere, 4πMsis the saturation magnetization, Huniis the in-plane uniaxial anisotropy,\n4πMeffis the effective magnetization, Hf= 2πf/γ, and ∆His the linewidth of the spectrum\n– the last term is of key importance to determine the damping parame ter. As shown by\nsolid lines in Fig. 7(a) and (b), the spectra can be fitted very well by adding a backgr ound,\na drift proportional to time, and a phase factor.11,22The field linewidth as a function of\nfrequency – ∆ H(f) is plotted in Fig. 7(c). The data points fall on a straight line. The\ndamping parameter αGL= 0.012±0.001 is therefore determined by the slope through9,23:\n∆H=4π\nγαGLf+∆H0 (2)\nThe error bar here is calculated from the confidence interval of th e fit.\nWe have also tested the setup with a magnetic field applied perpendicu lar to the sample\nplane. The results for a MgO(3nm)—Co 40Fe40B20(1.5nm)—MgO(3nm) stack deposited on\nsilicon substrate are shown in Fig. 8. Comparing the spectra obtained at different tempera-\ntures and fixed frequency, two observations are evident. First, the FMR peak position shifts\nto higher field as the temperature is lowered due to changes in the eff ective magnetization.\n9FIG. 7. FMR data of a Py thin film of thickness 5nm measured at 4. 4K with magnetic field\napplied parallel to the sample plane. (a) Real and (b) Imagin ary parts of transmitted signal S 21\nat selected frequencies. The data are normalized and the rel ative strength between the spectra at\ndifferent frequencies are kept. (c) FMR linewidth as a functio n of frequency. The damping was\ncalculated to be 0.012 ±0.001, using a linear fit.\nSecond, the FMR linewidth increases with decreasing temperature. Although the interfacial\nanisotropy can be determined by fitting the FMR peak positions to th e Kittel formula,24\nhere, we are more interested in the damping parameter as a functio n of temperature. The\ndynamic susceptibility in this configuration is25:\nχOP=4πMs(H−4πMeff−i∆H/2)\n(H−4πMeff)2−H2\nf+i∆H·(H−4πMeff)(3)\nFollowing the same procedure as for Py, the real and imaginary part of the spectra are\nfitted simultaneously to obtain the linewidth. In Fig. 8(b), we plot the linewidth as a\nfunction of frequency at the two boundaries of our measured tem peratures. Although the\nmeasured linewidth at lower temperature is larger, the slope of the t wo curves is in good\nagreement. The additional linewidth at 6K is primarily due to zero freq uency broadening,\nwhich quantifies the magnitude of dispersion of the effective magnet ization. The results\nare summarized in Fig. 8(c). Gilbert damping is essentially independent of temperature\nalthough there is a minimum at 40 K. The room temperature value obta ined is in agreement\nwith the value for a thicker CoFeB.21,26On the other hand, the inhomogeneous broaden-\ning increases with lowering temperatue. The value at 6K is more than d ouble compared\nto room temperatue. Notably, neglecting the zero-frequency off set ∆H(0), arising due to\ninhomogeneity, would give rise to an enhanced effective damping comp ared to the intrinsic\ncontribution. Cavity based, angulardependent FMRmayalso disting uish theGilbertdamp-\ning from inhomogeneity effects. A shortcoming however, is the need to take into account\nthe possible contribution of two magnon scattering, which causes in creased complications\nin the analysis of the data.27,28On the other hand, broadband FMR using a dipper probe\nwith the applied magnetic field in the perpendicular configuration, rule s out two magnon\nscattering making this technique relatively straightforward to imple ment.29\nThedipperprobediscussedhereisnotlimitedtomeasurementsofth edampingcoefficient.\nThe broadband design is also useful for time-domain measurements .30Furthermore, a spin\n10FIG. 8. Temperature dependent FMR measurement for a CoFeB th in film of thickness 1.5nm\nwith the magnetic field applied perpendicular to the film plan e. (a) Transmitted FMR signal at\n20GHz obtained at different temperatures. (b) FMR linewidth a s a function of frequency at 6K\nand 280K. (c) Damping constant and inhomogeneous broadenin g as a function of temperature.\nThe solid lines are the guides for the eye.\ntransfer torque ferromagnetic resonance31measurement on a single device can be performed\nwith variable temperature using bias Tee and a separate sample holde r.\nIV. CONCLUSION\nWe have developed a variable temperature FMR to measure the magn etic damping pa-\nrameter in ultra thin films. The 3-stage dipper and FMR head with a spr ing-loaded sample\nholder design have a temperature stability of milli Kelvin during the FMR measurements.\nThis apparatus demonstrates improved signal stability compared t o traditional flip-sample\nmounting. The results for Py and CoFeB thin films show that the FMR d ipper can measure\nthe damping parameter of ultra thin films with: Field parallel and perpe ndicular to the film\nplane in the temperature range 4.2-300K and frequency up to at lea st 26 GHz.\nACKNOWLEDGMENTS\nTheauthorsaregratefultoSzeTerLimatDataStorageInstitut eforpreparingtheCoFeB\nsamples. We acknowledge Singapore Ministry of Education (MOE), Ac ademic Research\nFund Tier 2 (Reference No: MOE2014-T2-1-050) and National Res earch Foundation (NRF)\nof Singapore, NRF-Investigatorship (Reference No: NRF-NRFI2 015-04) for the funding of\nthis research.\nREFERENCES\n1J. C. Slonczewski, Journal of Magnetism and Magnetic Materials 159, L1 (1996) .\n2J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ra lph,\nPhysical Review Letters 84, 3149 (2000) .\n113S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, a nd E. E.\nFullerton, Nat Mater 5, 210 (2006) .\n4N. Locatelli, V. Cros, and J. Grollier, Nat Mater 13, 11 (2014) .\n5A. Brataas, A. D. Kent, and H. Ohno, Nat Mater 11, 372 (2012) .\n6M. Farle, Reports on Progress in Physics 61, 755 (1998) .\n7B. Heinrich, Measurement Techniques and Novel Magnetic Properties (Springer\nBerlin Heidelberg, 2006).\n8B. Heinrich, G. Woltersdorf, R. Urban, and E. Simanek,\nJournal of Applied Physics 93, 7545 (2003) .\n9K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy, and A. J´ an ossy,\nPhysical Review B 73, 144424 (2006) .\n10B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y. S ong, Y. Sun,\nand M. Wu, Physical Review Letters 107, 066604 (2011) .\n11S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P . Kabos, T. J.\nSilva, and J. P. Nibarger, Journal of Applied Physics 99, 093909 (2006) .\n12I. Harward, T. O’Keevan, A. Hutchison, V. Zagorodnii, and Z. Celins ki,\nReview of Scientific Instruments 82, 095115 (2011) .\n13J. Wei, J. Wang, Q. Liu, X. Li, D. Cao, and X. Sun,\nReview of Scientific Instruments 85, 054705 (2014) .\n14E. Montoya, T. McKinnon, A. Zamani, E. Girt, and B. Heinrich,\nJournal of Magnetism and Magnetic Materials 356, 12 (2014) .\n15H. Gowiski, M. Schmidt, I. Gociaska, J.-P. Ansermet, and J. Dubowik ,\nJournal of Applied Physics 116, 053901 (2014) .\n16M. Bailleul, Applied Physics Letters 103, 192405 (2013) .\n17C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P . P. Freitas,\nJournal of Applied Physics 101, 074505 (2007) .\n18I. Neudecker, K. Perzlmaier, F. Hoffmann, G. Woltersdorf, M. Bue ss, D. Weiss, and\nC. H. Back, Physical Review B 73, 134426 (2006) .\n19V.P.DenysenkovandA.M.Grishin, Review of Scientific Instruments 74, 3400 (2003) .\n20C. Bell, S. Milikisyants, M. Huber, and J. Aarts,\nPhysical Review Letters 100, 047002 (2008) .\n21C. Bilzer, T. Devolder, J.-V. Kim, G. Counil, C. Chappert, S. Cardoso , and P. P.\nFreitas,Journal of Applied Physics 100, 053903 (2006) .\n22H. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider, M. J. Carey , S. Maat, and\nJ. R. Childress, Physical Review B 84, 054424 (2011) .\n23Z. Celinski and B. Heinrich, Journal of Applied Physics 70, 5935 (1991) .\n24M. Sparks, R. Loudon, and C. Kittel, Physical Review 122, 791 (1961) .\n25T. Devolder, P.-H. Ducrot, J.-P. Adam, I. Barisic, N. Vernier, J.-V. Kim, B. Ockert,\nand D. Ravelosona, Applied Physics Letters 102, 022407 (2013) .\n26S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai,\nJ. Hayakawa, F. Matsukura, and H. Ohno, Nat Mater 9, 721 (2010) .\n27S. Mizukami, Y. Ando, and T. Miyazaki, Physical Review B 66, 104413 (2002) .\n1228J.Lindner, I.Barsukov, C.Raeder,C.Hassel, O.Posth, R.Meck enstock, P.Landeros,\nand D. L. Mills, Physical Review B 80, 224421 (2009) .\n29R. Arias and D. L. Mills, Physical Review B 60, 7395 (1999) .\n30Y.-T. Cui, G. Finocchio, C. Wang, J. A. Katine, R. A. Buhrman, and D. C. Ralph,\nPhys. Rev. Lett. 104, 097201 (2010) .\n31L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011) .\n13"    },    {        "title": "2010.00144v1.Quantum_hydrodynamics_of_spin_winding.pdf",        "content": "Quantum hydrodynamics of spin winding\nYaroslav Tserkovnyak,1Ji Zou,1Se Kwon Kim,2and So Takei3\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea\n3Department of Physics, Queens College of the City University of New York, Queens, New York 11367, USA\nAn easy-plane spin winding in a quantum spin chain can be treated as a transport quantity,\nwhich propagates along the chain but has a \fnite lifetime due to phase slips. In a hydrodynamic\nformulation for the winding dynamics, the quantum continuity equation acquires a source term due\nto the transverse vorticity \row. The latter re\rects the phase slips and generally compromises the\nglobal conservation law. A linear-response formalism for the nonlocal winding transport then reduces\nto a Kubo response for the winding \row along the spin chain, in conjunction with the parasitic\nvorticity \row transverse to it. One-dimensional topological hydrodynamics can be recovered when\nthe vorticity \row is asymptotically small. Starting with a microscopic spin-chain formulation, we\nfocus on the asymptotic behavior of the winding transport based on the renormalized sine-Gordon\nequation, incorporating phase slips as well as Gilbert damping. A generic electrical device is proposed\nto manifest this physics. We thus suggest winding conductivity as a tangible concept that can\ncharacterize low-energy dynamics in a broad class of quantum magnets.\nI. INTRODUCTION\nIn addition to e\u000ecient heat transport carried by spin\ndynamics along electrically-insulating spin chains,1there\nhas also been much interest in their transmission of spin\nsignals.2In the case of spin currents polarized along a di-\nrection of axial symmetry, the spin signals can propagate\nballistically or di\u000busively, while generally also undergoing\ndecay due to spin-nonconserving perturbations. Alterna-\ntively, transport based on collective order-parameter dy-\nnamics and rooted in topological conservation laws has\nbeen suggested for potentially more robust propagation\nof signals.3\nThe winding dynamics of planar spins in an easy-plane\n(anti)ferromagnet is one ready example of this. Extend-\ning the natural super\ruid analogy for the SO(2) order\nparameter to the nonequilibrium setting, scenarios for\nspin super\ruidity have been proposed4and experimen-\ntally pursued.5The spontaneously broken U(1) symme-\ntry is replaced here by the axial symmetry (say along the\nzaxis) of the easy-plane spin winding (in the xyplane).\nIf the latter experiences some anisotropies within the xy\nplane, however, the associated SO(2) symmetry gets bro-\nken directly, invalidating spin conservation and possibly\npinning the conjugate phase (i.e., the winding angle) al-\ntogether.\nWhile the spin density \u001azis then no longer acting as\na long-wavelength transport quantity, the winding den-\nsity\u001a/@x'(xbeing the spatial coordinate along the\ntransport channel and 'the azimuthal angle of the order\nparameter in the xyplane) obeys a continuity equation\n(with the associated \rux j/\u0000@t'), irrespective of the\nanisotropies. This is crucially contingent on the abil-\nity to unambiguously de\fne '(x;t) along the channel,\nat all times, which is compromised whenever a vecto-\nrial order parameter traverses one of the poles along the\nhard (z) axis. Such processes could be visualized as vor-\ntices in the (1 + 1)-dimensional space-time, realizing a\nvorticity \row transverse to the xaxis. In analogy to sim-ilar parasitic events in low-dimensional super\ruids and\nsuperconductors,6these can be called phase slips.7,8\nIn this paper, we set out to formulate a rigorous mi-\ncroscopic formalism to address these issues, in regard to\nquantum winding hydrodynamics, at an arbitrary tem-\nperature. Once the formal framework is in place, our fo-\ncus is going to be on the role of anisotropies, phase slips,\nand general magnetic damping, in relation to spatiotem-\nporal transport properties of the spin-winding \rows. In\nparticular, we wish to establish regimes, where the no-\ntion of a winding conductivity can be meaningful both\ntheoretically and experimentally.\nOur discussion is structured as follows. We start, in\nSec. II, by recapping vorticity dynamics in two spatial\ndimensions. The notion of a topological conservation law\nis introduced for a classical theory in Sec. II A, which is\nthen discretized and quantized into an exact quantum\nformulation on a generic spin lattice, in Sec. II B. A simi-\nlar procedure is then attempted for the winding dynamics\nin Sec. III, where the quantum \row of spin winding along\na spin chain gets supplemented with vorticity \row trans-\nverse to it. Here, we develop a quantum Kubo formalism\nfor the winding transport, and establish boundary con-\nditions that could allow us to read it out electrically. In\nSec. IV, a sine-Gordon model is treated systematically,\nin order to study the interplay of the winding \row, phase\nslips, and other sources of dissipation associated with col-\nlective dynamics, both at zero and \fnite temperatures.\nA summary and outlook are o\u000bered in Sec. V.\nII. 2D VORTICITY (HYDRO)DYNAMICS\nA. Classical vorticity dynamics\nA three-component real vector \feld m= (mx;my;mz)\nresiding in 2+1 dimensions, m(r;t), realizes an R2!R3\nmapping, at any given time t. These spatial \feld textures\nare devoid of point defects, as the fundamental homo-arXiv:2010.00144v1  [cond-mat.mes-hall]  30 Sep 20202\ntopy group of the order-parameter space mis trivial:\n\u00191(R3) = 1. Such two-dimensional textures are, fur-\nthermore, all topologically equivalent, having \fxed the\nboundary pro\fle of mon a connected patch of R2, which\nis re\rected in the fact that \u00192(R3) = 1. Despite this,\na smooth vector \feld de\fnes a topological hydrodynam-\nicsgoverned by the continuity equation @\u0016j\u0016= 0 (with\nthe Einstein summation implied over the Greek letters:\n\u0016= 0;1;2!t;x;y ), where9\nj\u0016\u0011\u000f\u0016\u0017\u0018z\u0001@\u0017m\u0002@\u0018m\n2\u0019: (1)\nHere, zis the z-axis unit vector and \u000f\u0016\u0017\u0018is the Levi-\nCivita symbol.\nFor the special case of a rigid texture sliding at a veloc-\nityv, for example: j=\u001av, where\u001a\u0011j0andj= (jx;jy).\nFor another special case of a sharp vortex in a strongly\neasy-plane magnet with the planar order parameter nor-\nmalized to unity, jmj! 1:\u001a\u0019\u000e(r\u0000r0), where r0is\nthe position at which mtilts out of the plane (over an\nappropriate healing length de\fning the size of the core).\nThese examples intuitively suggest a \ruid whose density\nis given by the distribution of vorticity in the system.\nWhile in the extreme easy-plane case, a vortex core car-\nries a quantized topological charge, we do not generally\nassume this special limit.\nThe above conserved quantity j0can be recast as a\n\fctitious \rux\n\u001a=z\u0001r\u0002A\n2\u0019(2)\nassociated with the gauge \feld\nA=mxrmy\u0000myrmx: (3)\nApplying Green's theorem, we then see that the con-\nserved topological charge within a patch \n,\nQ\u0011Z\n\nd2r\u001a=I\n@\ndr\u0001A\n2\u0019=I\n@\nd\u001e\n2\u0019m2\nk; (4)\nis associated with the order-parameter winding around\nits boundary @\n.mkis the \feld's projection onto the\nxyplane (within the order-parameter space) and \u001eis the\nassociated azimuthal angle. This reveals the geometri-\ncal meaning of the conservation law: The charge Qin\nthe bulk can change only in response to a vorticity \row\nthrough the boundary.\nB. Quantum vorticity dynamics\nTo construct a simple quantum theory, which repro-\nduces the above classical hydrodynamics of vorticity in\nthe classical limit of ~!0, let us consider a square lat-\ntice model sketched in Fig. 1. We label each vertex of the\nlattice by two integer indices: \u0010 (along the xaxis) and|\n(along theyaxis). The same indices are used to label thesquare plaquettes, according to their lower left corner, as\nwell as the vertical links going upward and the horizon-\ntal links to the right of the site \u0010 |. Each site contains a\nquantum spin S= (Sx;Sy;Sz), of magnitude S(in units\nof~), characterized by the standard angular-momentum\nalgebra [Sa;Sb] =i\u000fabcSc.\n⇢ı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>x(ı)\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>y(|)\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>jxı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>jyı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>jyı˜|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>jx˜ı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>Sı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>S˜ı˜|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>S˜ı|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>Sı˜|\n<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>\nFIG. 1. The quantum spin lattice described by an arbitrary\nHamiltonian H.S\u0010|is the spin operator at site \u0010 |, with index\n\u0010 (|) running along the x(y) axis. ~ \u0010 = \u0010 + 1 and ~ |=|+ 1.\u001a\u0010|\nis the conserved topological charge per plaquette \u0010 |,jx\n\u0010|(jy\n\u0010|) is\nthe \rux per vertical (horizontal) link \u0010 |, which together satisfy\nthe quantum continuity equation (10).\nWe associate a charge density\n\u001a\u0010|\u0011Ax\n\u0010|\u0000Ax\n\u0010~|+Ay\n~ \u0010|\u0000Ay\n\u0010|\n2\u0019a(5)\nto each plaquette, where ais the lattice spacing. Here,\n~ \u0010\u0011\u0010 + 1 and ~|\u0011|+ 1, and\nAx\n\u0010|=z\u0001(S~ \u0010|+S\u0010|)\u0002(S~ \u0010|\u0000S\u0010|)\n4aS2+ H:c:=z\u0001S\u0010|\u0002S~ \u0010|\naS2;\nAy\n\u0010|=z\u0001(S\u0010~|+S\u0010|)\u0002(S\u0010~|\u0000S\u0010|)\n4aS2+ H:c:=z\u0001S\u0010|\u0002S\u0010~|\naS2;\n(6)\nwhich we assign formally to the corresponding horizontal\nand vertical sides of the plaquette, respectively. These\nde\fnitions mimic Eqs. (2) and (3), respectively, and\nshould reproduce them by coarse graining the magnetic\ntextures in the classical limit of S!1 .\nAccording to these de\fnitions,\n\u001a{|=z\u0001c{|\n2\u0019a2;where c{|\u00111\nS2X\nlSl\u0002S~l(7)\nis the vector chirality of the corresponding plaquette,\nwith the sum running over the four vertices labelled by\nl(~lbeing the vertex next to l, in the counterclockwise\ndirection).10We also see [according to Eq. (5)] that\nQ=X\n\u0010|\u001a\u0010| (8)3\nvanishes in the bulk and reduces to the boundary terms,\nwhich we can interpret as the quantum version of the\nvorticity (4). This suggests a conservation law with the\nboundary \ruxes corresponding to the vorticity \row. In-\ndeed, according to the Heisenberg equation of motion\n(for Hamiltonian Hand an arbitrary time-independent\noperatorO),\n@tO\u0011i\n~[H;O]; (9)\nthe quantum vorticity density \u001a\u0010|is seen to satisfy the\ncontinuity equation:\n@t\u001a\u0010|+jx\n~ \u0010|\u0000jx\n\u0010|+jy\n\u0010~|\u0000jy\n\u0010|\na= 0: (10)\nThe \ruxes in the second term are consistent with quan-\ntizing Eq. (1):\njx\n\u0010|=z\u0001(S\u0010~|\u0000S\u0010|)\u0002@t(S\u0010~|+S\u0010|)\n4\u0019aS2+ H:c:; (11)\nand similarly for the other components.\nIt is useful to emphasize that the associated conserva-\ntion law is not rooted in any speci\fc symmetry of the\nsystem. Indeed, the form of the Hamiltonian Hstill re-\nmains arbitrary. The continuity is rather dictated by the\ntopology associated with the vorticity (hydro)dynamics\nin the interior of the system. Speci\fcally, an arbitrary\nlocal deformation of the \feld in the bulk yields the same\nnet vorticity, irrespective of the details of the dynamics.\nIII. 1D WINDING DYNAMICS\nIn contrast to the vorticity \row, winding dynamics\nin, e.g., one-dimensional (1D) super\ruids6or magnets11\nobey the conservation law only approximately. In these\nsystems, the underlying topological invariant relies on\na nonlinear constraint applied to the order parameter,\nwhich ultimately makes the conserved quantity vulnera-\nble to thermal \ructuations. This leads to phase slips,7\nwhich are detrimental to the topological conservation\nlaw.\nThese issues carry over to the quantum regime, where\nquantum phase slips arise due to tunneling.8Supposing\nthese could be neglected, in an appropriate limit, we wish\nto formulate a Kubo approach for the topological quan-\ntum \row in terms of the corresponding current autocor-\nrelator.\nA. Quantum winding dynamics\nLet us illustrate these points by considering winding\ndynamics along a 1D quantum lattice, with an easy-plane\nanisotropy in spin space, which constrains the (ferro- or\nantiferro-)magnetic dynamics to lie close to the xyplane.\nAs we have already mentioned, a coarse-grained classicalhydrodynamics can be formulated in terms of the den-\nsity\u001a=\u0000@x'=\u0019 and \ruxj=@t'=\u0019, where'is the az-\nimuthal angle of the order parameter in the xyplane,11\nsuch that@t\u001a+@xj= 0.\nAllowing for arbitrary (unconstrained) spin dynamics,\nwe now formulate a quantum theory on a lattice through\nthe de\fnitions\n\u001a\u0010=z\u0001S~ \u0010\u0002S\u0010\n\u0019aS2; j\u0010=z\u0001S\u0010\u0002@tS\u0010\n2\u0019S2+ H:c:; (12)\nwhere, as before, @tshould be understood according to\nthe Heisenberg equation of motion (9) (which depends on\na concrete Hamiltonian, to be speci\fed later). Since these\nreduce to the winding density and \rux, in the appropri-\nate coarse-grained classical limit, we may expect them to\napproximately obey the continuity equation (when the\nphase slips can be disregarded). Indeed,\n@t\u001a\u0010+j~ \u0010\u0000j\u0010\na=z\u0001(S~ \u0010\u0000S\u0010)\u0002@t(S~ \u0010+S\u0010)\n2\u0019aS2+ H:c::(13)\nThe term on the right-hand side (RHS), which spoils the\nexact conservation law, can be recognized to be exactly\n(twice) the vorticity \row transverse to the spin chain,\ncf. Eq. (11). If it can be neglected, we would recover\nthe continuity equation and with it the Kubo formula\n(26) that governs the topological \row and the electrical\ntransconductance, to be discussed below.\nB. Boundary conditions\nIn order to place the spin chain (of length L) into a\nmeasurable external circuit, let us suppose it is biased by\nspin torques (polarized along the zaxis)\u001cL(R)at its left\n(right) ends. The (semiclassical) work associated with\nthe left torque is\n\u0001WL=Z\ndt\u001cL@t'=\u0019\u001cLZ\ndtj=\u0019\u001cL\u0001QL;(14)\nwhere \u0001QLis the topological charge transfer into the\nchain through the left end. This translates into the ef-\nfective chemical potential bias at the left end given by\n\u0016L=\u0001WL\n\u0001QL=\u0019\u001cL: (15)\nSimilarly for the right end, we get\n\u0016R=\u0001WR\n\u0001QR=\u0000\u0019\u001cR: (16)\nSuch torques can be induced, for example, by the spin\nHall e\u000bect triggered by an electrical current \rowing trans-\nverse to the chain.4See Fig. 2 below.\nReciprocally to these torques, the precessional dynam-\nics produces a transverse motive force on the electrons\nin the contacts, which can be used to detect the out\row\nof the topological charge through the ends.4,11We will\nreturn to discuss this in more detail in Sec. III D.4\nC. Kubo formula\nWe are now ready to de\fne the bulk impedance for\nthe topological \row, as an intrinsic property of the quan-\ntum magnet. Starting with a continuity equation for the\ncoarse-grained quantum dynamics in the bulk, we have\n@t\u001a+@xj= 0; (17)\nwhere the conserved density and current are obtained\nfrom Eqs. (12), and we neglect the RHS of Eq. (13), i.e.,\nphase slips, for now. We recall that the time derivatives\nare obtained in the Heisenberg picture. If we perturb the\nsystem by a scalar potential \u001e(x;t) that couples linearlyto the topological charge, the Hamiltonian becomes\nH!H+Z\ndx\u001e(x;t)\u001a(x): (18)\nNote that the topological density is even under time re-\nversal, while the \rux is odd (supposing the Hamiltonian\nis time-reversal invariant), so it vanishes in equilibrium,\nwhen\u001e\u00110. For a \fnite time-dependent potential \u001e, on\nthe other hand, the linear response dictates\nj(x;t) =1\n~Z\ndx0dt0G(x\u0000x0;t\u0000t0)\u001e(x0;t0); (19)\nwhere\nG(x\u0000x0;t\u0000t0)\u0011\u0000i\u0012(t\u0000t0)[j(x;t);\u001a(x0;t0)];(20)\naccording to the Kubo formula (with the equilibrium ex-\npectation value implicit on the right-hand side).\nTo invoke the continuity equation, we di\u000berentiate the\nresponse function in time:\n@tG(x\u0000x0;t\u0000t0) =i\u0012(t\u0000t0)[j(x;t);@t0\u001a(x0;t0)]\u0000i\u000e(t\u0000t0)[j(x);\u001a(x0)]\n=\u0000i\u0012(t\u0000t0)[j(x;t);@0\nxj(x0;t0)] +\u000e(t\u0000t0)@0\nxp(x\u0000x0);(21)\nwhere the auxiliary function p(x\u0000x0) is obtained by in-\ntegrating\n@0\nxp(x\u0000x0) =\u0000i[j(x);\u001a(x0)]: (22)\nFourier transforming in time, j(!) =R\ndtei!tj(t) etc., we\n\fnally get\nj(x;!) =i\n~!Z\ndx0&(x\u0000x0;!)\"(x0;!); (23)\nwhere\n&(x\u0000x0;t\u0000t0)\u0011\u0000i\u0012(t\u0000t0)[j(x;t);j(x0;t0)]\n+\u000e(t\u0000t0)p(x\u0000x0)(24)\ninvolves the current autocorrelator and\n\"\u0011\u0000@x\u001e (25)\nis the e\u000bective electric \feld. This gives for the conduc-\ntivity relating j(k;!) to\"(k;!):\n\u001b(k;!) =i\n~!&(k;!); (26)\nhaving also Fourier transformed in real space,R\ndxe\u0000ikx.\nFor a torque-biased spin chain,\n\"=\u0016L\u0000\u0016R\nL=\u0019\u001cL+\u001cR\nL; (27)\nsupposing that the length of the topological transport\nchannelLis long enough, so that the bulk dominatesover the interfacial impedances and focusing on the DC\nlimit.3\nWhile evaluating the Kubo formula, we should in gen-\neral also calculate the phase-slip rate governed by the\nRHS of Eq. (13) (driven by the potential \u001egradient that\ncouples to the winding density \u001a). For the internal con-\nsistency of the hydrodynamic treatment, it must be small\ncompared to the induced winding \rux along the spin\nchain. Writing the phase-slip rate per unit length (i.e.,\nthe vorticity \rux transverse to the chain) as\nj\u001e=\u0014\"; (28)\nwhere\"=\u0000@x\u001eis the e\u000bective \feld that drives the wind-\ning \row, we thus require (taking the k!0 and!!0\nlimit for\u001b)\nL\u001c\u001b\n\u0014; (29)\nfor the validity of the (approximate) conservation law\n(13). At the same time, however, we should not forget\nthatLmust be long enough for us not to concern with\nthe e\u000bective interfacial resistance, if we want the overall\nimpedance to be governed by the bulk.\nD. Electrical transconductance\nIf the winding injection is performed electrically, so\nthat the e\u000bective chemical potential (conjugate to the5\ntopological charge) \u0016=\u0011I, whereIis the applied cur-\nrent, the Onsager reciprocity dictates the backaction mo-\ntive force on the electrons E=\u0011j(which translates di-\nrectly into a measurable voltage).3Putting this together,\nfor a circuit sketched in Fig. 2, we obtain the electrical\ntransconductance\nG=E\nI=\u00112g: (30)\nmediated by the winding \row across the chain. The (lin-\near response) e\u000bective winding conductance here,\ng\u0011j\n\u0016; (31)\nwould be given by \u001b=L in the absence of phase slips, but\nis degraded by them otherwise, as a function of L.12\nL<latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit><latexit sha1_base64=\"(null)\">(null)</latexit>µ=⌘I\n<latexit sha1_base64=\"IEi0mVf0ESla/g4vRm/6o17P7Dc=\">AAABtnicZU9NS8NAFHxbv2r9qnr0UszFU0m0aD0IhV70VsF+YBPLZvtal+5uluymWEL/heBJ/5f/RluDEDKnYebNYybUghvrut+ktLG5tb1T3q3s7R8cHlWPT3omSmKGXRaJKB6E1KDgCruWW4EDHSOVocB+OGuv/P4cY8Mj9WQXGgNJp4pPOKP2V3r2ZXLno6W1h1HVcevuGrUi8TLiQIbOqPrhjyOWSFSWCWrM0HO1DVIaW84ELit+YlBTNqNTHI7nXBtFJZogfVuXzvkplcYsZFgQJbWv+U+JnTSDlCudWFQsH1hMImXNcr3kdoXr/95F0ruse1f1xmPDabWzTWU4g3O4AA9uoAX30IEuMFDwDp/wRZrkhSCZ/p2WSJY5hRyI/gFN2nsP</latexit>I<latexit sha1_base64=\"mEWQgXUx631UH4ZJ5NRVx3GsqXE=\">AAABrXicZU9NT8JAFHyLX4hfVY9ejL14wlaJHzcSLnqDxAIJVLJdHrhhd7vpbolNwy/wqvG3+W8UbEwa5jSZefMyE2nBjfW8b1LZ2Nza3qnu1vb2Dw6PnOOTronThGHAYhEn/YgaFFxhYLkV2NcJUhkJ7EWz1tLvzTExPFbPNtMYSjpVfMIZtb9S52nkuF7dW+F8nfgFcaFAe+R8DccxSyUqywQ1ZuB72oY5TSxnAhe1YWpQUzajUxyM51wbRSWaMH9bdS35OZXGZDJaEyW1r+VPqZ3chzlXOrWoWDmQTWJlzWK15GGJ2//e66R7Xfdv6o1Ow222ik1VOIMLuAQf7qAJj9CGABggvMMHfJIrEpAhefk7rZAicwolkOkP2Gt3hA==</latexit>\nE<latexit sha1_base64=\"7KV0SvEuXic3tf7X+j2zg57pbPQ=\">AAABt3icZU/JTgJBFHyNG+KGevRi5OKJzChBuZEQE4+YyJLAhPQ0b6BDb073EMmEzzDe9Lv8GwUmJhPqVKl69VIVGsGt87wfUtjZ3ds/KB6Wjo5PTs/K5xddq5OYYYdpoeN+SC0KrrDjuBPYNzFSGQrshbPWyu/NMbZcq1e3MBhIOlE84oy6P2kwlNRNGRXp03JUrnhVb43rbeJnpAIZ2qPy53CsWSJROSaotQPfMy5Iaew4E7gsDROLhrIZneBgPOfGKirRBun7unXOT6m0diHDLXFVL/8pcdFjkHJlEoeK5QOLSCtnN0saK9T/e2+T7l3Vv6/WXmqVZivbVIQruIFb8OEBmvAMbegAAw0f8AXfpEFGJCLTzWmBZJlLyIG8/QK3xXwS</latexit>4\n\u0001\u0002\u0003\u0002\u0004\u0005\u0006\u0007\u0004-\b\u0003\u0002\u0004\u0005****\t\n\u000b\f\u0001\u0002\u0003\u0001\u0002\u0004\u0001\u0002\u0005\u0001\u0002\u0006\u0001\u0002\n\u0001\u0002 \u0007\u0002\u0003\u0002\t\n\u000b\f\n\u0001\u0001\n\u0001\u0002 \u0007\u0001\u0002 \u0003\u0001\u0002\u0001\u0002\u0003\u0001\u0001\u0004\u0001\u0005\u00013,41,2(a)\u0001\u0001\u0002\u0001\u0003\u0001(b)(c)\nFIG. 2. Leaky integrate-and-fire functionality of the neuron.(a) The washboard potential. The angular order parameter'of the neuron and its time derivative@t'are analogousto the position and speed of a particle. Inset: The tiltedwashboard potential where local minima become unstable.(b) The firing and non-firing diagram depending on the inputrate!and amplitude⌫0of the incoming train of angularmomentum kicks. Numerics are done with parameters!0=10 GHz and⌘= 5. (c) The time evolution of the orderparameter of the neuron'(t) simulated for samples markedin the diagram. In each simulation, 20 pulses are sent to theneuron until it fires.time, are much less thant0). Thus⌧(t) can be regardedas composed of delta pulses, each elevates the canonicalmomentumMby\u0000M, and, as manifested by Eq. (4),boosts@t'by⌫0=\u0000M/e\u0000.Take the example of an initially stationary statetrapped at'|t=0= 0, a sudden angular momentum kicksets the initial condition@t'|t=0+=⌫0for the relaxationprocess. As a train of topological charges coming in ata rate!, the state'wiggles gently as the e↵ect of eachkick fades, featuring a leaky integration. See Fig.2(c).Only with a su\u0000ciently high input rate and/or large am-plitude, can the signal pulses trigger the neuron to fire—it then overcomes the energy barrier and falls into theneighboring local minimum'=⇡. The dependence ofthe firing or non-firing result on the rate and amplitudeof pulses is plotted in Fig.2(b). When the neuron fires, itexperiences a phase jump of⇡, whereas the phase deep inthe axon remains unchanged. A domain wall with topo-logical charge⇣=+1 is thus created on a time scale oft0,entering the axon with a certain initial velocity, where-after it triggers further actions in the network branches.Another behavior of a biological neuron that we canimitate is the bursting. In a bursting state, a neuron firesrepeatedly in groups separated by quiescent periods [71].For our artificial neuron, a constant torque⌧0(such asa spin transfer torque used in Ref. [45]) e↵ectively tiltsthe washboard potential. This can cause repetitive fir-ing once the torque exceeds a threshold where the lo-cal minima become unstable. On average, the firingrate is proportional to the tilting,r=⌧0/⇡↵eJ. As thephase'winds forward continuously, the neuron repeat-edly switches between the two ground states.Synapse.—A nonvolatile analog memory with an ad-justable weight, as the key element of a synapse, is re-alized in our proposal by an antiferromagnetic nanos-trip (2) holding a metastable winding texture. Wedefine the dimensionless density of topological chargesw=N\u0000/Las the synaptic weight, whereNis the netnumber of topological charges andLis the length of thenanostrip, which is large enough (L\u0000\u0000) to allow usto considerwas an essentially continuous variable. Thenanostrip is a one-dimensional thermodynamic system ofinteracting topological charges, with chemical potentialµ=\u0000F/\u0000N,w h e r eFis its total free energy. The de-pendence of the chemical potentialµon the topological-charge densitywis highly nonlinear. See Fig.3for theplot ofµversuswat zero temperature, calculated for themetastable ground-state textures of the Hamiltonian (2),which are the solutions to the static sine-Gordon equa-tion [68]. The chemical potential determines the leakagerate of topological charges into the dendrite of the post-synaptic neuron when the synapse is activated by a signalfrom the presynaptic neuron. A higher chemical poten-tial results in a stronger synaptic connection.The right end of the synapse [see Fig.1(a)] is initiallysealed by a strong anisotropy—a washboard potential forthe local order parameter\u00002, with a tilting proportionalto the chemical potentialµin the synapse. A topologicalcharge fired by presynaptic neuron arrives from a telo-dendron attpre, and activates the dynamics of\u00002.T h eactivation can be a result of either heating, which softensthe anisotropy, or a kinetic perturbation, which brings\u00002\n\u0001\u0002\u0001\u0002\u0003\u0004\u0002\n\u0005 \u0005\u0002\u0001 \u0005\u0002\u0004\u0005\u0002\u0006 \u0005\u0002\u0007FIG. 3. The nonlinear dependence of the chemical potentialµon the synaptic weightw, extracted from the metastableground-state textures. Inset: the magnetic textures and theircos\u0000(x) for two values ofw. The texture becomes closer toa uniform spiral aswapproaches 1. The asymptote has theanalytical expressionµ=pAK⇡2w.\u0000<latexit sha1_base64=\"Q7hv0XOyK4CymOh9yD3xaeSYsVY=\">AAABsnicZU/LTgJBEOzBF+Jr1aMXIxdPZFeJjxsJF4+YyCPChvQODY7MzE52Zolkwz94M/pn/o2CG5MNdapUdXWqIiOFdb7/zUobm1vbO+Xdyt7+weGRd3zSsXGacGrzWMZJL0JLUmhqO+Ek9UxCqCJJ3WjaXPrdGSVWxPrJzQ2FCidajAVH9yt1BlZMFA69ql/zVzhfJ0FOqpCjNfQ+BqOYp4q04xKt7Qe+cWGGiRNc0qIySC0Z5FOcUH80E8ZqVGTD7G1VuOBnqKydq2hNVOheip9SN74LM6FN6kjzYmA+jrWzi9WS+yVu/nuvk85VLbiu1R/r1UYz31SGM7iASwjgFhrwAC1oA4dXeIdP+GJ19syQ8b/TEsszp1AAkz9WWnna</latexit><latexit sha1_base64=\"AFFCKZ1a2h2fzaNqbvTVGBae2Ts=\">AAABsnicZU/LTgJBEOzBF+Jr1aMXIxdPZFeJjxsJF4+YyCPChvQODY7MzE52Zolkwz94M/pn/o2CG5MNdapUdXWqIiOFdb7/zUobm1vbO+Xdyt7+weGRd3zSsXGacGrzWMZJL0JLUmhqO+Ek9UxCqCJJ3WjaXPrdGSVWxPrJzQ2FCidajAVH9yt1BlM0Bode1a/5K5yvkyAnVcjRGnofg1HMU0XacYnW9gPfuDDDxAkuaVEZpJYM8ilOqD+aCWM1KrJh9rYqXPAzVNbOVbQmKnQvxU+pG9+FmdAmdaR5MTAfx9rZxWrJ/RI3/73XSeeqFlzX6o/1aqOZbyrDGVzAJQRwCw14gBa0gcMrvMMnfLE6e2bI+N9pieWZUyiAyR9RE3nW</latexit>\nFIG. 2. Schematic of a winding \row along a (horizontal)\nspin chain. Transverse charge current Igenerates an e\u000bective\nchemical potential bias \u0016that couples to the winding density\nat the left end. \u0011is a contact-dependent conversion param-\neter, which relates the input current to the out-of-plane spin\nHall torque \u001c=\u0011I=\u0019 acting on the magnetic dynamics in the\nchain.4The injected winding \rux is governed by the winding\nconductivity \u001b, while being dissipated by the transverse vor-\ntex \row/\u0014. The net remaining winding out\row produces\na measurable transverse motive force Eat the right electrical\ncontact.\nIV. SINE-GORDON MODEL\nAs a concrete illustration of this general \feld-\ntheoretic formalism, let us now consider an ideal 1D\nspin chain with an easy-plane collinear order param-\neter parametrized by azimuthal angle '(x) and the\n(canonically-conjugate) out-of-plane spin density s(x).\nOur main focus is on the antiferromagnetic case (which\nwill a\u000bect the phase-slip considerations). The classical\nlow-energy dynamics are generated by the (sine-Gordon)\nHamiltonian density\nH=s2\n2\u001f+A(@x')2\n2+Kcos(2'); (32)along with the Poisson bracket f'(x);s(x0)g=\u000e(x\u0000x0).\nIt is assumed here that the order-parameter con\fguration\nis close to the easy plane, at all times. \u001f,A, andKare\nrespectively the out-of-plane spin susceptibility, order-\nparameter sti\u000bness, and the axial in-plane anisotropy.\nThe Hamiltonian is invariant under time reversal, when\ns!\u0000sand'!'+\u0019. The sign of Kis inconsequential,\nas it can be \ripped by a phase change, '!'+\u0019=2.\nA. Luttinger-liquid mapping\nThis description can be quantized by promoting the\nPoisson bracket to the commutator:\n['(x);s(x0)] =i~\u000e(x\u0000x0); (33)\nmaking the theory formally analogous to a spinless Lut-\ntinger liquid.13s(x) would then correspond to the linear-\nmomentum density \u0005 and the topological density to the\nparticle density @x\u001e=\u0019. Indeed, the (spinless) Luttinger-\nliquid Hamiltonian density is\nH=u\n2\u0014\u0019g\n~\u00052+~\n\u0019g(@x\u001e)2\u0015\n+Kcos(4\u001e); (34)\nwhere [\u001e(x);\u0005(x0)] =i~\u000e(x\u0000x0).uis the speed of sound\nandg > 0 is the interaction parameter ( u!vF, the\nFermi velocity, and g!1, for free electrons; g<1 signals\nelectron repulsion and g>1 attraction). Kparametrizes\numklapp scattering (which requires an appropriate lat-\ntice \flling factor).\nThe corresponding Euclidean Lagrangian density is\nL=~\n2\u0019g\u00141\nu(@\u001c')2+u(@x')2\u0015\n+k\n2(\u0019a)2cos(2');(35)\nabeing a short-distance cuto\u000b. We have rede\fned the\ndisplacement \feld \u001e!'=2 and appropriately rescaled\ng, in order to match the notation of our spin model\n(32). Under the Wilsonian renormalization-group (RG)\nrescaling,13we get the Kosterlitz-Thouless \row equa-\ntions:\ndy\ndl= (2\u0000g)y;dg\ndl=\u0000g3y2=8; (36)\nwherey\u0011k=\u0019~u(which we can take to be positive, with-\nout loss of generality), and we have omitted the nonuni-\nversal cuto\u000b-dependent numerical prefactor on the right-\nhand side of the second equation. The RG \row of ycor-\nresponds simply to the scaling dimension of the cosine\noperator.13The generic reduction in g, under the RG\n\row (36), corresponds to the sti\u000bening of the order pa-\nrameter'due to the cosine term /Kin the Lagrangian\n(35) (which tries to order the \feld ').\nFor our original spin system, Eq. (32), u=p\nA=\u001f,\ng=~=\u0019pA\u001f. We interpret the cuto\u000b as a\u0018p\nA=K?,14\nwhereK?\u001dKis a strong easy-plane anisotropy that\nkeeps spin dynamics close to the xyplane. The bare6\norder-parameter sti\u000bness is A\u0019S2Jaand spin suscep-\ntibility\u001f\u0019~2=4Ja, in the large-spin Heisenberg limit\n(which acquire some corrections due to quantum \ruc-\ntuations when S\u00181). These estimates boil down to:\nu\u0019Ja=~,g\u00192=\u0019S,y\u0018K=K?\u001c1. Going beyond\nthe Heisenberg limit, with a large easy-plane anisotropy,\nwould decrease \u001f, while not similarly a\u000becting A, and\nthus increase the value of g.\nExpanding gclose to its critical value gc= 2,g!2+g,\nwe get13\ndy\ndl=\u0000gy;dg\ndl=\u0000y2; (37)\nwhich parametrize hyperbolic trajectories with g2\u0000y2=\nconst. These equations \row rapidly to a noninteracting\n(gapless) \fxed point g\u0003>0 andy= 0, if 0<jyj< g.\nOtherwise, the \row takes us to a gapped strong coupling,\nwithjyj!1 , where the phase is pinned along the easy\naxis (i.e.,'!0 or\u0019). In this latter case, the elementary\ndynamics could be constructed out of the corresponding\ndomain walls or their composites.13\nB. Quantum phase slips\nIt is important to also make an internal consistency\ncheck to ensure that the original spin system can indeed\nbe described e\u000bectively by the sine-Gordon Lagrangian\n(35) for nonsingular \felds '(i.e., con\fgurations free\nof vortex singularities in the 1 + 1 space-time dimen-\nsions). In the absence of the anisotropy K(or when it\nrenormalizes down to zero due to quantum \ructuations),\nthe easy-plane antiferromagnetic spin chain undergoes a\n(Kosterlitz-Thouless type) spin super\ruid-insulator tran-\nsition atS\u00192 (1=2), for integer (half-integer) spin S,8\nwith larger Splacing us in the (gapless) super\ruid phase.\nIn the latter, the magnetic vortices and antivortices bind,\nand we restore a hydrodynamic easy-plane \u001bmodel.\nThe rate of quantum phase slips, \u0014, in this limit is\ngiven by8\n\u0014/\u001a\u0019q2S\u00003; (38)\nat zero temperature (with a similar scaling with tem-\nperature, at zero winding density \u001a), where the elemen-\ntary vorticity is q= 1 (2) for the integer (half-integer)\nspinS.15The suppression of the phase slips as \u001a!0,\nat low temperatures, gives us an algebraically-large win-\ndow, Eq. (29), for the spin-chain length Lwith conserved\nwinding transport.16\nSummarizing the requirements for the low-energy hy-\ndrodynamic regime described by the e\u000bective long-\nwavelength Hamiltonian\nH=s2\n2\u001f+A(@x')2\n2; (39)\nwith canonically conjugate sand'(and e\u000bective, renor-\nmalized\u001fandA), we needS&1=2 (for the half-integercase, where the leading-order phase slips are mediated by\ndouble vortices, due to destructive interference of single\nvortices,8) whileg\u00182=\u0019S > 2, andy=k=\u0019~u<g\u00002.\nThe generic mean-\feld parameters for a Heisenberg an-\ntiferromagnetic chain17thus place us at a cusp of the\ngapless (hydrodynamic) phase for S= 1=2 (with a clear\ntendency to \row to the gapped phase for a larger spin).\ngcan be decreased beyond the Heisenberg limit when the\neasy-plane anisotropy is large (i.e., comparable to the ex-\nchange interaction), which would suppress the spin sus-\nceptibility\u001f. A more detailed and careful analysis would\nbe needed, in principle, in order to establish the exact\nvalues of the parameters yielding the ballistic transport\nregime in this quantum limit of S\u00191=2. A fermionic\ntreatment in the Wigner-Jordan representation of spin\ndynamics18is one possible approach to that end.\nC. Winding hydrodynamics\nSupposing we wind up with a gapless regime described\nby the Hamiltonian (39) (if the anisotropy Kis weak\nand/or irrelevant), we are ready to formulate a transport\ntheory for the winding density \u001a=\u0000@x'=\u0019. The associ-\nated current operator, j=@t'=\u0019, becomes [cf. Eq. (33)]\nj=i\n\u0019~[H;'] =s\n\u0019\u001f: (40)\nFrom Eq. (22), we thus get\n@0\nxp(x\u0000x0) =\u0000i[j(x);\u001a(x0)] =\u0000~\n\u00192\u001f@0\nx\u000e(x\u0000x0);(41)\nso that\np(x\u0000x0) =~\n\u00192\u001f\u000e(x\u0000x0): (42)\nThe current-current correlator in Eq. (24) vanishes in the\nlimitk!0 (while maintaining a \fnite !),19so that we\nend up with the conductivity (26):\n\u001b(k!0;!) =i\n\u00192\u001f!=iA\n!; (43)\nwhereA= 1=\u00192\u001f.\nRegularizing this result at zero frequency, !!!+i0+,\nwe get Re\u001b=\u0019A\u000e(!). As expected, the static con-\nductivity diverges in the low-frequency limit. In this\ncase, the super\ruid bulk has no impedance and the\nwinding conductance of the entire structure needs to be\ndetermined by carefully considering the interfacial in-\njection physics, which is akin to the Andreev conduc-\ntance of normaljsuperconducting interfaces.20The resul-\ntant winding conductance of a clean \fnite-length spin\nchain should thus be governed by the contact impedance,\nsimilarly to the charge conductance of a ballistic elec-\ntronic wire (whose contact resistance depends both on\nthe details of the contact itself as well as the electron-\nelectron interactions in the wire.21)7\nIf, on the other hand, we add some spin-relaxing impu-\nrities to the Hamiltonian, so that the collective spin den-\nsity de\fning the topological current according to Eq. (40)\nis now allowed to relax, the dynamic spin susceptibility\ncan be obtained at k!0 according to the Bloch phe-\nnomenology:\nds(t)\ndt=\u0000s(t)\u0000s0(t)\n\u001c: (44)\nHeres0(t) =\u001fh(t) is the (instantaneous) equilibrium\nspin density, according to the Zeeman coupling to a mag-\nnetic \feld:H!H\u0000hs. The dynamic spin susceptibility\nis thus:\n\u001fs(!)\u0011s(!)\nh(!)=\u001f\n1\u0000i!\u001c\u0019\u001f(1 +i!\u001c); (45)\nat!\u001c1=\u001c. This, in turn gives us the current (40) self-\ncorrelator, from which we \fnd the low-frequency topo-\nlogical conductivity according to Eq. (24):\n\u001b(!) =i\n!\u0012\n\u0000\u001fs\n(\u0019\u001f)2+1\n\u00192\u001f\u0013\n=\u001c\n\u00192\u001f: (46)\nThe response is Drude-like, with a well-de\fned DC\nlimit and a Drude weight /\u001f\u00001(in the\u001c! 1 ).\nIn the Gilbert damping phenomenology of magnetic\ndynamics,22the spin-pumping rate (i.e., torque \u001c\u000b) into\nthe environment is given by\n\u0000ds\ndt\u0011\u001c\u000b=\u000b@t'=\u000bs\u0000s0\n\u001f; (47)\nwhich can be obtained by supplementing the Heisenberg-\nHamilton dynamics of the planar order parameter with\nthe Rayleigh dissipation function R=\u000b(@t')2=2.23We\nare de\fning the Gilbert constant \u000bhere in units of spin\ndensity. This gives \u001c=\u001f=\u000b for spin-relaxation time,\nresulting in the low-frequency winding conductivity of\n\u001b=1\n\u00192\u000b: (48)\nA larger damping implies lower winding conductivity.\nThe corresponding electrical transconductance (30),\nG/(\u000bL)\u00001, reproduces the spin-super\ruid mediated\ntransconductance (drag) derived in Ref. 4, when L!1 .\nIndeed, in the limit of no axial anisotropy Kin Eq. (32) or\nwhen it renormalizes down by quantum \ructuations, the\nout-of-plane spin sis conserved and follows a super\ruid-\nlike hydrodynamics (with spin \rux /@x'). This pro-\nvides a dual description for the long-range signal propa-\ngation along the spin chain. In general, however, when\nthe spin is not conserved, the winding hydrodynamics\nbased on Eq. (13) establishes a more universal framework\nfor low-frequency long-distance transport in anisotropic\nspin chains. This is illustrated in our next and \fnal ex-\nample.D. Finite temperatures\nIn the opposite, classical limit of S&1 and thusg<2\nin Eq. (36), the phase 'is pinned and the winding dy-\nnamics are gapped. In this case, the zero-temperature\ntransport is exponentially suppressed over a long spin\nchain.18At \fnite temperatures, however, we generally\nanticipate a di\u000busive regime for thermally-activated chi-\nral domain walls,11along with the parasitic thermal\nphase slips.7If the latter is governed by a larger energy\ngap, in comparison to the domain-wall energy (which is\nthe case when K?>K), we can have a meaningful trans-\nport scenario for the conserved winding carried by the\nBrownian motion of domain walls, along a \fnite-length\nspin chain.11\nIn the strong easy-plane limit, when the winding is\ncarried by a classical gas of stable solitonic domain\nwalls (of width \u0015=p\nA=K ) with quantized topologi-\ncal charge\u00061 and mobility M, the corresponding con-\nductivity is simply \u001b= 2nM, wherenis the density of\ndomain walls of a given chirality. The associated dif-\nfusion coe\u000ecient is given by D=kBTM, according to\nthe Einstein-Smoluchowski relation. Within the Gilbert-\ndamping phenomenology,22the mobility of a rigid soli-\nton is given by M\u0018\u0015=\u000b. Since\u001b/n/e\u0000\fE,11while\n\u0014/e\u0000\fF,7whereE= 4p\nAKis the domain-wall energy\nandF= 4pAK?\u001dEis the thermal phase-slip barrier,\nwe can easily satisfy Eq. (29) at low temperatures. In the\nlimit ofK!0 (and/or high temperature, kBT&E), the\ndomain walls coalesce and we reproduce the conductivity\n(48).11\nE. Noise and quantum relaxometry\nIn addition to an electrical measurement of wind-\ning transport, as sketched in Fig. 2, it may be possi-\nble to investigate the associated topological transport\nproperties, such as winding conductivity, using quantum-\nimpurity (such as nitrogen-vacancy) relaxometry.24Sim-\nilarly to the Johnson-Nyquist noise generally associated\nwith charge conductivity, the winding transport is noisy.\nIn particular, the out of the easy-plane spin \ructuations\n(being canonically conjugate to the planar spin preces-\nsion, irrespective of the nature of the spin order), should\nproduce a detectable magnetostatic signal.25We expect\nit to re\rect similar winding transport properties as the\nelectrical setup of Fig. 2 (without the issues pertaining\nto the contacts), in the long-wavelength low-frequency\nlimit of the dynamics. The latter can be controlled by\nthe quantum-impurity positioning and applied magnetic\n\feld (e.g., Zeeman splitting of the nitrogen-vacancy spin\nstates), respectively.268\nV. DISCUSSION\nIn summary, we have constructed a general framework\nto study winding dynamics in spin chains, from the per-\nspective of a transport phenomenon. Motivated by the\nmean-\feld considerations that draw on the notions of\nspin super\rows along the chain and parasitic vorticity\n\rows transverse to it, we developed a fully quantum the-\nory, where both the winding transport and its relaxation\nby phase slips can be treated systematically by \feld-\ntheoretical approaches.\nWe illustrated the general formalism by specializing to\nthe case of antiferromagnetic easy-plane dynamics, whose\nsalient features can be captured by a sine-Gordon model.\nTwo distinct scenarios then arise concerning the winding\n\rows: the spin-super\ruid regime, where the parasitic in-\nplane anisotropy is washed out by quantum (or thermal)\n\ructuations, and the solitonic regime, where chiral do-\nmain walls carry conserved winding density by Brownian\nmotion (at \fnite temperatures). Both of these limits are\naddressed within our general Kubo formalism, reproduc-\ning and complimenting the pertinent special cases known\nin the literature. We see that rather generically, in the\npresence of magnetic damping, the winding \row can ex-\nhibit Drude-like dynamic response. This corresponds to\nan e\u000bectively metallic behavior of the conserved winding\ntransport. The key internal-consistency check for these\n\fndings concerns the transverse vorticity \row, which re-\n\rects phase slips and needs to be smaller than the wind-\ning \row along the spin chain.\nOne of our central motivations for this work is the po-\ntential ability to detect the topological transport, either\nin an electrical device (cf. Fig. 2) or by a nonintrusive\nquantum-impurity relaxometry (cf. Sec. IV E). The \feld-\ntheoretical framework combined with the experimental\ntangibility should open gates for nonelectrical transport-\nbased investigations of correlated magnetic materials. It\nis useful to add, furthermore, that a long-range order ofany kind is neither assumed nor needed for the emergence\nof topological hydrodynamics. Our microscopic quantum\nformulation, which we have explicitly constructed for vor-\nticity and winding \rows, furthermore, does not even rely\non a local ordering or any semiclassical approximations.\nWe have largely left out the contact-impedance consid-\nerations in our device concept sketched in Fig. 2. This\nmay be justi\fed when there a \fnite bulk resistivity for\nthe topological \row. In the opposite, clean limit, the\ntransport physics would, however, generally be domi-\nnated by the contacts and, at low temperatures, poten-\ntially strongly dependent on the many-body e\u000bects away\nfrom the contacts. These aspects are left for future work.\nNo attempt has been made to classify scenarios of topo-\nlogical hydrodynamics for general quantum magnets in\narbitrary dimensions, which also goes beyond our scope\nhere. Quantum skyrmions in two spatial dimensions27\nand hedgehogs in three dimensions28provide other in-\nteresting examples, with skyrmions, like winding, obey-\ning only an approximate continuity equation. We thus\nanticipate rich possibilities for topological hydrodynam-\nics in magnetic materials, with implications for novel\nprobes and device concepts that do not rely on electronic\n(charge) transport.\nACKNOWLEDGMENTS\nWe thank Michael Mulligan for insightful discussions.\nThe work was supported by NSF under Grant No. DMR-\n1742928 (Y.T. and J.Z.). S.K.K. was supported by\nBrain Pool Plus Program through the National Research\nFoundation of Korea funded by the Ministry of Sci-\nence and ICT (Grant No. NRF-2020H1D3A2A03099291)\nand by the National Research Foundation of Korea\nfunded by the Korea Government via the SRC Center\nfor Quantum Coherence in Condensed Matter (Grant No.\nNRF-2016R1A5A1008184). S.T. is supported by CUNY\nResearch Foundation Project #90922-07 10 and PSC-\nCUNY Research Award Program #63515-00 51.\n1X. F. Sun, W. Tao, X. M. Wang, and C. Fan, Phys. Rev.\nLett. 102, 167202 (2009); G. T. Hohensee, R. B. Wilson,\nJ. P. Feser, and D. G. Cahill, Phys. Rev. B 89, 024422\n(2014); A. L. Chernyshev and A. V. Rozhkov, Phys. Rev.\nLett. 116, 017204 (2016); X. Chen, J. Kim, Q. Jia, S. E.\nSullivan, Y. Xu, K. Jarvis, J. Zhou, and L. Shi, Adv. Func.\nMater. 30, 2001637 (2020).\n2F. Meier and D. Loss, Phys. Rev. Lett. 90, 167204 (2003);\nJ. Sirker, R. G. Pereira, and I. A\u000feck, Phys. Rev. B\n83, 035115 (2011); C. Karrasch, D. M. Kennes, and\nF. Heidrich-Meisner, Phys. Rev. B 91, 115130 (2015);\nF. Lange, S. Ejima, T. Shirakawa, S. Yunoki, and\nH. Fehske, Phys. Rev. B 97, 245124 (2018); V. B.\nBulchandani, R. Vasseur, C. Karrasch, and J. E. Moore,\nPhys. Rev. B 97, 045407 (2018); A. Biella, M. Collura,\nD. Rossini, A. De Luca, and L. Mazza, Nat. Comm. 10,4820 (2019).\n3Y. 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Zou, S. K. Kim, and Y. Tserkovnyak, Phys. Rev. B 99,\n180402(R) (2019).\n10By extending this de\fnition of the topological charge den-\nsity in terms of the vector chirality to arbitrarily-shaped\nplaquettes, the theory is naturally generalized to arbitrary\nspin lattices.\n11S. K. Kim, S. Takei, and Y. Tserkovnyak, Phys. Rev. B\n92, 220409(R) (2015).\n12For the simplest scenario with j\u001e/j, the ensuing signal\ndecay would be exponential, which we can see by adding\nthisj\u001eto the RHS of Eq. (17), in the steady state with\n@t\u001a= 0.\n13T. Giamarchi, Quantum Physics in One Dimension (Ox-\nford University Press, Oxford, 2004).\n14This cuto\u000b governs the Landau stability criterion for the\neasy-plane texture.29In the case of easy-plane ferromag-\nnets, the magnon dispersions become nonlinear at wave\nnumbersq\u00181=a. For antiferromagnets, at these wave\nnumbers, the higher-energy magnon branch, with out-of-\nplane N\u0013 eel dynamics, mixes with and modi\fes the sine-\nGordon description of the planar N\u0013 eel \ructuations.\n15In a related context, such topological quantum spin-parity\ne\u000bects were investigated also for chirality tunneling of lo-\ncalized antiferromagmetic domain walls.30.\n16Note that the phase-slip rate \u0014quoted in Eq. (38) was\nevaluated in Ref. 8 with respect to the winding density \u001a\nrather than the e\u000bective \feld \u000fas in our present notation\n[cf. Eq. (28)]. Since in a generic system, they are linearly re-\nlated, in linear response, the basic argument should stand.\nThe key observation here is that the phase slips vanish in\nlinear response in the spin-super\ruid phase.\n17In the case of a ferromagnetic chain, \u001f/1=K?, whichis typically larger than in the antiferromagnetic case, if\nthe easy-plane anisotropy K?is weaker than the exchange\ninteraction. This would reduce g, pushing us deeper into\nthe insulating regime.\n18S. Ho\u000bman, D. Loss, and Y. Tserkovnyak, \\Super\ruid\ntransport in quantum spin chains,\" ArXiv:1810.11470.\n19Y. Tserkovnyak and J. Zou, Phys. Rev. Res. 1, 033071\n(2019).\n20Y. V. Nazarov and Y. M. Blanter, Quantum Transport\n(Cambridge University Press, Cambridge, 2009).\n21D. L. Maslov and M. Stone, Phys. Rev. B 52, R5539\n(1995).\n22T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n23Y. Tserkovnyak and H. Ochoa, Phys. Rev. B 96, 100402(R)\n(2017).\n24C. Du, T. Van der Sar, T. X. Zhou, P. Upadhyaya,\nF. Casola, H. Zhang, M. C. Onbasli, C. A. Ross, R. L.\nWalsworth, Y. Tserkovnyak, and A. Yacoby, Science 357,\n195 (2017); F. Casola, T. van der Sar, and A. Yacoby,\nNat. Rev. Mater. 3, 17088 (2018).\n25Speci\fcally, according to Eq. (40), the winding \rux, /@t'\nis generally proportional to the local spin density. Fluc-\ntuations of the latter, in turn, emit a detectable magne-\ntostatic noise,24which contains direct information about\nthe topological current autocorrelator. According to the\n\ructuation-dissipation theorem,26furthermore, this could\nbe directly related to the (causal) linear response and thus\nthe winding conductivity.\n26B. Flebus and Y. Tserkovnyak, Phys. Rev. Lett. 121,\n187204(R) (2018).\n27H. Ochoa and Y. Tserkovnyak, Inter. J. Mod. Phys. B 33,\n1930005 (2019).\n28J. Zou, S. Zhang, and Y. Tserkovnyak, \\Topolog-\nical transport of decon\fned hedgehogs in magnets,\"\nArXiv:2006.10910.\n29E. B. Sonin, Adv. Phys. 59, 181 (2010).\n30B. A. Ivanov, A. K. Kolezhuk, and V. E. Kireev, Phys.\nRev. B 58, 11514 (1998)."    },    {        "title": "0905.0112v2.Spin_excitations_in_a_monolayer_scanned_by_a_magnetic_tip.pdf",        "content": "arXiv:0905.0112v2  [cond-mat.other]  3 Aug 2009Spin excitations in a monolayer scanned by a magnetic tip\nM.P. Magiera1, L. Brendel1, D.E. Wolf1andU. Nowak2\n1Department of Physics and CeNIDE, University of Duisburg-E ssen, D-47048 Duisburg, Germany, EU\n2Theoretical Physics, University of Konstanz, D-78457 Kons tanz, Germany, EU\nPACS68.35.Af – Atomic scale friction\nPACS75.10.Hk – Classical spin models\nPACS75.70.Rf – Surface Magnetism\nAbstract. - Energy dissipation via spin excitations is investigated f or a hard ferromagnetic tip\nscanning a soft magnetic monolayer. We use the classical Hei senberg model with Landau-Lifshitz-\nGilbert (LLG) dynamics including a stochastic field represe nting finite temperatures. The friction\nforce depends linearly on the velocity (provided it is small enough) for all temperatures. For\nlow temperatures, the corresponding friction coefficient is proportional to the phenomenological\ndamping constant of the LLG equation. This dependence is los t at high temperatures, where the\nfriction coefficient decreases exponentially. These finding s can be explained by properties of the\nspin polarisation cloud dragged along with the tip.\nIntroduction. – While on the macroscopic scale the\nphenomenology of friction is well known, several new as-\npects are currently being investigated on the micron and\nnanometer scale [1,2]. During the last two decades, the\nresearch on microscopic friction phenomena has advanced\nenormously, thanks to the development of Atomic Force\nMicroscopy (AFM, [3]), which allows to measure energy\ndissipation caused by relative motion of a tip with respect\nto a substrate.\nRecently the contribution of magnetic degrees of free-\ndom to energy dissipation processes has attracted increas-\ning interest [4–8]. Today, magnetic materials can be con-\ntrolleddowntothenanometerscale. Newdevelopmentsin\nthe data storage industry, spintronics and quantum com-\nputing require a better understanding of tribological phe-\nnomena in magnetic systems. For example, the reduction\nof heat generation in reading heads of hard disks which\nworkat nanometerdistancesis animportant issue, as heat\ncan cause data loss.\nMagnetic Force Microscopy(MFM), where both tip and\nsurface are magnetic, is used to investigate surface mag-\nnetism and to visualise domain walls. Although recent\nstudies have attempted to measure energy dissipation be-\ntween an oscillating tip and a magnetic sample [9,10], the\ndependenceofthefrictionforceonthetip’sslidingvelocity\nhasnot been consideredyet apartfroma workby C. Fusco\net al.[8] which is extended by the present work to temper-\naturesT∝negationslash=0. The relative motion of the tip with respect\nto the surface can lead to the creation of spin waves whichpropagate inside the sample and dissipate energy, giving\nrise to magnetic friction.\nWe will firstpresent asimulationmodel anddefine mag-\nnetic friction. The model contains classical Heisenberg\nspins located on a rigid lattice which interact by exchange\ninteraction with each other. Analogous to the reading\nhead of a hard disc or a MFM tip, an external fixed\nmagnetic moment is moved across the substrate. Using\nLangevin dynamics and damping, it is possible to simu-\nlate systems at finite temperatures. The main new results\nconcern the temperature dependence of magnetic friction.\nSimulation model and friction definition. – To\nsimulate a solid magnetic monolayer (on a nonmagnetic\nsubstrate), we consider a two-dimensional rigid Lx×Ly\nlattice of classical normalised dipole moments (“spins”)\nSi=µi/µs, where µsdenotes the material-dependent\nmagnetic saturationmoment(typically afew Bohrmagne-\ntons). These spins, located at z= 0 and with lattice spac-\ninga, represent the magnetic moments of single atoms.\nThey can change their orientation but not their absolute\nvalue, so that there are two degrees of freedom per spin.\nWeuseopenboundaryconditions. Aconstantpointdipole\nStippointing in the z-direction and located at z= 2arep-\nresentsthemagnetictip. Itismovedparalleltothesurface\nwith constant velocity v.\nThis model has only magnetic degrees of freedom and\nthus focusses on their contributions to friction. For a\nreal tip one could expect that magnetic, just like nonmag-\np-1M.P. Magiera 1 L. Brendel 1 D.E. Wolf 1 U. Nowak 2\nnetic [11,12] interactions might also lead to atomic stick-\nslip behaviour, and hence to phononic dissipation with\na velocity-independent friction contribution as described\nby the Prandtl-Tomlinson model [13,14]. However, this\nrequires a periodic potential between tip and substrate,\nthat is strong enough compared to the elastic deformation\nenergytoallowformultiple localpotentialenergyminima.\nThe magnetic tip-substrate interactions are unlikely to be\nstrong enough.\nThe Hamiltionan consists of two parts:\nH=Hsub+Hsub−tip. (1)\nThe first one represents the internal ferromagnetic short-\nrange interaction within the substrate. The second one\ndescribes the long-range coupling between the substrate\nand the tip.\nThe interaction between the substrate moments is mod-\neled by the anisotropic classical Heisenberg model,\nHsub=−J/summationdisplay\n/angbracketlefti,j/angbracketrightSi·Sj−dzN/summationdisplay\ni=1S2\ni,z. (2)\nJ >0 describes the ferromagnetic exchange interaction\nbetween two nearest neighbours, expressed by the an-\ngular brackets ∝angbracketlefti,j∝angbracketright.dz<0 quantifies the anisotropy,\nwhich prefers in-plane orientations of the spins. The\ndipole-dipole-interaction between the substrate spins is\nneglected, because it is much weaker than the exchange\ninteraction. A quantitative comparison of our simulation\nresults with the ones obtained in [8], where the dipole-\ndipole-interaction inside the substrate was taken into ac-\ncount, justifies this approximation, which reduces simula-\ntion time enormously.\nThe long-range interaction between substrate and tip is\ndescribed by a dipole-dipole interaction term\nHsub−tip=−wN/summationdisplay\ni=13 (Si·ei)(Stip·ei)−Si·Stip\nR3\ni,(3)\nwhereRi=|Ri|denotes the norm of the distance vector\nRi=ri−rtip, andeiits unit vector ei=Ri/Ri.riand\nrtipdenote the position vectors of the substrate spins and\nthe tip respectively. wquantifies the dipole-dipole cou-\npling of the substrate and the tip. Note, however, that\nthe results of the present study only depend on the com-\nbination w|Stip|, which is the true control parameter for\nthe tip-substrate coupling.\nTheproperequationofmotionofthemagneticmoments\nis theLandau-Lifshitz-Gilbert (LLG, [15]) equation,\n∂\n∂tSi=−γ\n(1+α2)µs[Si×hi+αSi×(Si×hi)],(4)\nwhich is equivalent to the Gilbert equation of motion [16]:\n∂\n∂tSi=−γ\nµsSi×/bracketleftbigg\nhi−αµs\nγ∂Si\n∂t/bracketrightbigg\n. (5)The first term on the right-hand side of eq. (4) describes\nthe dissipationless precession of each spin in the effective\nfieldhi(to be specified below). The second term de-\nscribes the relaxation of the spin towards the direction\nofhi.γdenotes the gyromagnetic ratio (for free electrons\nγ= 1.76086×1011s−1T−1), andαis a phenomenological,\ndimensionless damping parameter.\nThe effective field contains contributions from the tip\nand from the exchange interaction, as well as a thermally\nfluctuating term ζi[17,18],\nhi=−∂H\n∂Si+ζi(t). (6)\nThe stochastic, local and time-dependent vector ζi(t) ex-\npresses a “Brownian rotation”, which is caused by the\nheat-bath connected to each magnetic moment. In our\nsimulations this vector is realised by uncorrelated random\nnumbers with a Gaussian distribution, which satisfy the\nrelations\n∝angbracketleftζi(t)∝angbracketright= 0 and (7)\n∝angbracketleftζκ\ni(t)ζλ\nj(t′)∝angbracketright= 2αµs\nγkBTδi,jδκ,λδ(t−t′),(8)\nwhereTis the temperature, δi,jexpresses that the noise\nat different lattice sites is uncorrelated, and δκ,λrefers to\nthe absence of correlations among different coordinates.\nTofind aquantitywhichexpressesthe friction occurring\nin the system, it is helpful to discuss energy transfers be-\ntween tip, substrate and heat-bath first. It is straightfor-\nward to separate the time derivative of the system energy,\neq. (1), into an explicit and an implicit one,\ndH\ndt=∂H\n∂t+N/summationdisplay\ni=1∂H\n∂Si·∂Si\n∂t. (9)\nThe explicit time dependence is exclusively due to the tip\nmotion. The energy transfer between the tip and the sub-\nstrate is expressed by the first term of eq. (9), which jus-\ntifies to call it the “pumping power” Ppump:\nPpump=∂H\n∂t=∂Hsub−tip\n∂rtip·v\n=w/summationdisplay\nαvαN/summationdisplay\ni=13\nR4\ni/braceleftbigg\n(Si·eiei,α−Si,α)(Stip·ei) (10)\n+(Stip·eiei,α−Stip,α)(Si·ei)\n−Si·Stipei,α+3ei,α(Si·ei)(Stip·ei)/bracerightbigg\nAt any instance, the substrate exerts a force −∂Hsub−tip\n∂rtip\non the tip. Due to Newton’s third law, Ppumpis the work\nper unit time done by the tip on the substrate. Its time\nand thermal average ∝angbracketleftPpump∝angbracketrightis the average rate at which\nenergy is pumped into the spin system. In a steady state\nit must be equal to the average dissipation rate, i.e.to\np-2Spin excitations in a monolayer scanned by a magnetic tip\nh\nhSω\nδθδϕ\nFigure1: Asingle spininamagnetic fieldrotatingwithangul ar\nvelocity ω, is dragged along with a phase shift δϕand aquires\nan out of plane component δθ.\nthe net energy transferred to the heat bath per unit time\ndue to spin relaxation. The magnetic friction force can\ntherefore be calculated by\nF=∝angbracketleftPpump∝angbracketright\nv. (11)\nThe second term of eq. (9) describes the energy transfer\nbetween the spin system and the heat bath. Inserting eq.\n(5) intoPdiss=−/summationtextN\ni=1∂H\n∂Si·∂Si\n∂tleads to\nPdiss=N/summationdisplay\ni=1∂H\n∂Si·/bracketleftbiggγ\nµsSi×ζi−αSi×∂Si\n∂t/bracketrightbigg\n=−Ptherm+Prelax. (12)\nThe first term, Pthermcontaining ζi, describes, how much\nenergy is transferred into the spin system due to the ther-\nmal perturbation by the heat bath. The second term,\nPrelaxproportional to the damping constant α, describes\nthe rate of energy transfer into the heat bath due to the\nrelaxation of the spins.\nAtT= 0,Pthermis zero. The spins are only perturbed\nby the external pumping at v∝negationslash= 0. Then\nPrelax=Pdiss=γα\nµs(1+α2)N/summationdisplay\ni=1(Si×hi)2,(T= 0),\n(13)\nwhere for the last transformation we used eq. (4) in order\ntoshowexplicitlytherelationshipbetweendissipationrate\nand misalignment between spins and local fields.\nIt will be instructive to compare the magnetic substrate\nscanned by a dipolar tip with a much simpler system, inwhich the substrate is replaced by a single spin Ssub-\njected to an external field h(t) that rotates in the plane\nperpendicular to a constant angular velocity ω(replac-\ning the tip velocity). It is straight forward to obtain the\nsteady state solution for T= 0, where in the co-rotating\nframeSis at rest. Slags behind h/hby an angle δϕand\ngets a component δθinω-direction (cf. fig. 1), which are\nin first order given by\nδϕ\nα=δθ=ωµs\nhγ+O/parenleftBigg/parenleftbiggωµs\nhγ/parenrightbigg3/parenrightBigg\n.(14)\nInserting this into the ( N=1)-case of eq. (13) yields a dis-\nsipation rate of Pdiss=αω2µs/γ, which corresponds to a\n“viscous” friction F=Pdiss/ω∝αω.\nIt is instructive to give a simple physical explanation\nfor eq. (14), instead of presenting the general solution,\nwhich can be found in [19]. Two timescales exist in the\nsystem, which can be readily obtained from eq. (4); first,\nthe inverse Lamor frequency τLamor= (1+α2)µs/γh, and\nsecond, the relaxation time τrelax=τLamor/α. They govern\nthe time evolution of δϕandδθ. In leading order,\nδ˙θ=δϕ\nτLamor−δθ\nτrelax, (15)\nδ˙ϕ=ω−δθ\nτLamor−δϕ\nτrelax. (16)\nThefirstequationdescribes, how δθwouldincreasebypre-\ncession of the spin around the direction of the field, which\nis counteracted by relaxation back towards the equator.\nThe second equation describes that without relaxation\ninto the field direction, δϕwould increase with velocity\nωminus the azimuthal component of the precession veloc-\nity, which is in leading order proportional to δθ. Setting\nthe left hand sides to zeroin the steady state, immediately\ngives the solution (14).\nInthe (T= 0)-study[8], the timeaverage ∝angbracketleftPrelax∝angbracketright/vwas\nused to calculate the friction force. As pointed out above,\nthis quantity agrees with (11) in the steady state. For\nfinite temperatures, however, (11) is numerically better\nbehaved than ∝angbracketleftPrelax∝angbracketright/v. The reason is the following:\nForT∝negationslash= 0, the spins are also thermally agitated, even\nwithout external pumping, when the dissipation rate Pdiss\nvanishes. This shows that the two terms PthermandPrelax\nlargely compensate each other, and only their difference is\nthe dissipation rate we are interested in. This fact makes\nit difficult to evaluate (12) and is the reasonwhy we prefer\nto work with (11) as the definition of the friction force.\nWe have also analyzed the fluctuations of the friction\nforce (11). The power spectrum has a distinct peak at\nfrequency v/a, which means that the dominant temporal\nfluctuations are due to the lattice periodicity with lattice\nconstant a. A more complete investigation of the fluctu-\nations, which should also take into account, how thermal\npositional fluctuations influence the friction force, remains\nto be done.\np-3M.P. Magiera 1 L. Brendel 1 D.E. Wolf 1 U. Nowak 2\nTechnical remarks. – Because of the vector pro-\nduct in eq. (4), the noise ζienters in a multiplicative\nway, calling for special attention to the interpretation of\nthis stochastic differential equation (Stratonovich vs. Itˆ o\nsense). The physical origin of the noise renders it generi-\ncallycoloured and thus selects the Stratonovich interpre-\ntation as the appropriate one (Wong-Zakai theorem [20]),\nin which its appearance as white noise is an idealisation.\nAccordingly we employ the Heun integration scheme [21].\nAfter each time step the spins are rescaled so that their\nlength remains unchanged.\nTo get meaningful results, it is of prime importance to\nreach a steady state. The initial configuration turned out\nto be a crucial factor for achieving this within acceptable\ncomputing time. Therefore a long initialisation run is per-\nformed, before the tip motion starts.\nMoreover, the system size is another limiting factor. In\norder to avoid that the tip reaches the system boundary\nbefore the steady state is reached, we use a “conveyorbelt\nmethod” allowing to do the simulation in the comoving\nframe of the tip. The tip is placed in a central point above\nthe substrate plane, e.g.at ((Lx+1)/2,(Ly+1)/2). After\nan equilibration time, the tip starts to move with fixed\nvelocity in x-direction. When it passed exactly one lattice\nconstant a, the front line (at x=Lx) is duplicated and\nadded to the lattice at x=Lx+ 1. Simultaneously, the\nline at the opposite boundary of the system (at x= 1) is\ndeleted, so that the simulation cell is of fixed size and con-\nsists of the Lx×Lyspins centered around the tip position,\nwith open boundaries. Note that this is different from pe-\nriodicboundaryconditions, becausethespinconfiguration\ndeleted at one side is different from the one added at the\nopposite side. We compared the results obtained for a\nsmall system in the co-moving frame of the tip with some\nruns for a system that was long enough in x-direction that\nthe steady state could be reached in the rest frame of the\nsample, and confirmed that the same friction results and\nsteady state properties could be obtained with drastically\nreduced computation time.\nIt is convenient to rewrite the equations of motion in\nnatural units. An energy unit is prescribed by the ex-\nchange energy Jof two magnetic moments. It rescales the\nenergy related parameters dzandwas well as the simu-\nlated temperature,\nkBT′=kBT\nJ. (17)\nThe rescaled time further depends on the material con-\nstantsµsandγ,\nt′=Jγ\nµst. (18)\nA length scale is given by the lattice constant a, so a nat-\nural velocity for the system can be defined,\nv′=µs\nγJav. (19)-1-0.5 0 0.5 1\n-4-3-2-1 0 1 2 3 4mx(x)\nx-x0α = 0.3, v = 0.05\nα = 0.3, v = 0.10\nα = 0.5, v = 0.10\ntanh(0.75 x)-0.2 0\n 0  0.2 vx0/αα = 1.0\nα = 0.7\nα = 0.5\nFigure 2: Local magnetisation ( x-component) at T=0 along\nthe lattice axes in x-direction which are closest to the tip ( i.e.\naty=±0.5). Analogous to a domain wall one finds a tanh x-\nprofile. Depending on the damping constant αand the velocity\nv, its zero is shifted backwards from the tip position by a valu e\nx0≈−0.88αv, as shown in the inset.\nFrom the natural length and energy, the natural force re-\nsults to:\nF′=a\nJF (20)\nFrom now on, all variables are given in natural units, and\nwe will drop the primes for simplicity. The typical exten-\nsion of the simulated lattices is 50 ×30,which was checked\nto be big enough to exclude finite size-effects. For the tip\ncoupling, we chose large values ( e.g.wStip= (0,0,−10)),\nto get a large effective field on the substrate. Usually it\nis assumed that the dipole-dipole coupling constant has a\nvalue of about w= 0.01, which means that the magnetic\nmoment of the tip is chosen a factor of 1000 times larger\nthan the individual substrate moments. The anisotropy\nconstant is set to dz=−0.1 in all simulations. The damp-\ning constant αis varied from 0 .1 to the quite large value 1.\nAt finite temperatures typically 50 simulation runs with\ndifferent random number seeds are performed to get reli-\nable ensemble averages.\nSimulation results. – In [8] it was found that the\nmagnetic friction force depends linearly on the scanning\nvelocityvand the damping constant αfor small velocities\n(v≤0.3). For higher velocities the friction force reaches a\nmaximum and then decreases. In this work we focus on\nthe low-velocity regime with the intention to shed more\nlight on the friction mechanism and its temperature de-\npendence.1\nAdiabatic approximation at T=0.If we assume the\nfieldhfor each spin to vary slowly enough to allow the so-\nlution (14) to be attained as adiabatic approximation, the\nlinear dependence F∝αvfrom [8] follows immediately:\nAt every point, the temporal change of the direction of\nh, defining a local ωfor (14), is proportional to v. We\n1It should be noted that the smallest tip velocities we simula ted,\nare of the order of 10−2(aJγ/µ s), which is still fast compared to\ntypical velocities in friction force microscopy experimen ts.\np-4Spin excitations in a monolayer scanned by a magnetic tip\n-1-0.5 0 0.5 1\n-20-15-10-5 0 5 10 15 20mx\nx0.11.0\n2345678910|m|\nr0.11.0\n2345678910|m|\nrkBT=0.1\nkBT=0.5\nkBT=0.9\nkBT=1.5\nFigure 3: Left: Magnetisation profiles as in fig. 2 for several temperatures with wStip= 10,α= 0.5 andv= 0.01. Middle and\nright: Absolute value of the average magnetisation as a func tion of the distance rfrom (x,y) = (0,0), directly underneath the\ntip. For small temperatures (upper two curves) it decreases with a power law (cf. double-logarithmic plot, middle), for high\ntemperatures (lower two curves) it decreases exponentiall y (cf. semi-logarithmic plot, right).\nconfirmed the validity of the adiabatic approximation nu-\nmerically by decomposing S−h/hwith respect to the\nlocal basis vectors ∂t(h/h),h, and their cross-product, all\nof them appropriately normalized. In other words, we ex-\ntractedδϕandδθdirectly and found them in excellent\nagreement with (14).\nThe lag of Swith respect to hmanifests itself also\nmacroscopically in the magnetisation field as we will show\nnow. Thetip-dipoleisstrongenoughtoalignthesubstrate\nspins to nearly cylindrical symmetry. Since the anisotropy\nis chosen to generate an easy plane ( dz<0), spins far away\nfrom the tip try to lie in the xy-plane, while close to the\ntip they tilt into the z-direction. This is displayed in\nfig. 2 where the x-component of the local magnetisation is\nshown along a line in x-direction for a fixed y-coordinate.\nRemarkably, these magnetisation profiles for different val-\nues ofvandαcollapse onto a unique curve, if they are\nshifted by corresponding offsets x0with respect to the tip\nposition. As expected from (14), the magnetisation profile\nstays behind the tip by a ( y-dependent) shift x0∝αv(cf.\ninset of fig. 2).\nFriction at T>0.With increasing temperature the\nmagnetisation induced by the tip becomes smaller, as\nshown in fig. 3. One can distinguish a low tempera-\nture regime, where the local magnetisation decreases alge-\nbraically with the distance from the tip, and a high tem-\nperature regime, where it decreases exponentially. The\ntransitionbetween these regimeshappens around T≈0.7.\nFor all temperatures the friction force Fturns out to be\nproportional to the velocity (up to v≈0.3), as for T=0,\nwith a temperature dependent friction coefficient F/v.\nThe two temperature regimes manifest themselves also\nhere, as shown in fig. 4: For low temperatures the fric-\ntion coefficients depend nearly linearly on α, reflecting the\nT=0 behaviour. Towards the high temperature regime,\nhowever, the α-dependence vanishes, and all friction coef-\nficients merge into a single exponentially decreasing func- 0 2 4 6 8 10\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6F/v\nkBTwStip = 10, α=0.5\nwStip = 10, α=0.7\nwStip =  5, α=0.5\nwStip =  5, α=0.7\nFigure 4: Friction coefficients for different α,wStipandkBT.\nOne can distinguish between a low temperature regime, where\nthe friction coefficient depends on αbut not on wStip, and a\nhigh temperature regime, where it depends on wStipbut not\nonα.\ntion.\nThe low temperature behaviour can be understood es-\nsentially along the lines worked out for T=0, as result of\na delayed, deterministic response (precession and relax-\nation) to the time dependent tip field. At high tempera-\ntures, however,frictionresultsfromtheabilityofthetipto\npropagate partial order through the thermally disorderd\nmedium. The magnetisationpattern in the wakeofthe tip\nno longer adapts adiabatically to the dwindling influence\nofthe tip, but decaysdue to thermal disorder. Then, aris-\ningtemperatureletstheorderedareaaroundthetipshrink\nwhichleadstotheexponentialdecreaseofthefrictioncoef-\nficient. However, it increases with the tip strength, wStip,\nas stronger order can be temporarily forced upon the re-\ngion around the tip. By contrast, the tip strength looses\nits influence on friction in the limit T→0, because the\nsubstrate spins are maximally polarised in the tip field .\nThis picture of the two temperature regimes is sup-\np-5M.P. Magiera 1 L. Brendel 1 D.E. Wolf 1 U. Nowak 2\n-3-2.5-2-1.5-1-0.5 0\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6x0/v\nkBTwStip = 10, α=0.5\nwStip = 10, α=0.7\nwStip =  5, α=0.5\nwStip =  5, α=0.7\nFigure 5: Distance x0by which the magnetisation pattern lags\nbehind the tip is proportional to vfor all temperatures. The\nproportionality constant depends on αonly in the low temper-\nature regime.\nported by the distance x0, by which the magnetisation\npattern lags behind the tip. It is proportional to vfor all\ntemperatures, but α-dependent only in the low tempera-\nture regime, cf. fig. 5.\nConclusion and outlook. – In this work, we could\nexplain the low-velocity, zero-temperature findings from\n[8], namely that the magnetic friction force in the Heisen-\nberg model has a linear velocity dependence with a coeffi-\ncient proportional to the damping constant α. In the spin\npolarisation cloud dragged along with the tip, each sub-\nstratespin followsthe localfield with a lagproportionalto\nthe frequency of the field change and to α. Moreover, the\nmagnetisationpattern aroundthe tip getsdistorted due to\nprecession. These effects directly give rise to the observed\nmagneticfriction andcould be evaluated quantitativelyby\nmeans of a single spin model.\nSecond, for the first time the temperature dependence\nof magnetic friction in the Heisenberg model was investi-\ngated in the framework of Landau-Lifshitz-Gilbert (LLG)\ndynamics with a stochastic contribution to the magnetic\nfield. Two regimes were found, which can be charac-\nterised by their different relaxation behaviour. While in\nthe low-temperature regime the response of the system on\nthe perturbation due to the moving tip is dominated by\nthe deterministic precession and relaxation terms in the\nLLG equation, thermal perturbations competing with the\none caused by the moving tip are essential in the high-\ntemperature regime. This explains, why magnetic friction\ndepends on αbut noton wStipforlowtemperatures, while\nit depends on wStipbut not on αfor high temperatures\nwhere it decreases exponentially with T.\nImportant extensions of the present investigation in-\nclude the effects ofa tip magnetisationpointing in a differ-\nent than the z-direction, of the strength and sign of spin\nanisotropy, dz, or of the thickness of the magnetic layer.\nBoth, spin anisotropyand lattice dimension will be crucial\nfor the critical behaviour, as well as for the critical tem-perature itself. Studies dealing with these quantities are\nalready in progress and will be reported in a future work.\n∗∗∗\nThis work was supported by the German Research\nFoundation (DFG) via SFB 616 “Energy dissipation at\nsurfaces”. Computation time was granted in J¨ ulich by the\nJohn-von-Neumann Institute of Computing (NIC).\nReferences\n[1]Persson B. ,Sliding Friction (Springer, Berlin, Heidel-\nberg, New York) 1998.\n[2]Urbakh M., Klafter J., Gourdon D. andIs-\nraelachvili J. ,Nature,430(2004) 525.\n[3]Gnecco E., Bennewitz R., Gyalog T. andMeyer E. ,\nJ. Phys.: Condens. Matter ,13(2001) R619.\n[4]Acharyya M. andChakrabarti B. K. ,Phys. Rev. B ,\n52(1995) 6550.\n[5]Ort´ın J.andGoicoechea J. ,Phys. Rev. B ,58(1998)\n5628.\n[6]Corberi F., Gonnella G. andLamura A. ,Phys. Rev.\nLett.,81(1998) 3852.\n[7]Kadau D., Hucht A. andWolf D. E. ,Phys. Rev. Lett. ,\n101(2008) 137205.\n[8]Fusco C., Wolf D. E. andNowak U. ,Phys. Rev. B ,\n77(2008) 174426.\n[9]Grutter P., Liu Y., LeBlanc P. andDurig U. ,Appl.\nPhys. Lett. ,71(1997) 279.\n[10]Schmidt R., Lazo C., Holscher H., Pi U. H., Caciuc\nV., Schwarz A., Wiesendanger R. andHeinze S. ,\nNano Letters ,9(2009) 200.\n[11]Zw¨orner O., H ¨olscher H., Schwarz U. D. and\nWiesendanger R. ,Appl. Phys. A ,66(1998) S263.\n[12]Gnecco E., Bennewitz R., Gyalog T., Loppacher\nC., Bammerlin M., Meyer E. andG¨untherodt H.-\nJ.,Phys. Rev. Lett. ,84(2000) 1172.\n[13]Prandtl L. ,Zs. f. angew. Math. u. Mech. ,8(1928) 85.\n[14]Tomlinson G. A. ,Philos. Mag. ,7(1929) 905.\n[15]Landau L. D. andLifshitz E. M. ,Phys. Z. Sowjetunion ,\n8(1935) 153.\n[16]Gilbert T. L. ,IEEE Trans. Magn. ,40(2004) 3443.\n[17]N´eel L.,C. R. Acad. Sc. Paris ,228(1949) 664.\n[18]Brown W. F. ,Phys. Rev. ,130(1963) 1677.\n[19]Magiera M. P. ,Computer simulation of magnetic fric-\ntionDiploma Thesis, Univ. of Duisburg-Essen (2008).\n[20]Horsthemke W. andLefever R. ,Noise-Induced Tran-\nsitions(Springer) 1983.\n[21]Garc´ıa-Palacios J. L. andL´azaro F. J. ,Phys. Rev.\nB,58(1998) 14937.\np-6"    },    {        "title": "0811.2235v2.Intrinsic_Coupling_between_Current_and_Domain_Wall_Motion_in__Ga_Mn_As.pdf",        "content": "arXiv:0811.2235v2  [cond-mat.mes-hall]  27 Jun 2009Intrinsic Coupling between Current and Domain Wall Motion i n (Ga,Mn)As\nKjetil Magne Dørheim Hals, Anh Kiet Nguyen, and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway\nWe consider current-induceddomain wall motion and, therec iprocal process, movingdomain wall-\ninduced current. The associated Onsager coefficients are exp ressed in terms of scattering matrices.\nUncommonly, in (Ga,Mn)As, the effective Gilbert damping coe fficientαwand the effective out-of-\nplane spin transfer torqueparameter βware dominated byspin-orbit interaction incombination wit h\nscattering off the domain wall, and not scattering off extrins ic impurities. Numerical calculations\ngiveαw∼0.01 andβw∼1 in dirty (Ga,Mn)As. The extraordinary large βwparameter allows\nexperimental detection of current or voltage induced by dom ain wall motion in (Ga,Mn)As.\nThe principle of giant magneto resistance is used to\ndetect magnetic information. Large currents in mag-\nnetic nanostructures can manipulate the magnetization\nvia spin transfer torques [1]. A deeper knowledge of the\ncoupled out-of-equilibrium quasi-particle and magnetiza-\ntion dynamics is needed to precisely control and utilize\ncurrent-induced spin transfer torques.\nThemagnetizationrelaxestowardsits equilibriumcon-\nfiguration by releasing magnetic moments and energy\ninto reservoirs. This friction process is usually described\nby the Gilbert damping constant αin the Landau-\nLifshitz-Gilbert (LLG) equation. Spins traversingamag-\nnetic domain wall exert an in-plane and an out-of-plane\ntorqueonthe wall[2]. In dirtysystems, when the domain\nwall is wider than the mean-free-path, the out-of-plane\ntorque, often denoted the non-adiabatic torque, is pa-\nrameterized by the so-called β-factor [2]. The Gilbert\ndamping coefficient α, the in-plane spin-transfer torque,\nand the out-of-plane torque coefficient βdetermine how\nthe magnetization is influenced by an applied current,\ne.g. the current-induced Walker domain wall drift veloc-\nity is proportional to β/α[2, 3, 4]. Scattering off impu-\nrities are important for αandβ[2, 3, 4]. Additionally,\ndomain wall scattering can contribute to αandβ. In\nballistic(Ga,Mn)As, intrinsic spin-orbit coupling causes\nsignificant hole reflection at the domain wall, even in the\nadiabatic limit when the wall is much thicker than the\nFermi wavelength [5]. This grossly increases the out-of-\nplane spin-transfer torque, and consequently the current-\ndriven domain wall mobility. So far, there are no inves-\ntigations on the effect of these domain wall induced hole\nreflections on the effective Gilbert damping constant α.\nExperimental(Ga,Mn)As samples aredirty sothat the\neffectofdisorderontheeffectiveGilbertdampingandthe\nout-of-plane spin transfer torque should be taken into\naccount. We find surprisingly that, in systems with a\nlargeintrinsic spin-orbitcoupling, domainwallscattering\ncontributesdominantlyto αandβeveninthedirtylimit.\nIntrinsic current-domain wall motion coupling is robust\nagainst impurity scattering.\nCurrent-induced domain-wall motion has been seen in\nmany experiments [3]. The reciprocaleffect, domain-wall\nmotion induced current, is currentlytheoreticallyinvesti-gated[6, 7], andseenexperimentally[8]. Aprecessingdo-\nmain wall induces a charge current in ferromagnetic met-\nals [6] similar to spin-pumping in layered ferromagnet-\nnormal metal systems [9]. For rigid domain wall motion,\nthe induced chargecurrentis proportionalto β/α[7]. We\nfind that βandβ/αin (Ga,Mn)As are so large that the\ncurrent, or equivalently, the voltage induced by a moving\ndomain wall is experimentally measurable.\nOnsager’s reciprocity relations dictate that response\ncoefficients of domain wall motion induced current and\ncurrent induced domain wall motion are related. In dirty\nsystems, these relations have been discussed in Ref. [7].\nRef.[7]alsousedthescatteringtheoryofadiabaticpump-\ning to evaluate the non-adiabatic spin-transfer torque in\nballistic systems without intrinsic spin-orbit interaction.\nWe first extend the pumping approach to (Ga,Mn)As\nwith strong intrinsic spin-orbit interaction, and second,\nalso evaluate the Onsager coefficient as a function of\nsample disorder. In determining all Onsager coefficients,\nmagnetization friction must be evaluated on the same\nfooting. To this end, we generalize the energy pump-\ning scattering theory of Gilbert damping [10] to domain\nwall motion. Our numerical calculation demonstrates,\nfor the first time, that domain wall scattering is typi-\ncally more important than impurity scattering for the\neffective domain wall motion friction in systems with a\nstrongintrinsicspin-orbitinteraction. OurnovelOnsager\nscattering approach can also be used to compute the ef-\nfective rigid domain wall motion αandβparameters in\nrealistic materials like Fe, Ni, Co, and alloysthereof from\nfirst-principles.\nLet us discuss in more detail the Onsager reciprocity\nrelations in our system. The magnetic field is a thermo-\ndynamic force for the magnetization since it can move\ndomainwalls. The electric field is athermodynamic force\nfor the charges as it induces currents. In systems where\ncharge carriers also carry spin, the magnetic and charge\nsystems are coupled. Through this coupling, the elec-\ntric field can move a domain wall and, vice versa, the\nmagnetic field can induce a current. This phenomenon,\nwhere the thermodynamic force of one system can induce\na flux in another system is well-known in thermodynam-\nics [11]: Assume a system described by the quantities2\n{qi},Xidenotes the thermodynamic force, and Jithe\nflux associated with the quantity qi. In linear response,\nJi=/summationtext\njLijXj, whereLijare the Onsager coefficients.\nOnsager’s reciprocity principle dictates Lij=ǫiǫjLji,\nwhereǫi= 1 (ǫi=−1) ifqiis even (odd) under time-\nreversal [11]. Fluxes and forces are not uniquely defined,\nbut the Onsager reciprocity relations are valid when the\nentropy generation is ˙S=/summationtext\niJiXi[11].\nWe first derive expressions for the Onsager coefficients\nand determine the Onsager reciprocity relations between\na charge current and a moving domain wall in terms\nof the scattering matrix. Subsequently, we derive the\nrelation between the Onsager coefficients and the effec-\ntive Gilbert damping parameter αwand the out-of-plane\ntorque parameter βwfor domain wall motion. Finally,\nwe numerically compute αwandβwfor (Ga,Mn)As.\nWe start the derivation of the Onsager coefficients in\nterms of the scattering matrix by assuming the following\nfree energy functional for the magnetic system\nF[M] =Ms/integraldisplay\ndr/parenleftbiggJ\n2[(∇θ)2+sin2(θ)(∇φ)2]+\nK⊥\n2sin2(θ)sin2(φ)−Kz\n2cos2(θ)−Hextcos(θ)/parenrightbigg\n,(1)\nwhereMs,JandHextare the saturation magnetization,\nspin-stiffness and external magnetic field, respectively,\nandKzandK⊥are magnetic anisotropy constants. The\nlocal magnetization angles θandφare defined with re-\nspect to the z- andx-axis, respectively. The system con-\ntains a Bloch wall rotating in the (transverse) x-zplane,\ncos(θ) = tanh([ y−rw]/λw), sin(θ) = 1/cosh([y−rw]/λw),\nwhererwis the position of the wall, and λwis the wall\nwidth. We assume the external magnetic field is lower\nthan the Walker threshold, so that the wall rigidly moves\n(˙φ= 0) with a constant drift velocity. In this case rw\nandφcompletely characterize the magnetic system, and\nλw=/radicalBig\nJ/(Kz+K⊥sin2(φ)) [4]. The current is along\nthey-axis.\nTheheatdissipatedperunittimefromachargecurrent\nJcis˙Q=Jc(VL−VR), where VL(VR) is the voltage in\nthe left (right) reservoir. Using the relation dS=dQ/T,\nthis implies an entropy generation ˙S=Jc(VL−VR)/T.\nThus,Xc≡(VL−VR)/Tis the thermodynamic force\ninducing the flux Jc. We assume the magnetic system\nto be at constant temperature, which means that the\nheat transported out of the magnetic system as the do-\nmain wall moves equals the loss of free energy. This\nimplies an entropy generation ˙S=˙Q/T=−˙F/T=\n(−∂F[rw,φ]/T∂rw) ˙rw=XwJw, where we have defined\nthe force Xw≡ −∂F[rw,φ]/T∂rwand flux Jw≡˙rw. Us-\ning Eq. (1), wefind Xw=−2AMsHext/T, whereAis the\nconductor’s cross-section. Fluxes are related to forces by\nJw=LwwXw+LwcXc (2)\nJc=LccXc+LcwXw, (3)whereLcc=GTandGis the conductance. Lww(Lwc)\ndetermine the induced domain wall velocity by an exter-\nnal magnetic field (a current). The induced current by a\nmoving domain wall caused by an external magnetic field\nHextis controlled by Lcw. Both charge and rware even\nunder time-reversal so that Lcw=Lwc[12].\nThe current induced by a moving domain wall is para-\nmetric pumping in terms of the scattering matrix [9]:\nJc,α=e˙rw\n2π/summationdisplay\nβ=1,2ℑm/braceleftbigg\nTr/bracketleftbigg∂Sαβ\n∂rwS†\nαβ/bracketrightbigg/bracerightbigg\n,(4)\nwhereSαβis the scattering matrix between transverse\nmodes in lead βto transverse modes in lead α. The\nsystem has two leads ( α,β∈ {1,2}). The trace is over\nall propagating modes at the Fermi energy EF. From\nEqs. (2) and (3) we find Jc=Lcw˙rw/Lww.\nWe considertransportwellbelowthe criticaltransition\ntemperature in (Ga,Mn)As, which is relatively low, and\nassume the energy loss in the magnetic system is trans-\nferred into the leads by holes. Generalizing Ref. [10] to\ndomain wall motion, this energy-flux is related to the\nscattering matrix:\nJE=¯h\n4πTr/braceleftbiggdS\ndtdS†\ndt/bracerightbigg\n=¯h˙r2\nw\n4πTr/braceleftbigg∂S\n∂rw∂S†\n∂rw/bracerightbigg\n.(5)\nFor a domain wall moved by an external magnetic field,\nwe then find that XwJw=J2\nw/Lww=JE/T. In sum-\nmary, the Onsager coefficients in Eq. (2) and Eq. (3) are\nLww=/parenleftbigg¯h\n4πTr/braceleftbigg∂S\n∂rw∂S†\n∂rw/bracerightbigg/parenrightbigg−1\n, (6)\nLcw=2e\n¯h/summationtext\nβ=1,2ℑm/braceleftBig\nTr/bracketleftBig\n∂Sαβ\n∂rwS†\nαβ/bracketrightBig/bracerightBig\nTr/braceleftBig\n∂S\n∂rw∂S†\n∂rw/bracerightBig ,(7)\nLcc=e2\nhTr/braceleftbig\nt†t/bracerightbig\n, (8)\nwheretis the transmission coefficient in the scatter-\ning matrix. We have omitted the temperature factor\nin the coefficients (6), (7), and (8) since it cancels with\nthe temperature factor in the forces, i.e.we transform\nL→L/TandX→TX. The Onsager coefficient ex-\npressions in terms of the scattering matrix are valid irre-\nspectiveofimpuritydisorderandspin-orbitinteractionin\nthe band structures or during scattering events, and can\ntreat transport both in ballistic and diffusive regimes.\nLet us compare the global Onsager cofficients (6), (7),\nand (8) with the local Onsager coefficients in the dirty\nlimit to gainadditionalunderstanding. In the dirtylimit,\nall Onsager cofficients become local and the magnetiza-\ntion dynamics can be described by the following phe-\nnomenological local LLG equation [2, 3]:\n˙m=−γm×Heff+αm×˙m\n−(1−βm×)(vs·∇)m, (9)3\nwheremis the magnetization direction, Heffis the\neffective magnetic field, γis the gyromagnetic ratio,\nvs=−¯hPj/(eS0),S0=Ms/γ,Msthe magnetization,\nαthe Gilbert damping constant, Pthe spin-polarization\nalong−mof the charge carriers [13], and βis the out-\nof-plane spin-transfer torque parameter. Substituting\na Walker ansatz into Eq. (9) gives below the Walker\nthreshold [4]: α˙rw/λw=−γHext−¯hβPj/(eS0λw). In\ndirty, local, systems this equation determines the rela-\ntion between the flux Jwand the forces XwandXc\nasLww=λw/(2AS0α) andLwc=−¯hβPG/(eαS0A),\nwhere we have used j=σ(VL−VR)/L, andG=σA/L.\nHere,Lis the length of the conductor, ethe electron\ncharge, and σthe conductivity. This motivates defining\nthe following dimensionless global coefficients:\nαw≡λw\n2AS0Lww, βw≡ −λwe\n2¯hPGLwc\nLww.\nαwis the effective Gilbert damping coefficient and βwis\nthe effective out-of-plane torque on the domain wall.\nWe will in the following investigate αwandβwfor\n(Ga,Mn)As by calculating the scattering matrix expres-\nsions in Eq. (6) and Eq. (7). We use the following Hamil-\ntonian to model quantum transport of itinerant holes:\nH=HL+h(r)·J+V(r). (10)\nHere,HLis the 4×4 Luttinger Hamiltonian (parame-\nterized by γ1andγ2) for zincblende semiconductors in\nthe spherical approximation, while h·Jdescribes the\nexchange interaction between the itinerant holes and the\nlocal magnetic moment of the Mn dopants. We introduce\nAnderson impurities as V(r) =/summationtext\niViδr,Ri, whereRiis\nthe position of impurity i,Viits impurity strength, and\nδthe Kroneckerdelta. More details about the model and\nthe numericalmethod used can be found in Refs. [14, 15].\nWe consider a discrete conductor with transverse di-\nmensions Lx= 23nm,Lz= 17nmand length Ly=\n400nm. The lattice constant is 1 nm, much less than the\ntypical Fermi wavelength λF∼8nm. The Fermi energy\nEF= 82meVis measured from the bottom of the lowest\nsubband. |h|= 0.5×10−20Jandγ1= 7. The typical\nmean-free path for the systems studied ranges from the\ndiffusive to the ballistic regime l∼23nm→ ∞, and we\nare in the metallic regime kFl≫1. The domain wall\nlength is λw= 40nm. The spin-density S0from the local\nmagneticmoments is S0= 10¯hx/a3\nGaAs,aGaAsthe lattice\nconstant for GaAs, and x= 0.05 the doping level[14].\nFig. 1a shows the computed effective Gilbert damping\ncoefficient αwversusλw/lfor (Ga,Mn)As containing one\nBloch wall. Note the relatively high αw∼5×10−3in\nthe ballistic limit. Additional impurities, in combination\nwith the spin-orbit coupling, assist in releasing energy\nand angular momentum into the reservoirs and increase\nαw. However, as shown in Fig. 1a, impurities contribute\nonly about 20% to αweven when the domain wall is two0 0.5 1 1.50246x 10−3\n00.5 11.5 22.5024x 10−3\n0 0.5 1 1.502468\n00.5 11.5 22.50123λw/l\nλw/lαw βw\nαw βwγ2\nγ2(a)\n(b)\nFIG. 1: (a): Effective Gilbert damping αwas function of\nλw/l, whereλwis the domain wall length and lis the mean\nfree path when γ2= 2.5. Here,λwis kept fixed, and lis\nvaried. Inset: αwas a function of spin-orbit coupling γ2for a\nclean system, l=∞.(b):βwas a function of λw/l, whereλw\nis the domain wall length and lis the mean free path when\nγ2= 2.5. Here,λwis kept fixed, and lis varied. Inset: βwas\na function of spin-orbit coupling γ2for a clean system, l=∞.\nIn all plots, line is guide to the eye.\ntimes longer than the mean free path. Due to the strong\nspin-orbit coupling, ballistic domain walls have a large\nintrinsic resistance [5] that survives the adiabatic limit.\nWhen itinerant holes scatter off the domain wall their\nmomentum changes and through the spin-orbit coupling\ntheir spin also changes. This is the dominate process for\nreleasing energy and magnetization into the reservoirs.\nThe saturated value αw∼6×10−3is of the same order\nas the estimates in Ref. [16] for bulk (Ga,Mn)As. The\ninset in Fig. 1a shows the domain-wall contribution to\nαwversus the spin-orbit coupling for a clean system with\nno impurities. αwmonotonicallydecreasesfor decreasing\nγ2and vanish for γ2→0. Since, λw/λF∼5, itinerant\nholes will, without spin-orbit coupling, traverse the do-4\nmain wall adiabatically.\nFig. 1b shows βwversusλw/l.βwdecreases with\nincreasing disorder strength. This somewhat counter\nintuitive result stem from the fact that domain walls\nin systems with spin-orbit coupling have a large intrin-\nsic domain wall resistance [5] which originates from the\nanisotropy in the distribution of conducting channels [5].\nThe reflected spins do not follow the magnetization of\nthe domain wall, and thereby cause a large out-of-plane\ntorque [2]. This causes the large βwin the ballistic limit.\nScalar, rotational symmetric impurities tend to reduce\nthe anisotropy in the conducting channels, and thereby\nreduce the intrinsic domain wall resistance and conse-\nquently reduce βw. Deeper into the diffusive regime, β\nsaturates. Here, the domain wall resistance and βware\nkept at high levels due to the increase in the spin-flip\nrate caused by impurity scattering. The saturated value\nisβ∼1. For even dirtier systems than a reasonable\ncomputing time allows, we expect a further increase in\nβw. In comparison, simple microscopic theories for fer-\nromagnetic metals where one disregards the spin-orbit\ncoupling in the band structure predict β∼0.001−0.01\n[2, 3, 4]. Similar to the Gilbert damping, in ballistic sys-\ntemsβwincreaseswith spin-orbit coupling because ofthe\nincreased domain wall scattering [5], see Fig. 1b inset.\nβwcan be measured experimentally by the induced\ncurrent or voltage from a domain wall moved by an ex-\nternal magnetic field as a function of the domain wall\nvelocity [7]. From the Onsager relations we have that\nJc=LcwXw. UsingXw=Jw/Lww, the induced current\nand voltage are [7]:\nJc=−2β¯hPG\neλw˙rw⇒V=−2βw¯hP\neλw˙rw.(11)\nAn estimate of the maximum velocity of a domain wall\nmoved by an external magnetic field below the Walker\ntreshold is ˙ rw∼10m/s[17]. With λw= 40nmand\nP= 0.66 this indicates an experimentally measurable\nvoltageV∼0.2µV.\nInconclusion, wehavederivedOnsagercoefficientsand\nreciprocity relations between current and domain wall\nmotion in terms of scattering matrices. In (Ga,Mn)As,\nwe find the effective Gilbert damping constant αw∼0.01\nand out-of-plane spin transfer torque parameter βw∼1.\nIn contrast to ferromagnetic metals, the main contribu-\ntions to αwandβwin (Ga,Mn)As are intrinsic, and in-\nduced by scattering off the domain wall, while impurity\nscattering is less important. The large βwparameter im-\nplies a measurable moving domain wall induced voltage.\nThis work was supported in part by the Re-search Council of Norway, Grants Nos. 158518/143 and\n158547/431, computing time through the Notur project\nand EC Contract IST-033749 ”DynaMax”.\n[1] J.C. Slonczewski,J. Magn. Magn. Mater. 159, L1 (1996);\nL. Berger,Phys. Rev. B 54, 9353 (1996).\n[2] G. Tatara, H. Kohno,Phys. Rev. Lett. 92,086601 (2004);\nS. 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Sinova et al., Phys. Rev. B 69, 085209 (2004); Y.\nTserkovnyak, G. A. Fiete and B. I. Halperin,Appl. Phys.\nLett.84, 5234 (2004); I. Garate and A. MacDonald,\narXiv:0808.3923v1.\n[17] A. Dourlat, V. Jeudy, A. Lemaitre and C. Gourdon,\nPhys. Rev. B 78, 161303(R) (2008)."    },    {        "title": "2111.08768v1.Ultrathin_ferrimagnetic_GdFeCo_films_with_very_low_damping.pdf",        "content": "Ultrathin ferrimagnetic GdFeCo \flms with very low damping\nLakhan Bainsla*,1,a)Akash Kumar,1Ahmad A. Awad,1Chunlei Wang,2Mohammad Zahedinejad,1Nilamani\nBehera,1Himanshu Fulara,1Roman Khymyn,1Afshin Houshang,1Jonas Weissenrieder,2and J. \u0017Akerman1,b)\n1)Physics Department, University of Gothenburg, 412 96 Gothenburg, Sweden.\n2)Department of Applied Physics, KTH Royal Institute of Technology, 106 91 Stockholm,\nSweden\nFerromagnetic materials dominate as the magnetically active element in spintronic devices, but come with\ndrawbacks such as large stray \felds, and low operational frequencies. Compensated ferrimagnets provide an\nalternative as they combine the ultrafast magnetization dynamics of antiferromagnets with a ferromagnet-like\nspin-orbit-torque (SOT) behavior. However to use ferrimagnets in spintronic devices their advantageous prop-\nerties must be retained also in ultrathin \flms ( t<10 nm). In this study, ferrimagnetic Gd x(Fe87:5Co12:5)1\u0000x\nthin \flms in the thickness range t= 2{20 nm were grown on high resistance Si(100) substrates and studied\nusing broadband ferromagnetic resonance measurements at room temperature. By tuning their stoichiometry,\na nearly compensated behavior is observed in 2 nm Gd x(Fe87:5Co12:5)1\u0000xultrathin \flms for the \frst time,\nwith an e\u000bective magnetization of Me\u000b= 0.02 T and a low e\u000bective Gilbert damping constant of \u000b= 0.0078,\ncomparable to the lowest values reported so far in 30 nm \flms. These results show great promise for the\ndevelopment of ultrafast and energy e\u000ecient ferrimagnetic spintronic devices.\nI. INTRODUCTION\nSpintronic devices utilize the spin degree of freedom for\ndata storage, information processing, and sensing1,2with\ncommercial applications such as hard drives, magnetic\nrandom access memories, and sensors. Besides conven-\ntional memory applications based on quasi-static opera-\ntion of magnetic tunnel junctions, high frequency spin-\ntronic oscillators3,4have recently been demonstrated for\nanalog computing applications such as bio-inspired neu-\nromorphic computing5,6, logic operations, energy har-\nvesting and Ising Machines.7For the \frst time, such oscil-\nlators are now used in commercial magnetic hard drives\nto facilitate writing to the disc.8The key challenges in\ndeveloping such devices is to \fnd material combinations\nwhich allow for fast operation, low-power consumption,\nnon-volatility, and high endurance. Due to their nat-\nural spin polarization and easy manipulation, ferromag-\nnetic materials (FM) dominate as active elements in these\ndevices.4However, FMs come with drawbacks such as:\n(i) large magnetic stray \felds a\u000becting the operation of\nneighbouring devices; (ii) limited scalability of magnetic\nbits in memory devices; (iii) the operating frequency of\nspin-based oscillators limited by ferromagnetic resonance\nfrequency, and (iv) slow synchronization of such oscilla-\ntors. These shortcomings drive researchers to \fnd more\nsuitable materials for future spintronic devices.\nVery recently, the interest in antiferromagnetic (AFM)\nspintronics9{11increased rapidly, as AFM materials have\nno stray \felds and can o\u000ber ultrafast spin dynamics, in-\ncluding AFM resonance frequencies in the THz region.\nIt was theoretically shown that such high-frequency ex-\ncitations are possible to achieve without any applied\nmagnetic \feld by injecting spin currents into AFM\na)Electronic mail: lakhan.bainsla@physics.gu.se\nb)Electronic mail: johan.akerman@physics.gu.sematerials.12{15Experiments have since demonstrated\npossible THz writing/reading capabilities.16However,\nthe absence of a net magnetic moment in AFMs leads\nto di\u000eculties in the read-out of the spin dynamics, in-\ncluding any microwave output signal from the AFM\noscillators.13{15\nA possible solution is presented by ferrimagnets\n(FiMs), which combine the properties of FMs and AFMs.\nFiMs posses magnetic sub-lattices in the same way as\nAFMs do, but their sub-lattices are inequivalent. The\nmagnetic sub-lattices in FiMs often consist of di\u000berent\nmagnetic ions, such as rare earth (e.g. Gd) and transi-\ntion metal (e.g. Fe, Co) alloys (RE-TM) such as CoGd,\nand as a result, a large residual magnetization remains\ndespite the two opposing sub-magnetizations. The tem-\nperature dependence of RE and TM sub-magnetizations\nin FiM can be quite di\u000berent which result in magneti-\nzations that can increase, and even change sign, with\ntemperature17,18, in stark contrast to the non-monotonic\ndecreasing temperature dependence for FMs and AFMs.\nSimilar e\u000bects could also be seen by varying the com-\nposition of ferrimagnetic alloys instead of changing the\ntemperature.19In addition, the di\u000berent properties of the\ntwo magnetic sub-lattices also results in two compen-\nsation points, namely the magnetization compensation\npointTmand the angular compensation point Ta. AtTm,\nthe two magnetic sub-lattices cancel each other, which re-\nsults in a zero net magnetic moment, while at T a, their\nnet angular momentum vanishes, as in AFMs. Therefore,\natTa, FiMs can have a near-THz resonance as in AFMs,\nwhile still having a net magnetic moment which can lead\nto strong read-out signals, including e\u000ecient microwave\nsignal output from FiM-based oscillators20, as well as ef-\n\fcient control and excitation. FiMs also show high spin\npolarization which also make them suitable candidate for\ne\u000ecient magnetic tunnel junctions.21\nDue to these unique properties, research in FiMs for\nspintronic applications is intensifying22, focusing mainly\non RE-TM based systems such as CoTb23, CoGd24, andarXiv:2111.08768v1  [cond-mat.mtrl-sci]  16 Nov 20212\nFigure 1. (a) Schematic illustration of the coplanar waveguide (CPW), the thin \flm sample and its orientation, the directions\nof the applied magnetic \feld H, the microwave \feld hrf, and the e\u000bective magnetic \feld He\u000bduring FMR measurements.\nInset shows the \flm stack. (b) FMR response (derivative of the FMR absorption) for a 10 nm Gd 12:5Fe76:1Co11:4\flm (S2)\nrecorded at di\u000berent frequencies and \ftted (solid lines) to Eq. 1. While FMR curves were recorded at 1 GHz frequency intervals\nthroughout this study, \fgure (b) only shows curves with \u0001 f= 2 GHz for clarity.\nGdFeCo25and Mn 3\u0000xPtxGa26,27based Heusler alloy.\nAmong these, GdFeCo has been studied the most with\ndemonstrations of fast domain wall motion28and ultra-\nfast spin dynamics17nearTa, large spin-orbit torques\nand their sign reversal,25,29low magnetic damping in\nthick 30 nm \flms,30and sub-picosecond magnetization\nreversal,31to name a few. What is missing, however, is a\ndemonstration that these unique material properties per-\nsist down to much thinner \flms, which will ultimately be\nneeded if FiMs are to be used in spin-Hall nano oscillators\n(SHNOs).4\nIn the present study, we systematically study the\ngrowth and functional properties of ultrathin ferrimag-\nnetic Gd x(Fe87:5Co12:5)1\u0000xthin \flms [referred to as\nGdx(FeCo) 1\u0000xhereafter]. GdFeCo thin \flms in the\nthickness range of 2{20 nm were grown on high resis-\ntance silicon (HR-Si) substrate. The atomic composi-\ntion of Gd x(FeCo) 1\u0000xwas controlled using co-sputtering\nand determined using inductively coupled plasma optical\nemission spectroscopy (ICP-OES). The magnetic prop-\nerties and Gilbert damping were studied using broad-\nband ferromagnetic resonance (FMR) measurements. We\nalso demonstrate ultra low Gilbert damping for 2 nm\nGdFeCo, near the compensation point of Gd x(FeCo) 1\u0000x.\nThese results paves the way for integration of FiMs into\nvarious spintronic devices and applications.\nII. RESULTS AND DISCUSSION\nThe growth conditions for GdFeCo were \frst optimized\nby growing four 10 nm thick Gd 12:5Fe76:1Co11:4\flms on\nHR-Si (100) substrates using di\u000berent MgO seed layer\nthicknesses: 0 nm (S1), 6 nm (S2), 10 nm (S3 & S4);in S4, the seed was annealed at 600C for 1 hour prior\nto GdFeCo deposition to check the e\u000bect of MgO crys-\ntallinity. MgO was chosen as seed since it is insulating\nand therefore will not contribute any spin sinking to the\nmagnetic damping.32\nA. Seed layer dependence on 10nm thick\nGd12:5Fe76:1Co11:4\flms\nFurther details of the growth conditions are given in\nthe experimental section. FMR measurements, on 6 \u00023\nmm2rectangular pieces cut from these \flms, were then\nperformed using a NanOsc PhaseFMR-40 FMR Spec-\ntrometer. The sample orientation on the coplanar waveg-\nuide (CPW), together with the directions of the applied\n\feld, the microwave excitation \feld hrf, and the e\u000bec-\ntive magnetic \feld He\u000b, are shown in Fig. 1(a). Typical\n(derivative) FMR absorption spectra obtained for S2 are\nshown in \fgure 1(b) together with \fts to a sum of sym-\nmetric and anti-symmetric Lorentzian derivatives:33\ndP\ndH(H) =\u00008C1\u0001H(H\u0000HR)\n[\u0001H2+ 4(H\u0000HR)2]2+2C2(\u0001H2\u00004(H\u0000HR)2)\n[\u0001H2+ 4(H\u0000HR)2]2\n(1)\nwhereHR, \u0001H,C1, andC2represent the resonance \feld,\nthe full width at half maximum (FWHM) of the FMR ab-\nsorption, and the symmetric and anti-symmetric \ftting\nparameters of the Lorentzian derivatives, respectively.\nThe extracted values of HRvs.fare shown in \fgure\n2 (b) together with \fts to Kittel's equation:34\nf=\r\u00160\n2\u0019q\n(HR\u0000Hk)(HR\u0000Hk+Meff) (2)3\nFigure 2. (a) Seed layer dependence of frequency vs resonance \feld of the 10 nm thick Gd 12:5Fe76:1Co11:4\flms, here solid\nsymbols and solid lines are the experimental data points and \ftting with equation (2), respectively. (b) Resonance linewidth\n(\u0001H)vs.frequency of the 10 nm thick Gd 12:5Fe76:1Co11:4\flms, here solid symbols and solid lines are the experimental data\npoints and \ftting with equation (3), respectively. The e\u000bective Gilbert damping constant values of all the samples are given in\n\fgure 2 (b). The black and violet dotted lines in \fgure 2(b) shows the \ftting of equation (3) in low and high frequency regions,\nrespectively.\nwhere,\r,HkandMe\u000bare the gyromagnetic ratio, the\nin-plane magnetic anisotropy \feld, and the e\u000bective mag-\nnetization of the sample, respectively, all allowed to be\nfree \ftting parameters. Values for \randHkonly showed\nminor variation between the four samples, with \r=2\u0019=\n29.4-30.0 GHz/T and Hk= 66-104 Oe. Me\u000bvaried more\nstrongly, with values of 0.79, 1.19, 0.71 and 0.76 T ob-\ntained for S1, S2, S3 and S4, respectively.\nThe e\u000bective Gilbert damping constant \u000bcan then be\nobtained from \fts of \u0001 Hvs.fto:35\n\u0001H= \u0001H0+4\u0019\u000bf\n\r\u00160(3)\nwhere the o\u000bset \u0001 H0represents the inhomogeneous\nbroadening. Equation (3) is well \ftted to the experimen-\ntal values, using \u0001 H0and\u000bas adjustable \ftting param-\neters for all the four samples, as shown in the \fgure 2(b).\n\u0001H0= 2{4 mT is essentially sample independent within\nthe measurement accuracy. In contrast, the obtained val-\nues of\u000bvary quite strongly and are given inside \fgure\n2(b). The GdFeCo grown with 6 nm MgO seed layer (S2)\nclearly shows the lowest value of \u000b= 0:0055, although\nthis might be a\u000bected by the slight non-linear behavior\naround 10 to 15 GHz. However, when only the high-\feld\ndata is \ftted, the extracted damping of \u000b= 0.0076 is\nstill the lowest and at all frequencies the linewidth of S2\nlies well below all the other samples. As damping is one\nof the most important parameters for spintronic devices,\nwe hence chose the growth conditions of S2 for all subse-\nquent \flms in this study.B. Thickness dependence on Gd 12:5Fe76:1Co11:4\flms\nAfter optimizing the growth conditions for\nGd12:5Fe76:1Co11:4, the thickness dependence of the\n\flms was studied with the same composition using the\ngrowth conditions of sample S2. The FMR linewidth\n\u0001Hvs. f is shown in \fgure 3(a) and exhibits a relatively\nstrong dependence on thickness. It is noteworthy\nthat the 4 nm \flm shows the narrowest linewidth at\nall frequencies, clearly demonstrating that very low\ndamping can be achieved also in ultra-thin GdFeCo.\nThe extracted Me\u000band\u000bare shown vs.thickness in\n\fgure 3(b), both showing a strong thickness dependence.\nDamping as low as \u000b= 0:0055 is obtained for the 10\nnm thick \flms. If only the high-\feld portion of the data\nis \ftted, the extracted damping increases to 0.0076,\nwhich is still about an order of magnitude lower than\nany literature value on 10 or 30 nm \flms.19,36Both\nthe 10 and 20 nm \flms showed a minor nonlinearity in\n\u0001Hvs.fdata and were therefore analysed by \ftting\nthe data in both the low and the high \feld regions\nseparately, as shown by the dotted lines in \fgure 3(a).\nThe\u000bvalue for the 20 nm \flm increased slightly from\n0.0098 to 0.0109 if only high \feld data is used for\nanalysis. The relatively higher damping for the 20 nm\n\flm might be due to the radiative damping mechanism\nwhich increases proportionally with magnetic layer\nthickness.37We conclude that 2 nm ultrathin \flms can\nindeed be grown with reasonably low damping. Since\nthe damping is strongly thickness dependent in this\nregime, the optimum thickness for devices may likely be\nfound in the 2{4 nm range.4\nFigure 3. (a) FMR linewidth \u0001 Hvs.ffor four Gd 12:5Fe76:1Co11:4\flms with di\u000berent thicknesses, together with linear \fts to\nequation (3). The dotted lines show \fts for the 20 nm \flm in its low and high frequency regions, respectively. (b) E\u000bective\nmagnetization and e\u000bective Gilbert damping constant vs.thickness; lines are guides to the eye.\nC. Composition dependence on 2nm thick \flms\nTo \fnally investigate whether we can achieve a com-\npensated ferrimagnetic behavior also in ultra-thin \flms,\nwe grew 2 nm Gd x(FeCo) 1\u0000x\flms in the composition\nrange 12{27 at.% Gd. The \flms were characterized using\nFMR spectrometry as described above and the extracted\nresults are shown in \fgure 4.\nThe extracted Me\u000band\u000bfollow a similar trend as re-\nported earlier for one order of magnitude thicker GdFeCo\n\flms characterized using an all-optical pump-probe tech-\nnique.17We \frst note that we can indeed reach an es-\nsentially fully compensated antiferromagnetic behavior in\ntwo \flms around a composition of 25 at.% Gd. We have\nmarked this compensation point with xmand a dashed\nline in \fgure 4 (c). Both \flms show very low damping of\n0.0078 and 0.009 respectively. However, just below this\ncomposition, the damping shows a peak, which is con-\nsistent with an angular compensation point, which we\ndenote byxa. It is noteworthy that the extracted damp-\ning value of \u000b= 0.0142 is still more than an order of\nmagnitude lower than \u000b= 0.45 of 30 nm \flms measured\nusing FMR spectrometry19and\u000b= 0.20 of 20 nm \flms\nmeasured using an optical pump-probe technique.17\nIII. CONCLUSION\nIn view of the potential application of compensated\nferrimagnets to spintronic devices, we prepared ferri-\nmagnetic thin \flms of Gd x(FeCo) 1\u0000xon high resistance\nSi(100) substrates and studied them using the FMR mea-\nsurements. Their growth conditions were optimized us-\ning 10 nm thick Gd 12:5Fe76:1Co11:4\flms, after which\nthickness dependent studies were done on the same com-\nposition in the thickness range of 2{20 nm. Composi-\ntion dependence studies were \fnally done on 2 nm thick\nGdx(FeCo) 1\u0000x\flms and an essentially compensated fer-rimagnetic behavior was observed for the \frst time in\nultrathin 2 nm \flms. The angular momentum compensa-\ntion and magnetic compensation points observed in this\nwork are very close to those reported earlier on much\nthicker \flms in the literature. A record low \u000bvalue of\nabout 0.0078 is obtained near the magnetic compensa-\ntion point, which is an order of magnitude lower than\nthe values reported in the literature using similar analysis\nmethods. The observation of compensated ferrimagnetic\nbehavior in ultrathin \flms together with very low value\nof\u000bare promising results for the future development of\nultrafast and energy e\u000ecient ferrimagnetic spintronic de-\nvices.\nEXPERIMENTAL SECTION\nA. Thin \flms growth and composition analysis\nAll the samples were prepared on high resistivity\nSi(100) substrates using a magnetron sputtering sys-\ntem with a base pressure of less than 2 \u000210\u00008torr.\nThin \flms of Gd x(FeCo) 1\u0000xwere deposited using the\nco-sputtering of high purity (more than 99.95%) Gd\nand Fe 87:5Co12:5targets, and composition analysis\nwas done using the inductively coupled plasma mass\nspectroscopy (ICP-MS). Thin \flms stacking structure\nof Si(100)/MgO(t)/Gd 12:5Fe76:1Co11:4(10)/SiO 2(4)\nwere used for seed layer dependence studies, here,\nthe number in the bracket is the thickness of the\nlayer in nm, where t=0, 6 and 10 nm. Four sam-\nples, namely S1 to S4 were prepared to obtain the\nbest conditions to grow Gd 12:5Fe76:1Co11:4(10) \flms.\nFor S1, Gd 12:5Fe76:1Co11:4(10) was grown directly\nover HR-Si (100) substrates, while in both S2 and\nS3 Gd 12:5Fe76:1Co11:4were grown with MgO seed\nlayer of 6 and 10 nm, respectively. All the lay-\ners in S1-S3 were grown at room temperature and5\nFigure 4. (a) Frequency vs.resonance \feld and (b) resonance linewidth vs.frequency, of 2 nm thick Gd x(FeCo) 1\u0000x\flms as\na function of Gd content in atomic %. (c) E\u000bective magnetization and e\u000bective Gilbert damping constant vs.Gd content.\nSolid symbols represent the values obtained by \ftting the experimental FMR data in (a) and (b) using the equation (2) and\n(3), respectively; solid lines in (c) are guides to the eye. xaand xmshow the angular and magnetic compensation points,\nrespectively, obtained from the literature17,19.\nno further heat treatment was given to them. In\nS4, 10 nm MgO seed layer were grown over HR\nSi(100) substrates at RT and followed by a in-situ\npost-annealing at 600C for 1 hour, and after that\nGd12:5Fe76:1Co11:4were deposited. The stacking struc-\nture of Si(100)/MgO(6)/Gd 12:5Fe76:1Co11:4(m)/SiO 2(4)\nwere used for thickness dependence studies, where\nm is the thickness of Gd 12:5Fe76:1Co11:4layer,\nand varied from 2 to 20 nm. For composi-\ntion dependence studies, stacking structure of\nSi(100)/MgO(6)/Gd x(FeCo) 1\u0000x(2)/SiO 2(4) were used,\nwhere xvaried from 12.5 to 26.7. The composition of\nGdx(FeCo) 1\u0000x\flms was varied by changing the sput-\ntering rate of Fe 87:5Co12:5target, while keeping the Gd\nsputtering rate \fxed for most \flms. All the samples for\nthickness dependence and composition dependence were\ngrown at room temperature and no post-annealing was\nused. Layer thicknesses were determined by estimating\nthe growth rate using the Dektak pro\fler on more than\n100 nm thick \flms.B. Inductively coupled plasma mass spectroscopy\n(ICP-MS) measurements\nThe elemental composition (Co, Fe, and Gd) of the\nthin \flm samples was determined by inductively coupled\nplasma optical emission spectroscopy (ICP-OES) using a\nThermo Fisher Scienti\fc iCAP 6000 Series spectrometer.\nEach thin \flm sample was exhaustively extracted in 5 mL\nHNO3 (65%, Supelco, Merck KgaA, Sigma-Aldrich) for\na duration of 30 min. 5 mL ultrapure MilliQ-water (18\nM\ncm) was added to the solution and the extract was al-\nlowed to rest for 30 minutes. The extract was transferred\nto a 100 mL volumetric \rask. The extracted sample was\nthen rinsed for several cycles in ultrapure water. The\nwater used for rinsing was transferred to the same volu-\nmetric \rask. The extract was diluted to 100 mL for ICP\nanalysis. ICP check standards were prepared from stan-\ndard solutions (Co and Fe: Merck, Germany; Ga: Accu-\nstandard, USA). The relative standard deviation (from\nthree individual injections) were within 1%.6\nTable I. The obtained values of e\u000bective Gilbert damping constant \u000bat room temperature (RT) in this work and comparison\nwith the lowest values reported so far in the literature at RT and also at their respective angular momentum compensation\n(Ta) and magnetic compensation (T m) points.\nFilm composition Film thickness \u000b Measurement technique Analysis method Reference\nGd23:5Fe68:9Co7:6 30 \u00180.45 (at RT) FMR Kittel's FMR19\n\u00180.35 (at RT) Pump-probe\nGd22Fe74:6Co3:4 20 \u00180.21 (at T a) Pump-probe -do-17\n\u00180.13 (at T m)\nGd25Fe65:6Co9:4 10 \u00180.07 (at RT) Spin torque FMR -do-36\n\u00190.01 (at RT) Spin torque FMR Ferrimagnetc resonance\nGd23:5Fe66:9Co9:6 30 0.0072 (at RT) Domain wall (DW) Field driven DW30\nmotion mobility\nGd12:5Fe76:1Co11:4 10 0.0055 Broadband FMR Kittel's FMR This work\n0.0076 (HF data) -do- -do- This work\nGd12:5Fe76:1Co11:4 4 0.0064 -do- -do- This work\nGd12:5Fe76:1Co11:4 2 0.0101 -do- -do- This work\nGd23:4Fe67:0Co9:6 2 0.0141 -do- -do- This work\nGd24:4Fe66:1Co9:5 2 0.0078 -do- -do- This work\nC. Ferromagnetic resonance (FMR) measurements\nRectangular pieces of about 6 \u00023 mm2were cut from\nthe blanket \flms and broadband FMR spectroscopy was\nperformed using a NanOsc Phase FMR (40 GHz) system\nwith a co-planar waveguide for microwave \feld excita-\ntion. Microwave excitation \felds hrfwith frequencies up\nto 30 GHz were applied in the \flm plane, and perpendic-\nular to the applied in-plane dc magnetic \feld H. All the\nFMR measurements were performed at the room tem-\nperature. The schematic of FMR measurement setup is\nshown in 1(a), and further details about the measure-\nments are given in Section 2 (results and discussions).\nSUPPORTING INFORMATION\nSupporting Information is available from the Wiley\nOnline Library or from the corresponding author.ACKNOWLEDGEMENTS\nLakhan Bainsla thanks MSCA - European Commission\nfor Marie Curie Individual Fellowship (MSCA-IF Grant\nNo. 896307). This work was also partially supported\nby the Swedish Research Council (VR Grant No. 2016-\n05980) and the Horizon 2020 research and innovation\nprogramme (ERC Advanced Grant No. 835068 \"TOP-\nSPIN\").\nCONFLICT OF INTEREST\nThe authors declare no con\rict of interest.\nAUTHOR CONTRIBUTIONS\nL.B. and J. \u0017A. planned the study. L.B. grew the \flms,\nperformed the FMR measurements and analysed the ob-\ntained FMR data. J.W. helped with ICP-MS measure-\nments and analysis. L.B. wrote the original draft of the\npaper. J. \u0017A. coordinated and supervised the work. All\nauthors contributed to the data analysis and co-wrote\nthe manuscript.7\nDATA AVAILABILITY STATEMENT\nThe data that support the \fndings of this study are\navailable from the corresponding author on reasonable\nrequest.\nREFERENCES\n1S. Wolf, D. Awschalom, R. Buhrman, J. Daughton, v. S. von\nMoln\u0013 ar, M. Roukes, A. Y. Chtchelkanova, and D. 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Shaw, Nature Physics\n12, 839 (2016)."    },    {        "title": "1604.07552v1.First_principles_studies_of_the_Gilbert_damping_and_exchange_interactions_for_half_metallic_Heuslers_alloys.pdf",        "content": "arXiv:1604.07552v1  [cond-mat.mtrl-sci]  26 Apr 2016First principles studies of the Gilbert damping and exchang e interactions for\nhalf-metallic Heuslers alloys\nJonathan Chico,1,∗Samara Keshavarz,1Yaroslav Kvashnin,1Manuel Pereiro,1Igor\nDi Marco,1Corina Etz,2Olle Eriksson,1Anders Bergman,1and Lars Bergqvist3,4\n1Department of Physics and Astronomy, Materials Theory Divi sion,\nUppsala University, Box 516, SE-75120 Uppsala, Sweden\n2Department of Engineering Sciences and Mathematics,\nMaterials Science Division, Lule˚ a University of Technolo gy, Lule˚ a, Sweden\n3Department of Materials and Nano Physics, School of Informa tion and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, SE-16440 K ista, Sweden\n4SeRC (Swedish e-Science Research Center), KTH Royal Instit ute of Technology, SE-10044 Stockholm, Sweden\n(Dated: September 28, 2018)\nHeusler alloys havebeen intensivelystudied dueto thewide varietyof properties thatthey exhibit.\nOne of these properties is of particular interest for techno logical applications, i.e. the fact that some\nHeusler alloys are half-metallic. In the following, a syste matic study of the magnetic properties\nof three different Heusler families Co 2MnZ, Co 2FeZ and Mn 2VZ with Z = (Al, Si, Ga, Ge) is per-\nformed. A key aspect is the determination of the Gilbert damp ing from first principles calculations,\nwith special focus on the role played by different approximat ions, the effect that substitutional\ndisorder and temperature effects. Heisenberg exchange inte ractions and critical temperature for\nthe alloys are also calculated as well as magnon dispersion r elations for representative systems,\nthe ferromagnetic Co 2FeSi and the ferrimagnetic Mn 2VAl. Correlations effects beyond standard\ndensity-functional theory are treated using both the local spin density approximation including the\nHubbard Uand the local spin density approximation plus dynamical mea n field theory approx-\nimation, which allows to determine if dynamical self-energ y corrections can remedy some of the\ninconsistencies which were previously reported for these a lloys.\nI. INTRODUCTION\nThe limitations presented by traditional electronic de-\nvices, such as Joule heating, which leads to higher en-\nergyconsumption, leakagecurrentsandpoorscalingwith\nsize amongothers1, havesparkedprofoundinterest in the\nfields of spintronics and magnonics. Spintronics applica-\ntions rely in the transmission of information in both spin\nand charge degrees of freedom of the electron, whilst in\nmagnonics information is transmitted via magnetic exci-\ntations, spin waves or magnons. Half-metallic materials\nwith a large Curie temperature are of great interest for\nthese applications. Due to the fact that they are con-\nductors in only one of the spin channels makes them\nideal candidates for possible devices2. Half-metals also\nhave certain advantages for magnonic applications, due\nto the fact that they are insulators in a spin channel and\nthus can have a smaller total density of states at the\nFermi energy than metals. This can result into a small\nGilbert damping, which is an instrumental prerequisite\nfor magnonic applications3.\nThe name “full Heusler alloys”refer to a set of com-\npounds with formula X 2YZ with X and Y typically being\ntransition metals4. The interest in them stems from the\nfactthattheirpropertiescanbecompletelydifferentfrom\nthose of their constituents. Heusler compounds can be\nsuperconducting5(Pd2YSn), semiconductors6(TiCoSb),\nhalf-metallic7(Co2MnSi), and can show a wide array of\nmagnetic configurations: ferromagnetic7(Co2FeSi), fer-\nrimagnetic8(Mn2VAl) or antiferromagnetic9(CrMnSb).\nDue to such a wide variety of behaviours, full Heusleralloys have been studied in great detail since their dis-\ncovery in 1903, leading to the discovery of new Heusler\nfamilies such as the half-Heuslers, with formula XYZ,\nand the inverse Heuslers, with formula X 2YZ. The lat-\nter tend to exhibit a different crystal structure and have\nbeen predicted to show quite remarkable properties10.\nMany Heusler alloys have also been predicted to be\nhalf-metallic, in particular Co 2MnSi has been the focus\nofmany theoreticaland experimental works7,11,12, due to\nits large Curie temperature of 985 K13, half-metallicity\nand low damping parameter, which makes it an ideal\ncandidate for possible spintronic applications. Despite\nthe large amount of research devoted to the half-metallic\nHeusleralloys,suchasCo 2MnSi, onlyrecentlytheoretical\npredictions of the Gilbert damping parameter have been\nmade for some Heusler alloys14,15.\nIn the present work first principle calculations of the\nfull Heusler families Co 2MnZ, Co 2FeZ and Mn 2VZ with\nZ = (Al, Si, Ga, Ge) are performed, with special empha-\nsis on the determination of the Gilbert damping and the\ninteratomic exchange interactions. A study treatment of\nthesystemswithdifferentexchangecorrelationpotentials\nis also performed.\nThe paper is organized as follows, in section II the\ncomputational methods used are presented. Then, in\nsection III, magnetic moments and spectral properties\nare discussed. In section IV the results for the exchange\nstiffness parameter, the critical temperature obtained via\nMonteCarlosimulationsandmagnondispersionrelations\nare presented. Finally in section V, the calculated damp-\ning parameter for the different Heusler is presented and2\ndiscussed.\nII. COMPUTATIONAL METHODS\nThe full Heusler alloys(X 2YZ) havea crystalstructure\ngiven by the space group Fm-3m with X occupying the\nWyckoffposition 8c (1\n4,1\n4,1\n4), while Ysits in the 4a(0,0,0)\nand Z in the 4b (1\n2,1\n2,1\n2).\nTo determine the properties of the systems first prin-\nciples electronic structure calculations were performed.\nThey were mainly done by means of the Korringa-Kohn-\nRostocker Green’s function formalism as implemented in\nthe SPR-KKRpackage16. The shape ofthe potential was\nconsidered by using both the Atomic Sphere Approxi-\nmation (ASA) and a full potential (FP) scheme. The\ncalculations of exchange interactions were performed in\nscalar relativistic approximation while the full relativis-\ntic Dirac equation was used in the damping calculations.\nThe exchange correlation functional was treated using\nboth the Local Spin Density Approximation (LSDA), as\nconsidered by Vosko, Wilk, and Nusair (VWN)17, and\nthe Generalized Gradient Approximation (GGA), as de-\nvised by Perdew, Burke and Ernzerhof (PBE)18. For\ncases in which substitutional disorder is considered, the\nCoherent Potential Approximation (CPA) is used19,20.\nStatic correlation effects beyond LSDA or GGA are\ntaken into account by using the LSDA+ Uapproach,\nwherethe Kohn-ShamHamiltonianissupplemented with\nan additional term describing local Hubbard interac-\ntions21, for thed-states of Co, Mn and Fe. The U-matrix\ndescribing this on-site interactions was parametrized\nthrough the Hubbard parameter Uand the Hund ex-\nchangeJ, using values UCo=UMn=UFe= 3 eV and\nJCo=JMn=JFe= 0.8 eV, which are in the range of the\nvalues considered in previous theoretical studies13,22–24.\nThis approach is used for the Heusler alloys families\nCo2MnZ and Co 2FeZ, as previous studies have shown\nthat for systems such as Co 2FeSi it might be necessary to\nreproduce several experimental observations, although,\nthis topic is still up for debate23. Since part of correla-\ntioneffectsofthe3 dorbitalsisalreadyincludedinLSDA,\ntheir contribution has to be subtracted before adding the\n+Uself-energy. This contribution to be removed is usu-\nally called “double-counting”(DC) correction and there\nis no unique way of defining it (see e.g. Ref. 25). We\nhave used two of the most widely used schemes for the\nDC, namely the Atomic Limit (AL), also known as Fully\nLocalized Limit (FLL)26, and the Around Mean Field\n(AMF)27. The dependence of the results on this choice\nwill be extensively discussed in the following sections.\nIn order to shine some light on the importance of\nthe dynamical correlations for the magnetic properties\nof the selected Heusler alloys, a series of calculations\nwere performed in the framework of DFT plus Dynami-\ncal Mean Field Theory (DMFT)28,29, as implemented in\nthe full-potential linear muffin-tin orbital (FP-LMTO)\ncode RSPt30. As for LSDA+ U, the DMFT calculationsare performed for a selected set of metal 3 dorbitals on\ntop of the LSDA solution in a fully charge self-consistent\nmanner.31,32Theeffectiveimpurityproblem, whichisthe\ncore of the DMFT, is solved through the spin-polarized\nT-matrix fluctuation-exchange (SPTF) solver33. This\ntype of solver is perturbative and is appropriate for the\nsystems with moderate correlationeffects, where U/W <\n1 (Wdenotes the bandwidth).34Contrary to the prior\nDMFT studies35,36, we have performed the perturba-\ntion expansion of the Hartree-Fock-renormalizedGreen’s\nfunction ( GHF) and not of the bare one. Concerning the\nDC correction, we here use both the FLL approach, de-\nscribed above, as well as the so-called “Σ(0)”correction.\nIn the latter case, the orbitally-averaged static part of\nthe DMFT self-energy is removed, which is often a good\nchoice for metals29,37. Finally, in order to extract infor-\nmationaboutthemagneticexcitationsin thesystems, we\nhave performed a mapping onto an effective Heisenberg\nHamiltonian\nˆH=−/summationdisplay\ni/negationslash=jJij/vector ei/vector ej, (1)\nwhereJijis anexchangeinteractionbetweenthe spinslo-\ncated at site iandj, while the /vector ei(/vector ej) representsthe unity\nvectoralongthe magnetizationdirectionatsite i (j). The\nexchange parameters then are computed by making use\nof the well established LKAG (Liechtenstein, Katsnel-\nson, Antropov, and Gubanov) formalism, which is based\non the magnetic force theorem38–40. More specific de-\ntails about the implementation of the LKAG formalism\nin RSPt can be found in Ref. 41. We also note that the\nperformance of the RSPt method was recently published\nin Ref.42and it was found that the accuracy was similar\nto that of augmented plane wave methods.\nFrom the exchange interactions between magnetic\natoms, it is possible to obtain the spin wave stiffness,\nD, which, for cubic systems is written as43\nD=2\n3/summationdisplay\ni,jJij√mimj|rij|2exp/parenleftbigg\n−ηrij\nalat/parenrightbigg\n,(2)\nwhere the mi’s are the magnetic moments of a given\natom,rijisthedistancebetweenthetwoconsideredmag-\nneticmoments, alatisthelatticeparameter, ηisaconver-\ngence parameter used to ensure the convergence of Eq. 2,\nthe value of Dis taken under the limit η→0. To ensure\nthe convergence of the summation, it is also important\nto take into consideration long range interactions. Hence\nthe exchange interactions are considered up to 6 lattice\nconstants from the central atom.\nThe obtained exchange interactions were then used to\ncalculate the critical temperature by making use of the\nBindercumulant, obtainedfromMonteCarlosimulations\nas implemented in the UppASD package44. This was\ncalculated for three different number of cell repetitions\n(10x10x10, 15x15x15 and 20x20x20), with the intersec-\ntion point determining the critical temperature of the\nsystem45.3\nThe Gilbert damping, α, is calculated via linear re-\nsponse theory46. Temperature effects in the scattering\nprocess of electrons are taken into account by consider-\ning an alloy analogy model within CPA with respect to\nthe atomic displacements and thermal fluctuations of the\nspin moments47. Vertex corrections are also considered\nhere, because they provide the “scattering in”term of the\nBoltzmann equation and it corrects significant error in\nthe damping, whenever there is an appreciable s-p or s-d\nscattering in the system16,48.\nFrom the calculated exchange interactions, the adia-\nbatic magnon spectra (AMS) can be determined by cal-\nculating the Fourier transform of the interatomic ex-\nchange interactions49. This is determined for selected\ncases and is compared with the magnon dispersion re-\nlation obtained from the dynamical structure factor,\nSk(q,ω), resulting fromspin dynamics calculations. The\nSk(q,ω) is obtained from the Fourier transform of the\ntime and spatially displaced spin-spin correlation func-\ntion,Ck(r−r′,t)50\nSk(q,ω) =1√\n2πN/summationdisplay\nr,r′eiq·(r−r′)/integraldisplay∞\n−∞eiωtCk(r−r′,t)dt.\n(3)\nThe advantage of using the dynamical structure factor\nover the adiabatic magnon spectra is the capability of\nstudying temperature effects as well as the influence of\nthe damping parameter determined from first principles\ncalculations or from experimental measurements.\nIII. ELECTRONIC STRUCTURE\nThe calculated spin magnetic moments for the selected\nsystems are reported in Table I. These values are ob-\ntained from SPR-KKR with various approximations of\nthe exchange correlation potential and for different geo-\nmetrical shapes of the potential itself. For the Co 2MnZ\nfamily, when Z = (Si ,Ge), the obtained spin mag-\nnetic moments do not seem to be heavily influenced by\nthe choice of exchange correlation potential or potential\nshape. However, for Z = (Al ,Ga) a large variation is\nobserved in the spin moment when one includes the Hub-\nbard parameter U.\nFor the Co 2FeZ systems, a pronounced difference can\nbe observed in the magnetic moments between the LSDA\nand the experimental values for Z = (Si ,Ge). Previ-\nous theoretical works13,22,24suggested that the inclusion\nof a +Uterm is necessary to obtain the expected spin\nmagnetic moments, but such a conclusion has been re-\ncently questioned23. To estimate which double counting\nschemewould be most suitableto treatcorrelationeffects\nin this class of systems, an interpolation scheme between\nthe FLL and AMF treatments was tested, as described\nin Ref. 59 and implemented in the FP-LAPW package\nElk60. It was found that both Co 2MnSi and Co 2FeSi\nare better described with the AMF scheme, as indicatedby their small αUparameter of ∼0.1 for both materials\n(αU= 0denotes completeAMF and αU= 1FLL), which\nis in agreement with the recent work by Tsirogiannis and\nGalanakis61.\nTo test whether a more sophisticated way to treat cor-\nrelation effects improves the description of these mate-\nrials, electronic structure calculations for Co 2MnSi and\nCo2FeSi using the DMFT scheme were performed. The\nLSDA+DMFT[Σ(0)] calculations yielded total spin mo-\nments of 5.00 µBand 5.34 µBfor respectively Co 2MnSi\nand Co 2FeSi. These values are almost equal to those ob-\ntained in LSDA, which is also the case in elemental tran-\nsition metals32. As mentioned above for LSDA+ U, the\nchoice of the DC is crucial for these systems. The main\nreason why no significant differences are found between\nDMFT and LSDA values is that the employed “Σ(0)”DC\nalmost entirely preserves the static part of the exchange\nsplitting obtained in LSDA62. For instance, by using\nFLL DC, we obtained a total magnetization of 5.00 µB\nand 5.61 µBin Co2MnSi and Co 2FeSi, respectively. We\nnote that the spin moment of Co 2FeSi still does not reach\nthe value expected from the Slater-Pauling rule, but the\nDMFT modifies it in a right direction, if albeit to a\nsmaller degree that the LSDA+ Uschemes.\nAnother important aspect of the presently studied sys-\ntems is the fact that they are predicted to be half-\nmetallic. In Fig. 1, the density of states (DOS) for\nboth Co 2MnSi and Co 2FeSi is presented using LSDA and\nLSDA+U. For Co 2MnSi, the DOS at the Fermi energy\nis observed to exhibit a very clear gap in one of the spin\nchannels, in agreement with previous theoretical works7.\nFor Co 2FeSi, instead a small pseudo-gap region is ob-\nserved in one of the spin channels, but the Fermi level\nis located just at the edge of the boundary as shown in\nprevious works24. Panels a) and b) of Fig. 1 also show\nthat some small differences arise depending on the ASA\nor FP treatment. In particular, the gap in the minority\nspin channel is slightly reduced in ASA.\nWhen correlation effects are considered within the\nLSDA+Umethod, the observed band gap for Co 2MnSi\nbecomes larger, while the Fermi level is shifted and still\nremainsin the gap. When applyingLSDA+ Uto Co2FeSi\nin the FLL scheme, EFis shifted farther away from the\nedgeofthe gap, whichexplainswhythemoment becomes\nalmostanintegerasexpected fromtheSlater-Paulingbe-\nhaviour7,24,63. Moreover,onecanseethatinASAthegap\nin the spin down channel is much smaller in comparison\nto the results obtained in FP.\nWhen the dynamical correlation effects are considered\nvia DMFT, the overall shape of DOS remains to be quite\nsimilartothatofbareLSDA,especiallyclosetotheFermi\nlevel, as seen in Fig. A.1 in the Appendix A. This is re-\nlated to the fact that we use a perturbative treatment\nof the many-body effects, which favours Fermi-liquid be-\nhaviour. Similarly to LSDA+ U, the LSDA+DMFT cal-\nculations result in the increased spin-down gaps, but the\nproducedshiftofthebandsisnotaslargeasinLSDA+ U.\nThis is quite natural, since the inclusion ofthe dynamical4\nTABLE I. Summary of the spin magnetic moments obtained using different approximations as obtained from SPR-KKR for the\nCo2MnZ and Co 2FeZ families with Z = (Al ,Si,Ga,Ge). Different exchange correlation potential approximati ons and shapes of\nthe potential have been used. The symbol†signifies that the Fermi energy is located at a gap in one of the spin channels.\nQuantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe\nalat[˚A] 5.75515.77515.65525.743535.730515.737515.640235.75054\nmASA\nLDA[µB] 4.04†4.09†4.99†4.94†4.86†4.93†5.09 5.29\nmASA\nGGA[µB] 4.09†4.15†4.99†4.96†4.93†5.00†5.37 5.53\nmASA\nLDA+UAMF [µB] 4.02†4.08 4.98†4.98†4.94†4.99†5.19 5.30\nmASA\nLDA+UFLL [µB] 4.77 4.90 5.02†5.11 5.22 5.36 5.86†5.94†\nmFP\nLDA[µB] 4.02†4.08†4.98†4.98†4.91†4.97†5.28 5.42\nmFP\nGGA[µB] 4.03†4.11 4.98†4.99†4.98†5.01†5.55 5.70\nmFP\nLDA+UAMF [µB] 4.59 4.99 4.98†5.13 5.12 5.40 5.98†5.98†\nmFP\nLDA+UFLL [µB] 4.03†4.17 4.99†4.99†4.99†5.09 5.86†5.98†\nmexp[µB] 4.04554.09564.96574.84574.96555.15576.00245.7458\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]ASA\nFPa)\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]ASA\nFPb )\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]FP [FLL]\nFP [AMF]c)\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EFASA [AMF]\nFP [FLL]\nFP [AMF]d )\n[eV]\nFIG. 1. (Color online) Total density of states for different e xchange correlation potentials with the dashed line indica ting the\nFermi energy, sub-figures a) and b) when LSDA is used for Co 2MnSi and Co 2FeSi respectively. Sub-figures c) and d) show the\nDOS when the systems (Co 2MnSi and Co 2FeSi respectively) are treated with LSDA+ U. It can be seen that the half metalicity\nof the materials can be affected by the shape of the potential a nd the choice of exchange correlation potential chosen.\ncorrelations usually tends to screen the static contribu-\ntions coming from LSDA+ U.\nAccording to Ref. 35 taking into account dynami-\ncal correlations in Co 2MnSi results in the emergence of\nthe non-quasiparticle states (NQS’s) inside the minority-\nspin gap, which at finite temperature tend to decrease\nthe spin polarisation at the Fermi level. These NQS’s\nwere first predicted theoretically for model systems64and stem from the electron-magnon interactions, which\nare accounted in DMFT (for review, see Ref. 2). Our\nLSDA+DMFT results for Co 2MnSi indeed show the ap-\npearance of the NQS’s, as evident from the pronounced\nimaginary part of the self-energy at the bottom of the\nconduction minority-spin band (see Appendix B). An\nanalysis of the orbital decomposition of the self-energy\nreveals that the largest contribution to the NQS’s comes5\nfrom the Mn- TEgstates. However, in our calculations,\nwhere the temperature was set to 300K, the NQS’s ap-\npeared above Fermi level and did not contribute to the\nsystem’s depolarization, in agreementwith the recent ex-\nperimental study12.\nWe note that a half-metallic state with a magnetic\nmoment of around 6 µBfor Co 2FeSi was reported in a\nprevious LSDA+DMFT[FLL] study by Chadov et al.36.\nIn their calculations, both LSDA+ Uand LSDA+DMFT\ncalculations resulted in practically the same positions of\nthe unoccupied spin-down bands, shifted to the higher\nenergies as compared to LSDA. This is due to techni-\ncal differences in the treatment of the Hartree-Fock con-\ntributions to the SPTF self-energy, which in Ref. 36 is\ndone separately from the dynamical contributions, while\nin this study a unified approach is used. Overall, the\nimprovements in computational accuracy with respect to\npreviousimplementationscouldberesponsiblefortheob-\ntained qualitative disagreement with respect to Refs. 35\nand 36. Moreover, given that the results qualitatively\ndepend on the choice of the DC term, the description of\nthe electronic structure of Co 2FeSi is not conclusive.\nThe discrepancies in the magnetic moments presented\nin Table I with respect to the experimental values can in\npart be traced back to details of the density of the states\naround the Fermi energy. The studied Heusler alloys are\nthought to be half-metallic, which in turn lead to inte-\nger moments following the Slater-Pauling rule7. There-\nfore, any approximation that destroys half-metallicity\nwill have a profound effect on their magnetic properties7.\nFor example, for Co 2FeAl when the potential is treated\nin LSDA+ U[FLL] with ASA the Fermi energy is located\nat a sharp peak close to the edge of the band gap, de-\nstroyingthehalf-metallicstate(Seesupplementarymate-\nrial Fig.1). A similar situation occurs in LSDA+ U[AMF]\nwith a full potential scheme. It is also worth mention-\ning that despite the fact that the Fermi energy for many\nof these alloys is located inside the pseudo-gap in one of\nthe spin channels, this does not ensure a full spin po-\nlarization, which is instead observed in systems as e.g.\nCo2MnSi. Another important factor is the fact that EF\ncan be close to the edge of the gap as in Co 2MnGa when\nthe shape of the potential is considered to be given by\nASA and the exchange correlation potential is dictated\nby LSDA, hence the half-metallicity of these alloys could\nbe destroyed due to temperature effects.\nThe other Heusler family investigated here is the ferri-\nmagnetic Mn 2VZ with Z = (Al ,Si,Ga,Ge). The lattice\nconstants used in the simulations correspond to either\nexperimental or previous theoretical works. These data\nare reported in Table II together with appropriate ref-\nerences. Table II also illustrates the magnetic moments\ncalculated using different exchange correlation potentials\nand shapes of the potential. It can be seen that in gen-\neral there is a good agreement with previous works, re-\nsulting in spin moments which obey the Slater-Pauling\nbehaviour.\nFor these systems, the Mn atoms align themselves inTABLE II. Lattice constants used for the electronic struc-\nture calculations and summary of the magnetic properties fo r\nMn2VZ with Z = (Al ,Si,Ga,Ge). As for the ferromagnetic\nfamilies, different shapes of the potential and exchange cor -\nrelations potential functionals were used. The magnetic mo -\nments follow quite well the Slater-Pauling behavior with al l\nthe studied exchange correlation potentials. The symbol†\nsignifies that the Fermi energy is located at a gap in one of\nthe spin channels.\nQuantity Mn2VAl Mn 2VGa Mn 2VSi Mn 2VGe\nalat[˚A] 5.687655.905666.06656.09567\nmASA\nLDA[µB] 1.87 1.97†1.00†0.99†\nmASA\nGGA[µB] 1.99†2.04†1.01†1.00†\nmFP\nLDA[µB] 1.92 1.95†0.99†0.99\nmFP\nGGA[µB] 1.98†2.02†0.99†0.99†\nmexp[µB] — 1.8666— —\nan anti-parallel orientation with respect to the V mo-\nments, resulting in a ferrimagnetic ground state. As for\nthe ferromagnetic compounds, the DOS shows a pseu-\ndogap in one of the spin channels (see supplementary\nmaterial Fig.8-9) indicating that at T= 0 K these com-\npoundscouldbehalf-metallic. An importantfactoristhe\nfact that the spin polarization for these systems is usu-\nally considered to be in the opposite spin channel than\nfor the ferromagneticalloys presently studied, henceforth\nthe total magnetic moment is usually assigned to a neg-\native sign such that it complies with the Slater-Pauling\nrule7,65.\nIV. EXCHANGE INTERACTIONS AND\nMAGNONS\nIn this section, the effects that different exchange cor-\nrelation potentials and geometrical shapes of the poten-\ntial haveoverthe exchangeinteractionswill be discussed.\nA. Ferromagnetic Co 2MnZ and Co 2FeZ with\nZ= (Al,Si,Ga,Ge)\nIn Table III the calculated spin wave stiffness, D, is\nshown. In general there is a good agreement between\nthe calculated values for the Co 2MnZ family, with the\nobtained values using LSDA or GGA being somewhat\nlarger than the experimental measurements. This is in\nagreement with the observations in the previous section,\nin which the same exchange correlation potentials were\nfound to be able to reproducethe magnetic moments and\nhalf-metallicbehaviourfortheCo 2MnZfamily. Inpartic-\nular, for Co 2MnSi the ASA calculations are in agreement\nwith experiments68,69and previous theoretical calcula-\ntions70. It is important to notice that the experimen-\ntal measurements are performed at room temperature,\nwhich can lead to softening of the magnon spectra, lead-\ning to a reduced spin wave stiffness.6\nHowever, for the Co 2FeZ family neither LSDA or GGA\ncan consistently predict the spin wave stiffness, with\nZ=(Al, Ga) resulting in an overestimated value of D,\nwhile for Co 2FeSi the obtained value is severely underes-\ntimated. However, for some materials in this family, e.g.\nCo2FeGathespinwavestiffnessagreeswith previousthe-\noretical results70. These data reflect the influence that\ncertain approximations have on the location of the Fermi\nlevel, which previously has been shown to have profound\neffects on the magnitude of the exchange interactions71.\nThis can be observed in the half-metallic Co 2MnSi; when\nit is treated with LSDA+ U[FLL] in ASA the Fermi level\nis located at the edge of the gap (see Fig. 1c). Result-\ning in a severely underestimated spin wave stiffness with\nrespect to both the LSDA value and the experimental\nmeasurements (see Table III). The great importance of\nthe location of the Fermi energy on the magnetic proper-\nties can be seen in the cases of Co 2MnAl and Co 2MnGa.\nIn LSDA+ U[FLL], these systems show non integer mo-\nments which are overestimated with respect to the ex-\nperimental measurements (see Table I), but also results\nin the exchange interactions of the system preferring a\nferrimagnetic alignment. Even more the exchange inter-\nactions can be severely suppressed when the Hubbard U\nisused. Forexample, forCo 2MnGe inASAthe dominant\ninteraction is between the Co-Mn moments, in LSDA the\nobtainedvalueis0.79mRy, while inLSDA+ U[FLL]isre-\nduced to 0.34 mRy, also, the nearest neighbour Co 1-Co2\nexchange interaction changes from ferromagnetic to anti-\nferromagnetic when going from LSDA to LSDA+ U[FLL]\nwhich lead the low values obtainedfor the spin wavestiff-\nness. As will be discussed below also for the low Tcfor\nsome of these systems.\nIt is important to notice, that the systems that exhibit\nthe largest deviation from the experimental values, are\nusually those that under a certain exchange correlation\npotential and potential geometry loosetheir half-metallic\ncharacter. Such effect are specially noticeable when one\ncompares LSDA+ U[FLL] results in ASA and FP, where\nhalf-metallicity is more easily lost in ASA due to the\nfact that the pseudogap is much smaller under this ap-\nproximation than under FP (see Fig. 1). In general, it\nis important to notice that under ASA the geometry of\nthe potential is imposed, that is non-spherical contribu-\ntions to the potential are neglected. While this has been\nshown to be very successful to describe many properties,\nit does introduce an additional approximation which can\nlead to anill treatment ofthe properties ofsome systems.\nHence, care must be placed when one is considering an\nASA treatment for the potential geometry, since it can\nlead to large variations of the exchange interactions and\nthus is one of the causes of the large spread on the values\nobserved in Table III for the exchange stiffness and in\nTable IV for the Curie temperature.\nOne of the key factors behind the small values of the\nspin stiffness for Co 2FeSi and Co 2FeGe, in comparison\nwith the rest of the Co 2FeZ family, lies in the fact that\nin LSDA and GGA an antiferromagnetic long-range Fe-Fe interaction is present (see Fig. C.2 in Appendix C).\nAs the magnitude of the Fe-Fe interaction decreases the\nexchange stiffness increases, e.g. as in LSDA+ U[AMF]\nwith afull potential scheme. Theseexchangeinteractions\nare one of the factors behind the reduced value of the\nstiffness, this is evident when comparing with Co 2FeAl,\nwhich while having similar nearest neighbour Co-Fe ex-\nchange interactions, overall displays a much larger spin\nwave stiffness for most of the studied exchange correla-\ntion potentials.\nUsing LSDA+DMFT[Σ(0)] for Co 2MnSi and Co 2FeSi,\nthe obtained stiffness is 580 meV ˚A2and 280 meV ˚A2re-\nspectively, whilst in LSDA+DMFT[FLL] for Co 2MnSi\nthestiffnessis630meV ˚A2andforCo 2FeSiis282meV ˚A2.\nAs can be seen for Co 2MnSi there is a good agree-\nment between the KKR LSDA+ U[FLL], the FP-LMTO\nLSDA+DMFT[FLL] and the experimental values.\nThe agreement with experiments is particularly good\nwhen correlation effects are considered as in the\nLSDA+DMFT[Σ(0)] approach. On the other hand, for\nCo2FeSi the spin wave stiffness is severely underesti-\nmated which is once again consistent with what is shown\nin Table III.\nUsing the calculated exchange interactions, the criti-\ncal temperature, Tc, for each system can be calculated.\nUsing the ASA, the Tcof both the Co 2MnZ and Co 2FeZ\nsystems is consistently underestimated with respect to\nexperimental results, as shown in Table IV. The same\nunderestimation has been observed in previous theo-\nretical studies78, for systems such as Co 2Fe(Al,Si) and\nCo2Mn(Al,Si). However, using a full potential scheme\ninstead leads to Curie temperatures in better agreement\nwith the experimental values, specially when the ex-\nchange correlation potential is considered to be given by\nthe GGA (see Table IV). Such observation is consistent\nwith what was previouslymentioned, regardingthe effect\nofthe ASA treatmentonthe spin wavestiffness andmag-\nnetic moments, where in certain cases, ASA was found to\nnot be the best treatment to reproduce the experimen-\ntal measurements. As mentioned above, this is strongly\nrelated to the fact that in general ASA yields a smaller\npseudogapin the half-metallic materials, leading to mod-\nification of the exchange interactions. Thus, in general, a\nfullpotentialapproachseemstobeabletobetterdescribe\nthe magnetic properties in the present systems, since the\npseudogaparoundthe Fermienergyisbetter describedin\na FP approach for a given choice of exchange correlation\npotential.\nThe inclusionofcorrelationeffects forthe Co 2FeZfam-\nily, lead to an increase of the Curie temperature, as for\nthe spin stiffness. This is related to the enhancement of\nthe interatomic exchange interactions as exemplified in\nthe case of Co 2FeSi. However, the choice of DC once\nmore is shown to greatly influence the magnetic proper-\nties. For the Co 2FeZ family, AMF results in much larger\nTcthan the FLL scheme, whilst for Co 2MnZ the dif-\nferences are smaller, with the exception of Z=Al. All\nthese results showcase how important a proper descrip-7\nTABLE III. Summary of the spin wave stiffness, Dfor Co 2MnZ and Co 2FeZ with Z = (Al ,Si,Ga,Ge). For the Co 2MnZ family\nboth LSDA and GGA exchange correlation potentials yield val ues close to the experimental measurements. However, for th e\nCo2FeZ family a larger data spread is observed. The symbol∗implies that the ground state for these systems was found to b e\nFerri-magnetic from Monte-Carlo techniques and the critic al temperature presented here is calculated from the ferri- magnetic\nground state.\nQuantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe\nDASA\nLDA[meV˚A2] 282 291 516 500 644 616 251 206\nDASA\nGGA[meV˚A2] 269 268 538 515 675 415 267 257\nDASA\nLDA+UFLL [meV ˚A2] 29∗487∗205 94 289 289 314 173\nDASA\nLDA+UAMF [meV ˚A2] 259 318 443 417 553 588 235 214\nDFP\nLDA[meV˚A2] 433 405 613 624 692 623 223 275\nDFP\nGGA[meV˚A2] 483 452 691 694 740 730 323 344\nDFP\nLDA+UFLL [meV ˚A2] 447 400 632 577 652 611 461 436\nDFP\nLDA+UAMF [meV ˚A2] 216 348 583 579 771 690 557 563\nDexp[meV˚A2] 190722647357568-5346941374370754967671577—\ntion of the pseudogap region is in determining the mag-\nnetic properties of the system.\nAnother observation, is the fact that even if a given\ncombination of exchange correlation potential and geo-\nmetrical treatment of the potential can yield a value of\nTcin agreementwith experiments, it does not necessarily\nmeans that the spin wave stiffness is correctly predicted\n(see Table III and Table IV).\nWhen considering the LSDA+DMFT[Σ(0)] scheme,\ncritical temperatures of 688 K and 663 K are ob-\ntained for Co 2MnSi and Co 2FeSi, respectively. Thus,\nthe values of the Tcare underestimated in compari-\nson with the LSDA+ Uor LSDA results. The reason\nfor such behaviour becomes clear when one looks di-\nrectly on the Jij’s, computed with the different schemes,\nwhich are shown in Appendix C. These results sug-\ngest that taking into account the dynamical correlations\n(LSDA+DMFT[Σ(0)]) slightly suppresses most of the\nJij’s as compared to the LSDA outcome. This is an\nexpected result, since the employed choice of DC correc-\ntion preserves the exchange splitting obtained in LSDA,\nwhile the dynamical self-energy, entering the Green’s\nfunction, tends to lower its magnitude. Since these two\nquantities are the key ingredients defining the strength\nof the exchange couplings, the Jij’s obtained in DMFT\nare very similar to those of LSDA (see e.g. Refs. 41\nand 81). The situation is a bit different if one employs\nFLL DC, since an additional static correction enhances\nthe local exchange splitting.82For instance, in case of\nCo2MnSi the LSDA+DMFT[FLL] scheme provided a Tc\nof 764 K, which is closer to the experiment. The con-\nsistently better agreement of the LSDA+ U[FLL] and\nLSDA+DMFT[FLL] estimates of the Tcwith experimen-\ntal values might indicate that explicit account for static\nlocal correlations is important for the all considered sys-\ntems.\nUsing the calculated exchange interactions, it is also\npossible to determine the adiabatic magnon spectra\n(AMS). In Fig. 2 is shown the effect that different ex-\nchange correlation potentials have overthe description ofthe magnon dispersion relation of Co 2FeSi is shown. The\nmost noticeable effect between different treatments of\nthe exchange correlation potential is shifting the magnon\nspectra, while its overall shape seems to be conserved.\nThis is a direct result from the enhancement of nearest\nneighbour interactions (see Fig. C.2).\nWhen comparing the AMS treatment with the dy-\nnamical structure factor, S(q,ω), atT= 300 K and\ndamping parameter αLSDA= 0.004, obtained from first\nprinciples calculations (details explained in section V),\na good agreement at the long wavelength limit is found.\nHowever, a slight softening can be observed compared\nto the AMS. Such differences can be explained due to\ntemperature effects included in the spin dynamics sim-\nulations. Due to the fact that the critical temperature\nof the system is much larger than T= 300 K (see Ta-\nble IV), temperature effects are quite small. The high\nenergy optical branches are also softened and in general\nare much less visible. This is expected since the correla-\ntion was studied using only vectors in the first Brillouin\nzone and as has been shown in previous works50, a phase\nshift is sometimes necessary to properly reproduce the\noptical branches, implying the need of vectors outside\nthe first Brillouin zone. Also, Stoner excitations dealing\nwith electron-holeexcitations arenot included in this ap-\nproach,whichresultintheLandaudampingwhichaffects\nthe intensity of the optical branches. Such effects are not\ncaptured by the present approach, but can be studied\nby other methods such as time dependent DFT83. The\nshape of the dispersion relationalong the path Γ −Xalso\ncorresponds quite well with previous theoretical calcula-\ntions performed by K¨ ubler84.\nB. Ferrimagnetic Mn 2VZ with Z = (Al,Si,Ga,Ge)\nAsmentionedabove,theMnbasedMn 2VZfullHeusler\nfamily has a ferrimagnetic ground state, with the Mn\natoms orienting parallel to each other and anti-parallel\nwith respect to the V moments. For all the studied sys-8\nTABLE IV. Summary of the critical temperature for Co 2MnZ and Co 2FeZ with Z = (Al ,Si,Ga,Ge), with different exchange\ncorrelation potentials and shape of the potentials. The sym bol∗implies that the ground state for these systems was found to b e\nFerri-magnetic from Monte-Carlo techniques and the critic al temperature presented here is calculated from the ferri- magnetic\nground state.\nQuantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe\nTLDA\ncASA [K] 360 350 750 700 913 917 655 650\nTGGA\ncASA [K] 350 300 763 700 975 973 800 750\nTLDA+U\ncASAFLL[K] 50∗625∗125 225 575 550 994 475\nTLDA+U\ncASAAMF[K] 325 425 650 600 950 950 650 625\nTLDA\ncFP [K] 525 475 875 825 1050 975 750 750\nTGGA\ncFP [K] 600 525 1000 925 1150 1100 900 875\nTLDA+U\ncFPFLL[K] 525 475 950 875 1050 975 1050 1075\nTLDA+U\ncFPAMF[K] 450 450 1000 875 1275 1225 1450 1350\nTexp\nc[K] 69778694 98513905 10007910938011002498158\nTABLE V. Summary of the spin wave stiffness, D, and the\ncritical temperature for Mn 2VZ with Z = (Al ,Si,Ga,Ge) for\ndifferent shapes of the potential and exchange correlation p o-\ntentials.\nQuantity Mn2VAl Mn 2VGa Mn 2VSi Mn 2VGe\nDASA\nLDA[meV˚A2] 314 114 147\nDASA\nGGA[meV˚A2] 324 73 149\nDFP\nLDA[meV˚A2] 421 206 191\nDFP\nGGA[meV˚A2] 415 91 162\nDexp[meV˚A2] 53485— — —\nTLDA\ncASA [K] 275 350 150 147\nTGGA\ncASA [K] 425 425 250 250\nTLDA\ncFP [K] 425 450 200 200\nTGGA\ncFP [K] 600 500 350 350\nTexp\nc[K] 7688578366— —\ntemstheMn-Mnnearestneighbourexchangeinteractions\ndominates. In Table V the obtained spin wave stiffness,\nD, and critical temperature Tcare shown. For Mn 2VAl,\nit can be seen that the spin wave stiffness is trend when\ncompared to the experimental value. The same under-\nestimation can be observed in the critical temperature.\nFor Mn 2VAl, one may notice that the best agreement\nwith experiments is obtained for GGA in FP. An inter-\nesting aspect of the high Tcobserved in these materials\nis the fact that the magnetic order is stabilized due to\nthe anti-ferromagnetic interaction between the Mn and\nV sublattices, since the Mn-Mn interaction is in general\nmuch smaller than the Co-Co, Co-Mn and Co-Fe inter-\nactions present in the previously studied ferromagnetic\nmaterials.\nFor these systems it can be seen that in general the FP\ndescriptionyields Tc’swhichareinbetter agreementwith\nexperiment, albeit if the values are still underestimated.\nAs for the Co based systems the full potential technique\nimproves the description of the pseudogap, it is impor-\ntant to notice that for most systems both in ASA and\nFP the half-metallic characteris preserved. However, the\ndensity of states at the Fermi level changes which could\nlead to changes in the exchange interactions.As for the ferromagnetic systems one can calculate the\nmagnon dispersion relation and it is reported in Fig. 3\nfor Mn 2VAl. A comparison with Fig. 2 illustrates some\nof the differences between the dispersion relation of a fer-\nromagnet and of a ferrimagnetic material. In Fig. 3 some\noverlap between the acoustic and optical branches is ob-\nserved, as well as a quite flat dispersion relation for one\nof the optical branches. Such an effect is not observed in\nthe studied ferromagnetic cases. In general the different\nexchange correlation potentials only tend to shift the en-\nergy of the magnetic excitations, while the overall shape\nof the dispersion does not change noticeably, which is\nconsistent with what was seen in the ferromagnetic case.\nThe observed differences between the LSDA and GGA\nresults in the small qlimit, corresponds quite well with\nwhatisobservedinTableV, wherethe spinwavestiffness\nfor GGA with the potential given by ASA is somewhat\nlargerthan the LSDA case. This is directly related to the\nobservation that the nearest neighbour Mn-Mn and Mn-\nV interactions are large in GGA than in LSDA. Again,\nsuch observation is tied to the DOS at the Fermi level,\nsince Mn 2VAl is not half-metallic in LSDA, on the other\nhand in GGA the half-metallic state is obtained (see Ta-\nble. II.\nV. GILBERT DAMPING\nThe Gilbert damping is calculated for all the previ-\nously studied systems using ASA and a fully relativistic\ntreatment. In Fig. 4, the temperature dependence of the\nGilbert damping for Co 2MnSi is reported for different\nexchange-correlationpotentials. Whencorrelationeffects\nare neglected or included via the LSDA+ U[AMF], the\ndampingincreaseswith temperature. Onthe otherhand,\nin the LSDA+ U[FLL] scheme, the damping decreases as\na function of temperature, and its overall magnitude is\nmuch larger. Such observation can be explained from the\nfact that in this approximation a small amount of states\nexists at the Fermi energyin the pseudogapregion, hence\nresulting in a larger damping than in the half-metallic9\n0100200300400500\nΓ X W L ΓEnergy [meV]FP-LSDA\nDMFT[Σ(0)]a)\nFIG. 2. (Color online) a) Adiabatic magnon spectra for\nCo2FeSi for different exchange correlation potentials. In the\ncase of FP-LSDA and LSDA+DMFT[Σ(0)] the larger devia-\ntionsareobservedinthecase ofhighenergies, withtheDMFT\ncurve having a lower maximum than the LSDA results. In b)\na comparison of the adiabatic magnon spectra (solid lines)\nwith the dynamical structure factor S(q,ω) atT= 300 K,\nwhen the shape of the potential is considered to be given\nby the atomic sphere approximation and the exchange cor-\nrelation potential to be given by LSDA, some softening can\nbe observed due to temperature effects specially observed at\nhigher q-points.\ncases(see Fig. 1c).\nIn general the magnitude of the damping, αLSDA=\n7.4×10−4, is underestimated with respect to older ex-\nperimental measurements at room temperature, which\nyielded values of α= [0.003−0.006]86andα∼0.025\nfor polycrystalline samples87, whilst it agrees with previ-\nously performed theoretical calculations14. Such discrep-\nancy between the experimental and theoretical results\ncould stem from the fact that in the theoretical calcula-\ntions only the intrinsic damping is calculated, while in\nexperimental measurements in addition extrinsic effects\nsuch as eddy currents and magnon-magnon scattering\ncan affect the obtained values. It is also known that sam-FIG. 3. (Color online) Adiabatic magnon dispersion relatio n\nfor Mn 2VAl when different exchange correlation potentials\nare considered. In general only a shift in energy is observed\nwhen considering LSDA or GGA with the overall shape being\nconserved.\n00.511.522.533.54\n50 100 150 200 250 300 350 400 450 500Gilbert damping (10-3)\nTemperature [K]LSDA\nGGA\nLSDA+U [FLL]\nLSDA+U [AMF]\nFIG.4. (Color online)TemperaturedependenceoftheGilber t\ndamping for Co 2MnSi for different exchange correlation po-\ntentials. For LSDA, GGA and LSDA+ U[AMF] exchange cor-\nrelation potentials the damping increases with temperatur e,\nwhilst for LSDA+ U[FLL]thedampingdecreases as afunction\nof temperature.\nple capping or sample termination, can have profound ef-\nfects over the half-metallicity of Co 2MnSi88. Recent ex-\nperiments showed that ultra-low damping, α= 7×10−4,\nfor Co 1.9Mn1.1Si can be measured when the capping\nis chosen such that the half-metallicity is preserved89,\nwhich is in very good agreement with the present theo-\nretical calculations.\nIn Fig. 5, the Gilbert damping at T= 300 K for the\ndifferent Heusler alloys as a function of the density of\nstates at the Fermi level is presented. As expected, the\nincreased density of states at the Fermi energy results in10\nFIG. 5. (Color online) Gilbert damping for different Heusler\nalloys at T= 300 K as a function of density of states at the\nFermi energy for LSDA exchange correlation potential. In\ngeneral the damping increases as the density of states at the\nFermi Energy increases (the dotted line is to guide the eyes) .\nan increased damping. Also it can be seen that in gen-\neral, alloys belonging to a given family have quite similar\ndamping parameter, except for Co 2FeSi and Co 2FeGe.\nTheir anomalous behaviour, stems from the fact that\nin the LSDA approach both Co 2FeSi and Co 2FeGe are\nnot half-metals. Such clear dependence on the density of\nstates is expected, since the spin orbit coupling is small\nfor these materials, meaning that the dominating con-\ntribution to the damping comes from the details of the\ndensity of states around the Fermi energy90,91.\n1. Effects of substitutional disorder\nIn order to investigate the possibility to influ-\nence the damping, we performed calculations for the\nchemically disordered Heusler alloys Co 2Mn1−xFexSi,\nCo2MeAl1−xSixand Co 2MeGa 1−xGexwhere Me =\n(Mn,Fe).\nDue to the small difference between the lattice param-\neters of Co 2MnSi and Co 2FeSi, the lattice constant is\nunchanged when varying the concentration of Fe. This\nis expected to play a minor role on the following results.\nWhen one considers only atomic displacement contribu-\ntions to the damping (see Fig. 6a), the obtained values\nare clearlyunderestimated in comparisonwith the exper-\nimental measurements at room temperature92. Under\nthe LSDA, GGA and LSDA+ U[AMF] treatments, the\ndamping is shown to increase with increasing concentra-\ntion of Fe. On the other hand, in LSDA+ U[FLL] the\ndamping at low concentrations of Fe is much larger than\ninthe othercases, andit decreaseswith Feconcentration,\nuntil a minima is found at Fe concentration of x∼0.8.\nThis increase can be related to the DOS at the Fermi\nenergy, which is reported in Fig. 1c for Co 2MnSi. One00.511.522.533.544.55\n0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3)\nFe concentrationLSDA\nGGA\nLSDA+ U[FLL]\nLSDA+ U[AMF]a)\n00.511.522.533.544.5\n0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3)\nFe concentrationLSDA\nGGA\nLSDA+ U[FLL]\nLSDA+ U[AMF]b)\nFIG. 6. (Color online) Gilbert damping for the random alloy\nCo2Mn1−xFexSi as a function of the Fe concentration at T=\n300 K when a) only atomic deisplacements are considered and\nb) when both atomic displacements and spin fluctuations are\nconsidered.\ncan observe a small amount of states at EF, which could\nlead to increased values of the damping in comparison\nwith the ones obtained in traditional LSDA. As for the\npure alloys, a general trend relating the variation of the\nDOS at the Fermi level and the damping with respect to\nthe variation of Fe concentration can be obtained, anal-\nogous to the results shown in Fig. 5.\nWhen spin fluctuations are considered in addition to\nthe atomic displacements contribution, the magnitude of\nthe damping increases considerably, as shown in Fig. 6b.\nThis is specially noticeable at low concentrations of Fe.\nMn rich alloys have a Tclower than the Fe rich ones,\nthus resulting in larger spin fluctuations at T= 300 K.\nThe overall trend for LSDA and GGA is modified at low\nconcentrations of Fe when spin fluctuations are consid-\nered, whilst for LSDA+ U[FLL] the changes in the trends\noccur mostly at concentrations between x= [0.3−0.8].\nAn important aspect is the overall good agreement of11\nLSDA, GGA and LSDA+ U[AMF]. Instead results ob-\ntained in LSDA+ U[FLL] stand out as different from the\nrest. This is is expected since as was previously men-\ntioned the FLL DC is not the most appropriate scheme\nto treat these systems. An example of such inadequacy\ncan clearly be seen in Fig. 6b for Mn rich concentrations,\nwhere the damping is much larger with respect to the\nother curves. As mentioned above, this could result from\nthe appearance of states at the Fermi level.\nOverall the magnitude of the intrinsic damping pre-\nsented here is smaller than the values reported in experi-\nments92, whichreportvaluesforthedampingofCo 2MnSi\nofα∼0.005 and α∼0.020 for Co 2FeSi, in comparison\nwith the calculated values of αLSDA= 7.4×10−4and\nαLSDA= 4.1×10−3for Co 2MnSi and Co 2FeSi, respec-\ntively. In experiments also a minimum at the concentra-\ntion of Fe of x∼0.4 is present, while such minima is not\nseen in the present calculations. However, similar trends\nas those reported here (for LSDA and GGA) are seen in\nthe work by Oogane and Mizukami15. A possible reason\nbehind the discrepancy between theory and experiment,\ncould stem from the fact that as the Fe concentration\nincreases, correlation effects also increase in relative im-\nportance. Such a situation cannot be easily described\nthrough the computational techniques used in this work,\nandwill affectthe detailsofthe DOSatthe Fermienergy,\nwhich in turn could modify the damping. Another im-\nportant factor influencing the agreement between theory\nand experiments arise form the difficulties in separating\nextrinsic and intrinsic damping in experiments93. This,\ncombined with the large spread in the values reported in\nvarious experimental studies87,94,95, points towards the\nneed of improving both theoretical and experimental ap-\nproaches,ifoneintendstodeterminetheminimumdamp-\ning attainable for these alloys with sufficient accuracy.\nUp until now in the present work, disorder effects\nhave been considered at the Y site of the Heusler struc-\nture. In the following chemical disorder will be consid-\nered on the Z site instead. Hence, the chemical structure\nchanges to the type Co 2MeZA\n1−xZB\nx(Me=Fe,Mn). The\nalloys Co 2MeAlxSi1−xand Co 2MeGa xGe1−xare consid-\nered. The lattice constant for the off stoichiometric com-\npositions is treated using Vegard’s law96, interpolating\nbetween the values given in Table I.\nIn Fig. 7 the dependence of the damping on the con-\ncentration of defects is reported, as obtained in LSDA.\nFor Co 2FeGaxGe1−xas the concentration of defects in-\ncreases the damping decreases. Such a behaviour can\nbe understood by inspecting the density of states at the\nFermi level which follows the same trend, it is important\nto notice that Co 2FeGa is a half-metallic system, while\nCo2FeGe is not (see table I). On the other hand, for\nCo2FeAlxSi1−x, the damping increases slightly with Al\nconcentration, however, for the stoichiometric Co 2FeAl\nis reached the damping decreases suddenly, as in the pre-\nvious case. This is a direct consequence of the fact that\nCo2FeAl is a half metal and Co 2FeSi is not, hence when\nthe half-metallic state is reached a sudden decrease ofthe damping is observed. For the Mn based systems, as\nthe concentration of defects increases the damping in-\ncreases, this stark difference with the Fe based systems.\nFor Co 2MnAlxSi1−xthis is related to the fact that both\nCo2MnAl and Co 2MnSi are half-metals in LSDA, hence,\nthe increase is only related to the fact that the damp-\ning for Co 2MnAl is larger than the one of Co 2MnSi, it\nis also relevant to mention, that the trend obtained here\ncorresponds quite well with what is observed in both ex-\nperimental and theoretical results in Ref.86. A similar\nexplanation can be used for the Co 2MnGa xGe1−xalloys,\nas both are half-metallic in LSDA. As expected, the half\nmetallic Heuslers have a lower Gilbert damping than the\nother ones, as shown in Fig. 7.\n00.511.522.533.544.5\n0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3)\nConcentration of defectsCo2FeAlxSi1-xCo2FeGaxGe1-xCo2MnAlxSi1-xCo2MnGaxGe1-x\nFIG. 7. (color online) Dependence of the Gilbert damping\nfor the alloys Co 2MeAlxSi1−xand Co 2MeGa xGe1−xwith Me\ndenoting Mn or Fe under the LSDA exchange correlation po-\ntential.\nVI. CONCLUSIONS\nThe treatment of several families of half-metallic\nHeusler alloys has been systematically investigated us-\ning several approximations for the exchange correlation\npotential, as well as for the shape of the potential. Spe-\ncial care has been paid to the calculation of their mag-\nnetic properties, such as the Heisenberg exchange inter-\nactions and the Gilbert damping. Profound differences\nhave been found in the description of the systems de-\npending on the choice of exchange correlation potentials,\nspeciallyforsystems in whichcorrelationeffects might be\nnecessarytoproperlydescribethepresumedhalf-metallic\nnature of the studied alloy.\nIn general, no single combination of exchange correla-\ntion potential and potential geometry was found to be\nable to reproduce all the experimentally measured mag-\nnetic properties of a given system simultaneously. Two\nof the key contributing factors are the exchange correla-12\ntion potential and the double counting scheme used to\ntreat correlation effects. The destruction of the half-\nmetallicity of any alloy within the study has profound\neffects on the critical temperature and spin wave stiff-\nness. A clear indication of this fact is that even if the\nFLL double counting scheme may result in a correct de-\nscription of the magnetic moments of the system, the\nexchange interactions may be severely suppressed. For\nthe systems studied with DMFT techniques either mi-\nnor improvement or results similar to the ones obtained\nfrom LSDA is observed. This is consistent with the in-\nclusion of local d−dscreening, which effectively dimin-\nishes the strength of the effective Coulomb interaction\nwith respect to LSDA+ U(for the same Hubbard param-\neterU). In general, as expected, the more sophisticated\ntreatment forthe geometricalshape ofthe potential, that\nis a full potential scheme, yields results closer to experi-\nments, which in these systems, is intrinsically related to\nthe description of the pseudogap region.\nFinally, the Gilbert damping is underestimated with\nrespect to experimental measurements, but in good\nagreement with previous theoretical calculations. One of\nthe possible reasons being the difficulty from the experi-\nmental point of view of separating intrinsic and extrinsic\ncontributions to the damping, as well as the strong de-\npendence of the damping on the crystalline structure.\nA clear correlation between the density of states at the\nFermi level and the damping is also observed, which is\nrelated to the presence of a small spin orbit coupling\nin these systems. This highlights the importance that\nhalf-metallic materials, and their alloys, have in possible\nspintronic and magnonic applications due to their low in-\ntrinsic damping, and tunable magnetodynamic variables.\nThese results could spark interest from the experimental\ncommunity due to the possibility of obtaining ultra-low\ndamping in half-metallic Heusler alloys.\nVII. ACKNOWLEDGEMENTS\nThe authors acknowledge valuable discussions with\nM.I. Katsnelsson and A.I. Lichtenstein. The work was\nfinanced through the VR (Swedish Research Council)\nand GGS (G¨ oran Gustafssons Foundation). O.E. ac-\nknowledges support form the KAW foundation (grants\n2013.0020 and 2012.0031). O.E. and A.B acknowledge\neSSENCE. L.B acknowledge support from the Swedish\ne-Science Research Centre (SeRC). The computer sim-\nulations were performed on resources provided by the\nSwedish National Infrastructure for Computing (SNIC)\nat the National Supercomputer Centre (NSC) and High\nPerformance Computing Center North (HPC2N).\nAppendix A: DOS from LSDA+DMFT\nHere we show the DOS in Co 2MnSi and Co 2FeSi ob-\ntained from LSDA and LSDA+DMFT calculations. The0369n↑tot[sts./eV]\n0\n3\n6\n9-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]LSDA\n0369n↑tot[sts./eV]\n0\n3\n6\n9-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]LSDA\nDMFT[Σ(0)]\nDMFT[FLL]\nFIG. A.1. (color online) DOS in Co 2FeSi (top panel) and\nCo2MnSi (bottom panel) obtained in different computational\nsetups.\nresults shown in Fig. A.1 indicate that the DMFT in-\ncreases the spin-down (pseudo-)gap in both Co 2FeSi and\nCo2MnSi. In the latter casethe shift ofthe bands is more\npronounced. InCo 2FeSiitmanifestsitselfinanenhanced\nvalue of the total magnetization. For both studied sys-\ntems, the FLL DC results in relatively larger values of\nthe gaps as compared with the “Σ(0)”estimates. How-\never, for the same choice of the DC this gap appears to\nbe smaller in LSDA+DMFT than in LSDA+ U. Present\nconclusion is valid for both Co 2FeSi and Co 2MnSi (see\nFig. 1 for comparison.)\nAppendix B: NQS in Co 2MnSi\nHere we show the calculated spectral functions in\nCo2MnSi obtained with LSDA+DMFT[Σ(0)] approach.\nAs discussed in the main text, the overall shape of DOS\nis reminiscent of that obtained in LSDA. However, a cer-\ntain amount of the spectral weight appears above the\nminority-spin gap. An inspection of the imaginary part\nof the self-energy in minority-spin channel, shown in the\nbottom panel of Fig. B.1, suggests a strong increase of\nMn spin-down contribution at the corresponding ener-\ngies, thus confirming the non-quasiparticle nature of the\nobtained states. We note that the use of FLL DC formu-\nlationresultsinanenhancedspin-downgapwhichpushes\nthe NQS to appear at even higher energies above EF(see\nAppendix A).13\n-30030PDOS [sts./Ry]\n-0.2-0.10Im [ Σ↑]\nCo Eg\nCo T2g\n-0.2 -0.1 0 0.1 0.2\nE-EF [Ry]-0.2-0.10Im [ Σ↓]\nMn Eg\nMn T2g\nFIG. B.1. (color online) Top panel: DOS in Co 2MnSi pro-\njected onto Mn and Co 3 dstates of different symmetry. Mid-\ndle and bottom panels: Orbital-resolved spin-up and spin-\ndown imaginary parts of the self-energy. The results are\nshown for the “Σ(0)”DC.\nAppendix C: Impact of correlation effects on the\nJij’s in Co 2MnSi and Co 2FeSi\nIn this section we present a comparison of the ex-\nchange parameters calculated in the framework of the\nLSDA+DMFT using different DC terms. The calculated\nJij’s between different magnetic atoms within the first\nfew coordination spheres are shown in Fig. C.1. One can\nsee that the leading interactions which stabilize the fer-\nromagnetism in these systems are the nearest-neighbour\nintra-sublattice couplings between Co and Fe(Mn) atoms\nand, to a lower extend, the interaction between two Co\natoms belonging to the different sublattices. This qual-\nitative behaviour is obtained independently of the em-\nployed method for treating correlation effects and is in\ngood agreement with prior DFT studies. As explained\nin the main text, the LSDA and LSDA+DMFT[Σ(0)] re-\nsults are more similar to each other, whereas most of the\nJij’s extracted from LSDA+DMFT[FLL] are relatively\nenhanced due to inclusion of an additional static contri-\nbution to the exchange splitting. This is also reflected in\nboth values of the spin stiffness and the Tc.\nIn order to have a further insight into the details of\nthe magnetic interactions in the system, we report here\nthe orbital-resolved Jij’s between the nearest-neighbours\nobtained with LSDA. The results, shown in Table. C.1,\nreveal few interesting observations. First of all, all the0.6 1.2 1.8 2.4-0.0500.050.1Jij[mRy]\n0.6 1.2 1.8 2.400.10.2\n0.6 1.2 1.8 2.4\nRij/aalat00.10.20.30.4Jij[mRy]\n0.6 1.2 1.8 2.4\nRij/aalat00.511.5\nLSDA\nLSDA+DMFT [ Σ0]\nLSDA+DMFT [FLL]Co1-Co1Mn-Mn\nCo1-Co2Co-Mn\nFIG. C.1. (color online) The calculated exchange parameter s\nin Co 2MnSi within LSDA and LSDA+DMFT for different\nchoice of DC.\nTABLEC.1. Orbital-resolved Jij’sbetweenthenearestneigh-\nbours in Co 2MnSi in mRy. In the case of Co 1-Co1, the second\nnearest neighbour value is given, due to smallness of the firs t\none. The results were obtained with LSDA.\nTotalEg−EgT2g−T2gEg−T2gT2g−Eg\nCo1-Co10.070 0.077 -0.003 -0.002 -0.002\nCo1-Co20.295 0.357 -0.058 -0.002 -0.002\nCo-Mn 1.237 0.422 -0.079 0.700 0.194\nMn-Mn 0.124 -0.082 0.118 0.044 0.044\nT2g-derived contributions are negligible for all the inter-\nactions involving Co atoms. This has to do with the\nfact that these orbitals are practicallyfilled and therefore\ncan not participate in the exchange interactions. As to\nthe most dominant Co-Mn interaction, the Eg−Egand\nEg−T2gcontributions are both strong and contribute\nto the total ferromagnetic coupling. This is related to\nstrong spin polarisation of the Mn- Egstates.\n00.050.1\n0.6 1.2 1.8 2.4Jij[mRy]\n-0.15-0.1-0.0500.050.1\n0.61.21.82.4\n00.20.40.6\n0.61.21.82.4Jij[mRy]\nRij/alat00.511.522.53\n0.61.21.82.4\nRij/alatLSDA\nLSDA+U[FLL]\nLDA+U[AMF]Co1-Co1 Fe-Fe\nCo1-Co2\nCo-Fe\nFIG. C.2. Exchange interactions for Co 2FeSi within LSDA\nand LSDA+ Uschemes and a full potential approach for dif-\nferent DC choices.14\nCorrelationeffectsalsohaveprofoundeffectsontheex-\nchange interactions of Co 2FeSi. In particular, the Fe-Fe\ninteractions can be dramatically changed when consid-\nering static correlation effects. It is specially noticeable\nhow the anti-ferromagneticexchangeinteractions can de-\ncreasesignificantlywhichcanaffecttheexchangestiffness\nand the critical temperature as described in the maintext. 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Phys. 5, 17 (1921), ISSN 0044-3328."    },    {        "title": "1110.5112v3.CoB_Ni_Based_Multilayer_Nanowire_with_High_Speed_Domain_Wall_Motion_under_Low_Current_Control.pdf",        "content": "1\nCoB/Ni-Based Multilayer Nanowire with High-Speed Domain Wall Motion \nunder Low Current Control \nDuc-The Ngo*, Norihito Watanabe, and Hiroyuki Awanoj \nInformation Storage Materials Laboratory, Toyota Technological Institute, Nagoya 468-\n8511, Japan \n \n The spin-transfer torque motion of magnetic DWs in a CoB/Ni-based nanowire driven by \na low current density of (1.12±0.8)×10\n11 A m-2 has been observed indirectly by \nmagnetotransport measurements. A high DW velocity of 85±4 m/s at zero field was measured at the threshold current density. Upon increasing the current density to 2.6×10\n11 \nA m-2, the DW velocity increases to 197±16 m/s before decreasing quickly in the high-\ncurrent-density regime owing to nonadiabatic spin-transfer torque at a low damping factor and weak pinning. The addition of B atoms to the Co layers decreased the level of \nsaturation magnetization, Gilbert damping fact or, and density of pinning sites, making the \nCoB/Ni multilayer nanowire favora ble for practical applications.  \n \n \n \n*Present address: Department of Electrical and Computer Engineering, National University of Singapore, 4 \nEngineering Drive 3, Singapore 117576.  \nj E-mail address: awano@toyota-ti.ac.jp. 2\n \n1. Introduction \nWhen a spin-polarized electron current hits a magnetic moment, it exerts a torque on the \nmoment, transfers its angular momentum to the moment, and thereby affects the precession motion and switching of the mo ment. This phenomenon was theoretically \npredicted by Berger [1] and Slonczewski [2], and subsequently was named the spin-\ntransfer torque (STT). The motion of magne tic domain walls (DWs) caused by an \nelectrical current in magnetic nanostructures is also a consequence of the SST in which \nthe spin-polarized current switc hes the magnetic moments in the wall. This is nowadays \nwidely applied in spintronic technology such  as the DW logic gate [3,4] and racetrack \nmemory [5, 6]. Over the last 15 years, most studies have focused on a NiFe patterned \nfilm, a typical soft magnetic material with in-plane magnetic anisotropy and nearly zero \nmagnetocrystalline anisotropy, as it is cheap and highly stable and it is easy to fabricate and control its composition and properties. The motion of DWs with a very high velocity, up to ~200 m/s, has been demonstrated in a number of NiFe-based nanowire devices [3-6]. However, the motion in such in-plane anisotropy films was controllable only at a \nrelatively high current density (~10\n12 A m−2) owing to a wide DW and a low spin-torque \nefficiency (it should be noted that the threshold current density is expected to be proportional to the wall width [7]). A high control current consumes much energy and the \nheat released from electrical current would sometimes degrade the performance of such \ndevices. Therefore, deceasing of current dens ity is one of the most important technical \nissues at the moment.  3\nPerpendicularly magnetized thin films have recently been proposed to replace the \nin-plane NiFe film [8,9] to realize this goa l. In the perpendicular magnetic anisotropy \nfilms, formation of Bloch-type walls that are 1-2 orders of magnitudes thinner than Néel-\ntype walls in the in-plane films and a high spin-torque efficiency would lower the intrinsic current density by one or even two orders of magnitude. Many authors [10-13] \nhave presented the decrease in threshold current to (2-5)×10\n11 A m−2 using multilayer \nnanowires, e.g., Co/Pt, CoFe/Pt, and Co/Ni in which the perpendicular magnetic anisotropy was one of the keys to decrease the critical current density. Nonetheless, layer \nthickness in those multilayers was normally ~3-20 Ä and might be badly influenced by \nthe heat from electrical current. The perpendicular anisotropy is logically threatened to disappear owing to the diffusion of the layers under Joule heating of electrical current.  \nAmong the researchers, Yamanouchi et al. [14] were successful in establishing \nthe motion of magnetic DWs in a perpendicularly magnetized ferromagnetic semiconductor (Ga,Mn)As with a very low current density of about 10\n9 A m−2. However, \nthis material (and most ferromagnetic semiconductors) has a Curie temperature far below room temperature and therefore is not realistic for room-temperature devices. \nIn this article, we present the enhancement of the motion of the magnetic DWs in \nthe CoB/Ni multilayer nanowire at a low current density. The addition of B atoms to the Co layers decreased the density of the pinning sites in the film, enhanced the DW motion, \nand improved the stability of the multilayer  by preventing the diffusion between the \nCo/Ni interfaces.  \n 4\n2. Experimental Methods \nA multilayer film of Pt 5 nm/[CoB 0.6 nm/Ni 1.1 nm]4/CoB 0.6 nm/Pt 1 nm was \nfabricated by radio-frequency (RF) magnetron sputtering using Ar gas. The base vacuum \nof the deposition chamber was 3×10−8 Torr whereas the Ar pressure was maintained at 5 \nmTorr during the deposition process. The com position of the CoB ta rget was chosen as \nCo80B20 (at.%). The film was grown on a naturally oxidized Si substrate. A nanowire \nwith 300 nm width and 150 µm length was subsequently patterned by electron beam \nlithography and ion beam etching (Fig. 1). The nanowire was modified to have a planar \nHall shape for magnetotransport measurements. A square pad was made at one end of the \nwire as a source for DW nucleation [15], and the shape of the other end of the wire was modified to be triangular to prohibit the propagation of DW [16]. A Ti/Au electrode \npattern produced by photolithography was mounted to the wire for magnetotransport \nmeasurements. The magnetic properties of the film specimen were measured using an alternating gradient magnetometer (AGM).  \nThe magnetic DWs were nucleated in the square pad by an Oersted field \ngenerated from the 30 ns width, 6.5 MHz pulse current flowing in the Ti/Au electrode \ndeposited on the pad (Fig. 1, electrodes A-B). The motion of DWs in the nanowire was \nthen driven by a DC current (J\nDC, electrodes G-B in Fig. 1). Anomalous Hall effect \n[17,18] measurement (either electrodes C-D or E-F) was carried out to detect the \npropagation of DW in the wire. Hysteresis loop measurement on the continuous film \nspecimen (data not shown) using the AGM confirmed that the film exhibited a strong perpendicular magnetic anisotropy with a saturation magnetization of M\ns=5.6×105 A/m 5\n(about 15% lower than that of a Co/Ni-based film [13]) and a uniaxial anisotropy \nconstant of K u=3.57×105 J/m3 (~8% higher than that of a Co/Ni film [12, 13]). \n \n3. Results and Discussion \nThe time-resolved Hall effect voltage signal obtained from the nanowire at a driven \ncurrent (DC current) of 0.59 mA corresponding to a current density of  jDC=1.12×1011 A \nm−2 and an external field of +5 mT is illustrat ed in Fig. 2(a) and represents the movement \nof DW along the wire [17,18]. Initially, the wire was magnetically saturated, then \nmagnetization reversal was induced by the Oersted field generated from the pulse current \nwith nucleation of a tiny domain at the region between the wire and the square pad. The DC current sequentially forced the domain (with two walls on two sides) moving along \nthe wire, toward the Hall bar. When the domain moved into the Hall bar, the presence of \nthe domain in the cross bar induced a change in  the Hall voltage signal, as seen in Fig. \n2(a). The Hall signal switched to a low valu e when the domain (w ith the two walls) had \npassed the Hall bar. The progress of the Ha ll signal could be interpreted approximately \non the basis of a simple schematic shown in Fig. 2(b). Because of a periodic pulse, the \ndomains were nucleated and driven to the wire periodically and the Hall voltage signal \nappeared to be a periodic pulse. This result looks similar to the DW motion observed previously [17,18]. A square Hall-voltage hysteresis loop [inset of Fig. 2(a)] exhibits a \nsharp change in the magnetization, proving a fast propagation of a domain through the \nHall bar. The square aspect of the hysteresis loop indicated that a reversal occurred through DW nucleation followed by easy DW propagation. It should be noted that the \nOersted field released from the driven current was estimated to be about 30 mT, which 6\nwas much smaller than the coercive field of  the sample (see the Hall effect hysteresis \nloop in the inset of Fig. 2). Therefore, the influence of the Oersted field on the motion of \nthe wall along the wire (described in Fig. 2) could be minor, whereas the effect of the \nspin-polarized current is essentially considered.  \nThe current dependence of the variation of normalized Hall resistance, ΔRHall, is \nshown in Fig. 3. The normalized Hall resistance here was defined as the change in the \nHall voltage signal when the domain propagated through the Hall bar [Fig. 2(b)]. Therefore, the normalized Hall resistance became high (1) above a threshold current \ndensity of 1.12×10\n11 A m−2, whereas this value was low (0) below the threshold current \ndensity. This indicates that the motion of the magnetic DWs, denoted by a change in Hall resistance, could be induced when the density of the spin-polarized current is above \n1.12×10\n11 A m−2, confirming that it is possible to drive the DW motion in the 300 nm \nwidth CoB/Ni nanowire with a threshold current density of 1.12×1011 A m−2 by the STT \nmechanism. It is important that the threshold current density obtained here was reasonably lower than either ~(1-3)×10\n12 A m−2 in the NiFe-based devices [6,7] or ~(2-\n5)×1011 A m−2 in a similar multilayer Co/Ni wire [12,13], or lower than the current \ndensity in a spin-valve nanowire [19] reported recently. Moreover, it is shown in Fig. 2 \nthat the pulse like signal of Hall voltage is periodic and coherent with the nucleation \npulse, presuming the continuous propagation of a multidomain similar to a shift register \nwriting process. \nFrom the pulse like Hall signal, the velo city of DW moving in the Hall bar could \nbe derived [13] from Fig. 2(b): T 1 was the time when the front-edge wall of the domain \nstarted coming to the Hall bar and T 2 was the time that it passed the Hall bar (in L = 500 7\nnm). Therefore, the velocity of the front-edge wall could be referred as L/ Δt1 (Δt1 = T 2-\nT1). On the other hand, the velocity of the rear-edge wall could be attained from the time \ninterval Δt2 = T 4 – T 3. On the other hand, from the phase delay between the signals at the \nC-D and E-F Hall bars that reflected the time of flight of the wall between two Hall bars, \nthe velocity of the wall in the straight wire  (from C to E) was de termined. Figure 4 shows \nthe wall velocity in the straight wire area as a function of external magnetic field \nmeasured at the threshold current density. The field dependence here is consistent with the following expression [20]:  \nv(H) = µ\nH(H - H 0) + v(J),   (1) \nwhere µ H(J) is the DW mobility, J is the current density, and H 0 is the “dynamic coercive \nforce”. The term v(J) - µ HH0 can be referred as the velocity at zero field.  \nUsing this linear dependence, a zero-field wall velocity of 86±5 m/s was \ncalculated at the critical current density (1.12×1011 A m−2) with a mobility of 2640±170 \n(m s−1 T−1). This matches well with the velocity measured directly at H = 0 (85±4 m/s). It \nis interesting to note that the field-free wa ll velocity here was much higher than that of \nthe Co/Ni wire [12,13] or TbFeCo nanowire [18,21]. Therefore, this aspect is very \npromising for high-speed devices. As DW moved in the region of the Hall bars, the wall \nvelocity (as defined above) was found to be sli ghtly lower than that in the straight wire \narea but only in the error scale of the measurement. The non-zero DW velocity and linear \ndependence of the wall velocity on the external field can be attributed to the motion \ndriven by the nonadiabatic torque [20]. \nThe velocities of the front-edge and rear-edge walls were perfectly identical to \neach other and remained invariable at positions of two Hall bars. These suggest that i) the 8\neffect of the pinning on the motion of th e walls along the wire was predominantly \ngoverned by the material rather than the geometry of the Hall bars and  ii) no distortion of \nthe domain and the wall geometry as the domain length was conserved when they were \nlocated in the Hall bars. Usually, the distortion  of the domain in the Hall bar, denoted by \nthe small difference between the velocities of the front-edge wall (faster) and the rear-\nedge wall (slower), was only observed at a high applied field (over 90 mT), and can be \nimagined similarly to the distortion of a balloon, as reported elsewhere [22]. \nRegarding other interesting points, the time interval T 3 - T 1 [see Fig. 2(b)] \nexpresses the period necessary for the whole domain to reach the rear side of the Hall bar, \nallowing domain size to be estimated. Using this relationship, the average size of the domain was calculated to be 900±35 nm at  the critical current density of 1.12×10\n11 A m−2 \nand zero field. Under an external field, the domain size was slightly reduced to 630±25 \nnm at the field of 40 mT, which was similar to theoretical prediction [23]. It is supposed that the external field in this case acted oppositely to the nucleated field from the pulse current, and compressed the domain when it was nucleated. It should be noted that the domain length was conserved when the domain was located in two Hall bars. \nThe dependence of wall velocity at zero field on controlled current density is \nshown in Fig. 5. The velocity firstly incr eased with current density from 85 m/s at the \nthreshold current density to a maximum value of 197±16 m/s at a current density of \n2.63×10\n11 Am−2, then markedly dropped at higher current densities. The variation of wall \nvelocity with current density in this case can be explained qualitatively by referring to the \nmodel given in refs.#7 and 24. The model proposed by Tatara et al. [24] predicted that \nthe trend of the wall velocity variation (including a linear increase at low current and a 9\ndecrease with increasing current at high cu rrents) is a consequence of the nonadiabatic \ntorque driving when the damping factor is low and the pinning effect is weak. In the low \ncurrent regime, DW velocity was linearly dependent on current density, which is in \naccordance with the zero-field velocity de scribed in eq. (1) and somehow similar to a \nprevious experimental observation [20]. At a high current density (above 2.63×1011 A \nm−2), wall velocity appeared to decrease, indicating that the nonadiabatic parameter β was \nnot zero and not equal to the Gilbert damping fa ctor. This led to a deformation of the wall \nstructure above the Walker breakdown current density [20]. This dependence and linear \nfield-velocity function discussed in previous paragraphs indicated that the motion of the \nwalls in our device was mainly governed by the nonadiabatic term. \nIn an attempt to explain the decrease in the threshold current density, theoretical \nmodels [7,24] are employed, in which the threshold (or intrinsic critical) current density \ncould be referred to as follows [7]: \n21~s\nc\nBeMJgP\n ,  (2) \nwhere α and β are the Gilbert damping factor and nonadiabatic spin-transfer torque \nparameter, respectively;  is the DW width; Ms is the saturation magnetization; P is the \nspin polarization of the material; g is the gyromagnetic ratio,  e is the electron charge, and \nB is the Bohr magneton.  \nAs discussed in the previous paragraphs, the magnitude of saturation \nmagnetization of the studied CoB/Ni multilayer film was decreased by ~15%, whereas \nuniaxial anisotropy was slightly enhanced which subsequently led to a thinner DW. \nAdditionally, the substitution of B for Co is expected to decrease the Gilbert damping \nfactor of the film. Hence, such decreases in saturation magnetization magnitude and wall 10\nwidth could result in a decrease in the threshold critical current. Moreover, the addition of \nB atoms on the other hand weakens the pinning of the DW by decreasing the number of \npinning sites [25] as mentioned in previous paragraphs. This effect also enhances the \nvelocity of the walls. It should be noted that the Walker current at which the wall velocity \ndropped as seen in Fig. 5, is also expected to be a function of the intrinsic parameters of \nthe materials [26,27]. \nAdditionally, the addition of B to Co would make the devices more stable. B atoms with a small atomic radius would locate at the vacancies in the Co lattice and increase the \nclosed-package degree of the lattice, thus preventing the diffusion between the layers and \npreserving the magnetic properties of the film under heating caused by the electrical current applied. The addition of B atoms to th e Co lattice also decreases the difference in \nthe lattice constant between Co/Ni and result s in a smooth Co/Ni interface. It should be \nnoted that 20% addition of B to the Co layers (Co\n80B20) only gives rise to ~12% of \nresistivity in comparison with pure Co laye rs. From the technical point of view, these \nbenefits would to enhance the working stability of the devices.  \n \n4. Conclusions \nThe motion of magnetic DW in the CoB/Ni multilayer nanowire with a very low current \ndensity of (1.12±0.8)×1011 A m-2 and a high DW velocity of 85±4 m/s has been \nsuccessfully induced. DW velocity can be raised up to 197±16 m/s by increasing the \ncurrent density to 2.63×1011 A m−2. The variation of wall velocity was consistent with the \nnonadiabatic STT mechanism. These advantages were attributed to the presence of CoB \nlayers with a low Gilbert damping factor, a low saturation magnetization, and a low 11\ndensity of pinning sites. 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D. Terris, and C. Chappert: Phys. Rev. \nLett. 95 (2005) 117203. \n[23] K.-J. Kim, K.-W. Moon, K.-S. Lee, and S.-B. Choe: Nanotechnology 22 (2011) \n025702. \n[24] G. Tatara, H. Kohno, and J. Shibata: J. Phys. Soc. Jpn. 77 (2008) 031003. \n[25] R. Lavrijsen, G. Malinowski, J. H. Franken, J. T. Kohlhepp, H. J. M. Swagten, B. \nKoopmans, M. Czapkiewicz, and T. Stobiecki: Appl. Phys. Lett. 96 (2010) 022501. 14\n[26] I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-Prejbeanu, S. Auffret, B. \nRodmacq, S. Pizzini, J. Vogel, M. Bonfim, A. Schuhl, and G. Gaudin: Nat. Mater. \n10 (2011) 419. \n[27] O. Boulle, G. Malinowski, and M. Cläui: Mater. Sci. Eng. R 72 (2011) 159. 15\nFigure captions \nFig. 1 . Electron microscopy image of the CoB/Ni nanowire with Ti/Au electrodes for \nmagnetotransport measurements. The plus sign denotes the direction of applied field.  \nFig. 2 . (a) Time-resolved Hall voltage signal measured at a driven current of 0.59 mA \n(j\nDC=1.12×1011 A.m-2) and external field of 5 mT; (b) Interpretation of the Hall-voltage \npulse as domain motion progresses in the Hall bar. The inset shows the field dependence \nof Hall voltage.  \nFig. 3 . Hall resistance changes as a function of driven current density. The inset shows a \ntime-resolved Hall voltage signal measure at the critical current density and zero field.  \nFig. 4 . External field dependence of the velocity of the wall measured at the threshold \ncurrent density (j\nDC=1.12×1011 A.m-2). \n \nFig. 5 . Variation of wall velocity as a function of driven current density at zero external \nfield. 16\n \n \nFigure 1 17\n \n \nFigure 2 18\n \n \nFigure 3 19\n \n \nFigure 4 20\n \n \nFigure 5 "    },    {        "title": "1010.0478v2.Thermal_fluctuation_field_for_current_induced_domain_wall_motion.pdf",        "content": "Thermal fluctuation field for current-induced domain wall motion\nKyoung-Whan Kim and Hyun-Woo Lee\nPCTP and Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea\n/H20849Received 18 May 2010; revised manuscript received 23 August 2010; published 20 October 2010 /H20850\nCurrent-induced domain wall motion in magnetic nanowires is affected by thermal fluctuation. In order to\naccount for this effect, the Landau-Lifshitz-Gilbert equation includes a thermal fluctuation field and literatureoften utilizes the fluctuation-dissipation theorem to characterize statistical properties of the thermal fluctuationfield. However, the theorem is not applicable to the system under finite current since it is not in equilibrium. Toexamine the effect of finite current on the thermal fluctuation, we adopt the influence functional formalismdeveloped by Feynman and Vernon, which is known to be a useful tool to analyze effects of dissipation andthermal fluctuation. For this purpose, we construct a quantum-mechanical effective Hamiltonian describingcurrent-induced domain wall motion by generalizing the Caldeira-Leggett description of quantum dissipation.We find that even for the current-induced domain wall motion, the statistical properties of the thermal noise isstill described by the fluctuation-dissipation theorem if the current density is sufficiently lower than theintrinsic critical current density and thus the domain wall tilting angle is sufficiently lower than\n/H9266/4. The\nrelation between our result and a recent result /H20851R. A. Duine, A. S. Núñez, J. Sinova, and A. H. MacDonald,\nPhys. Rev. B 75, 214420 /H208492007/H20850/H20852, which also addresses the thermal fluctuation, is discussed. We also find\ninteresting physical meanings of the Gilbert damping /H9251and the nonadiabaticy parameter /H9252; while /H9251charac-\nterizes the coupling strength between the magnetization dynamics /H20849the domain wall motion in this paper /H20850and\nthe thermal reservoir /H20849or environment /H20850,/H9252characterizes the coupling strength between the spin current and the\nthermal reservoir.\nDOI: 10.1103/PhysRevB.82.134431 PACS number /H20849s/H20850: 75.78.Fg, 75.60.Ch, 05.40.Ca\nI. INTRODUCTION\nCurrent-induced domain wall /H20849DW/H20850motion in a ferro-\nmagnetic nanowire is one of representative examples tostudy the effect of spin-transfer torque /H20849STT/H20850. The motion of\nDW is generated by the angular momentum transfer betweenspace-time-dependent magnetization m\n/H6023/H20849x,t/H20850and conduction\nelectrons, of which spins interact with m/H6023by the exchange\ncoupling. This system is usually described by the Landau-Lifshitz-Gilbert /H20849LLG/H20850equation,\n1–3\n/H11509m/H6023\n/H11509t=/H92530H/H6023eff/H11003m/H6023+/H9251\nmsm/H6023/H11003/H11509m/H6023\n/H11509t+jp/H9262B\nems/H20875/H11509m/H6023\n/H11509x−/H9252\nmsm/H6023/H11003/H11509m/H6023\n/H11509x/H20876,\n/H208491/H20850\nwhere /H92530is the gyromagnetic ratio, jpis the spin-current\ndensity, ms=/H20841m/H6023/H20841is the saturation magnetization, and /H9262Bis the\nBohr magneton. /H9251is the Gilbert damping coefficient, and /H9252\nis the nonadiabatic coefficient representing the magnitude ofthe nonadiabatic STT.\n4In Eq. /H208491/H20850, the effective magnetic\nfield Heffis given by\nH/H6023eff=A/H116122m/H6023+H/H6023ani+H/H6023th, /H208492/H20850\nwhere Ais stiffness constant, H/H6023anidescribes the effect of the\nmagnetic anisotropy, and H/H6023this the thermal fluctuation field\ndescribing the thermal noise. In equilibrium situations, the\nmagnitude and spatiotemporal correlation of H/H6023thare gov-\nerned by the fluctuation-dissipation theorem,5–7\n/H20855Hth,i/H20849x/H6023,t/H20850Hth,j/H20849x/H6023/H11032,t/H11032/H20850/H20856=4/H9251kBT\n/H6036/H9267/H9254/H20849x/H6023−x/H6023/H11032/H20850/H9254/H20849t−t/H11032/H20850/H9254ij,/H208493/H20850\nwhere /H20855¯/H20856represents the statistical average, i,jdenote x,y,\norzcomponent, kBis the Boltzmann constant, Tis the tem-perature, and /H9267=ms//H9262Bis the spin density. Equation /H208493/H20850\nplays an important role for the study of the magnetizationdynamics at finite temperature,\n8\nEquation /H208493/H20850has been also used in literature9–13to exam-\nine effects of thermal fluctuations on the current-inducedDW motion. In nonequilibrium situations, however, the\nfluctuation-dissipation theorem does not hold generally.Since the system is not in equilibrium any more when thecurrent is applied, it is not clear whether Eq. /H208493/H20850may be still\nused. Recalling that H\n/H6023this estimated to affect the magnetiza-\ntion dynamics considerably in many experimentalsituations\n14–17of the current-driven DW motion, it is highly\ndesired to properly characterize H/H6023thin situations with non-\nzero jp. Recently, Duine18attempted this characterization and\nshowed that Eq. /H208493/H20850is not altered by the spin current up to\nfirst order in the spin-current magnitude. This analysis how-ever is limited to situations where the spin-flip scattering isthe main mechanism responsible for\n/H9252. In this paper, we\ngeneralize this analysis by using a completely different ap-proach which does not assume any specific physical origin of\n/H9252.\nHtharises from extra degrees of freedom /H20849other than mag-\nnetization /H20850, which are not included in the LLG equation. The\nextra degrees of freedom /H20849phonons for instance /H20850usually have\nmuch larger number of degrees of freedom than magnetiza-tion and thus form a heat reservoir. Thus properties of H\nthare\ndetermined by the heat reservoir. The heat reservoir playsanother role. In the absence of the extra degrees of freedom,the Gilbert damping coefficient\n/H9251should be zero since the\ntotal energy should be conserved when all degrees of free-dom are taken into account. Thus the heat reservoir is re-sponsible also for finite\n/H9251. These dual roles of the heat res-\nervoir are the main idea behind the Einstein’s theory of thePHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n1098-0121/2010/82 /H2084913/H20850/134431 /H2084916/H20850 ©2010 The American Physical Society 134431-1Brownian motion.19There are also claims that /H9251is correlated\nwith/H9252/H20849Refs. 18and20–22/H20850in the sense that mechanisms,\nwhich generate /H9252, also contribute to /H9251. Thus the issue of H/H6023th\nand the issue of /H9251and/H9252are mutually connected. Recalling\nthat the main mechanism responsible for /H9251varies from ma-\nterial to material, it is reasonable to expect that the main\nmechanism for H/H6023thand/H9252may also vary from material to\nmaterial. Recently, various mechanisms of /H9252were examined\nsuch as momentum transfer,23–25spin mistracking,26,27spin-\nflip scattering,18,21,22,25,28and the influence of a transport\ncurrent.29This diversity of mechanisms will probably apply\ntoH/H6023thas well.\nInstead of examining each mechanism of H/H6023thone by one,\nwe take an alternative approach to address this issue. In1963, Feynman and Vernon\n30proposed the so-called influ-\nence functional formalism, which allows one to take accountof damping effects without detailed accounts of dampingmechanisms. This formalism was later generalized by Smith\nand Caldeira.\n31This formalism has been demonstrated to be\na useful tool to address dissipation effects /H20849without specific\naccounts of detailed damping mechanisms /H20850on, for instance,\nquantum tunneling,32nonequilibrium dynamic Coulomb\nblockade,33and quantum noise.34To take account of damp-\ning effects which are energy nonconserving processes in gen-eral, the basic idea of the influence functional formalism is tointroduce infinite number of degrees of freedom /H20849called en-\nvironment /H20850behaves like harmonic oscillators which couple\nwith the damped system. /H20851See Eq. /H2084913/H20850./H20852Caldeira and\nLeggett\n32suggested the structure of the spectrum of environ-\nment Eq. /H2084913/H20850and integrated out the degrees of freedom of\nenvironment to find the effective Hamiltonian describing theclassical damping Eq. /H2084912/H20850. For readers who are not familiar\nwith the Caldeira-Leggett’s theory of quantum dissipation,we present the summary of details of the theory in Sec. II B.\nIn order to address the issue of H\n/H6023th, we follow the idea of\nthe influence functional formalism and construct an effectiveHamiltonian describing the magnetization dynamics. The ef-fective Hamiltonian describes not only energy-conservingprocesses but also energy-nonconserving processes such asdamping and STT. From this approach, we find that Eq. /H208493/H20850\nholds even in nonequilibrium situations with finite j\np, pro-\nvided that jpis sufficiently smaller than the so-called intrin-\nsic critical current density23so that the DW tilting angle /H9278\n/H20849to be defined below /H20850is sufficiently smaller than /H9266/4. We\nremark that in the special case where the spin-flip scattering\nmechanism of /H9252is the main mechanism of H/H6023th, our finding is\nconsistent with Ref. 18, which reports that the spin flip scat-\ntering mechanism does not alter Eq. /H208493/H20850at least up to the first\norder in jp. But our calculation indicates that Eq. /H208493/H20850holds\nnot only in situations where the spin flip scattering is the\ndominant mechanism of H/H6023thand/H9252but also in more diverse\nsituations as long as the heat reservoir can be described bybosonic excitations /H20849such as electron-hole pair excitations or\nphonon /H20850, i.e., the excitations effectively behave like har-\nmonic oscillators to be described by Caldeira-Leggett’stheory. We also remark that in addition to the derivation ofEq./H208493/H20850in nonequilibrium situations, our calculation also re-\nveals an interesting physical meaning of\n/H9252, which will be\ndetailed in Sec. III.This paper is organized as follows. In Sec. II, we first\nintroduce the Caldeira-Leggett’s version of the influencefunctional formalism and later generalize this formalism sothat it is applicable to our problem. This way, we construct aHamiltonian describing the DW motion. In Sec. III, some\nimplications of this model is discussed. First, a distinct in-sight on\n/H9252is emphasized. Second, as an application, statisti-\ncal properties of the thermal fluctuation field are calculatedin the presence of nonzero j\np, which verifies the validity of\nEq./H208493/H20850when jpis sufficiently smaller than the intrinsic criti-\ncal density. It is believed that many experiments16,17are in-\ndeed in this regime. Finally, in Sec. IV, we present some\nconcluding remarks. Technical details about the quantumtheory of the DW motion and methods to obtain solutions areincluded in Appendices.\nII. GENERALIZED CALDEIRA-LEGGETT DESCRIPTION\nA. Background\nInstead of full magnetization profile m/H6023/H20849x,t/H20850, the DW dy-\nnamics is often described2,23,35–37by two collective coordi-\nnates, DW position x/H20849t/H20850and DW tilting angle /H9278/H20849t/H20850. When\nexpressed in terms of these collective coordinates, the LLGEq./H208491/H20850reduces to the so-called Thiele equations,\ndx\ndt=jp/H9262B\nems+/H9251/H9261d/H9278\ndt+/H92530K/H9261\nmssin 2/H9278+/H9257x/H20849t/H20850,/H208494a/H20850\n/H9261d/H9278\ndt=−/H9251dx\ndt+/H9252jp/H9262B\nems+/H9257p/H20849t/H20850. /H208494b/H20850\nHere Kis the hard-axis anisotropy, /H9261is the DW thickness.\n/H9257x/H20849t/H20850and/H9257p/H20849t/H20850are functions describing thermal noise field\nHth,i/H20849x,t/H20850. By definition, the statistical average of the thermal\nnoise field Hth,i/H20849x,t/H20850is zero and similarly the statistical aver-\nages of /H9257x/H20849t/H20850and/H9257p/H20849t/H20850should also vanish regardless of\nwhether the system is in equilibrium. The question of theircorrelation function is not trivial however. If the thermalnoise field H\nth,i/H20849x,t/H20850satisfies the correlation in Eq. /H208493/H20850,i tc a n\nbe derived from Eq. /H208493/H20850that/H9257x/H20849t/H20850and/H9257p/H20849t/H20850satisfy the cor-\nrelation relation12\n/H20855/H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856/H11008/H9251kBT/H9254ij/H9254/H20849t−t/H11032/H20850, /H208495/H20850\nfor/H20853i,j/H20854=/H20853x,p/H20854. But as mentioned in Sec. I, Eq./H208493/H20850is not\nguaranteed generally in the presence of the nonzero current.Then Eq. /H208495/H20850is not guaranteed either. The question of what\nshould be the correlation function /H20855\n/H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856in such a situ-\nation will be discussed in Sec. III.\nWhen the spin-current density jpis sufficiently smaller\nthan the so-called intrinsic critical density /H20841e/H92530K/H9261//H9262B/H20841,23/H9278\nstays sufficiently smaller than /H9266/4. In many experimental\nsituations,38–40this is indeed the case,41so we will confine\nourselves to the small /H9278regime in this paper. Then, one can\napproximate sin 2 /H9278/H110152/H9278to convert the equations into the\nfollowing form:42\ndx\ndt=vs+/H9251S\n2KMdp\ndt+p\nM+/H92571/H20849t/H20850, /H208496a/H20850KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-2dp\ndt=−2/H9251KM\nSdx\ndt+2/H9252KM\nSvs+/H92572/H20849t/H20850, /H208496b/H20850\nwhere p=2KM/H9261/H9278/S,Sis the spin angular momentum at\neach individual magnetic site, and vs=jp/H9262B/emsis the adia-\nbatic velocity,43which is a constant of velocity dimension\nand proportional to jp. The yet undetermined constant Mis\nthe effective DW mass42–44which will be fixed so that the\nnew variable pbecomes the canonical conjugate to x./H92571/H20849t/H20850\nand/H92572/H20849t/H20850are the same as /H9257x/H20849t/H20850and/H9257p/H20849t/H20850except for propor-\ntionality constants.\nWhen the thermal noises /H92571/H20849t/H20850and/H92572/H20849t/H20850are ignored, one\nobtains from Eq. /H208496/H20850the time dependence of the DW posi-\ntion,\nx/H20849t/H20850=x/H208490/H20850+/H9252\n/H9251vst+S\n2KM/H92512/H208491−e−2K/H9251t/S/H208491+/H92512/H20850/H20850\n/H11003/H20851/H9251p/H208490/H20850−Mvs/H20849/H9251−/H9252/H20850/H20852. /H208497/H20850\nNote that after a short transient time, the DW speed ap-\nproaches the terminal velocity /H9252vs//H9251. Thus the ratio /H9252//H9251is\nan important parameter for the DW motion. When the ther-mal noises are considered, they generate a correction to Eq./H208497/H20850. However, from Eq. /H208496/H20850, it is evident that the statistical\naverage of x/H20849t/H20850should still follow Eq. /H208497/H20850. Thus as far as the\ntemporal evolution of the statistical average is concerned, wemay ignore the thermal noises. In the rest of Sec. II,w ea i m\nto derive a quantum mechanical Hamiltonian, which repro-duces the same temporal evolution as Eq. /H208497/H20850in the statistical\naverage level. In Sec. III, we use the Hamiltonian to derive\nthe correlation function /H20855\n/H9257i/H20849t/H20850/H9257j/H20849t/H11032/H20850/H20856in the presence of the\nnonzero current.\nNow, we begin our attempt to construct an effective\nHamiltonian that reproduces the DW dynamics Eq. /H208496/H20850/H20851or\nequivalently Eq. /H208497/H20850/H20852. We first begin with the microscopic\nquantum-mechanical Hamiltonian Hs-d,\nHs-d=−J/H20858\niS/H6023i·S/H6023i+1−A/H20858\ni/H20849S/H6023i·zˆ/H208502+K/H20858\ni/H20849S/H6023i·yˆ/H208502+HcS,\n/H208498/H20850\nwhich has been used in previous studies20of the DW dynam-\nics. Here Jrepresents the ferromagnetic exchange interac-\ntion, AandKrepresent longitudinal /H20849easy-axis /H20850and trans-\nverse/H20849hard-axis /H20850anisotropy, respectively. The last term HcS\nrepresents the coupling of the spin system with the spin-\npolarized current,\nHcS=−/H20858\ni,/H9251=↑,↓/H20851t/H20849ci/H9251†ci+1/H9251+ci+1/H9251†ci/H9251/H20850−/H9262ci/H9251†ci/H9251/H20852−JH/H20858\niS/H6023ci·S/H6023i,\n/H208499/H20850\nwhere JHis the exchange interaction between conduction\nelectron and the localized spins, ci/H9251is the annihilation opera-\ntor of the conduction electron at the site i,S/H6023ciis the electron-\nspin operator, tis the hopping integral, and /H9262is the chemical\npotential of the system.Recently Kim et al.43analyzed Hs-din detail in the small\ntilting angle regime and found that Hs-dcontains gapless\nlow-lying excitations and also high-energy excitations with afinite energy gap. The gapless excitations of H\ns-dare de-\nscribed by a simple Hamiltonian H0,\nH0=vsP+P2\n2M/H2084910/H20850\nwhile the high-energy excitations have a finite energy gap\n2S/H20881A/H20849A+K/H20850.I nE q . /H2084910/H20850,Pis the canonical momentum of\nthe DW position operator Q, and M=/H60362\nK/H208812A\nJa4is the effective\nDW mass called Döring mass.44Here, ais the lattice spacing\nbetween two neighboring spins. /H20849See, for details, Appendix\nA./H20850Below we will neglect the high energy excitations and\nfocus on the low-lying excitations described by Eq. /H2084910/H20850. For\nthe analysis of the high-energy excitation effects on the DW,See Ref. 42.\nFrom Eq. /H2084910/H20850, one obtains the following Heisenberg’s\nequation of motion:\ndQ\ndt=vs+P\nM, /H2084911a/H20850\ndP\ndt=0 . /H2084911b/H20850\nNote that the current /H20849proportional to vs/H20850appears in the equa-\ntion fordQ\ndt. Thus the current affects the DW dynamics by\nintroducing a difference between the canonical momentum P\nand the kinematic momentum P+Mvs. In this sense, the ef-\nfect of the current is similar to a vector potential /H20851canonical\nmomentum P/H6023and kinematic momentum P/H6023+/H20849e/c/H20850A/H6023/H20852. The\nvector potential /H20849difference between the canonical momen-\ntum and the kinetic momentum /H20850allows the system in the\ninitially zero momentum state to move without breaking thetranslational symmetry of the system. In other words, thecurrent-induced DW motion is generated without any forceterm in Eq. /H2084911b/H20850violating the translational symmetry of the\nsystem. This should be contrasted with the effect of the mag-netic field or magnetic defects, which generates a force termin Eq. /H2084911b/H20850.\nThe solution of Eq. /H2084911/H20850is trivial, /H20855Q/H20849t/H20850/H20856=/H20855Q/H208490/H20850/H20856\n+/H20849/H20855P/H208490/H20850/H20856/M+\nvs/H20850t. Here, the statistical average /H20855¯/H20856is de-\nfined as /H20855¯/H20856=Tr/H20849/H9267¯/H20850/Tr/H20849/H9267/H20850, where /H9267denotes the density\nmatrix at t=0. Associating /H20855Q/H20849t/H20850/H20856=x/H20849t/H20850,/H20855P/H20849t/H20850/H20856=p/H20849t/H20850, one\nfinds that Eq. /H2084911/H20850is identical to Eq. /H208496/H20850if/H9251=/H9252=0. This\nimplies that the effective Hamiltonian H0/H20851Eq./H2084910/H20850/H20852fails to\ncapture effects of nonzero /H9251and/H9252. In the next three sections,\nwe attempt to resolve this problem.\nB. Caldeira-Leggett description of damping\nTo solve the problem, one should first find a way to de-\nscribe damping. A convenient way to describe finite dampingwithin the effective Hamiltonian approach is to adopt theCaldeira-Leggett description\n32of the damping. Its main idea\nis to introduce a collection of additional degrees of freedom/H20849called environment /H20850and couple them to the original dy-\nnamic variables so that energy of the dynamic variables canTHERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-3be transferred to the environment. For instance, for a one-\ndimensional /H208491D/H20850particle subject to damped dynamics,\ndx\ndt=p\nM, /H2084912a/H20850\ndp\ndt=−dV/H20849x/H20850\ndx−/H9253dx\ndt. /H2084912b/H20850\nCaldeira and Leggett32demonstrated that its quantum-\nmechanical Hamiltonian can be constructed by adding damp-ing Hamiltonian H\n1to the undamped Hamiltonian H0\n=P2/2M+V/H20849Q/H20850. The damping Hamiltonian H1contains a\ncollection of environmental degrees of freedom /H20853xi,pi/H20854be-\nhaving like harmonic oscillators /H20851see Eq. /H2084914/H20850/H20852, which couple\nto the particle through the linear coupling term /H20858iCixiQbe-\ntween Qand the environmental variables xi. Here, Ciis the\ncoupling constant between xiandQ. The implication of the\ncoupling is twofold: /H20849i/H20850the coupling to the environment gen-\nerates damping, whose precise form depends on Ci,mi, and\n/H9275i. It is demonstrated in Ref. 32that the coupling generates\nthe simple damping of the form in Eq. /H2084912b/H20850ifCi,mi, and/H9275i\nsatisfy the following relation of the spectral function J/H20849/H9275/H20850:\nJ/H20849/H9275/H20850/H11013/H9266\n2/H20858\niCi2\nmi/H9275i/H9254/H20849/H9275−/H9275i/H20850=/H9253/H9275. /H2084913/H20850\n/H20849ii/H20850The coupling also modified the potential Vby generating\nan additional contribution − /H20858iCi2Q2/2mi/H9275i2. This implies that\nV/H20849x/H20850in Eq. /H2084912b/H20850should not be identified with V/H20849Q/H20850inH0\n/H20849even though the same symbol Vis used /H20850but should be iden-\ntified instead with the total potential that includes the contri-bution from the environmental coupling. If we express thetotal Hamiltonian Hin terms of the effective V/H20849x/H20850that ap-\npears in Eq. /H2084912b/H20850, it reads\nH=H\n0+H1, /H2084914a/H20850\nH0=P2\n2M+V/H20849Q/H20850, /H2084914b/H20850\nH1=/H20858\ni/H20875pi2\n2mi+1\n2mi/H9275i2/H20873xi+Ci\nmi/H9275i2Q/H208742/H20876./H2084914c/H20850\nBy identifying x/H20849t/H20850=/H20855Q/H20849t/H20850/H20856,p/H20849t/H20850=/H20855P/H20849t/H20850/H20856, the equations of\nmotion obtained from Eqs. /H2084913/H20850and/H2084914/H20850reproduce Eq. /H2084912/H20850.\nC. Generalization to the DW motion: /H9251term\nHere we aim to apply the Caldeira-Leggett approach to\nconstruct an effective Hamiltonian of the DW dynamics sub-ject to finite damping /H20849\n/H9251/HS110050/H20850. To simplify the problem, we\nfirst focus on a situation, where only /H9251is relevant and /H9252is\nirrelevant. This situation occurs if there is no current /H20849vs\n=0/H20850. Then Eq. /H208496/H20850reduces to\ndx\ndt=/H9251S\n2KMdp\ndt+p\nM, /H2084915a/H20850dp\ndt=−2/H9251KM\nSdx\ndt. /H2084915b/H20850\nNote that /H9252does not appear. Note also that these equations\nare slightly different from Eq. /H2084912/H20850, where a damping term is\ncontained only in the equation ofdp\ndt. However, in the equa-\ntions of the DW /H20851Eq./H2084915/H20850/H20852, damping terms appear not only in\nthe equation ofdp\ndt/H20851Eq./H2084915b/H20850/H20852but also in the equation ofdx\ndt\n/H20851Eq./H2084915a/H20850/H20852.\nThus the Caldeira-Leggett description in the preceding\nsection is not directly applicable and should be generalized.To get a hint, it is useful to recall the conjugate relation\nbetween QandP. The equations ofdQ\ndtanddP\ndtare obtained\nby differentiating Hwith respect to Pand − Q, respectively.\nOf course, it holds for /H20849xi,pi/H20850, also. Thus, one can obtain\nanother set of Heisenberg’s equation of motion by exchang-ing/H20849Q,x\ni/H20850↔/H20849−P,−pi/H20850. By this canonical transformation, the\nposition coupling /H20858iCixiQchanges to a momentum coupling\nterm, and the damping term in the equation ofdP\ndtis now in\nthat ofdQ\ndt. This mathematical relation that the momentum\ncoupling generates a damping term in the equation ofdQ\ndt\nmakes it reasonable to expect that the momentum coupling\n/H20858iDipiPis needed45to generate the damping in the equation\nfordQ\ndt. Here Diis the coupling constant between Pandpi.\nThe reason why, in the standard Caldeira-Leggett approach,the damping term appears only in Eq. /H2084912b/H20850is that Eq. /H2084914/H20850\ncontains only position coupling terms /H20858\niCixiQ. It can be eas-\nily verified that the implications of the momentum couplingare again twofold: /H20849i/H20850the coupling indeed introduces the\ndamping term in the equation ofdQ\ndt./H20849ii/H20850it modifies the DW\nmass. The mass renormalization arises from the fact that inthe presence of the momentum coupling /H20858\niDipiP, the kine-\nmatic momentum midxi\ndtof an environmental degree of free-\ndom xiis now given by /H20849pi+DimiP/H20850instead of pi. Then the\nterm/H20858i/H20851pi2\n2mi+DipiP/H20852can be decomposed into two pieces,\n/H20858i/H20849pi+DimiP/H208502\n2mi, which is the kinetic energy associated with xi,\nand/H20851−/H20858iDi2mi\n2/H20852P2. Note that the second piece has the same\nform as the DW kinetic termP2\n2M. Thus this second piece\ngenerates the renormalization of the DW mass. Due to thismass renormalization effect, Min Eq. /H2084915/H20850should be inter-\npreted as the renormalized mass that contains the contribu-tion from the environmental coupling. If MinH\n0in Eq. /H2084910/H20850\nis interpreted as the renormalized mass, the environmentHamiltonian H\n2for the DW dynamics becomes\nH2=/H20858\ni/H208751\n2mi/H20849pi+DimiP/H208502+1\n2mi/H9275i2/H20873xi+Ci\nmi/H9275i2Q/H208742/H20876.\n/H2084916/H20850\nHere,/H20858i/H20849pi+DimiP/H208502/2micoupling is equivalent to the origi-\nnal form /H20858i/H20849pi2/2mi+DipiP/H20850under the mass renormalization\n1/M→1/M−/H20858iDi2mi/2. Note that in H2, the collective co-\nordinates QandPof the DW couple to the environmental\ndegrees of freedom /H20853xi,pi/H20854through two types of coupling,\n/H20858iCixiQand/H20858iDipiP.\nFinally, one obtains the total Hamiltonian describing the\nDW motion in the absence of the current,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-4H=H0/H20841vs=0+H2=P2\n2M+/H20858\ni/H208751\n2mi/H20849pi+DimiP/H208502\n+1\n2mi/H9275i2/H20873xi+Ci\nmi/H9275i2Q/H208742/H20876. /H2084917/H20850\nNow, the renormalized mass Min the above equation is iden-\ntical to the mass in Eq. /H2084915/H20850. To make the physical meaning\nofxiclearer, we perform the canonical transformation,\nxi→−Ci\nmi/H9275i2xi,pi→−mi/H9275i2\nCipi. /H2084918/H20850\nDefining /H9253i=CiDi\n/H9275i2, and redefining a new miasmi/H20849new/H20850=Ci2\nmi/H9275i4,\nthe Hamiltonian becomes simpler as\nH=P2\n2M+/H20858\ni/H208751\n2mi/H20849pi−/H9253iP/H208502+1\n2mi/H9275i2/H20849xi−Q/H208502/H20876.\n/H2084919/H20850\nNow, the translational symmetry of the system and the physi-\ncal meaning of xibecome obvious.\nThe next step is to impose proper constraints on /H9253iandmi,\nso that the damping terms arising from Eq. /H2084919/H20850agree exactly\nwith those in Eq. /H2084915/H20850. For this purpose, it is convenient to\nintroduce Laplace transformed variables Q˜/H20849/H9261/H20850,P˜/H20849/H9261/H20850,x˜i/H20849/H9261/H20850,\np˜i/H20849/H9261/H20850, where Q˜/H20849/H9261/H20850=/H208480/H11009e−/H9261t/H20855Q/H20849t/H20850/H20856dt, and other transformed\nvariables are defined in a similar way. Then the variables x˜i\nandp˜ican be integrated out easily /H20849see Appendix B /H20850. After\nsome tedious but straightforward algebra, it is verified thatwhen the following three constraints on\n/H9253i,/H9275i,miare satis-\nfied for any positive /H9261,\n/H20858\ni/H9253i/H9275i2\n/H92612+/H9275i2=0 , /H2084920a/H20850\n/H20858\ni/H9253i2/H9261\nmi/H20849/H92612+/H9275i2/H20850=/H9251S\n2KM, /H2084920b/H20850\n/H20858\nimi/H9275i2/H9261\n/H92612+/H9275i2=2/H9251KM\nS, /H2084920c/H20850\nthe DW dynamics satisfies the following equation:\n/H20898/H9261 −1\nM−/H9251S/H9261\n2KM\n2/H9251KM\nS/H9261 /H9261/H20899/H20873Q˜\nP˜/H20874\n=/H20873/H20855Q/H208490/H20850/H20856\n/H20855P/H208490/H20850/H20856/H20874+/H20898−/H9251S\n2KM/H20855P/H208490/H20850/H20856\n2/H9251KM\nS/H20855Q/H208490/H20850/H20856/H20899, /H2084921/H20850\nwhich is nothing but the Laplace transformation of the DW\nequation /H20851Eq./H2084915/H20850/H20852if/H20855Q/H20856and/H20855P/H20856are identified with xandp.\nThus we verify that the Hamiltonian Hin Eq. /H2084919/H20850indeed\nprovides a generalized Caldeira-Leggett-type quantumHamiltonian for the DW motion. As a passing remark, we\nmention that in the derivation of Eq. /H2084921/H20850, the environmental\ndegrees of freedom at the initial moment /H20849t=0/H20850are assumed\nto be in their thermal equilibrium so that\n/H20855xi/H208490/H20850/H20856=/H20855Q/H208490/H20850/H20856, /H2084922a/H20850\n/H20855pi/H208490/H20850/H20856=/H9253i/H20855P/H208490/H20850/H20856. /H2084922b/H20850\nEquation /H2084922/H20850can be understood as follows. First, one ob-\ntains Eq. /H2084922/H20850by following Appendix D which describes the\nstatistical properties of Eq. /H2084919/H20850at high temperature. In Ap-\npendix D, /H20855xi/H208490/H20850−Q/H208490/H20850/H20856=/H20855pi/H208490/H20850−/H9253iP/H208490/H20850/H20856is reduced to an in-\ntegration of an odd function so it is shown to vanish. Thesecond way is probably easier to understand and does notrequire the classical limit or high-temperature limit. TheHamiltonian /H20851Eq./H2084919/H20850/H20852is symmetric under the canonical\ntransformation Q/H208490/H20850→−Q/H208490/H20850,P/H208490/H20850→−P/H208490/H20850,x\ni/H208490/H20850→−xi/H208490/H20850,\nand pi/H208490/H20850→−pi/H208490/H20850. Due to this symmetry, one obtains\n/H20855xi/H208490/H20850−Q/H208490/H20850/H20856=/H20855Q/H208490/H20850−xi/H208490/H20850/H20856and/H20855pi/H208490/H20850−/H9253iP/H208490/H20850/H20856=/H20855/H9253iP/H208490/H20850\n−pi/H208490/H20850/H20856, which lead to /H20855xi/H208490/H20850/H20856=/H20855Q/H208490/H20850/H20856and/H20855pi/H208490/H20850/H20856=/H9253i/H20855P/H208490/H20850/H20856,\nrespectively.\nHere physical origin of the momentum coupling /H20849/H9253/H20850be-\ntween the DW and environment deserves some discussion.Equation /H2084919/H20850is reduced to the original Caldeira-Leggett\nHamiltonian if\n/H9253i=0. However, Eq. /H2084920b/H20850implies that the\nmomentum coupling as well as the position coupling is in-dispensable to describe the Gilbert damping. To understandthe origin of the momentum coupling\n/H9253i, it is useful to recall\nthat since P/H11008/H9278/H11008/H20849tilting/H20850, one can interpret Pand Qas\ntransverse and longitudinal spin fluctuation of the DW state,respectively. /H20849See, for explicit mathematical relation, Appen-\ndix A. /H20850Thus, if there is rotational symmetry on spin interac-\ntion with the heat bath /H20849or environment /H20850, the existence of the\nposition coupling requires the existence of the momentumcoupling. Thus the appearance of the damping terms both inEqs./H2084915a/H20850and/H2084915b/H20850is natural in view of the rotational sym-\nmetry of the spin exchange interaction and also in view ofthe physical meaning of PandQas transverse and longitu-\ndinal spin fluctuations.\nD. Coupling with the spin current: /H9252term\nIn this section, we aim to construct a Caldeira-Leggett-\ntype effective quantum Hamiltonian that takes account of notonly\n/H9251but also /H9252. Since /H9252becomes relevant only when there\nexists finite spin current, we have to deal with situations withfinite current /H20849\nvs/HS110050/H20850. Then the system is notin thermal equi-\nlibrium.\nAs demonstrated in Eq. /H2084910/H20850, the spin current couples with\nthe DW linear momentum, i.e., vsP. Here, adiabatic velocity\nvsacts as the coupling constant proportional to spin current.\nThe spin current may also couple directly to the environmen-tal degrees of freedom. Calling this coupling constant\nv, one\nintroduces the corresponding coupling term /H20858ivpi. Later we\nfind that this coupling is crucial to account for nonzero /H9252.A t\nthis point we will not specify the value of v. Now, the total\neffective Hamiltonian in the presence of the spin current ob-tained by adding the coupling term /H20858\nivpito Eq. /H2084919/H20850. Then,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-5Htot=H+Hcurrent =P2\n2M+vsP+/H20858\nivpi\n+/H20858\ni/H208751\n2mi/H20849pi−/H9253iP/H208502+1\n2mi/H9275i2/H20849xi−Q/H208502/H20876./H2084923/H20850\nIn order to illustrate the relation between Eqs. /H208496/H20850and\n/H2084923/H20850, we consider a situation, where the current is zero until\nt=0 and turned on at t=0 to a finite value. This situation is\ndescribed by the following time-dependent Hamiltonian:\nHtot=P2\n2M+vs/H20849t/H20850P+/H20858\niv/H20849t/H20850pi\n+/H20858\ni/H208751\n2mi/H20849pi−/H9253iP/H208502+1\n2mi/H9275i2/H20849xi−Q/H208502/H20876,/H2084924/H20850\nwhere vs/H20849t/H20850=vs/H9008/H20849t/H20850andv/H20849t/H20850=v/H9008/H20849t/H20850. And /H9008/H20849t/H20850is\n/H9008/H20849t/H20850=/H208771fort/H110220,\n0fort/H110210./H20878 /H2084925/H20850\nTo make a quantitative comparison between Eqs. /H208496/H20850and\n/H2084924/H20850, one needs to integrate out environmental degrees of\nfreedom /H20853xi,pi/H20854, which requires one to specify their initial\nconditions. Since the system is in thermal equilibrium untilt=0, we may still impose the constraint in Eq. /H2084922/H20850to exam-\nine the DW dynamics for t/H110220. By following a similar pro-\ncedure as in Sec. II C and by using the constraints in Eq.\n/H2084920/H20850,\n46one finds that the effective Hamiltonian H/H20851Eq./H2084924/H20850/H20852\npredicts/H20855Q/H20849t/H20850/H20856=/H20855Q/H208490/H20850/H20856+vt+S\n2KM/H92512/H208491−e−2K/H9251t/S/H208491+/H92512/H20850/H20850\n/H11003/H20851/H9251/H20855P/H208490/H20850/H20856−M/H9251/H20849vs−v/H20850/H20852. /H2084926/H20850\nThis is exactly the same as Eq. /H208497/H20850if\n/H9252\n/H9251=v\nvs. /H2084927/H20850\nSo by identifying vwith vs/H9252//H9251, we obtain a Caldeira-\nLeggett-type effective quantum Hamiltonian of the DW dy-namics.\nOne needs to consider an external force on Eq. /H208496b/H20850/H20851or\nEq./H208494b/H20850/H20852when the translational symmetry of the system is\nbroken by some factors such as external magnetic field andmagnetic defects. To describe this force, one can add a posi-tion dependent potential V/H20849Q/H20850/H20849Ref. 47/H20850to Eq. /H2084924/H20850. Consid-\nering the Heisenberg’s equation, the potential V/H20849Q/H20850generates\nthe term − V\n/H11032/H20849Q/H20850in Eq. /H208496b/H20850.\nIII. IMPLICATIONS\nA. Insights on the physical meaning of /H9252\nEquation /H2084927/H20850provides insights on the physical meaning\nof/H9252./H9252depends largely on the coupling between the envi-\nronment and current, not on the damping form. Recallingthat\nvsdescribes the coupling between the current and the\nDW, we find that /H9252//H9251, which describes the asymptotic be-\nhavior of the DW motion, is the ratio between the current-magnetization /H20849DW in the present case /H20850coupling and\ncurrent-environment coupling. That is,\n/H9252\n/H9251=/H20849Coupling between the current and the environment /H20850\n/H20849Coupling between the current and the DW /H20850. /H2084928/H20850\nTo make the physical meaning of Eq. /H2084928/H20850more transpar-\nent, it is useful to examine consequences of the nonzero cou-pling\nvbetween the current and the environment. One of the\nimmediate consequences of the nonzero vappears in the ve-\nlocities of the environmental degrees of freedom. It can beverified easily that the initial velocities of environmental co-\nordinates are given by exactly\nv,/H20855x˙i/H208490/H20850/H20856=v. Recalling that\nthe terminal velocity of the DW, /H20855Q˙/H20849t/H20850/H20856approaches vs/H9252//H9251,\none finds from Eq. /H2084927/H20850that the terminal velocity of the DW\nis nothing but the environment velocity. This result is verynatural since the total Hamiltonian H\ntot/H20851Eq./H2084923/H20850/H20852is Galilean\ninvariant and the total mass of the environment /H20849or reservoir /H20850\nis much larger than the DW mass.48A very similar conclu-\nsion is obtained by Garate et al.29By analyzing the Kamber-\nsky mechanism,49which is reported50to be the dominant\ndamping mechanism in transition metals such as Fe, Co, Ni,they found that the ratio\n/H9252//H9251is approximately given by the\nratio between the drift velocity of the Kohn-Sham quasipar-\nticles and vs. Since the collection of Kohn-Sham quasiparti-cles play the role of the environment in case of the Kamber-\nsky mechanism, the result in Ref. 29is consistent with ours.\nIt is interesting to note that our calculation, which is largelyindependent of details of damping mechanism, reproducesthe result for the specific case.\n29This implies that the result\nin Ref. 29can be generalized if the drift velocity of the\nKohn-Sham quasiparticles is replaced by the general cou-pling constant\nvbetween the current and the environment.\nOur claim that the origin of /H9252is the direct coupling be-\ntween the current and environment has an interesting con-ceptual consistency with the work by Zhang and Li.\n4Zhang\nand Li derived the nonadiabatic term by introducing a spin-relaxation term in the equation of motion of the conduction\nelectrons. A clear consistency arises from generalizing thespin relaxation in Ref. 4to the coupling with environment in\nour work. In Ref. 4, Gilbert damping /H20849\n/H9251/H20850and the nonadia-\nbatic STT /H20849/H9252/H20850are identified as the spin relaxation of magne-\ntization and conduction electrons, respectively. Generalizingthe spin relaxation to environmental coupling,\n/H9251and/H9252areKYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-6now identified as the coupling of the environment with the\nmagnetization /H20849i.e., the DW in our model /H20850and the coupling\nof the environment with current, respectively. It is exactlyhow we identified\n/H9251and/H9252, and this gives the conceptual\nconsistency between our work and Ref. 4. As an additional\ncomment, while some magnitudes and origins of /H9252claimed\nin different references, such as Refs. 4and29, seem to be\nbased on completely independent phenomena, our work andinterpretation on\n/H9252provide a connection between them\nthrough the environmental degrees of freedom.\nB. Effect of environment on stochastic forces\nUntil now, our considerations has been limited to the evo-\nlution of the expectation values /H20855Q/H20849t/H20850/H20856and/H20855P/H20849t/H20850/H20856and thus\nthermal fluctuation effects have been ignored. In this section,we address the issue of thermal fluctuations. For this pur-pose, we need to go beyond the expectation values and so wederive the following operator equations from the Hamil-\ntonian Eq. /H2084923/H20850:\nQ˙=\nvs+/H9251S\n2KMP˙+P\nM+/H92571/H20849t/H20850, /H2084929a/H20850\nP˙=−2/H9251KM\nSQ˙+2/H9251KM\nSv+/H92572/H20849t/H20850, /H2084929b/H20850\nwhere\n/H92571/H20849t/H20850=/H20858\ni/H9253i/H9275i/H20873/H9004xisin/H9275it−/H9004pi\nmi/H9275icos/H9275it/H20874,/H2084930a/H20850\n/H92572/H20849t/H20850=/H20858\ni/H20849mi/H9275i2/H9004xicos/H9275it+/H9275i/H9004pisin/H9275it/H20850./H2084930b/H20850\nHere/H9004xi/H11013xi/H208490/H20850−Q/H208490/H20850and/H9004pi/H11013pi/H208490/H20850−/H9253iP/H208490/H20850. The deriva-\ntion of Eqs. /H2084929/H20850and/H2084930/H20850utilizes constraints Eqs. /H2084920/H20850and\n/H2084927/H20850. We remark that the result in Sec. II D can be recovered\nfrom Eqs. /H2084929/H20850and/H2084930/H20850by taking the expectation values of\nthe operators. When Eq. /H2084929/H20850is compared to Eq. /H208496/H20850,i ti s\nevident that /H92571/H20849t/H20850and/H92572/H20849t/H20850defined in Eq. /H2084930/H20850carry the\ninformation about the thermal noise. It is easy to verify thatthe expectation values of\n/H92571/H20849t/H20850and/H92572/H20849t/H20850vanish, thus repro-\nducing the results in the earlier section. Here it should benoticed that Eq. /H2084930/H20850relates\n/H92571/H20849t/H20850and/H92572/H20849t/H20850in the nonequi-\nlibrium situations /H20849after the current is turned on or t/H110220/H20850to\nthe operators /H9004xiand/H9004pi, which are defined in the equilib-\nrium situation /H20849right before the current is turned on or t=0/H20850.\nThus by combining Eq. /H2084930/H20850with the equilibrium noise char-\nacteristics of /H9004xiand/H9004pi, we can determine the thermal\nnoise characteristic in the nonequilibrium situation /H20849t/H110220/H20850.\nTo extract information about the noise, one needs to\nevaluate the correlation functions /H20855/H20853/H9257i/H20849t/H20850,/H9257j/H20849t/H20850/H20854/H20856 /H20849i,j=1,2/H20850,\nwhere /H20853,/H20854denotes the anticommutator. Due to the relations in\nEq./H2084930/H20850, the evaluation of the correlation function reduces to\nthe expectation value evaluation of the operator products/H20853x\ni/H208490/H20850,pj/H208490/H20850/H20854,xi/H208490/H20850xj/H208490/H20850, and pi/H208490/H20850pj/H208490/H20850in the equilibrium\nsituation governed by the equilibrium Hamiltonian /H20851Eq.\n/H2084919/H20850/H20852.\nIn the classical limit /H20849/H6036→0, see the next paragraph to find\nout when the classical limit is applicable /H20850, Eq./H2084919/H20850is just acollection of independent harmonic oscillators of /H20853/H9004xi,/H9004pi/H20854.\nHence, the equipartition theorem determines their correla-tions,\n/H20855/H9004x\ni/H20856=/H20855/H9004pi/H20856=/H20855/H9004xi/H9004pi/H20856=0 , /H2084931a/H20850\n/H20855/H9004xi/H9004xj/H20856=kBT\nmi/H9275i2/H9254ij, /H2084931b/H20850\n/H20855/H9004pi/H9004pj/H20856=mikBT/H9254ij. /H2084931c/H20850\nEquation /H2084920/H20850and/H2084931/H20850give the correlations of /H92571/H20849t/H20850and\n/H92572/H20849t/H20850. After some algebra, one straightforwardly gets\n/H20855/H9257i/H20849t/H20850/H20856=0 , /H2084932a/H20850\n/H20855/H92571/H20849t/H20850/H92572/H20849t/H11032/H20850/H20856=0 , /H2084932b/H20850\n/H20855/H92571/H20849t/H20850/H92571/H20849t/H11032/H20850/H20856=/H9251S\n2KMkBT/H9254/H20849t−t/H11032/H20850, /H2084932c/H20850\n/H20855/H92572/H20849t/H20850/H92572/H20849t/H11032/H20850/H20856=2/H9251KM\nSkBT/H9254/H20849t−t/H11032/H20850. /H2084932d/H20850\nThese relations are consistent with Eq. /H208495/H20850when/H92571/H20849t/H20850and\n/H92572/H20849t/H20850in Eq. /H2084930/H20850are identified with those in Eq. /H208496/H20850. Thus\nthey confirm that the relations /H20851Eq./H2084932/H20850/H20852assumed in many\npapers9–13indeed hold rather generally in the regime where\nthe tilting angle remains sufficiently smaller than /H9266/4.\nNext we consider the regime where the condition of the\nclassical limit is valid. Since statistical properties of the sys-\ntem at finite temperature is determined bykBT\n/H6036, the classical\nlimit/H20849/H6036→0/H20850is equivalent to the high-temperature limit /H20849T\n→/H11009/H20850. Thus, in actual experimental situations, the above cor-\nrelation relations, Eq. /H2084932/H20850, will be satisfied at high tempera-\nture. In this respect, we find that most experimental situa-tions belong to the high-temperature regime. See AppendixD for the estimation of the “threshold” temperature, abovewhich Eq. /H2084932/H20850is applicable. In Appendix D, the correlations\nin the high temperatures are derived more rigorously.\nFinally we comment briefly on the low-temperature quan-\ntum regime. In this regime, one cannot use the equipartitiontheorem since the system is not composed of independentharmonic oscillators, that is, /H20851/H9004x\ni,/H9004pj/H20852=i/H6036/H20849/H9254ij+/H9253j/H20850. Note\nthat the commutator contains an additional term i/H6036/H9253j. Here,\nthe additional term i/H6036/H9253jcomes from the commutator /H20851−Q,\n−/H9253jP/H20852. Then, Eq. /H2084932/H20850, which is assumed in other papers,9–13\nis not guaranteed any more.\nIV . CONCLUSION\nIn this paper, we examine the effect of finite current on\nthermal fluctuation of current-induced DW motion by con-structing generalized Caldeira-Leggett-type Hamiltonian ofthe DW dynamics, which describes not only energy-conserving dynamics processes but also the Gilbert dampingand STT. Unlike the classical damping worked out by Cal-deira and Leggett,\n32the momentum coupling is indispensable\nto describe the Gilbert damping. This is also related to theTHERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-7rotational symmetry of spin-interaction nature. It is demon-\nstrated that the derived Caldeira-Leggett-type quantum-mechanical Hamiltonian reproduces the well-known DWequations of motion.\nOur Hamiltonian also illustrates that the nonadiabatic\nSTT is closely related with the coupling of the spin current tothe environment. Thus, the environmental degrees of free-dom are responsible for both the Gilbert damping /H20849\n/H9251/H20850and the\nnonadiabatic STT /H20849/H9252/H20850. By this process, the ratio of /H9252and/H9251\nwas derived to be the ratio of current-DW coupling and\ncurrent-environment coupling. The nonadiabatic term isnothing but the result of the direct coupling between thecurrent and environment in our theory.\nBy using the Calderia-Leggett-type Hamiltonian, which\ndescribes the time evolution of the system, we obtained theexpression of stochastic forces caused by thermal noise inthe presence of the finite current. By calculating the equilib-rium thermal fluctuation at high temperature, we verify thatwhen j\npis sufficiently smaller than the intrinsic critical den-\nsity, jpdoes not modify the correlation relations of thermal\nnoise unless the temperature is extremely low. The upperbound of the critical temperature, below which the aboveconclusion does not apply, is obtained by reexamining thesystem with Feynman path integral. The bound is muchlower than the temperature in most experimental situations.\nLastly we remark that the Joule heating\n51is an important\nfactor that affects the thermal fluctuation field since it raisesthe temperature of the nanowire. The degree of the tempera-ture rise depends on the thermal conductivities and heat ca-pacities of not only the nanowire but also its surroundingmaterials such as substrate layer materials of the nanowire.Such factors are not taken into account in this paper. Simul-taneous account of the Joule heating dynamics and the ther-mal fluctuation field /H20849in the presence of current /H20850goes beyond\nthe scope of the paper and may be a subject of future re-search.\nACKNOWLEDGMENTS\nWe acknowledge critical comment by M. Stiles, who\npointed out the importance of the momentum coupling andinformed us of Ref. 29. This work was financially supported\nby the NRF /H20849Grants No. 2007-0055184, No. 2009-0084542,\nand No. 2010-0014109 /H20850and BK21. K.W.K. acknowledges\nthe financial support by the TJ Park.\nAPPENDIX A: EFFECTIVE HAMILTONIAN OF THE DW\nMOTION FROM 1D s-dMODEL (Ref. 52)\nThe starting point is 1D s-dmodel,\nHs-d=−J/H20858\niS/H6023i·S/H6023i+1−A/H20858\ni/H20849S/H6023i·zˆ/H208502+K/H20858\ni/H20849S/H6023i·yˆ/H208502+HcS,\n/H20849A1/H20850\nas mentioned in Sec. II A.\nIn order to consider the DW dynamics, one first introduce\nthe classical DW profile initially given by\n/H20855S/H6023i·xˆ/H20856=Ssin/H9258/H20849zi/H20850, /H20849A2a/H20850/H20855S/H6023i·yˆ/H20856=0 , /H20849A2b/H20850\n/H20855S/H6023i·zˆ/H20856=Scos/H9258/H20849zi/H20850, /H20849A2c/H20850\nwhere ziis the position of the ith localized spin, and /H9258/H20849z/H20850\n=2 cot−1e−/H208812A/Ja2/H20849z−q/H20850. Here qis the classical position of the\nDW. Small quantum fluctuations of spins on top of the clas-sical DW profile can be described by the Holstein-Primakoffboson operator b\ni, to describe magnon excitations. Kim et\nal.43found eigenmodes of these quantum fluctuations in the\npresence of the classical DW background, which amount toquantum mechanical version of the classical vibration eigen-modes in the presence of the DW background reported longtime ago by Winter.\n53The corresponding eigenstates of this\nHamiltonian are composed of spin-wave states with the finite\neigenenergy Ek=/H20881/H20849JSa2k2+2AS/H20850/H20849JSa2k2+2AS+2KS/H20850\n/H20849/H113502S/H20881A/H20849A+K/H20850/H20850and so-called bound magnon states with\nzero energy Ew=0. Here, kis the momentum of spin wave\nstates and ais the lattice spacing between two neighboring\nspins. Let akandbwdenote proper linear combinations of bi\nandbi†, which represent the boson annihilation operators of\nfinite-energy spin-wave states and zero-energy bound mag-non states, respectively. In terms of these operators, Eq. /H208498/H20850\nreduces to\nH\ns-d=P2\n2M+/H20858\nkEkak†ak+HcS, /H20849A3/H20850\nwhere higher-order processes describing magnon-magnon in-\nteractions are ignored. Here Mis the so-called Döring\nmass,44defined as M=/H60362\nK/H208812A\nJa4, and Pis defined as\n−i/H6036/H208492AS2\nJa4/H208501/4/H20849bw†−bw/H20850. According to Ref. 43,Pis a translation\ngenerator of the DW position, that is, exp /H20849iPq 0//H6036/H20850shifts the\nDW position by q0. Thus Pcan be interpreted as a canonical\nmomentum of the DW translational motion. The first term inEq./H20849A3/H20850, which amounts to the kinetic energy of the DW\ntranslational motion, implies that Mis the DW mass. We\nidentify this Mwith the undetermined constant Min Eq. /H208496/H20850.\nAccording to Ref. 43,Pis also proportional to the degree of\nthe DW tilting, that is, /H20849b\nw†−bw/H20850/H11008Siy.\nIn the adiabatic limit, that is, when the DW width /H9261is\nsufficiently large in view of the electron dynamics, the re-maining term H\ncScan be represented in a simple way in\nterms of the bound magnon operators and the adiabatic ve-locity of the DW,\n20,43\nHcS=vsP. /H20849A4/H20850\nThen the effective s-dHamiltonian of the DW motion be-\ncomes\nHs-d=P2\n2M+vsP+/H20858\nkEkak†ak. /H20849A5/H20850\nNote that the bound magnon part and the spin-wave part are\ncompletely decoupled in Eq. /H20849A5/H20850since Pcontains only the\nbound magnon operators, which commute with the spin-wave operators.\nThe DW position operator should satisfy the following\ntwo properties: geometrical relation /H20855Q/H20856−q=a\n2S/H20858i/H20855S/H6023i·zˆ/H20856andKYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-8canonical relation /H20851Q,P/H20852=i/H6036. Then, one can show that Q\n=q−/H20849Ja4\n32AS2/H208501/4/H20849bw†+bw/H20850satisfies these two properties. Note\nthatQis expressed in terms of the bound magnon operators.\nThen as far as the Heisenberg equations of motion for Qand\nPare concerned, the last term in Eq. /H20849A5/H20850does not play any\nrole. This term will be ignored from now on. Thus, the ef-fective Hamiltonian for the DW motion is reduced to\nH\n0=P2\n2M+vsP /H20849A6/H20850\nso we got Eq. /H2084910/H20850, the effective Hamiltonian of the DW\nmotion.\nAPPENDIX B: SOLUTION FOR A GENERAL QUADRATIC\nDAMPING\nThis section provides the solution of the equation of mo-\ntion for a general quadratic damping. This is applicable notonly for the generalized Caldeira-Leggett description in thispaper but also for any damping type which quadraticallyinteracts with the DW.\nIn general, let us consider a general quadratic damping\nHamiltonian,\nH=P\n2\n2M+vsP+/H20858\ni/H9273iTAi/H9273i, /H20849B1/H20850\nwhere /H9273i=/H20849QPx ipi/H20850T, and Aii sa4/H110034 Hermitian matrix.\nNow, one straightforwardly gets the corresponding coupledequations,\ndQ\ndt=P\nM+vs+/H20858\ni/H20849B21iQ+B22iP+B23ixi+B24ipi/H20850,\n/H20849B2a/H20850\ndP\ndt=−/H20858\ni/H20849B11iQ+B12iP+B13ixi+B14ipi/H20850,/H20849B2b/H20850\ndxi\ndt=B41iQ+B42iP+B43ixi+B44ipi, /H20849B2c/H20850\ndpi\ndt=−/H20849B31iQ+B32iP+B33ixi+B34ipi/H20850./H20849B2d/H20850\nHere, Bii sa4 /H110034 real symmetric matrix defined as Bi\n=2 Re /H20851Ai/H20852, and Bjkiis the element of Biinjth row and kth\ncolumn.\nWith the Laplace transform of the expectation values of\neach operator, for example,\nQ˜/H20849/H9261/H20850/H11013L/H20851Q/H20849t/H20850/H20852/H20849/H9261/H20850=/H20885\n0/H11009\n/H20855Q/H20849t/H20850/H20856e−/H9261tdt, /H20849B3/H20850\nthe set of coupled equations transforms as\n/H9261Q˜−/H20855Q/H208490/H20850/H20856=P˜\nM+vs\n/H9261+/H20858\ni/H20849B21iQ˜+B22iP˜+B23ix˜i+B24ip˜i/H20850,\n/H20849B4a/H20850/H9261P˜−/H20855P/H208490/H20850/H20856=−/H20858\ni/H20849B11iQ˜+B12iP˜+B13ix˜i+B14ip˜i/H20850,\n/H20849B4b/H20850\n/H9261x˜i−/H20855xi/H208490/H20850/H20856=B41iQ˜+B42iP˜+B43ix˜i+B44ip˜i,/H20849B4c/H20850\n/H9261p˜i−/H20855pi/H208490/H20850/H20856=−/H20849B31iQ˜+B32iP˜+B33ix˜i+B34ip˜i/H20850.\n/H20849B4d/H20850\nRewriting these in matrix forms, the equations become sim-\npler as\n/H9261/H20873Q˜\nP˜/H20874−/H20898/H20855Q/H208490/H20850/H20856+vs\n/H9261\n/H20855P/H208490/H20850/H20856/H20899=/H20902/H2089801\nM\n00/H20899+/H20858\ni/H20873B21iB22i\n−B11i−B12i/H20874/H20903\n/H11003/H20873Q˜\nP˜/H20874+/H20858\ni/H20873B23iB24i\n−B13i−B14i/H20874\n/H11003/H20873x˜i\np˜i/H20874, /H20849B5a/H20850\n/H9261/H20873x˜i\np˜i/H20874−/H20873/H20855xi/H208490/H20850/H20856\n/H20855pi/H208490/H20850/H20856/H20874=/H20873B41iB42i\n−B31i−B32i/H20874/H20873Q˜\nP˜/H20874\n+/H20873B43iB44i\n−B33i−B34i/H20874/H20873x˜i\np˜i/H20874.\n/H20849B5b/H20850\nFrom Eq. /H20849B5b/H20850, one can calculate /H20849x˜ip˜i/H20850Tin terms of Q˜and\nP˜,\n/H20873x˜i\np˜i/H20874=/H20873/H9261−B43i−B44i\nB33i/H9261+B34i/H20874−1/H20873B41iB42i\n−B31i−B32i/H20874/H20873Q˜\nP˜/H20874\n+/H20873/H9261−B43i−B44i\nB33i/H9261+B34i/H20874−1/H20873/H20855xi/H208490/H20850/H20856\n/H20855pi/H208490/H20850/H20856/H20874. /H20849B6/H20850\nFrom Eqs. /H20849B5a/H20850and/H20849B6/H20850, one finally gets the equation of\n/H20849Q˜P˜/H20850T,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-9/H20900/H20898/H9261−1\nM\n0/H9261/H20899−/H20858\ni/H20877/H20873B21iB22i\n−B11i−B12i/H20874+/H20873B23iB24i\n−B13i−B14i/H20874/H20873/H9261−B43i−B44i\nB33i/H9261+B34i/H20874−1/H20873B41iB42i\n−B31i−B32i/H20874/H20878/H20901/H20873Q˜\nP˜/H20874\n=/H20898/H20855Q/H208490/H20850/H20856+vs\n/H9261\n/H20855P/H208490/H20850/H20856/H20899+/H20858\ni/H20873B23iB24i\n−B13i−B14i/H20874/H20873/H9261−B43i−B44i\nB33i/H9261+B34i/H20874−1/H20873/H20855xi/H208490/H20850/H20856\n/H20855pi/H208490/H20850/H20856/H20874. /H20849B7/H20850\nInverting the matrix in front of /H20849Q˜P˜/H20850T, one can get the solu-\ntion of /H20849Q˜P˜/H20850T. Then, finally, the solution /H20849/H20855Q/H20856/H20855P/H20856/H20850Tis ob-\ntained by the inverse Laplace transform of /H20849Q˜P˜/H20850T,\n/H20873/H20855Q/H20849t/H20850/H20856\n/H20855P/H20849t/H20850/H20856/H20874=L−1/H20875/H20873Q˜/H20849/H9261/H20850\nP˜/H20849/H9261/H20850/H20874/H20876. /H20849B8/H20850\nAPPENDIX C: SOLUTION OF EQ. ( 24)\nIn the special case that current is applied at t=0,/H9008/H20849t/H20850in\nEq./H2084925/H20850becomes Heaviside step function. This is the case\nwe are interested in. In a real DW system, the DW velocityjumps from 0 to a finite value at the moment that the spincurrent starts to be applied. This jumping comes from thediscontinuity in Eq. /H2084925/H20850which makes the Hamiltonian dis-\ncontinuous. Right before the current is applied, the DW re-mains on the stable /H20849or equilibrium /H20850state described by Eq.\n/H2084922/H20850.\nSuppose that Eq. /H2084920a/H20850also holds for /H9261=0. Then, Eq. /H2084924/H20850\ntransforms as /H20849up to constant /H20850\nH\ntot=P2\n2M+vs/H20849t/H20850P+/H20858\ni/H208751\n2mi/H20849pi−/H9253iP+miv/H20849t/H20850/H208502/H20876\n+/H20858\ni1\n2mi/H9275i2/H20849xi−Q/H208502. /H20849C1/H20850\nPerforming the canonical transform pi→pi−miv/H20849t/H20850, one can\ntransform this Hamiltonian in the form of Eq. /H20849B1/H20850,\nHtot=P2\n2M+vs/H20849t/H20850P+/H20858\ni/H208751\n2mi/H20849pi−/H9253iP/H208502+1\n2mi/H9275i2/H20849xi−Q/H208502/H20876.\nHere, one of the constraints Eq. /H2084920a/H20850is generalized to hold\neven for /H9261=0, so that /H20858i/H9253i=0. Note that the discontinuity due\ntov/H20849t/H20850is absorbed in the new pi. Thus, Eq. /H2084922b/H20850should be\nwritten as\n/H20855pi/H208490+/H20850/H20856=/H20855pi/H208490−/H20850/H20856+miv=/H9253i/H20855P/H208490/H20850/H20856+miv./H20849C2/H20850\nThe initial condition of xiis the same as Eq. /H2084922a/H20850. Now,\nusing these initial conditions and Eqs. /H20849B7/H20850and/H20849B8/H20850under\nthe constraints in Eq. /H2084920/H20850, one gets the solution of this sys-\ntem as Eq. /H2084926/H20850.APPENDIX D: CORRELATIONS OF STOCHASTIC\nFORCES AT HIGH TEMPERATURE\nThis section provides the quantum derivation of correla-\ntion relations of stochastic forces at high temperature. Theclassical correlation relations in Eq. /H2084932/H20850are valid quantum\nmechanically at high temperature. Since Eq. /H2084931/H20850implies Eq.\n/H2084932/H20850, it suffices to show Eq. /H2084931/H20850in this section. The basic\nstrategy is studying statistical properties of the HamiltonianEq./H2084919/H20850/H20849under quadratic potential bQ\n2/H2085054by the Feynman\npath integral along the imaginary-time axis. The Feynmanpath integral of a system described by a quadratic Lagrang-ian is proportional to the exponential of the action valueevaluated at the classical solution. Hence, the key point ofthe procedure is to get the classical solution with imaginarytime.\n1. General relations\na. Classical action under high-temperature limit\nDefine a column vector /H9273=/H20849Qx1x2¯/H20850T. Let the Euclidean\nLagrangian of the system be LE=1\n2/H9273˙TA/H9273˙+1\n2/H9273B/H9273, where A\nandBare symmetric matrices. /H20849The symbols “ A” and “ B” are\nnot the same as those in Appendix B. /H20850Explicitly, L\n=1\n2/H20858nmx˙nAnmx˙m+1\n2/H20858nmxnBnmxm. Here x0/H11013Q./H11509LE\n/H11509x˙n=/H20858mAnmx˙m\n=A/H9273˙and/H11509LE\n/H11509xn=/H20858mBnmxm=B/H9273lead to the classical equation of\nmotion,\nA/H9273¨=B/H9273. /H20849D1/H20850\nThe classical action value Sc/H20849evaluated at the classical path /H20850\nis then, Sc=/H208480/H9270LEdt=1\n2/H208480/H9270/H20849/H9273˙TA/H9273˙+/H9273B/H9273/H20850dt=1\n2/H9273TA/H9273˙/H208410/H9270\n+/H208480/H9270/H20849−/H9273TA/H9273¨+/H9273B/H9273/H20850dt=1\n2/H9273TA/H9273˙/H208410/H9270. Here, /H9270=/H6036/kBT. Now, the\nonly thing one needs is to find /H9273˙at boundary points.\nIn the case of Eq. /H2084919/H20850,Ais invertible. Hence, the equa-\ntion becomes /H9273¨=A−1B/H9273. Suppose that A−1Bis diagonaliz-\nable, that is A−1B=C−1DC. Here Dnm=/H9261n/H9254nmis diagonal ma-\ntrix and /H9261nisnth eigenvalue of A−1B. Define a new vector\n/H9264=C/H9273. Finally, we get the equation,\n/H9264¨=/H20898/H926100¯\n0/H92611¯\n]]/GS/H20899/H9264. /H20849D2/H20850\nImposing the boundary condition /H9273/H208490/H20850=/H9273i,/H9273/H20849/H9270/H20850=/H9273fand de-\nfining the corresponding /H9264i=C/H9273i,/H9264f=C/H9264f, then one gets the\nsolution of /H9264and its derivative straightforwardly,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-10/H9264n=/H9264fn+/H9264in\n2cosh/H20881/H9261n/H20873t−/H9270\n2/H20874\ncosh/H20881/H9261n/H9270\n2+/H9264fn−/H9264in\n2sinh/H20881/H9261n/H20873t−/H9270\n2/H20874\nsinh/H20881/H9261n/H9270\n2,\n/H20849D3/H20850\n/H9264˙n=/H20881/H9261n/H20900/H9264fn+/H9264in\n2sinh/H20881/H9261n/H20873t−/H9270\n2/H20874\ncosh/H20881/H9261n/H9270\n2\n+/H9264fn−/H9264in\n2cosh/H20881/H9261n/H20873t−/H9270\n2/H20874\nsinh/H20881/H9261n/H9270\n2/H20901. /H20849D4/H20850\nNow, /H9264˙at boundary points are obtained as\n/H9264˙n/H208490/H20850=/H20881/H9261n/H20873−/H9264fn+/H9264in\n2tanh/H20881/H9261n/H9270\n2+/H9264fn−/H9264in\n2coth/H20881/H9261n/H9270\n2/H20874,\n/H20849D5/H20850\n/H9264˙n/H20849/H9270/H20850=/H20881/H9261n/H20873/H9264fn+/H9264in\n2tanh/H20881/H9261n/H9270\n2+/H9264fn−/H9264in\n2coth/H20881/H9261n/H9270\n2/H20874.\n/H20849D6/H20850\nIf/H20881/H20841/H9261n/H20841/H9270\n2=/H20881/H20841/H9261n/H20841/H6036\n2kBT/H112701, tanh/H20881/H9261n/H9270\n2/H11015/H20881/H9261n/H9270\n2. Then,\n/H9264˙n/H208490/H20850/H11015−/H9264fn+/H9264in\n2/H9261n/H9270\n2+/H9264fn−/H9264in\n/H9270, /H20849D7/H20850\n/H9264˙n/H20849/H9270/H20850/H11015/H9264fn+/H9264in\n2/H9261n/H9270\n2+/H9264fn−/H9264in\n/H9270. /H20849D8/H20850\nIn matrix form,\n/H9264˙/H208490/H20850/H11015−D/H9264f+/H9264i\n2/H9270\n2+/H9264f−/H9264i\n/H9270=−DC/H9273f+/H9273i\n2/H9270\n2+C/H9273f−/H9273i\n/H9270,\n/H20849D9/H20850\n/H9264˙/H20849/H9270/H20850/H11015D/H9264f+/H9264i\n2/H9270\n2+/H9264f−/H9264i\n/H9270=DC/H9273f+/H9273i\n2/H9270\n2+C/H9273f−/H9273i\n/H9270.\n/H20849D10/H20850\nUsing A−1B=C−1DC, it leads to\n/H9273˙/H208490/H20850/H11015−A−1B/H9273f+/H9273i\n2/H9270\n2+/H9273f−/H9273i\n/H9270, /H20849D11/H20850\n/H9273˙/H20849/H9270/H20850/H11015A−1B/H9273f+/H9273i\n2/H9270\n2+/H9273f−/H9273i\n/H9270. /H20849D12/H20850\nFinally one can obtain the classical action,Sc=1\n2/H9273TA/H9273˙/H208410/H9270=/H20873/H9273f+/H9273i\n2/H20874T\nB/H20873/H9273f+/H9273i\n2/H20874/H9270\n2\n+/H20873/H9273f−/H9273i\n2/H20874T\nA/H20873/H9273f−/H9273i\n2/H208742\n/H9270. /H20849D13/H20850\nThis is valid even if some eigenvalues are zero. /H20849By taking\nlimit of /H9261i→0, cosh and sinh becomes constant and linear,\nrespectively. /H20850\nb. Propagator and its derivatives\nThe propagator is given by the Feynman path integral,\nK/H20849/H9273f,/H9273i;/H9270/H20850=/H20855/H9273f/H20841e−H/kBT/H20841/H9273i/H20856=/H20848D/H9273e−/H20848LEdt//H6036, where D/H9273=/H20863iDxi.\nFor quadratic Lagrangian, it is well known that/H20848D\n/H9273e−/H20848LEdt/H6036=F/H20849/H9270/H20850e−Sc//H6036. Here F/H20849/H9270/H20850is a smooth function de-\npendent on /H9270only.\nNow we aim to calculate K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850. It is easy to\nobtain the corresponding classical action by replacing /H9273f\n=/H9273i+/H9254/H9273in Eq. /H20849D13/H20850,\nSc/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=/H9270\n2/H9273iTB/H9273i+/H9270\n2/H9273iTB/H9254/H9273+/H9270\n8/H9254/H9273TB/H9254/H9273\n+1\n2/H9270/H9254/H9273TA/H9254/H9273. /H20849D14/H20850\nThen, K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850is/H20849up to second order of /H9254/H9273/H20850,\nK/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=F/H20849/H9270/H20850e−Sc//H6036=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i\n/H11003/H208751−1\n/H6036/H20873/H9270\n2/H9273iTB/H9254/H9273+/H9270\n8/H9254/H9273TB/H9254/H9273\n+1\n2/H9270/H9254/H9273TA/H9254/H9273/H20874+1\n2/H60362/H20873/H9270\n2/H9273iTB/H9254/H9273/H208742/H20876.\nZeroth order: F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i.\nFirst order:−/H9270\n2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H9273iTB/H9254/H9273\n=−/H9270\n2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20858nmxinBnm/H9254xm.\nSecond order: F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1\n/H6036/H20873/H9270\n8/H9254/H9273TB/H9254/H9273\n+1\n2/H9270/H9254/H9273TA/H9254/H9273/H20874+1\n2/H60362/H20873/H9270\n2/H9273iTB/H9254/H9273/H208742/H20878\n=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1\n2/H6036/H20858\nnm/H9254xn/H20873/H9270\n4Bnm+1\n/H9270Anm/H20874/H9254xm\n+/H92702\n8/H60362/H20873/H20858\nklmnxikBkn/H9254xnxilBlm/H9254xm/H20874/H20878. /H20849D15/H20850\nBy the relation, K/H20849/H9273i+/H9254/H9273,/H9273i;/H9270/H20850=K/H20849/H9273i,/H9273i;/H9270/H20850+/H20858m/H11509K\n/H11509xfm/H9254xm\n+/H20858nm1\n2/H115092K\n/H11509xfn/H11509xfm/H9254xn/H9254xm+O/H20849/H9254/H92733/H20850,\nK/H20849/H9273i,/H9273i;/H9270/H20850=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i, /H20849D16/H20850THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-11/H20879/H11509K\n/H11509xfm/H20879\n/H9273i=/H9273f=−/H9270\n2/H6036F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20858\nnBnmxin,/H20849D17/H20850\n/H20879/H115092K\n/H11509xfn/H11509xfm/H20879\n/H9273i=/H9273f=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1\n/H6036/H20873/H9270\n4Bnm+1\n/H9270Anm/H20874\n+/H92702\n4/H60362/H20873/H20858\nklBknxikBlmxil/H20874/H20878\n=F/H20849/H9270/H20850e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273i/H20877−1\n/H6036/H20873/H9270\n4Bnm+1\n/H9270Anm/H20874\n+/H92702\n4/H60362/H20873/H20858\nkBknxik/H20874/H20873/H20858\nkBkmxik/H20874/H20878./H20849D18/H20850\nc. Correlations\nStatistical average of an operator Ais given byTr/H20849Ae−H/kBT/H20850\nTr/H20849e−H/kBT/H20850.\nWhat we want to find are the averages of /H9004xn/H9004xm,/H9004pn/H9004pm,\nand/H20853/H9004xn,/H9004pm/H20854for/H9004xn/H11013xn−Qand/H9004pn/H11013pn−/H9253nP,\nTr/H20849/H9004xn/H9004xme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004xn/H9004xme−H/kBT/H20841/H9273i/H20856\n=/H20885d/H9273i/H20849xin−Qi/H20850/H20849xim−Qi/H20850/H20855/H9273i/H20841e−H/kBT/H20841/H9273i/H20856\n=/H20885d/H9273i/H20849xin−Qi/H20850/H20849xim−Qi/H20850K/H20849/H9273i,/H9273i;/H9270/H20850,\n/H20849D19/H20850\nTr/H20849/H9004pn/H9004pme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004pn/H9004pme−H/kBT/H20841/H9273i/H20856\n=−/H60362/H20885/H20879d/H9273i/H20873/H11509\n/H11509xfn−/H9253n/H11509\n/H11509Qf/H20874\n/H11003/H20873/H11509\n/H11509xfm−/H9253m/H11509\n/H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879\n/H9273i=/H9273f,\n/H20849D20/H20850\nTr/H20849/H9004xn/H9004pme−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841/H9004xn/H9004pme−H/kBT/H20841/H9273i/H20856\n=−i/H6036/H20885/H20879d/H9273i/H20849xin−Qi/H20850/H20873/H11509\n/H11509xfm\n−/H9253m/H11509\n/H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879\n/H9273i=/H9273f,\n/H20849D21/H20850\nTr/H20849e−H/kBT/H20850=/H20885d/H9273i/H20855/H9273i/H20841e−H/kBT/H20841/H9273i/H20856=/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850,\n/H20849D22/H20850\nwhere d/H9273i=/H20863ndxin.2. Correlations under quadratic potential\nUnder potential bQ2, the matrices AandBcorresponding\nthe Hamiltonian Eq. /H2084919/H20850are\nA=/H20898MM /H92531 M/H92532 ¯\nM/H92531M/H925312+m1M/H92531/H92532¯\nM/H92532M/H92532/H92531M/H925322+m2¯\n]] ] /GS /H20899,/H20849D23/H20850\nB=/H20898b+/H20858\nnmn/H9275n2\n−m1/H927512−m2/H927522¯\n−m1/H927512m1/H927512 0 ¯\n−m2/H9275220 m2/H927522¯\n]] ] /GS/H20899./H20849D24/H20850\nThen, e−/H20849/H9270/2/H6036/H20850/H9273iTB/H9273iis written as e−/H20849/H9270/2/H6036/H20850/H20851/H20858nmn/H9275n2/H20849Qi−xin/H208502+bQi2/H20852.\na. x-x correlations\nSince K/H20849/H9273,/H9273;/H9270/H20850is an even function of /H20849xn−Qi/H20850,i ti s\ntrivial that Tr /H20849/H9004xn/H9004xme−H/kBT/H20850=0 unless n=m.\nFor n=m,T r /H20849/H9004xn2e−H/kBT/H20850=/H20848d/H9273i/H20849xin−Qi/H208502K/H20849/H9273i,/H9273i;/H9270/H20850.\nThus,\nTr/H20849/H9004xn2e−H/kBT/H20850\nTr/H20849e−H/kBT/H20850=/H20885dxin/H20849xin−Qi/H208502e−/H20849/H9270/2/H6036/H20850mnwn2/H20849Qi−xin/H208502\n/H20885dxine−/H20849/H9270/2/H6036/H20850mnwn2/H20849Qi−xin/H208502\n=/H6036\n/H9270mnwn2=kBT\nmnwn2. /H20849D25/H20850\nSo, finally one gets /H20855/H9004xn/H9004xm/H20856=kBT\nmnwn2/H9254nm.\nb. x-p correlations\nExplicitly rewriting the derivative of K,\n/H20879/H11509K\n/H11509xfm/H20879\n/H9273i=/H9273f=−/H9270\n2/H6036K/H20849/H9273i,/H9273i;/H9270/H20850mm/H9275m2/H20849xim−Qi/H20850for/H20849m\n/HS110050/H20850, /H20849D26/H20850\n/H20879/H11509K\n/H11509Qf/H20879\n/H9273i=/H9273f=−/H9270\n2/H6036K/H20849/H9273i,/H9273i;/H9270/H20850/H20877bQi2+/H20858\nnmn/H9275n2/H20849Qi−xin/H20850/H20878.\n/H20849D27/H20850\nUsing the above relations,KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-12Tr/H20849/H9004xn/H9004pme−H/kBT/H20850=−i/H6036/H20885/H20879d/H9273i/H20849xin−Qi/H20850/H20873/H11509\n/H11509xfm−/H9253m/H11509\n/H11509Qf/H20874K/H20849/H9273f,/H9273i;/H9270/H20850/H20879\n/H9273i=/H9273f\n=−i/H9270\n2/H20885d/H9273i/H20849xin−Qi/H20850/H20877/H9253mbQi2+/H9253m/H20858\nlml/H9275l2/H20849Qi−xil/H20850+mm/H9275m2/H20849Qi−xim/H20850/H20878K/H20849/H9273i,/H9273i;/H9270/H20850\n=−i/H9270\n2/H20885d/H9273i/H20849xin−Qi/H20850/H20877/H9253m/H20858\nlml/H9275l2/H20849Qi−xil/H20850+mm/H9275m2/H20849Qi−xim/H20850/H20878K/H20849/H9273i,/H9273i;/H9270/H20850\n=i/H9270\n2/H9253m/H20858\nlml/H9275l2Tr/H20849/H20849/H9004xin/H9004xile−H/kBT/H20850/H20850+mm/H9275m2Tr/H20849/H9004xin/H9004xime−H/kBT/H20850. /H20849D28/H20850\nIn the third line, it is used that /H20848dxin/H20849xin−Qi/H20850\n/H11003/H20851even function of /H20849xin−Qi/H20850/H20852=0.\nOne can now write the x-pcorrelations in terms of x-x\ncorrelations.\n/H20855/H9004xn/H9004pm/H20856=i/H9270\n2/H20873/H9253m/H20858\nlml/H9275l2/H20855/H9004xin/H9004xil/H20856+mm/H9275m2/H20855/H9004xin/H9004xim/H20856/H20874\n=i/H9270kBT\n2/H20873/H9253m/H20858\nl/H9254nl+/H9254nm/H20874=i/H6036\n2/H20849/H9253m+/H9254nm/H20850,/H20849D29/H20850\nwhich is purely imaginary. Thus, /H20855/H20853/H9004xn,/H9004pm/H20854/H20856=/H20855/H9004xn/H9004pm/H20856\n+/H20855/H9004xn/H9004pm/H20856/H11569=0.\nc. p-p correlations\nIt is convenient to calculate /H20848d/H9273i/H115092K\n/H11509xfn/H11509xfm/H20841/H9273i=/H9273f. The trickiest\npart is /H20848d/H9273i/H20858kBknxik/H20858kBkmxikK/H20849/H9273i,/H9273i;/H9270/H20850,\nn/HS110050,m/HS110050:/H20858\nkBknxik/H20858\nkBkmxik=mn/H9275n2/H20849xin−Qi/H20850mm/H9275m2/H20849xim\n−Qi/H20850,\nn=0 , m/HS110050:/H20858\nkBknxik/H20858\nkBkmxik=/H20873/H20858\nkmk/H9275k2/H20849Qi−xik/H20850\n+bQi/H20874mm/H9275m2/H20849xim−Qi/H20850,\nn=0 , m=0 :/H20858\nkBknxik/H20858\nkBkmxik=/H20873/H20858\nkmk/H9275k2/H20849Qi−xik/H20850+bQi/H20874\n/H11003/H20873/H20858\nkmk/H9275k2/H20849Qi−xik/H20850+bQi/H20874.\nAfter integrating over xik, odd terms with respect to /H20849xik\n−Q/H20850vanish. Taking only even terms, one obtains\nn/HS110050,m/HS110050→mn2/H9275n4/H20849xin−Qi/H208502/H9254nm=mm/H9275m2/H20849xin−Qi/H208502Bnm,\nn=0 , m/HS110050→−mm2/H9275m4/H20849Qi−xim/H208502=mm/H9275m2/H20849xim−Qi/H208502Bnm,n=0 , m=0→/H20858\nkmk2/H9275k4/H20849Qi−xik/H208502+b2Qi2.\nIntegrating out and using the identity /H20848duu2e−u2/2/H9251\n=/H9251/H20848due−u2/2/H9251for/H9251/H110220, one finds\nn/HS110050,m/HS110050:/H20885d/H9273imm/H9275m2/H20849xin−Qi/H208502BnmK/H20849/H9273i,/H9273i;/H9270/H20850\n=/H6036\n/H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850,\nn=0 , m/HS110050:/H20885d/H9273imm/H9275m2/H20849xim−Qi/H208502BnmK/H20849/H9273i,/H9273i;/H9270/H20850\n=/H6036\n/H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850,\nn=0 , m=0 :/H20885d/H9273i/H20875/H20858\nkmk2/H9275k4/H20849Qi−xik/H208502+b2Qi2/H20876K/H20849/H9273i,/H9273i;/H9270/H20850\n=/H6036\n/H9270/H20873/H20858\nkmk/H9275k2+b/H20874/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850\n=/H6036\n/H9270Bnm/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850.\nThe result is/H6036\n/H9270Bnm/H20848d/H9273iKindependent of the cases. Finally,\none can obtain\n/H20885/H20879d/H9273i/H115092K\n/H11509xfn/H11509xfm/H20879\n/H9273i=/H9273f=/H20877−1\n/H6036/H20873/H9270\n4Bnm+1\n/H9270Anm/H20874\n+/H9270\n4/H6036Bnm/H20878/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850\n=−Anm\n/H9270/H6036/H20885d/H9273iK/H20849/H9273i,/H9273i;/H9270/H20850,/H20849D30/H20850\nor equivalently,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-13/H20883/H115092\n/H11509xn/H11509xm/H20884=−kBT\n/H60362Anm=−kBT\n/H60362/H20849M/H9253n/H9253m+mn/H9254nm/H20850,\n/H20849D31/H20850\nwhere /H92530=1,m0=0. Finally, p-pcorrelation is obtained\n/H20855/H9004pn/H9004pm/H20856=−/H60362/H20883/H20873/H11509\n/H11509xn−/H9253n/H11509\n/H11509Q/H20874/H20873/H11509\n/H11509xm−/H9253m/H11509\n/H11509Q/H20874/H20884\n=kBT/H20849Anm−/H9253mAn0−/H9253nAm0+/H9253n/H9253mA00/H20850\n=kBT/H20849M/H9253n/H9253m+mn/H9254mn−M/H9253m/H9253n−M/H9253n/H9253m\n+M/H9253n/H9253m/H20850=mnkBT/H9254mn. /H20849D32/H20850\nThe above three results of x-x,x-p, and p-pcorrelations\nare the same as Eq. /H2084931/H20850.3. Sufficient condition for “high” temperature\nWe assumed the high-temperature approximationkBT\n/H6036\n/H11271/H20881/H20841/H9261n/H20841\n2. Indeed, the temperature should satisfykBT\n/H6036/H11271/H20881/H9261M\n2,\nwhere /H9261Mis the absolute value of maximum eigenvalue of\nA−1B. It is known that, for eigenvalue /H9261of a matrix A,/H20841/H9261/H20841is\nnot greater than maximum column /H20849or row /H20850sum,55\n/H20841/H9261/H20841/H11349max\nj/H20858\ni/H20841aij/H20841/H11013/H20648A/H20648. /H20849D33/H20850\nAccording to the above definition of /H20648·/H20648, It is not hard to see\nthat/H20648AB/H20648/H11349/H20648A/H20648/H20648B/H20648.\nThe above argument says\n/H9261M/H11349/H20648A−1B/H20648/H11349/H20648A−1/H20648/H20648B/H20648. /H20849D34/H20850\nIt is not hard to obtain A−1with the following LDU factorization.\n/H20898MM /H92531 M/H92532 ¯\nM/H92531M/H925312+m1M/H92531/H92532¯\nM/H92532M/H92532/H92531M/H925322+m2¯\n]] ] /GS /H20899=/H2089810 0 ¯\n/H9253110 ¯\n/H9253201 ¯\n]] ]/GS/H20899/H20898M 0 0¯\n0m10¯\n00 m2¯\n]]]/GS/H20899/H208981/H92531/H92532¯\n01 0 ¯\n00 1 ¯\n]] ]/GS/H20899. /H20849D35/H20850\nInverting the factorized matrices,\nA−1=/H208981/H92531/H92532¯\n01 0 ¯\n00 1 ¯\n]] ]/GS/H20899−1\n/H20898M 0 0¯\n0m10¯\n00 m2¯\n]]]/GS/H20899−1\n/H2089810 0 ¯\n/H9253110 ¯\n/H9253201 ¯\n]] ]/GS/H20899−1\n=/H208981−/H92531−/H92532¯\n01 0 ¯\n00 1 ¯\n]] ]/GS/H20899/H208981\nM0 0¯\n01\nm10¯\n001\nm2¯\n]]] /GS/H20899\n/H11003/H2089810 0 ¯\n−/H9253110 ¯\n−/H9253201 ¯\n]] ] /GS/H20899=/H208981\nM+/H20858\nn/H9253n2\nmn−/H92531\nm1−/H92532\nm2¯\n−/H92531\nm11\nm10¯\n−/H92532\nm201\nm2¯\n]] ] /GS/H20899. /H20849D36/H20850KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-14Thus, the maximum column sum of A−1is\n/H20648A−1/H20648= max\nn/H208731\nM+/H20858\ni/H9253i2\nmi+/H20858\ni/H20841/H9253i/H20841\nmi,1+/H20841/H9253n/H20841\nmn/H20874./H20849D37/H20850\nIf/H9253iare on the order of 1 or larger,1\nM+/H20858i/H9253i2\nmi+/H20858i/H20841/H9253i/H20841\nmiis the\nmaximum value. And, in this limit, it is smaller than1\nM\n+2/H20858i/H9253i2\nmi. So one can get\n/H20648A/H20648/H113491\nM+2/H20858\ni/H9253i2\nmi. /H20849D38/H20850\nSince Bis given by\nB=/H20898b+/H20858\nnmn/H9275n2\n−m1/H927512−m2/H927522¯\n−m1/H927512m1/H927512 0 ¯\n−m2/H9275220 m2/H927522¯\n]] ] /GS/H20899,/H20849D39/H20850\nthe maximum column sum of Bis\n/H20648B/H206481= max\nn/H20873b+2/H20858\nimiwi2,2mnwn2/H20874/H11349/H20841b/H20841+2/H20858\nimiwi2.\n/H20849D40/H20850\nFinally, one obtains the upper bound of /H9261M,\n/H9261M/H11349/H20648A−1/H20648/H20648B/H20648/H11349/H208731\nM+2/H20858\ni/H9253i2\nmi/H20874/H20873/H20841b/H20841+2/H20858\nimi/H9275i2/H20874.\n/H20849D41/H20850\nIn order to evaluate the expression on the right-hand side of\nthe inequality Eq. /H20849D41/H20850, we use the constraints Eq. /H2084920/H20850.T o\nconvert the summations to known quantities, we generalizethe constraint to the Caldeira-Legget-type continuous formwith the following definitions of spectral functions,\nJ\np/H20849/H9275/H20850/H11013/H9266\n2/H20858\ni/H9253i2/H9275i\nmi/H9254/H20849/H9275i−/H9275/H20850=/H9251S\n2KM/H9275,/H20849D42/H20850\nJx/H20849/H9275/H20850/H11013/H9266\n2/H20858\nimi/H9275i3/H9254/H20849/H9275i−/H9275/H20850=2/H9251KM\nS/H9275./H20849D43/H20850\nChecking the constraints,/H20858\ni/H9253i2/H9261\nmi/H20849/H92612+/H9275i2/H20850=2/H9261\n/H9266/H20885d/H9275Jp/H20849/H9275/H20850\n/H9275/H20849/H92612+/H92752/H20850\n=2/H9261\n/H9266/H9251S\n2KM/H20885d/H92751\n/H92612+/H92752=/H9251S\n2KM,\n/H20849D44/H20850\n/H20858\nimi/H9275i2/H9261\n/H92612+/H9275i2=2/H9261\n/H9266/H20885d/H9275Jx/H20849/H9275/H20850\n/H9275/H20849/H92612+/H92752/H20850\n=2/H9261\n/H92662/H9251KM\nS/H20885d/H92751\n/H92612+/H92752=2/H9251KM\nS.\n/H20849D45/H20850\nFinally,\n/H20858\ni/H9253i2\nmi=2\n/H9266/H20885d/H9275/H9251S\n2KM=/H9251S\n/H9266KM/H9275c, /H20849D46/H20850\n/H20858\nimi/H9275i2=2\n/H9266/H20885d/H92752/H9251KM\nS=4/H9251KM\nS/H9266/H9275c,/H20849D47/H20850\nwhere /H9275cis the critical frequency of the environmental exci-\ntations.\nTherefore, /H9261M/H11349/H208491\nM+2/H9251S\n/H9266KM/H9275c/H20850/H20849/H20841b/H20841+8/H9251KM\nS/H9266/H9275c/H20850. Hence, one fi-\nnally finds that the sufficient condition of the high tempera-\nture is T/H11271Tc, where the critical temperature Tcis defined as\nTc/H11013/H6036\n2kB/H20881/H208731\nM+2/H9251S\n/H9266KM/H9275c/H20874/H20873/H20841b/H20841+8/H9251KM\nS/H9266/H9275c/H20874.\n/H20849D48/H20850\nNow, we check if the above condition is satisfied in ex-\nperimental situations. Ignoring /H20841b/H20841, the critical temperature\nbecomes /H20881/H208491+2/H9251S\n/H9266K/H9275c/H208502/H9251K\nS/H9266/H9275c. Since the environmental excita-\ntion is caused by magnetization dynamics, one can note thatthere is no need to consider the environmental excitationwith frequencies far exceeding the frequency scale of mag-netization dynamics. This concludes that\n/H9275cis on the order\nof the frequency of magnetization dynamics, which is knownas about 10 GHz or less.\n56,57With conventional scale /H9251\n/H110110.01, K/H1101110−4eV, and 2 S/H11011/H6036, the critical temperature is\nestimated as Tc/H1101130 mK. Therefore, our calculation is con-\ncluded to be well satisfied in most experimental situation.\n1A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki, Europhys.\nLett. 69, 990/H208492005/H20850.\n2G. Tatara, T. Takayama, H. Kohno, J. Shibata, Y . Nakatani, and\nH. Fukuyama, J. Phys. Soc. Jpn. 75, 064708 /H208492006/H20850.\n3M. D. Stiles, W. M. Saslow, M. J. Donahue, and A. Zangwill,\nPhys. Rev. B 75, 214423 /H208492007/H20850.\n4S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 /H208492004/H20850.\n5W. F. Brown, Jr., Phys. Rev. 130, 1677 /H208491963/H20850.\n6R. Kubo and N. Hashitsume, Suppl. Prog. Theor. Phys. 46, 210/H208491970/H20850.\n7J. Foros, A. Brataas, Y . Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. B 78, 140402 /H20849R/H20850/H208492008/H20850.\n8One of the well-known situations where thermal fluctuations\nplays a crucial role is the DW creep. See, for instance, S. Le-merle, J. Ferre, C. Chappert, V . Mathet, T. Giamarchi, and P. LeDoussal, Phys. Rev. Lett. 80, 849/H208491998/H20850; P. Chauve, T. Giama-\nrchi, and P. Le Doussal, Phys. Rev. B 62, 6241 /H208492000/H20850; K. Kim,\nJ.-C. Lee, S.-M. Ahn, K.-S. Lee, C.-W. Lee, Y . J. Cho, S. Seo,THERMAL FLUCTUATION FIELD FOR CURRENT-INDUCED … PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-15K.-H. Shin, S.-B. Choe, and H.-W. Lee, Nature /H20849London /H20850458,\n740/H208492009/H20850.\n9R. A. Duine, A. S. Núñez, and A. H. MacDonald, Phys. Rev.\nLett. 98, 056605 /H208492007/H20850.\n10R. A. Duine and C. M. Smith, Phys. Rev. B 77, 094434 /H208492008/H20850.\n11J.-V . Kim and C. Burrowes, Phys. Rev. B 80, 214424 /H208492009/H20850.\n12E. Martinez, L. Lopez-Diaz, L. Torres, C. Tristan, and O. Alejos,\nPhys. Rev. 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Brataas, and G. E. W. Bauer,\nPhys. Rev. B 74, 144405 /H208492006/H20850.\n23G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 /H208492004/H20850.\n24G. Tatara, H. Kohno, J. Shibata, Y . Lemaho, and K.-J. Lee, J.\nPhys. Soc. Jpn. 76, 054707 /H208492007/H20850.\n25G. Tatara, H. Kohno, and J. Shibata, J. Phys. Soc. Jpn. 77,\n031003 /H208492008/H20850.\n26X. Waintal and M. Viret, Europhys. Lett. 65, 427/H208492004/H20850.\n27A. Vanhaverbeke and M. Viret, Phys. Rev. B 75, 024411 /H208492007/H20850.\n28M. Thorwart and R. Egger, Phys. Rev. B 76, 214418 /H208492007/H20850.\n29I. Garate, K. Gilmore, M. D. Stiles, and A. H. MacDonald, Phys.\nRev. B 79, 104416 /H208492009/H20850.\n30R. P. Feynman and F. L. Vernon, Jr., Ann. Phys. /H20849N.Y ./H2085024,1 1 8\n/H208491963/H20850.\n31C. M. Smith and A. O. Caldeira, Phys. Rev. A 36, 3509 /H208491987/H20850.\n32A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46,2 1 1\n/H208491981/H20850.\n33G. L. Ingold and Yu. V . Nazarov, in Single Charge Tunneling\nCoulomb Blockade Phenomena in Nanostructures , edited by H.\nGrabert and M. Devoret /H20849Plenum, New York, 1992 /H20850.\n34H. Lee and L. S. Levitov, Phys. Rev. B 53, 7383 /H208491996/H20850.\n35N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 /H208491974/H20850.\n36A. P. Malozemoff and J. C. Slonczewski, Magnetic Domains\nWalls in Bubble Materials /H20849Academic, New York, 1979 /H20850.\n37S.-W. Jung, W. Kim, T.-D. Lee, K.-J. Lee, and H.-W. Lee, Appl.Phys. Lett. 92, 202508 /H208492008/H20850; J. Ryu and H.-W. Lee, J. Appl.\nPhys. 105, 093929 /H208492009/H20850.\n38M. Kläui, P.-O. Jubert, R. Allenspach, A. Bischof, J. A. C. Bland,\nG. Faini, U. Rüdiger, C. A. F. Vaz, L. Vila, and C. V ouille, Phys.\nRev. Lett. 95, 026601 /H208492005/H20850.\n39G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L. Ersk-\nine,Phys. Rev. Lett. 97, 057203 /H208492006/H20850.\n40M. Hayashi, L. Thomas, C. Rettner, R. Moriya, Y . B. Bazaliy,\nand S. S. P. Parkin, Phys. Rev. Lett. 98, 037204 /H208492007/H20850.\n41For permalloy, /H20841e/H92530K/H9261//H9262B/H20841/H11011109A/cm2, which is about an or-\nder larger than the current density of /H11011108A/cm2used in many\nexperiments /H20849Refs. 38–40/H20850.\n42Y . Le Maho, J.-V . Kim, and G. Tatara, Phys. Rev. B 79, 174404\n/H208492009/H20850.\n43T. Kim, J. Ieda, and S. Maekawa, arXiv:0901.3066 /H20849unpub-\nlished/H20850.\n44V . W. Döring, Z. Naturforsch. A 3A, 373/H208491948/H20850.\n45We thank M. Stiles for pointing out this point.\n46To solve this system, one of the constaints Eq. /H2084920a/H20850is general-\nized to hold even for /H9261=0. That is, /H20858i/H9253i=0. See, for a detail,\nAppendix C.\n47To consider a force on Eq. /H208496a/H20850, the potential should be general-\nized to depend on the momentum.\n48Forv=0, the terminal velocity of the DW vanishes indepen-\ndently of its the initial velocity since the environmental mass ismuch larger than the DW mass. With\nv/H110220, one can perform the\nGalilean transformation to make /H20855x˙i/H208490/H20850/H20856=0 instead of /H20855x˙i/H208490/H20850/H20856\n=v. Since the system is Galilean invariant, one expect that the\nDW also stops in this frame, just as v=0. It implies that the\nterminal velocity of the DW in the lab frame is also v.\n49V . Kamberský, Czech. J. Phys., Sect. B 26, 1366 /H208491976/H20850; Can. J.\nPhys. 48, 2906 /H208491970/H20850;Czech. J. Phys., Sect. B 34, 1111\n/H208491984/H20850.\n50K. Gilmore, Y . U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99,\n027204 /H208492007/H20850.\n51C.-Y . You, I. M. Sung, and B.-K. Joe, Appl. Phys. Lett. 89,\n222513 /H208492006/H20850; C.-Y . You and S.-S. Ha, ibid. 91, 022507\n/H208492007/H20850.\n52This section summarizes the work by Kim et al./H20849Ref. 43/H20850.\n53J. M. Winter, Phys. Rev. 124, 452/H208491961/H20850.\n54By the same argument, Eq. /H2084932/H20850is obtained under an arbitrary\npotential V/H20849Q/H20850. Since the system was in equilibrium before ap-\nplying current, we assume V/H11032/H20849Q/H20850=0. At high temperature limit,\n/H9273moves in very short /H20849imaginary /H20850time interval. Therefore, we\ncan take quadratic approximation and V/H20849Q/H20850to be the form of\nbQ2.\n55See, for example, G. Strang, Linear Algebra and its Applications\n/H20849Thomson, USA, 1988 /H20850, Chap. 7.\n56A. Mourachkine, O. V . Yazyev, C. Ducati, and J.-Ph. Ansermet,\nNano Lett. 8, 3683 /H208492008/H20850.\n57C. Boone, J. A. Katine, J. R. Childress, J. Zhu, X. Cheng, and I.\nN. Krivorotov, Phys. Rev. B 79, 140404 /H20849R/H20850/H208492009/H20850.KYOUNG-WHAN KIM AND HYUN-WOO LEE PHYSICAL REVIEW B 82, 134431 /H208492010/H20850\n134431-16"    },    {        "title": "1911.12017v2.Ellipticity_and_Dissipation_Effects_in_Magnon_Spin_Valves.pdf",        "content": "Ellipticity and Dissipation E\u000bects in Magnon Spin Valves\nJiansen Zheng,1Andreas R uckriegel,1Scott A. Bender,1and Rembert A. Duine1, 2, 3\n1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,\nUtrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands\n2Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n3Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: March 2, 2020)\nWe consider alignment-dependent spin and heat transport across a magnon spin valve in the\ntunneling regime, i.e., a junction consisting of two weakly coupled ferromagnetic insulators. We\ndetermine the di\u000berence in spin and heat conductance between the parallel and antiparallel con\fg-\nuration of the magnetization direction. The dependence of these conductances on both the Gilbert\ndamping and ellipticity is studied. We \fnd that both magnon ellipticity and dissipation open\nchannels for magnons to tunnel through in the antiparallel con\fguration. Our results highlight an\nimportant di\u000berence between electronic and magnon spin transport in spin-valve structures and may\nbe important for the development of devices based on magnetic insulators.\nI. INTRODUCTION\nSpintronics based on spin-polarized charge currents has\nled to a boost in information storage technology with the\ndiscovery of giant magnetoresistance (GMR) in antiferro-\nmagnetically coupled Fe =Cr superlattices1,2. GMR arises\nfrom the spin-dependent transmission of the conduction\nelectron. Magnons, the quanta of collective excitations in\nmagnetically ordered systems, can carry spin current in\nmagnetic insulators in the absence of any charge current,\ne.g., in the spin Seebeck e\u000bect3, where the magnons are\ndriven by a thermal bias, or in nonlocal setups in which\nthe magnons are biased electrically using the spin Hall ef-\nfect in adjacent normal metals4. Magnon spin transport\nis promising, for example, to improve the power e\u000eciency\nof logic devices5{7and for neuromorhpic computing8,9.\nTo \fnd analogies of GMR in magnon spin trans-\nport, spin-valve structures that encompass magnetic in-\nsulators have recently been studied experimentally and\ntheoretically10{12. Wu et al. have observed that the spin\nSeebeck e\u000bect of a heterostructure made of two ferro-\nmagnetic insulators, namely yttrium iron garnet (YIG),\nseparated by a nonmagnetic heavy metal layer, depends\non the relative orientation of the magnetizations of the\ntwo magnetic insulators13. The di\u000berence in spin Seebeck\nsignal between parallel and antiparallel con\fgurations is\nobserved to decrease signi\fcantly as the temperature is\nlowered. In a recent theoretical work, a Green's function\nformalism for magnon tunneling driven by a temperature\nbias across a ferromagnetic junction has been developed\nand applied to compute the diode properties of the tun-\nneling magnon current14. A key aspect of this study is\nthe inclusion of magnon-magnon interactions that are ex-\nploited for the recti\fcation and negative di\u000berential spin\nSeebeck e\u000bects. Furthermore, a tunable spin Seebeck\ndiode based on a magnetic junction structure in which\nthe tunneling spin current can be turned on and o\u000b by\ncontrolling the magnetization orientation has also been\ntheoretically proposed15.In this work, we study the alignment dependence of\nmagnon heat and spin transport across a heterostruc-\nture consisting of two ferromagnetic insulators that are\nweakly exchange coupled, e.g., by a nonmagnetic spacer\nlayer that mediates exchange interactions. The setup we\nconsider is illustrated in Fig. I. The ferromagnetic in-\nsulators act as reservoirs for magnons. The magnons\ncan be coherently driven by ferromagnetic resonance,\nor incoherently generated with an electrical or ther-\nmal bias using adjacent normal metals16. We focus on\nthe e\u000bect of the ellipticity of the magnetization pre-\ncession, which is usually caused by anisotropies, and\nalso on the e\u000bects of dissipation that we parametrize\nwith a Gilbert damping constant. The latter is a phe-\nnomenological parameter that characterizes the decay of\nmagnons. The ellipticity of precession has been shown\nto strongly a\u000bect the parametric excitation of magnons\nin ferromagnetic resonance experiments17, and plays a\nrole in Rayleigh-Jeans condensation of pumped magnons\nin thin-\flm ferromagnets18. Moreover, at the quantum-\nmechanical level, the ellipticity leads to squeezed ground\nstates and, in the case of antiferromagnets, entanglement\nbetween di\u000berent sublattice magnons19,20. Meanwhile, a\nlow Gilbert damping has also been demonstrated to en-\nable long-distance spin transport in magnetic insulators\nsuch as yttrium iron garnet4. Neither the in\ruence of\ndamping nor ellipticity has, however, been considered for\nmagnon tunneling in the insulating spin valve structure\nconsidered here.\nIn our setup, magnons tunnel between the ferromag-\nnets due to the weak exchange coupling and carry heat\nand spin currents in response to applied temperature or\nmagnon chemical potential di\u000berences. For circularly po-\nlarized magnons, conservation of spin forbids magnon\ntunneling in the antiparallel con\fguration. We \fnd that\nanisotropies and dissipation, both of which break spin\nconservation, lead to tunneling currents even in the an-\ntiparallel con\fguration. The di\u000berence between circular\nand elliptical case has no straightforward analog in elec-arXiv:1911.12017v2  [cond-mat.mes-hall]  3 Mar 20202\nFIG. 1. Illustration of a magnetic junction consisting of two\nferromagnetic insulators, which interact with each other with\nweak exchange coupling U. The left (right) insulator has tem-\nperatureTL(TR) and spin accumulation \u0016L(\u0016R). The layer\nin green is a nonmagnetic insulator and is thin enough for the\nmagnons to tunnel across it. The left insulator is the \fxed\nlayer, with the red arrow denoting an up spin, while the right\ninsulator is a free layer in which the magnetization can be\ntuned from a parallel con\fguration (denoted with the red ar-\nrow) to an anti-parallel con\fguration (denoted with the blue\narrow).\ntronic spin valves, as in these latter systems the spin of\nthe electrons at the Fermi level is usually approximately\nconserved, and tunneling current is determined by their\nspin-dependent density of states. Our result therefore\nshows that the di\u000berence between magnon currents in\nthe parallel and antiparallel con\fguration is governed by\ndi\u000berent physics than for metallic structures, which may\nbe useful in designing devices that exploit the tunability\nof the tunneling current.\nThe remainder of the paper is organized as follows:\nIn Sec. II, we introduce the model that we use for our\nsetup. The magnon tunneling currents in the presence of\nboth precession ellipticity and Gilbert damping for both\nparallel and antiparallel con\fgurations are calculated in\nSec. III. In Sec. IV, we numerically calculate the tun-\nneling conductances and discuss their dependencies on\nthe ellipticity and dissipation. Section V summarizes our\nmain \fndings and conclusions. Lastly, the Appendix out-\nlines the derivation of the tunneling currents using rate\nequations.II. MODEL HAMILTONIAN\nA. Lead Hamiltonians\nThe magnetization dynamics in the bulk of each insu-\nlating lead is modeled by the Hamiltonian\nHX=\u00001\n2X\nijJX;ijSX;i\u0001SX;j\u0000~\rX\u00160HXX\niSz\nX;i\n\u00001\n2X\ni\u0014\nKX;x\u0000\nSx\nX;i\u00012+KX;y\u0010\nSy\nX;i\u00112\u0015\n;(1)\nwhereX=L=R denotes the left/right lead and i;jlabel\nthe lattice sites. Here, JX;ij are nearest-neighbor ex-\nchange interactions with strength JX>0, whereasKX;x\nandKX;yare anisotropy constants. Lastly, \u00160HXare the\nmagnetic \felds in the bulk of the leads with gyromagnetic\nratio\rX.\nThe spin operators SX;iare bosonized via a Holstein-\nPrimako\u000b transformation21. ForHX>0 we assume that\nthe magnetic order parameter points in zdirection, so\nthat\nS+\nX;i=Sx\nX;i+iSy\nX;i=p\n2SX\u0002\nbX;i+O(S\u00001\nX)\u0003\n;(2a)\nSz\nX;i=SX\u0000by\nX;ibX;i; (2b)\nwhereSXis the spin quantum number of the magnetic\nmoments in lead X, andbX;iandby\nX;iare the magnon an-\nnihilation and creation operators that satisfy the bosonic\ncommutation relations [ bX;i;by\nX0;j] =\u000eX;X0\u000ei;j. Con-\nversely, for HX<0 we assume that the magnetic order\nparameter points in \u0000zdirection and apply the following\nHolstein-Primako\u000b transformation:\nS+\nX;i=Sx\nX;i+iSy\nX;i=p\n2SXh\nby\nX;i+O(S\u00001\nX)i\n;(3a)\nSz\nX;i=\u0000SX+by\nX;ibX;i: (3b)\nIn the bulk of each lead, we may expand the magnon\noperators in a Fourier series as\nbX;i=1pNXX\nkeik\u0001RibX;k; (4)\nwhereNXdenotes the number of magnetic moments in\nthe leadX. Then the spin Hamiltonian (1) becomes\nHX=X\nk\u0014\nAX;kby\nX;kbX;k+BX\n2\u0010\nby\nX;kby\nX;\u0000k+ h.c.\u0011\u0015\n;\n(5)\nwhere we dropped a constant contribution to the ground\nstate energy as well as O(S\u00001=2\nX) corrections containing\nhigher powers of the magnon operators. The coe\u000ecients\nof the Hamiltonian (5) are given by\nAX;k=~\rX\u00160jHXj+JXSXa2\nXk2\u0000SX\n2(KX;x+KX;y);\n(6a)\nBX=\u0000SX\n2(KX;x\u0000KX;y); (6b)3\nFIG. 2. Semiclassical depiction of a spin Sthat precesses\nelliptically around a magnetic \feld H. The solid blue line is\nthe elliptical precession, whereas the dashed gray line corre-\nsponds to a circular precession. Because the length of the spin\nis conserved, its projection onto the direction of the magnetic\n\feld is not constant during the elliptical precession. There-\nfore, this spin projection is no longer a good quantum number\nfor elliptical magnons.\nregardless of whether we assume HX>0 orHX<0\nand employ the respective Holstein-Primako\u000b transfor-\nmation (2) or (3). In writing down Eq. (6a), we further-\nmore assumed that only long-wavelength magnons with\naXjkj\u001c1 are relevant, where aXis the lattice constant\nof leadX. The quadratic magnon Hamiltonian (5) is\ndiagonalized via a Bogoliubov transformation:\n\u0012bX;k\nby\nX;\u0000k\u0013\n=\u0012\nuX;k\u0000vX;k\n\u0000vX;kuX;k\u0013\u0012\fX;k\n\fy\nX;\u0000k\u0013\n; (7)\nwhere\nuX;k=s\nAX;k+EX;k\n2EX;k; (8a)\nvX;k=BX\njBXjs\nAX;k\u0000EX;k\n2EX;k; (8b)\nwhere\nEX;k=q\n(AX;k+BX) (AX;k\u0000BX): (9)\nThe operators \fX;kand\fy\nX;kcreate and destroy Bogoli-\nubov quasiparticles and obey the bosonic commutation\nrelations [\fX;k;\fy\nX0;k0] =\u000eX;X0\u000ek;k0. Semiclassically, a\nBogoliubov quasiparticle created by \fy\nX;kcorresponds to\nan elliptical spin wave; in contrast, a magnon created\nbyby\nX;kcorresponds to a circular spin wave. The Bogoli-\nubov quasiparticles are often also referred to as magnons,\nor as elliptical or squeezed magnons19,20. For an ellipti-\ncal spin wave, the zcomponent of the spin is not con-\nserved and hence not a good quantum number, unlike\nfor a circular spin wave. This is illustrated semiclassi-\ncally in Fig. 2. With the Bogoliubov transformation (7),the magnon Hamiltonian (5) becomes\nHX=X\nk\u0014\nEX;k\fy\nX;k\fX;k+1\n2(EX;k\u0000AX;k)\u0015\n:(10)\nNote that this Hamiltonian is only valid as long as the\nquasiparticle dispersion (9) is real, i.e., for ~\rX\u00160jHXj>\nSKX;x;SKX;y. If this is not satis\fed, our original as-\nsumption that the magnetic order points in \u0006zdirection\nis not correct and we have to expand around a di\u000berent\nground state. However, for the remainder of this work we\nwill assume that ~\rX\u00160jHXj> SKX;x;SKX;y, so that\nthe quasiparticle Hamiltonian (10) is stable.\nB. Tunneling Hamiltonian\nThe tunneling between the leads is facilitated by a\nlead-lead exchange interaction of the form\nHT=\u0000X\nijUijSL;i\u0001SR;j; (11)\nwhere the exchange coupling Uijis assumed to be small\ncompared to the bulk energy scales and only \fnite for i;j\nclose to the interface. The microscopic origin of such an\ninteraction could either be direct exchange mediated by\nthe conduction electrons in the normal metal or an indi-\nrect superexchange interaction via the ions in the non-\nmagnetic spacer layer.\n1. Parallel Con\fguration\nIn the parallel con\fguration, we take the form (2) for\nboth leads. Then the tunneling Hamiltonian (11) be-\ncomes\nHP\nT=\u0000p\nSLSRX\nijUij\u0010\nby\nL;ibR;j+by\nR;jbL;i\u0011\n;(12)\nwhere we dropped constants and higher order magnon\ncorrections as before, as well as an on-site energy shift\nfor the magnons in each lead. This is justi\fed by our as-\nsumption that the lead-lead exchange is small compared\nto the bulk energy scales. After applying both Fourier\nand Bogoliubov transformations, Eqs. (4) and (7) respec-\ntively, we \fnd\nHP\nT=\u0000r\nSLSR\nNLNRX\nkk0\n\u0002\u0012\nV(n)\nk;k0\fy\nL;k\fR;k0\u0000V(a)\nk;k0\fy\nL;k\fy\nR;\u0000k0+ h.c.\u0013\n;\n(13)\nwhere\nV(n)\nk;k0=Uk;\u0000k0(uL;kuR;k0+vL;kvR;k0); (14a)\nV(a)\nk;k0=Uk;\u0000k0(uL;kvR;k0+vL;kuR;k0) (14b)4\nare the normal ( n) and anomalous ( a) tunneling am-\nplitudes, with the Fourier transform of the lead-\nlead exchange coupling Uk;k0= (U\u0000k;\u0000k0)\u0003=P\ni2LP\nj2Re\u0000ik\u0001Ri\u0000ik0\u0001RjUij. Note that the anomalous\ncoupling (14b) is only \fnite when the magnon ellipticity\nis \fnite, leading to qualitatively new physics in this case.\n2. Antiparallel Con\fguration\nIn the antiparallel con\fguration, we take the Holstein-\nPrimako\u000b transformations (2) for the left and (3) for the\nright lead, yielding\nHAP\nT=\u0000p\nSLSRX\nijUij\u0010\nby\nL;iby\nR;j+bL;ibR;j\u0011\n;(15)\nwithin the same approximations as for the parallel con-\n\fguration considered in the preceding Sec. II B 1. As be-\nfore, we apply the Fourier and Bogoliubov transforma-\ntions given, respectively, in Eqs. (4) and (7) to obtain\nHAP\nT=\u0000r\nSLSR\nNLNRX\nkk0\n\u0002\u0012\nV(n)\nk;k0\fy\nL;k\fy\nR;\u0000k0\u0000V(a)\nk;k0\fy\nL;k\fR;k0+ h.c.\u0013\n(16)\nNote that from the comparison of the magnon tunneling\nHamiltonian between the parallel and antiparallel case,\nEq. (12) and Eq. (15), respectively, it is clear that these\ntwo situations di\u000ber qualitatively. In the parallel case,\none deals with a tunneling Hamiltonian that is also en-\ncountered in the electron transport, whereas in the an-\ntiparallel case, the tunneling corresponds to creation or\ndestruction of a pair of circular magnons. We also stress\nthat the magnon ellipticity, being related to the breaking\nof magnon number, i.e., spin conservation, has no ana-\nlog in electronic systems, where the electron number is\nalways conserved. Therefore, the e\u000bects discussed here\nhave no direct analog in electronic valves. From Eq. (15)\nit is furthermore obvious that there is no spin transport\nin the antiparallel con\fguration without breaking of the\ntotal spin conservation, either by anisotropies or damp-\ning. In this respect, the undamped, circular magnon spin\nvalve resembles a half-metallic system that only trans-\nmits spin when the magnetizations of the two magnets\nare aligned parallel.\nIII. TUNNELING CURRENTS\nThe tunneling currents can be obtained from the rate\nequations for the distribution function of the Bogoliubov\nquasiparticles in each lead,\nnX;k=D\n\fy\nX;k\fX;kE\n=fB\u0012EX;k\u0000\u0016X\nkBTX\u0013\n(17)wherefB(x) = 1=(ex\u00001) is the Bose function, and the\nsecond equality holds in a steady state in which lead X\nis kept at temperature TXand chemical potential \u0016X.\nTo allow for \fnite damping in each lead, we recast the\nsteady state distribution function (17) as\nnX;k=Z1\n\u00001d\u000f\u000e(\u000f\u0000EX;k)fB\u0012\u000f\u0000\u0016X\nkBTX\u0013\n: (18)\nWithin the Gilbert damping phenomenology, we may\nthen add dissipation by broadening the Dirac distribu-\ntions according to14,22\n\u000e(\u000f\u0000EX;k)!A(\u000f\u0000EX;k)\u00111\n\u0019\u000b\u000f\n(\u000f\u0000EX;k)2+ (\u000b\u000f)2;\n(19)\nwhere\u000bis the bulk Gilbert damping parameter.\nDetails of the derivation of the tunneling currents from\nkinetic equations for the quasiparticle distribution func-\ntions can be found in the Appendix; here we only state\nthe results. Labeling the parallel/antiparallel con\fgura-\ntions withY=P=AP , we \fnd the following expressions\nfor the energy current\nIY\nE=2\u0019\n~Z1\n\u00001d\u000f\u000f\n\u0002(\nDY\nE(\u000f)\u0014\nfB\u0012\u000f\u0000\u0016L\nkBTL\u0013\n\u0000fB\u0012\u000f\u0000\u0016R\nkBTR\u0013\u0015\n+~DY\nE(\u000f)\u0014\nfB\u0012\u000f\u0000\u0016L\nkBTL\u0013\n\u0000fB\u0012\u000f+\u0016R\nkBTR\u0013\u0015)\n;\n(20)\nand the spin current\nIY\nS=2\u0019Z1\n\u00001d\u000f\n\u0002(\nDY\nS(\u000f)\u0014\nfB\u0012\u000f\u0000\u0016L\nkBTL\u0013\n\u0000fB\u0012\u000f\u0000\u0016R\nkBTR\u0013\u0015\n+~DY\nS(\u000f)\u0014\nfB\u0012\u000f\u0000\u0016L\nkBTL\u0013\n\u0000fB\u0012\u000f+\u0016R\nkBTR\u0013\u0015)\n;\n(21)\n\rowing from the left to the right lead. Here, DP=AP\nE=S(\u000f)\nare the normal tunneling densities of state, explicitly\ngiven by\n(\nDP=AP\nE (\u000f)\nDP=AP\nS (\u000f))\n=SLSR\nNRNLX\nkk0\f\f\fV(n=a)\nk;k0\f\f\f2(1\u0010\nu2\nR;k0+v2\nR;k0\u0011)\n\u0002A(\u000f\u0000EL;k)A(\u000f\u0000ER;k0): (22)\nNote that their contributions to currents (20) and (21)\nvanish if both leads are at the same temperature and\nchemical potential. On the other hand, ~DP=AP\nE=S(\u000f) are\nanomalous tunneling densities of state that arise because5\nthe Gilbert damping breaks the number conservation of\nthe Bogoliubov quasiparticles. Hence it gives rise to a\nspin current even when both leads are at the same tem-\nperature and chemical potential; it vanishes only if both\nleads are in true thermal equilibrium at the same temper-\nature and vanishing chemical potential. These anomalous\ntunneling densities of state are\n(\n~DP=AP\nE (\u000f)\n~DP=AP\nS (\u000f))\n=SLSR\nNRNLX\nkk0\f\f\fV(a=n)\nk;k0\f\f\f2(1\u0010\nu2\nR;k0+v2\nR;k0\u0011)\n\u0002A(\u000f\u0000EL;k)A(\u000f+ER;k0): (23)\nIt is instructive to consider the limit of conserved quasi-\nparticles (\u000b= 0+) as well as the limit of circular magnons\nin more detail. If there is no dissipation, the anomalous\ncontributions to the currents vanish because energy con-\nservation strictly demands EL;k+ER;k0= 0, which can\nnever be satis\fed since both of these energies are posi-\ntive. Finite damping softens this restriction by allowing\nenergy (and spin) transfer to a thermal bath, thereby\nopening up another channel for energy and spin transfer\nbetween the leads. In the limit of circular magnons, i.e.,\nwhen there are no anisotropies that break rotation sym-\nmetry around the zdirection, we may set uX;k= 1 and\nvX;k= 0; see Eqs. (8). Then ~DP\nE=S(\u000f) = 0 =DAP\nE=S(\u000f).\nThis re\rects the conservation of the total spin Sz\nL+Sz\nR\nin the absence of anisotropies. In the parallel con\fgura-\ntion, tunneling is in this case only allowed for the process\nin which a magnon carrying spin \u0000~is destroyed in one\nlead and another magnon carrying spin \u0000~is created in\nthe other lead. Conversely, in the antiparallel con\fgura-\ntion magnons carry spin \u0000~in the left lead and + ~in the\nright lead. Thus spin conservation only allows anomalous\nprocesses in which magnon pairs in the left and right lead\nare simultaneously destroyed or created. As this process\nviolates energy conservation, it is only possible in the\npresence of dissipation. Therefore, there are no energy\nand spin currents in the antiparallel con\fguration with-\nout either damping (enabling pair creation/annihilation\nprocesses) or breaking of spin conservation (enabling nor-\nmal hopping). This is further illustrated in Fig. 3.\nA. Tunneling Conductances\nIf the biasing is su\u000eciently small, i.e., \u0001 T=TL\u0000TR\u001c\nT, whereT=1\n2(TL+TR) is the average temperature,\nand\u0016L=R\u001cEL=R;k=0, we may linearize the Bose func-\ntions appearing in the currents (20) and (21), yielding\nIY\nE=\u0014Y\u0001T+ \u0005Y(\u0016L\u0000\u0016R) +\rY\nE(\u0016L+\u0016R);(24)\nIY\nS=LY\u0001T+\u001bY(\u0016L\u0000\u0016R) +\rY\nS(\u0016L+\u0016R):(25)\nHere,\n\u0014Y=\u0019\n2~kBT2Z1\n\u00001d\u000f\u000f2\nsinh2\u0010\n\u000f\n2kBT\u0011h\nDY\nE(\u000f) +~DY\nE(\u000f)i\n;\n(26a)\n(a)\nspin current spin current(b)FIG. 3. Tunneling processes allowed by spin conservation for\ncircular magnons. (a) Hopping of a magnon carrying spin \u0000~\nfrom the right to the left lead in the parallel con\fguration.\nThe inverse process of hopping from left to right lead is also\npossible. (b) Pair creation of a magnon carrying spin \u0000~in\nthe left lead and a magnon carrying spin + ~in the right lead\nin the antiparallel con\fguration. The inverse process of pair\nannihilation is also allowed. However, while allowed by spin\nconservation, both pair creation and annihilation processes\nare forbidden by energy conservation if there is no dissipa-\ntion. The spin currents in (a) and (b) are polarized in the\nzdirection. For the inverse processes, the spin currents \row\nin the opposite direction. The blue (red) arrows indicate the\nspin change associated with the creation (annihilation) of a\ncircular magnon.\nis the thermal conductance,\nLY=\u0019\n2kBT2Z1\n\u00001d\u000f\u000f\nsinh2\u0010\n\u000f\n2kBT\u0011h\nDY\nS(\u000f) +~DY\nS(\u000f)i\n;\n(26b)\nis the spin Seebeck conductance,\n\u0005Y=\u0019\n2~kBTZ1\n\u00001d\u000f\u000f\nsinh2\u0010\n\u000f\n2kBT\u0011DY\nE(\u000f); (26c)\nis the spin Peltier conductance, and\n\u001bY=\u0019\n2kBTZ1\n\u00001d\u000f1\nsinh2\u0010\n\u000f\n2kBT\u0011DY\nS(\u000f); (26d)\nis the spin conductance. Lastly,\n\rY\nE=\u0019\n2~kBTZ1\n\u00001d\u000f\u000f\nsinh2\u0010\n\u000f\n2kBT\u0011~DY\nE(\u000f); (26e)\n\rY\nS=\u0019\n2kBTZ1\n\u00001d\u000f1\nsinh2\u0010\n\u000f\n2kBT\u0011~DY\nS(\u000f) (26f)\nare the additional energy and spin loss or gain terms aris-\ning because the \fnite damping breaks the number con-\nservation of Bogoliubov quasiparticles. Since they do not\nvanish when both leads are mutually equilibrated, these\nterms are not part of the transport current and should\nrather be identi\fed with the spin and energy lost to or\ngained from the thermal bath that provides the dissipa-\ntion, which is ultimately the crystal lattice. Microscop-\nically, they correspond to the simultaneous creation or6\nannihilation of a magnon in the left and a magnon in the\nright lead; the required energy and angular momentum\nis provided by the lattice. Therefore, these terms de-\nscribe energy and spin currents \rowing from the lattice\nto both leads, instead of currents \rowing from one lead\nto the other. Because the total angular momentum of\nspins and lattice is conserved, this additional spin trans-\nfer should be experimentally detectable as torques on the\nwhole sample.\nNote also that the spin Seebeck and Peltier conduc-\ntances, Eqs. (26b) and (26c), respectively, are not On-\nsager reciprocals of each other, ~\u0005Y6=TLY. There are\ntwo independent reasons for this: the breaking of time-\nreversal symmetry by the dissipation and the breaking\nof spin conservation by the anisotropies. While the for-\nmer opens up an new channel for bath-assisted energy\ntransfer, namely the pair creation/annihilation processes\ncontained in ~DY\nE=S(\u000f), the latter allows for changes in\nspin without accompanying changes in energy, resulting\ninDY\nE(\u000f)6=DY\nS(\u000f) and ~DY\nE(\u000f)6=~DY\nS(\u000f).\nIV. NUMERICAL RESULTS AND DISCUSSION\nIn realistic systems, the interfaces between layers of\ndi\u000berent materials are usually rough. Such rough inter-\nfaces break the momentum conservation of incident par-\nticles, e\u000bectively randomizing the momenta of the scat-\ntered particles. Therefore, we approximate the interface\ncoupling as Uk;k0\u0019U= const. Furthermore, we work\nin the thermodynamic limit where1\nNLP\nk=\u0000aL\n2\u0019\u00013R\nd3k\nand1\nNRP\nk0=\u0000aR\n2\u0019\u00013R\nd3k0, and take both leads to be\nof the same material, so that we can drop the L=R label.\nThen ~DP=AP\nE (\u000f) = ~DP=AP\nE (\u0000\u000f), resulting in \rP=AP\nE = 0\n[see Eqs. (23) and (26e)]; i.e., there is no additional en-\nergy transfer to the lattice. In keeping with the long-\nwavelength expansion used in Sec. II A, we only con-\nsider low temperatures T\u001cJS=kB. For yttrium iron\ngarnet23, this means T\u001c40 K.\nThe tunneling conductances (26) are displayed in Fig. 4\nas functions of the in-plane anisotropy, i.e., of the spin-\nwave ellipticity. In the parallel con\fguration shown in\nFig. 4(a), all conductances depend only weakly on the\nmagnitude of the anisotropy. With the exception of\nthe dissipation-assisted spin conductance \rP\nS, they de-\ncrease for hard-axis ( Ky<0) and increase for easy-plane\n(Ky>0) anisotropy. This can be attributed to the\nmagnon gap increasing or decreasing, respectively, which\nincreases or decreases the overall magnon population.\nThe strong increase and eventual divergence of the spin\nconductance for Ky!~\r\u00160Hsigni\fes the divergence of\nthe Bose distribution for vanishing spin-wave gap, and is\na precursor to the magnetic reorganization transition in\nwhich the magnetization tilts into the anisotropy plane.\nThe additional dissipation-assisted spin conductance \rP\nS\nmirrors this behavior, but also increases for hard-axis\nanisotropies, in contrast to all other conductances. Thereason for this is that it is an o\u000b-resonance process that\nis less sensitive to the exact value of the gap than the\nresonant ones, whereas its magnitude is determined by\nthe strength of the anisotropies. Because of this, it is\nalso three to four orders of magnitude smaller than the\nother conductances. Also, note that the breaking of the\nOnsager reciprocity of spin Seebeck and Peltier conduc-\ntances by the spin-wave ellipticity and the Gilbert damp-\ning is negligible in the parallel con\fguration.\nIn the antiparallel con\fguration displayed in Fig. 4(b),\non the other hand, the anisotropy dependence of the\nconductances is more pronounced. In agreement with\nthe discussion in Sec. III, spin and spin Peltier conduc-\ntances are in this case both zero if there is no anisotropy.\nHowever, chemical-potential driven spin transfer is still\npossible in this case because the dissipation-assisted spin\nconductance \rAP\nSis \fnite for Ky= 0. Apart from \rAP\nS,\nwhich decreases for Ky<0, all conductances increase\naway from Ky= 0, although they stay small compared\nto the conductances in the parallel con\fguration shown\nin Fig. 4. Note that, as the spin-wave gap closes, the\nspin conductance diverges and the breaking of Onsager\nreciprocity becomes visible.\nTo quantify the e\u000bect of Gilbert damping and spin-\nwave ellipticity on the magnon spin valve, we introduce\nmagnetothermal conductance (MTC), magnetospin con-\nductance (MSC), magneto-Seebeck conductance (MLC),\nand magneto-Peltier conductance (MPC) ratios as fol-\nlows:\nMTC =\u0014P\u0000\u0014AP\n\u0014P; (27a)\nMSC =\u001bP+\rP\nS\u0000\u001bAP\u0000\rAP\nS\n\u001bP+\rP\nS; (27b)\nMLC =LP\u0000LAP\nLP; (27c)\nMPC =\u0005P+\rP\nE\u0000\u0005AP\u0000\rAP\nE\n\u0005P+\rP\nE: (27d)\nIn the absence of dissipation and spin-wave ellipticity\nthere are no currents in the antiparallel con\fguration,\nhence these ratios reduce to 1. Their deviation from\n1 thus measures the magnitude of dissipation and spin-\nwave ellipticity e\u000bects on the magnon spin valve. The\nadditional energy and spin currents \rP=AP\nE and\rP=AP\nS\nare included in the ratios (27) because they a\u000bect the\nconductance ratios that will be measured in an experi-\nment, even though they originate from the lattice and\nnot from the magnons in the other lead.\nAs shown in Fig. 5(a), the in-plane anisotropy a\u000bects\nthe MTC ratio only negligibly. In the presence of large\nGilbert damping, on the other hand, the MTC ratio can\ndeviate from 1 by up to 10%. Responsible for this de-\ncrease are the dissipation-assisted pair creation and an-\nnihilation processes that enable energy transfer in the an-\ntiparallel con\fguration even when there is no spin-wave\nellipticity. The MSC ratio, displayed in Fig. 5(b), be-\nhaves similarly for most values of the anisotropy; when7\nκP/κ0\nσP/σ0\nLP/L0\nΠP/Π0\n103×γSP/γ0\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.0000.0050.0100.0150.0200.0250.0300.035\nKy/ℏγμ0H(a)\nκAP/κ0\nσAP/σ0\nLAP/L0\nΠAP/Π0\nγSAP/γ0\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.00000.00010.00020.00030.00040.00050.00060.0007\nKy/ℏγμ0H(b)\nFIG. 4. Tunneling conductances (26) in the (a) parallel and (b) antiparallel con\fgurations as functions of in-plane anisotropy\nKy, forKx= 0, temperature kBT= 10 \u0002~\r\u00160H, and Gilbert damping parameter \u000b= 10\u00002. The conductances are rescaled\nby the dimensionfull prefactors \u00140=U2p\nSk3\nBT=~2J3,\u001b0=\r0=~\u00140=k2\nBT,L0=~\u00140=kbT, and \u0005 0=TL0=~. With this\nrescaling, spin Seebeck and Peltier conductances, L=L 0and \u0005=\u00050, respectively, lie almost perfectly on top of each for most\nvalues of anisotropy Ky, re\recting Onsager reciprocity.\nα=5×10-3\nα=10-2\nα=5×10-2\nα=10-1\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.900.920.940.960.981.00\nKy/ℏγμ0HMTC(a)\nα=5×10-3\nα=10-2\nα=5×10-2\nα=10-1\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.650.700.750.800.850.900.951.00\nKy/ℏγμ0HMSC(b)\nα=5×10-3\nα=10-2\nα=5×10-2\nα=10-1\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.9850.9900.9951.000\nKy/ℏγμ0HMLC(c)\nα=5×10-3\nα=10-2\nα=5×10-2\nα=10-1\n-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.9900.9920.9940.9960.9981.000\nKy/ℏγμ0HMPC(d)\nFIG. 5. (a) Magnetothermal, (b) magnetospin, (c) magneto-Seebeck, and (d) magneto-Peltier conductance ratios as de\fned\nin Eqs. (27) as functions of in-plane anisotropy Ky, forKx= 0, temperature kBT= 10 \u0002~\r\u00160H, and various values of the\nGilbert damping parameter \u000b.8\nthe magnon gap closes for Ky!~\r\u00160H, however, it\nrapidly decreases because of the divergence of the spin\nconductance.\nThe magneto-Seebeck and Peltier conductance ratios,\nshown in Figs. 5(c) and 5(d), respectively, display an\nopposite behavior: they are sensitive to the anisotropy,\nbut only slightly a\u000bected by the Gilbert damping. They\nshow a decrease of the order of 1% when the magnon gap\ncloses forKy!~\r\u00160H, and also decrease, albeit more\nslowly, for increasing hard axis anisotropy; this re\rects\nthe increasing strength of the spin-conservation break-\ning in both cases. Unlike the MTC and MSC ratios,\nthe MLC and MPC ratios are actually increased by the\nGilbert damping for larger values of the anisotropy. Note\nalso that the Seebeck and Peltier ratios are both qualita-\ntively and quantitatively almost identical; the breaking of\nthe Onsager reciprocity is only apparent in the stronger\ndecrease of the MLC ratio for Ky!~\r\u00160H.\nV. CONCLUSIONS\nWe have studied the tunneling current in a magnon\nspin valve device. By applying the Holstein-Primako\u000b\ntransformation to the Heisenberg Hamiltonian, we de-\nrived the magnon Hamiltonian, in which transverse\nanisotropies introduce ellipticity of the magnons. We\nhave also added Gilbert damping to the magnon spectral\nfunction to study the e\u000bects of dissipation on the magnon\ntunneling. Both precession ellipticity and Gilbert damp-\ning are found to open new, Onsager-reciprocity breaking\nchannels for heat and spin transport across the junction,\nresulting in \fnite currents even when the magnetizations\nof both leads are aligned antiparallel. We have not only\nfound that dissipation and spin-wave ellipticity decrease\nthe spin and heat conductance ratios, but have also re-\nvealed a clear di\u000berence in the sensitivity of heat and\nspin currents to these two quantities. We hope that our\nresults provide useful guidance for the design and under-\nstanding of magnon spin valve devices.ACKNOWLEDGMENTS\nThis work is supported by the European Research\nCouncil via Consolidator Grant No. 725509 SPINBE-\nYOND. R.D. is a member of the D-ITP consortium, a\nprogram of the Netherlands Organisation for Scienti\fc\nResearch (NWO) that is funded by the Dutch Ministry\nof Education, Culture and Science (OCW). J.Z. would\nlike to thank the China Scholarship Council. This re-\nsearch was supported in part by the National Science\nFoundation under Grant No. NSF PHY-1748958.\nAPPENDIX: DERIVATION OF THE\nTUNNELING CURRENTS\nIn this Appendix, we outline the derivation of the tun-\nneling currents given in Sec. III. The total energy current\nfrom the left to the right lead is given by\nIP=AP\nE =@thHRi (28)\n=X\nk0ER;k0@tnR;k0; (29)\nwhereas the spin current from the left to right lead is\nIP\nS=\u0000@tX\nj~\nSz\nR;j\u000b\n(30)\n=~X\nk0\u0000\nu2\nR;k0+v2\nR;k0\u0001\n@tnR;k0; (31)\nin the parallel con\fguration, or\nIAP\nS=@tX\nj~\nSz\nR;j\u000b\n(32)\n=~X\nk0\u0000\nu2\nR;k0+v2\nR;k0\u0001\n@tnR;k0: (33)\nin the antiparallel con\fguration. Using Fermi's golden\nrule24, we \fnd the following kinetic equations for the\nquasiparticle distribution functions in the parallel con-\n\fguration:\n@tnL;k=2\u0019SLSR\n~NLNRX\nk0h\njUk;\u0000k0j2(uL;kuR;k0+vL;kvR;k0)2\u000e(EL;k\u0000ER;k0) (nR;k0\u0000nL;k)\n+jUk;k0j2(uL;kvR;k0+vL;kuR;k0)2\u000e(EL;k+ER;k0) (1 +nL;k+nR;k0)i\n; (34a)\n@tnR;k0=2\u0019SLSR\n~NLNRX\nkh\njUk;\u0000k0j2(uL;kuR;k0+vL;kvR;k0)2\u000e(EL;k\u0000ER;k0) (nL;k\u0000nR;k0)\n+jUk;k0j2(uL;kvR;k0+vL;kuR;k0)2\u000e(EL;k+ER;k0) (1 +nL;k+nR;k0)i\n: (34b)\nThe corresponding expressions in the antiparallel con- \fguration can be obtained from Eq. (34) by ex-9\nchanging the Bogoliubov-coe\u000ecient prefactors according\nto (uL;kuR;k0+vL;kvR;k0)2$(uL;kvR;k0+vL;kuR;k0)2.\nThe energy and spin currents (20) and (21) are obtained\nby inserting the kinetic equations (34) into their respec-tive de\fnitions (29) and (31) or (33), assuming a steady\nstate as in Eq. 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Here, we develop a fully ab-initio parameterized out-of-equilibrium\ntheory based on a quantum kinetic approach–termed (N+2) temperature model –that describes\nmagnon occupation dynamics due to electron-magnon scattering. We apply this model to per-\nform quantitative simulations on the ultrafast, laser-induced generation of magnons in iron and\ndemonstrate that on these timescales the magnon distribution is non-thermal: predominantly high-\nenergy magnons are created, while the magnon occupation close to the center of the Brillouin zone\neven decreases, due to a repopulation towards higher energy states via a so-far-overlooked scattering\nterm. We demonstrate that the simple relation between magnetization and temperature computed\nat equilibrium does not hold in the ultrafast regime and that the 3TM greatly overestimates the\ndemagnetization. The ensuing Gilbert damping becomes strongly magnon wavevector dependent\nand requires a description beyond the conventional Landau-Lifshitz-Gilbert spin dynamics. Our ab-\ninitio -parameterized calculations show that ultrafast generation of non-thermal magnons provides\na sizable demagnetization within 200fs in excellent comparison with experimentally observed laser-\ninduced demagnetizations. Our investigation emphasizes the importance of non-thermal magnon\nexcitations for the ultrafast demagnetization process.\nI. INTRODUCTION\nThe discovery that magnetic order can be manipu-\nlated on sub-picosecond timescales by femtosecond laser\npulses [1–3] has fueled the emergence of intensive exper-\nimental and theoretical research efforts in the field of ul-\ntrafast magnetization dynamics. What makes this field\nparticularly interesting, apart from its technological po-\ntential in future memory and spintronic devices [4, 5], is\nthat many well-established physical paradigms cannot be\nsimply transferred from the equilibrium to the ultrafast\nregime, due to its highly non-equilibrium nature. Relat-\nedly, albeit more than 25 years of intense research, the\nunderlying mechanisms of ultrafast demagnetization are\nstill heavily debated [6–8]: while some works [9–14] lean\ntowards longitudinal excitations – i.e., the reduction of\nthe magnetic moment carried by each atom due to the de-\ncrease of exchange splitting – others [15–19] hint at trans-\nverse spin excitations – a reduction of the average magne-\ntization due to the mutual tilting of the moments carried\nby different atoms – as the main contribution. Non-local\ncontributions due to superdiffusive spin currents [20, 21]\nare relevant in certain situations [22–25]. However, it has\nbecome evident that they are most likely not the only\nmechanism of ultrafast demagnetization [26, 27].\nTheoretical models describing ultrafast magnetization\ndynamics typically rely on a separation of electronic,\nphononic and – if magnetization dynamics are to be con-\nsidered separately – spin degrees of freedom. Beaurepaire\net al. [1] introduced the three-temperature model (3TM)\nto explain the flow of the energy transferred by the laser\n∗markus.weissenhofer@fu-berlin.deby assuming that each subsystem is internally in thermal\nequilibrium and the system can hence be described by\nthree temperatures (for electrons, phonons and spins), to-\ngether with the respective distributions (Fermi-Dirac and\nBose-Einstein). However, it was pointed out in numer-\nous investigations that the distributions are non-thermal\non ultrafast timescales [28–37]. Also, the 3TM discards\ncompletely the transfer of angular momentum due to de-\nmagnetization, which, according to recent experiments\n[38, 39], appears to be primarily to the lattice.\nTransverse demagnetization is often studied using\natomistic spin dynamics simulations based on the\nstochastic Landau-Lifshitz-Gilbert (LLG) equation to-\ngether with an extended Heisenberg model [40–42], which\ncan successfully reproduce experimentally measured de-\nmagnetization curves [43, 44]. The stochastic LLG is\na Langevin-type equation with a coupling to a heat\nbath with given temperature via a single parameter, the\nGilbert damping parameter. This parameter includes all\npossible contributions – Fermi surface breathing, crystal\ndefects, coupling to phonons, s−dcoupling, etc. [45–52] –\nto damping and while it can in principle be obtained from\nab initio calculations, in practice it is typically taken from\nexperimental measurements of ferromagnetic resonance\n(FMR) [53]. On the one hand, this ensures the versatility\nof atomistic spin dynamics simulations, but on the other\nhand, it obscures the details of the underlying micro-\nscopic energy and angular momentum transfer processes\n- which are crucial for understanding the fundamentals\nof ultrafast demagnetization. For this reason, steps have\nbeen taken in recent years to explicitly consider the cou-\npling of spins to phonons [54–62] and electrons [63–65].\nAlso, due to the classical nature of the commonly used\nstochastic LLG, the equilibrium magnon occupations cal-arXiv:2309.14167v3  [cond-mat.mtrl-sci]  15 Jan 20242\nculated by it follow Rayleigh-Jeans rather than Bose-\nEinstein statistics, henceforth leading to the wrong tem-\nperature scaling of the magnetization [66, 67]. Implemen-\ntation of quantum statistics in the spin-dynamics simu-\nlations can however provide the correct low-temperature\nscaling of the magnetization [68, 69].\nIn this work, we investigate the laser-induced gener-\nation of magnons, the low energy transverse excitations\nof the spin system, due to electron-magnon scattering.\nWe develop a quantum kinetic approach, which will be\ntermed (N+2)-temperature model [(N+2)TM], to per-\nform quantitative simulations of the time evolution of\nthe non-thermal magnon dynamics in bcc iron. Being\nbased on ab initio parameters and considering also non-\nthermal magnon distributions, our work goes well beyond\nwhat has been done in Refs. [63, 64, 70] and the conven-\ntional 3TM. In addition, we show that the 3TM and its\nrelevant parameters can be obtained from our (N+2)TM\nand, with that, from ab initio calculations. Importantly,\nusing ab initio calculated input parameters, our quantum\nkinetic theory predicts a sizable and ultrafast demagne-\ntization of iron within 200 fs, in excellent agreement with\nexperiments [15].\nII. OUT-OF-EQUILIBRIUM MAGNON\nDYNAMICS MODEL\nTo describe the time evolution of the ultrafast non-\nthermal magnon occupation dynamics, we assume that\ntheir creation and annihilation is dominated by electron-\nmagnon scattering processes. In this work, we use the\nsp−dmodel [71, 72] to describe such processes. The\nbasic idea of both s−dmodel and sp−dmodel is the\nseparation of electrons in localized ( dband) electrons and\nitinerant ( sband or sandpbands) electrons. The mag-\nnetic moments of the delectrons make up the Heisenberg-\ntype [73] magnetic moments of constant length, the small\nenergy excitations of which are the magnons. The itin-\nerant electrons are described within a Stoner-type model\n[74]. While an unambiguous identification of spandd\nelectrons as localized and itinerant is strictly speaking\nnot possible, it has nonetheless been established in liter-\nature that these models provide a suitable framework for\nthe description of electron-spin interaction in many phe-\nnomena relevant for spintronics, e.g. magnetic relaxation\n[75–77], ultrafast demagnetization [63–65, 70, 78–80] and\nspin torques [81].\nWe assume local exchange between the itinerant and\nlocalized spins, as given by the Hamiltonian ˆHem∼PN\ni=1ˆsitin·ˆSloc\ni, with Nbeing the number of atoms,\nandˆsitinand ˆSloc\nithe spin operators for itinerant ( sp)\nelectrons and localized ( d) electrons at atom i. In sec-\nond quantization and second order in magnon variables(details in Method Section V A), the Hamiltonian reads\nˆHem≈ −∆X\nkν\u0010\nˆc†\nkν↑ˆckν↑−ˆc†\nkν↓ˆckν↓\u0011\n−∆r\n2\nSNX\nkνν′,q\u0010\nˆc†\nk+qν↑ˆckν′↓ˆb†\n−q+ ˆc†\nk+qν↓ˆckν′↑ˆbq\u0011\n+∆\nSNX\nkνν′,qq′\u0010\nˆc†\nk−q+q′ν↑ˆckν′↑−ˆc†\nk−q+q′ν↓ˆckν′↓\u0011\nˆb†\nqˆbq′.\n(1)\nHere, ∆ is the sp−dexchange parameter, Sis the ab-\nsolute value of the localized spins, kandqare vectors in\nreciprocal space, ˆ c(†)\nkνσis the fermionic electron annihila-\ntion (creation) operator for the itinerant electrons – with\nνbeing the band index and σ∈ {↑,↓}– and ˆb(†)\nqis the\nbosonic magnon annihilation (creation) operator. The\nfirst term in Equation (1) describes the spin-splitting of\nthe itinerant electrons due to the exchange with the lo-\ncalized magnetic moments, the second one the excitation\n(annihilation) of a magnon due to a spin flip process and\nthe third one the spin-conserving scattering of a magnon\nand an electron from one state to another. It is worth\nnoting that the second term leads to a transfer of both\nenergy and angular momentum (i.e., spin) – since it can\nchange the total number of magnons – while the third\nterm can only transfer energy. For this reason, this term\nwas discarded earlier works [63–65], however, our quanti-\ntative analysis reveals that the energy transferred by this\nterm can exceed the energy transferred by the term first\norder in magnon operators.\nWe complete our Hamiltonian H=ˆHe+ˆHm+ˆHem\nby considering ˆHe=P\nkνσεkνσˆc†\nkνσˆckνσand ˆHm=P\nqℏωqˆb†\nqˆbq, with εkνσ=εkν−2∆δσ↑being the mode\nand spin dependent electron energies that are calcu-\nlated from first-principles calculations and ℏωqbeing the\nmagnon energies. Note that we have absorbed the term\nzero-th order in magnon variables in Equation (1) in the\notherwise spin-independent ˆHe.\nNext, we use the Hamiltonian introduced above to\nconstruct a quantum kinetic approach for the descrip-\ntion of the out-of-equilibrium dynamics of electrons and\nmagnons. We define the rates of energy exchange be-\ntween both subsystems as\n˙Em=X\nqℏωq˙nq (2)\n˙Ee=X\nkνσεkνσ˙fkνσ=−X\nqℏωq˙nq. (3)\nwhere the dot represents temporal derivative and with\nthe electron ( fkνσ) and magnon ( nq) occupation num-\nbers. The equivalence in Equation (3) results from the\nconservation of total energy. The time derivatives of\nthe occupation numbers can be calculated by applying\nFermi’s golden rule to the scattering Hamiltonian (1).3\nTo simplify the calculations, we further assume a ther-\nmal electron distribution and can hence introduce a sin-\ngle electronic temperature Tethat relates to the occu-\npation of electronic states via the Fermi-Dirac distribu-\ntion. This allows us to apply and also extend (by in-\ncluding terms second order in the bosonic operators) the\nideas laid out in Allen’s seminal work on electron-phonon\ninteraction [82] to electron-magnon scattering, yielding\n˙nq=\u0002\nnBE(ωq, Te)−nq\u0003\nγq+P\nq′\u0002\n(nq+ 1)nq′nBE(ωq−\nωq′, Te)+(q↔q′)\u0003\nΓqq′, with nBE(ωq, Te) = [eℏωq\nkBTe−1]−1\nbeing the Bose-Einstein distribution evaluated at the\nelectron temperature. The scattering rates are given by\nγq=4π∆2\nSNωqI↑↓(Te)X\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓),\n(4)\nΓqq′=2π∆2\nS2N2(ωq−ωq′)X\nσIσσ(Te)\n×X\nkνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ),(5)\nwith εFbeing the Fermi energy. The functions Iσσ′(Te)\nhave the property lim Te→0Iσσ′(Te) = 1 and account\nfor the smearing of the Fermi-Dirac distribution at high\nelectron temperatures, similar to what has been derived\nfor electron-phonon scattering [35]. The expression for\nIσσ′(Te) and details of the derivation of Equations (4)–\n(5) are in the Method Section V A. Note that a com-\nparison with linear spin-wave theory in the framework\nof the Landau-Lifshitz-Gilbert equation [83] reveals that\nγq/ωq=αqcan be viewed as a mode-dependent Gilbert\ndamping parameter.\nDue to the assumption that the electron occupation\nnumbers follow the Fermi-Dirac distribution at all times,\nthe change in electron energy is determined by the\nchange in Te, i.e., ˙Ee=P\nkνσεkνσ(∂fkνσ/∂T e)˙Te=\nCe˙Te, with the electronic heat capacity Ce=P\nkνσεkνσ(∂fkνσ/∂T e). By additionally considering the\nabsorption of a laser pulse with power P(t) by the\nelectrons and a coupling of the electrons to a phonon\nheat bath as in the 2TM, we finally obtain our out-of-\nequilibrium magnon dynamics model:\n˙nq=h\nnBE(ωq, Te)−nqi\nγq\n+X\nq′h\n(nq+ 1)nq′nBE(ωq−ωq′, Te) + (q↔q′)i\nΓqq′,\n(6)\n˙Te=1\nCeh\n−X\nqℏωq˙nq+Gep(Tp−Te) +P(t)i\n, (7)\n˙Tp=−Gep\nCp(Tp−Te). (8)\nHere, Tp,Cpand Gepare the phonon temperature\nand heat capacity and electron-phonon coupling con-stant, respectively. Note that we do not consider di-\nrect magnon-phonon coupling, which has been shown\nto be a reasonable approximation for 3 dferromagnets\n[43, 44]. We would like to point out that the non-\nthermal magnon occupations nqcan be translated to\nmode-specific temperatures via the Bose-Einstein distri-\nbution, Tq:=ℏωq/(kBln(n−1\nq+ 1)). Based on this – and\nin distinction from the 3TM – we term the framework\nprovided by Equations (6)-(8) the (N+2)-temperature\nmodel ((N+2)TM). Below, we reveal by solving these\ncoupled equations numerically that they provide a vi-\nable framework to describe laser-induced ultrafast mag-\nnetization dynamics and the generation of non-thermal\nmagnons, going beyond the well-established 3TM.\nBefore doing so, we want to shortly discuss the relation\nbetween the (N+2)TM introduced here and the 3TM. Al-\nbeit their phenomenological nature, the 2TM ( TeandTp)\nand the 3TM ( Te,TpandTm) have been successfully ap-\nplied to explain a plethora of phenomena [84], perhaps\nmost prominently by Beaurepaire et al. to describe the\nultrafast demagnetization of Ni [1]. Allen [82] and Man-\nchon et al. [78] demonstrated that the 2TM and the 3TM\ncan be derived from a microscopic out-of-equilibrium ap-\nproach similar to the one used here. By assuming instan-\ntaneous relaxation of the magnon occupation numbers\nto the Bose-Einstein distribution with a single magnon\ntemperature Tm, our (N+2)TM reduces to the 3TM (in\nabsence of magnon-phonon coupling),\nCm˙Tm=Gem(Te−Tm),\nCe˙Te=Gem(Tm−Te) +Gep(Tp−Te) +P(t),\nCp˙Tp=Gep(Te−Tp),(9)\nwith the magnon heat capacity Cm=P\nqCq=P\nqℏωq(∂nq/∂T m) and the electron-magnon coupling\nconstant\nGem=X\nqCqh\nγq+X\nq′kBTm\nℏωqΓqq′i\n. (10)\nDetails of the derivation are found in Method Section\nV B. The above expression goes beyond what was derived\nin Ref. [78] by including terms second order in magnon\nvariables and allows us to determine the electron-magnon\ncoupling fully based on ab initio parameters. We would\nlike to point out that it can be extended further by going\nto higher order in the magnon variables.\nIII. RESULTS\nA. Magnon lifetimes and Gilbert damping\nWe apply the (N+2)TM model defined by Equations\n(6)-(8) to bcc iron. To obtain a full solution of the out-\nof-equilibrium dynamics, it is required to calculate ma-\nterial specific quantities. First, we estimate ∆ ≈0.75 eV\nfrom the band structure and with that we compute the4\nΓ H N Γ P H020406080100120magnon frequency (THz)\n100101102103\nlifetime (ps)\nFigure 1. Magnon dispersion of bcc iron with lifetimes γ−1\nq\ngiven as color code, shown along high-symmetry lines of the\nBZ. The lifetimes are due to the first-order contribution to\nthe electron-magnon scattering.\nquantities γq, Γqq′andIσσ′(Te), both using the full-\npotential linear augmented plane wave code ELK [85]\n(details can be found in the Method Section V C). For\nbcc iron it turns out that Iσσ′(Te) only scales weakly\nwith temperature and hence we use the low tempera-\nture limit Iσσ′(Te) = 1 hereinafter. The parameters gov-\nerning the magnon energies ℏωq=S(2d+P\njJij[1−\nexp(−iq·(rj−ri))]) were taken from earlier works: the\nexchange constants Jijare from first-principles calcula-\ntions [86] and the magneto-crystalline anisotropy energy\nd= 6.97µeV per atom is from experiments [87]. Further,\nwe used the saturation moment µs= 2.2µBand spin\nS= 2.2/2. Based on these parameters and the formulas\nderived above, we get Cm= 5.720×104Jm−3K−1and\nGem= 6.796×1017Wm−3K−1atTm= 300 K. Notably,\nthe term first order in magnon variables leads to a con-\ntribution to Gemthat is one order of magnitude smaller\nthan the second-order term. We further use the room-\ntemperature values Ce= 1.013×105Jm−3K−1,Cp=\n3.177×106Jm−3K−1andGep= 1.051×1018Wm−3K−1\nthat were obtained in Refs. [35, 37] from first-principles\ncalculations. The influence of a temperature dependent\nelectronic heat capacity Ceon the demagnetization is dis-\ncussed in the Supporting Information.\nBoth ωqand the inverse of γq, i.e., the lifetime of\nmagnons due to the contribution to electron-magnon\nscattering linear in the magnon variables, are shown in\nFigure 1 along high-symmetry lines of the Brillouin zone\n(BZ). It can readily be observed that the lifetimes of\nhigh-frequency magnons are drastically reduced as com-\npared to the low energy ones. The q-dependent lifetimes\ngive rise to mode-specific Gilbert damping αq(=ωq/γq).\nOur finding of mode-dependent Gilbert damping is con-\nsistent with experiments [88] and also with a recent field-\ntheory derivation [89]. The computed αqvalues, shown\nin Method Section V C, range between 1 .5×10−3and\n1.08×10−2. These values are close to the experimentally\nobtained ones (via FMR measurements) for Fe ranging\nfrom 1 .9×10−3to 7.2×10−3[90–95], however with a\n0 1 2 3300400500600700temperature T (K)(a)\n0 1 2 3\ntimet(ps)−1.0−0.50.0∆M/M 0(%) (b)Te\nTp\n/angbracketleftTq/angbracketrightT3TM\ne\nT3TM\np\nT3TM\nm\n0.0 0.2 0.4 0.6 0.8 1.0\nlaser fluence (mJ /cm2)0510demagnetization (%)\n(c)Figure 2. Laser-induced ultrafast non-equilibrium dynamics\nof iron calculated from an ab initio parameterized model.\n(a) Temporal evolution of electron temperature Te, phonon\ntemperature Tpand average magnon temperature ⟨Tq⟩=\n1/NP\nqTqobtained by the (N+2)TM (solid lines). The\nblue shaded region indicates the temperature range within\nwhich all magnon temperatures are contained. Dashed lines\nshow the results of the 3TM solved with ab initio calcu-\nlated input parameters. (b) Relative change of total mag-\nnetization of the localized magnetic moments ∆ M/M 0=P\nq(ninit\nq−nq)/(NS−P\nqninit\nq), with ninit\nq=nBE(ωq,300 K)\nbeing the occupation number before the laser pulse. (c) De-\nmagnetization max( |∆M/M 0|) versus laser fluence computed\nfor a ferromagnetic layer with a thickness of 20 nm. The dot-\nted line serves as a guide to the eye.\nsomewhat larger variation with qas compared to what\nwas reported in Ref. [83].\nWe note that the q-dependent Gilbert damping goes\nbeyond the conventional LLG description which assumes\none single damping parameter for all spin dynamics.\nMoreover, a further distinction between the current the-\nory and the LLG framework is that, in the latter, there\nis a single damping term that governs both the energy\nand angular momentum transfer [96], whereas the cur-\nrent theory has two terms [see Equation (1)], one that\ntransfers energy and angular momentum and one that\ntransfers only energy. As shown in the following, this 2nd\nterm is found to be important for non-thermal magnon5\ngeneration.\nB. Ultrafast dynamics\nBased on the above-given parameters, we calculate\nthe coupled out-of-equilibrium magnon, electron, and\nphonon dynamics induced by a Gaussian laser pulse\nP(t) = A/p\n2πζ2exp[−(t/ζ)2/2] with A= 9.619×\n107Jm−3andζ= 60 fs for N= 203magnon modes. Note\nthat this value of Atranslates to an absorbed fluence of\n0.19 mJ /cm2for a ferromagnetic layer with thickness of\n20 nm, which is a typical thickness in ultrafast demagne-\ntization experiments [1].\nFigure 2(a) depicts the time evolution of electron,\nphonon and average magnon temperature – together with\nthe temperature range of all magnon temperatures – cal-\nculated using the (N+2)TM. The electron temperature\nreaches a maximum of 685 K at around 52 fs after the\nmaximum of the laser pulse (located at t= 0) and con-\nverges to the phonon temperature in less than 1 .5 ps. The\nmaximum of the average magnon temperature of 520 K\nis reached only slightly after the electronic one at around\n136 fs, followed by a convergence to the electronic and\nphononic temperature to a final temperature of around\n329 K at 3 ps, in agreement with what can be estimated\nfrom the energy supplied by the laser pulse and the in-\ndividual heat capacities via ∆ T=A/(Cm+Ce+Cp) =\n28.8 K. Notably, the magnon temperatures still cover a\nrange of around 50 K at this point in time. Our results\nclearly demonstrate the shortcomings of the conventional\n3TM (shown as dotted lines): While the initial increase\nof temperatures is comparable to the (N+2)TM, magnon\nthermalization happens much faster in the 3TM.\nIn Figure 2(b), we show the laser-induced change in\nmagnetization (associated with the localized magnetic\nmoments) due to the creation of additional magnons. We\nobserve ultrafast transversal demagnetization of around\none percent in less than 300 fs, demonstrating that the\ntimescales obtained by our ab initio based calculations\nare in reasonable agreement with experimental measure-\nments (see, e.g., [15, 97–99]). Notably, the minimum\nof the magnetization and the maximum in the average\nmagnon temperature computed by the (N+2)TM are at\ndifferent points in time. Also, the drop in the (localized)\nmagnetization is much less pronounced than expected\nfrom the increase in average temperature: in thermal\nequilibrium, a temperature increase from 300 K to above\n500 K approximately leads a demagnetization of 20% for\niron [100]. These observations clearly demonstrate the\nshortcomings of the 3TM – where a thermal magnon dis-\ntribution at all times is assumed – and underline the im-\nportance of treating the full, non-thermal magnon distri-\nbution in the ultrafast regime.\nFigure 2(c) depicts the maximum of the demagnetiza-\ntion versus laser fluence for an iron layer of 20 nm. We\nfind a nonlinear dependence, which is a result of the non-\nlinearity of our (N+2)TM, and a substantial demagneti-\n0 2 4 6 8 10\ntimet(ps)0.850.900.951.00M/M 0\n(N+2)TM\nexperimentFigure 3. Comparison between experiment and the (N+2)TM\ntheory for ultrafast demagnetization in iron. The experimen-\ntal data (symbols) are those of Carpene et al. [15] and the\nsolid lines are calculated from the ab initio parameterized\n(N+2)TM.\nzation of around ten percent at 0 .95 mJcm−2. We note\nthat for high fluences, higher-order magnon-magnon scat-\ntering terms that are not included in the current model\ncould start to play a role.\nThe obtained amount of demagnetization and the mag-\nnetization decay time (below 200 fs) for this fluence are\ncomparable with experiments, which suggests that ultra-\nfast magnon excitation [15–17] provides a viable mech-\nanism for ultrafast laser-induced demagnetization. It\nis also consistent with time-resolved extreme ultraviolet\nmagneto-optical and photoemission investigations that\ndetected magnon excitations during ultrafast demagneti-\nzation of elemental ferromagnetic samples [18, 101].\nFor a more precise examination of the predictions\nof the (N+2)TM, we compare the calculated time-\ndependent demagnetization with experimental data for\nFe in Figure 3. The experimental data were measured\nby Carpene et al. [15] on a 7-nm thin film, using the\ntime-resolved magneto-optical Kerr effect for two differ-\nent pump laser fluences of 1 .5 mJcm−2and 3 mJcm−2.\nIn the calculations we used an absorbed laser fluence\nthat is about five times lower, as the exact value of the\nabsorbed fluence in experiments is difficult to estimate\n(due to influence of optical losses, sample reflection,\netc). Specifically, in the simulations we used absorbed\nlaser energies of 433 Jcm−3and 693 Jcm−3in a 7-nm Fe\nfilm. Figure 3 exemplifies that not only the amount of\ndemagnetization but also the full time dependence of\nthe demagnetization predicted by the (N+2)TM is in\nremarkable agreement with experiments.\nC. Non-thermal magnon dynamics\nNext, we analyze the non-thermal magnon dynam-\nics in more detail in Figure 4. There, we show the\nmagnon temperatures versus frequency (a) and along\nhigh-symmetry lines of the BZ (b) at different points in\ntime. The laser pulse primarily heats up high energy\nmagnons, while the temperature of low energy magnons6\n300500\n−250 fs\n300500\n0 fs\n300500temperature (K)\n250 fs\n300500\n500 fs\n0 20 40 60 80 100 120\nmagnon frequency (THz)300500\n1000 fs\nΓ H N Γ P H−200 fs−100 fs0 fs100 fs200 fs300 fs400 fs500 fs600 fs700 fs800 fs900 fs1000 fs\n300350400450500550600\ntemperature (K)(a) (b)\nFigure 4. Magnon temperatures of iron during ultrafast laser excitation at different points in time (w.r.t. the maximum of\nthe laser pulse) calculated from the ab initio parameterized (N+2)TM. (a) Magnon temperatures (dots) versus frequency. The\nsolid line indicates the electron temperature. (b) Magnon dispersion and their temperatures, depicted by the color code, shown\nalong high-symmetry lines of the BZ.\nbarely changes and even decreases slightly in the vicinity\nof the Γ point (the temperatures drop by up to around\n2.5 K). This surprising observation is caused by a\nredistribution of magnons from this region to other\nparts of the BZ due to the term second order in the\nmagnon operators in Equation (1); the effective second\norder scattering rate γ(2)\nq:=P\nq′Γqq′is negative for low\nmagnon frequencies (more details can be found in the\nMethod Section V C). It is also observed that although\nthe magnon temperatures reached after the laser pulse\nare generally higher at higher frequencies, however, there\nis not necessarily a monotonous increase of temperature\nwith frequency at all times: e.g., at 100 fs after the laser\npulse [Figure 4(b)], the temperatures at the points H, N,\nand P is higher than in between these points. Notably,\nthe position of the maximum magnon temperature in\nthe BZ also varies with time.\nD. Discussion\nDifferent physical mechanisms have been proposed for\nultrafast demagnetization in elemental 3 dferromagnets\n[9, 11, 15, 18]. The preeminent mechanisms are Elliott-\nYafet (EY) electron-phonon spin-flip scattering [11, 13]\nand ultrafast magnon generation [15]. In the former, a\nStoner-type picture is used to model the longitudinal re-\nduction of the atomic moment due to electron-phonon\nspin-flip scattering, whereas the latter is based on length-\nconserving transverse spin-wave excitations. Experimen-\ntal indications of electron-phonon scattering [38, 39] aswell as of electron-magnon scattering have been reported\n[18, 101].\nThe strength of the different demagnetization channels\nis an important issue in the on-going discussion on the\ndominant origin of ultrafast demagnetization [7]. Ab ini-\ntiocalculated quantities such as the EY spin-flip prob-\nability are essential to achieve reliable estimates [102–\n104]. Griepe and Atxitia [14] recently employed the\nmicroscopic 3TM [11] and obtained quantitative agree-\nment with measured demagnetizations for the elemental\n3dferromagnets. They compared the fitted EY spin-\nflip probability αsfwith ab initio calculated values [104]\nand found these to be in good agreement, in support of\nan electron-phonon mechanism of ultrafast demagnetiza-\ntion. A drawback of their employed approach is how-\never that only magnetization reducing spin flips are in-\ncluded. EY spin flips that increase the magnetization are\nalso possible and, including these would lead to a signifi-\ncantly smaller demagnetization amplitude [104]. This in\nturn would question again what amount of demagneti-\nzation is precisely due to EY electron-phonon spin flip\nscattering. Conversely, in our non-thermal magnon ap-\nproach we employ ab initio calculated quantities without\nfit parameter. We find that the ab initio predicted ultra-\nfast demagnetization agrees accurately with experiments,\nwhich provides a strong support for the prominence of the\nnon-thermal magnon channel to the ultrafast demagne-\ntization process.7\nIV. CONCLUSIONS\nWe have developed an ab initio parameterized quan-\ntum kinetic approach to study the laser-induced gen-\neration of magnons due to electron-magnon scattering,\nwhich we applied to iron. Our results clearly demon-\nstrate that on ultrafast timescales the magnon distribu-\ntion is non-thermal and that henceforth the simple re-\nlation between magnetization and temperature via the\nM(T) curves computed at equilibrium does not hold:\nsince predominantly high-energy magnons are excited the\nenergy transferred from the laser-excited electrons cre-\nates relatively few magnons and hence the demagneti-\nzation (proportional to the total number of magnons) is\nmuch less pronounced than expected from the increaseof the average magnon temperature. Notably, the num-\nber of magnons actually decreases near the center of the\nBrillouin zone, which is due to the scattering from low to\nhigh energy magnons by a previously neglected scattering\nterm that can transfer energy but not angular momen-\ntum. This term, which is not included in LLG simula-\ntions, is a crucial quantity for out-of-equilibrium magnon\ndynamics.\nOurab initio -based calculations of the induced demag-\nnetization in iron furthermore provide strong evidence\nthat non-thermal magnons are excited fast and lead to\na sizable demagnetization within 200 fs. The result-\ning time-dependent demagnetization agrees remarkably\nwell with experiments, which establishes the relevance of\nmagnon excitations for the process of ultrafast optically\ninduced demagnetization.\nV. METHOD\nA. Derivation of electron-magnon scattering rates\nIn this Method Section we derive the (N+2)TM for the description of non-thermal magnons from a microscopic\nHamiltonian for electron-magnon scattering. We start with a local sp−dmodel Hamiltonian,\nˆHem=−Jsp−dX\niδ(r−ri)ˆsitin·Sloc\ni, (11)\nwith Jsp−dbeing the sp−dvolume interaction energy, ˆsitin=ˆσbeing the spin operators of itinerant ( sandp)\nelectrons and Sloc\nibeing the localized ( d) spins located at ri. For now, we treat the latter as classical vectors. The\nexpectation value for a given spin wave function Ψ(r) is given by\n⟨ˆHem⟩=−Jsp−dX\niZ\nΨ†(r)δ(r−ri)ˆsitin·Sloc\niΨ(r)dr (12)\n=−Jsp−dX\niZ\nδ(r−ri)\u0000Ψ∗\n↑(r),Ψ∗\n↓(r)\u0001\b\nˆσxSx\ni+ ˆσySy\ni+ ˆσzSz\ni\t\u0012\nΨ↑(r)\nΨ↓(r)\u0013\ndr (13)\n=−Jsp−dX\niZ\nδ(r−ri)(\nΨ∗\n↑(r)Ψ↓(r)S−\ni+ Ψ∗\n↓(r)Ψ↑(r)S+\ni+ (Ψ∗\n↑(r)Ψ↑(r)−Ψ∗\n↓(r)Ψ↓(r))Sz\ni)\ndr.(14)\nHere, we have introduced S±\ni=Sx\ni±iSy\ni. Next, we perform a plane wave expansion of the wave functions (for a\nsingle band of itinerant electrons),\nΨσ(r) =1√\nVX\nkeik·rckσ, (15)\nand a Holstein-Primakoff transformation of the localized spins,\nS+\ni=p\n2S−b∗\nibibi, S−\ni=b∗\nip\n2S−b∗\nibi, Sz\ni=S−b∗\nibi, (16)\ntogether with introducing the Fourier transform of the magnon amplitudes\nb∗\ni=1√\nNX\nqe−iq·rib∗\nq, b i=1√\nNX\nqeiq·ribq. (17)8\nInsertion of (15)–(17) into (14) and keeping terms up to second order in magnon variables, we get\n⟨ˆHem⟩=−Jsp−d\nVX\niX\nkk′(r\n2S\nNX\nqe−i(k−k′+q)·ric∗\nk↑ck′↓b∗\nq+r\n2S\nNX\nqe−i(k−k′−q)·ric∗\nk↓ck′↑bq\n+Se−i(k−k′)·ri(c∗\nk↑ck′↑−c∗\nk↓ck′↓)−1\nNX\nqq′e−i(k−k′+q−q′)·ri(c∗\nk↑ck′↑−c∗\nk↓ck′↓)b∗\nqbq′) (18)\n=−Jsp−dSN\nVX\nk(c∗\nk↑ck↑−c∗\nk↓ck↓)−Jsp−dSN\nVX\nkqr\n2\nSN\u0010\nc∗\nk+q↑ck↓b∗\n−q+c∗\nk+q↓ck↑bq\u0011\n+Jsp−d\nVX\nkqq′\u0010\nc∗\nk−q+q′↑ck↑−c∗\nk−q+q′↓ck↓\u0011\nb∗\nqbq′.(19)\nFor multiple itinerant bands and in second quantization we obtain\nˆHem=−∆X\nkν(ˆc†\nkν↑ˆckν↑−ˆc†\nkν↓ˆckν↓)−∆r\n2\nSNX\nkνν′,q\u0010\nˆc†\nk+qν↑ˆckν′↓ˆb†\n−q+ ˆc†\nk+qν↓ˆckν′↑ˆbq\u0011\n+∆\nSNX\nkνν′,qq′\u0010\nˆc†\nk−q+q′ν↑ˆckν′↑−ˆc†\nk−q+q′ν↓ˆckν′↓\u0011\nˆb†\nqˆbq′.(20)\nwhere we have introduced ∆ =Jsp−dSN\nV. Note that due to the plane wave ansatz we have implicitly assumed that\nthe itinerant electrons are completely delocalized and interband scattering (from νtoν′̸=ν) fully contributes to the\nelectron-magnon scattering.\nNext, we use Fermi’s golden rule to get the change of the magnon occupation number nq=⟨ˆb†\nqˆbq⟩. Fermi’s golden\nrule computes the probability W(i→f) for a small perturbation term in the Hamiltonian, ˆH′(in our specific case,\nˆHem) via\nW(i→f) =2π\nℏ|⟨f|ˆH′|i⟩|2δ(Ef−Ei), (21)\nwhere |i⟩and|f⟩denote the initial and final state, respectively.\nWe start with the term first order in the magnon variables,\n˙n(1)\nq=W(nq→nq+ 1)−W(nq→nq−1)\n=2π\nℏ2∆2\nSNX\nkνν′\b\n(1−fk−qν↑)fkν′↓−(fk−qν↑−fkν′↓)nq\t\nδ(εkν′↓−εk−qν↑−ℏωq),(22)\nwith fkνσ=⟨ˆc†\nkνσˆckνσ⟩andεkνσandℏωqbeing the eigenenergies of electrons and magnons, respectively.\nHereinafter, we make the assumption that due to the fast equilibration processes for electrons, they always follow\nthe Fermi-Dirac distribution, fFD(εkνσ, Te) = [e(εkνσ−εF)/kBTe+ 1]−1,with a single electron temperature Te. Before\nwe continue we need the following relation,\nfFD(εkν′↓, Te)(1−fFD(εk−qν↑, Te))δ(εkν′↓−εk−qν↑−ℏωq) =\n(fFD(εk−qν↑, Te)−fFD(εkν′↓, Te))nBE(ωq, Te)δ(εkν′↓−εk−qν↑−ℏωq)(23)\nwith nBE(ωq, Te) = [eℏωq\nkBTe−1]−1being the Bose-Einstein distribution evaluated at the electron temperature. Now\nwe can simplify Equation (22), yielding\n˙n(1)\nq≈2π\nℏ2∆2\nSNX\nkνν′\u0002\nnBE(ωq, Te)−nq\u0003\n(fFD(εk−qν↑, Te)−fFD(εkν′↓, Te))δ(εkν′↓−εk−qν↑−ℏωq)\n=\u0002\nnBE(ωq, Te)−nq\u0003\nγq. (24)9\nWith γqbeing the linewidth – i.e., the inverse lifetime – of the magnon due to the first order contribution to electron-\nmagnon scattering. Following the ideas laid out by Allen [82] and Maldonado et al. [35], it can be computed as\nγq=2π\nℏ2∆2\nSNX\nkνν′[fFD(εk−qν↑, Te)−fFD(εkν′↓, Te)]δ(εkν′↓−εk−qν↑−ℏωq) (25)\n=2π\nℏ2∆2\nSNX\nkνν′Z\ndε δ(ε−εk−qν↑)Z\ndε′δ(ε′−εkν′↓)[fFD(ε, Te)−fFD(ε′, Te)]δ(ε′−ε−ℏωq) (26)\n≈2π\nℏ2∆2\nSNX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z\ndεZ\ndε′[fFD(ε, Te)−fFD(ε′, Te)]δ(ε′−ε−ℏωq)g↑(ε)g↓(ε′)\ng↑(εF)g↓(εF)\n(27)\n≈2π\nℏ2∆2\nSNℏωqX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z\ndε(−1)∂fFD(ε, Te)\n∂εg↑(ε)g↓(ε+ℏωq)\ng↑(εF)g↓(εF)(28)\n≈2π\nℏ2∆2\nSNℏωqX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z\ndε(−1)∂fFD(ε, Te)\n∂εg↑(ε)g↓(ε)\ng↑(εF)g↓(εF)(29)\n=4π∆2\nNSωqX\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)I↑↓(Te) (30)\nwith εFbeing the Fermi energy, the spin-dependent density of states is gσ(ε) =P\nkνδ(ε−εkνσ) and the thermal\ncorrection factor given by\nIσσ′(Te) =Z\ndε(−1)∂fFD(ε, Te)\n∂εgσ(ε)g′\nσ(ε)\ngσ(εF)g′σ(εF). (31)\nIt is obvious that lim Te→0Iσσ′(Te) = 1. Note that we have used that the energy scale of magnons is much smaller\nthan the one of electrons, i.e., that ℏωq≪ε, ε′.\nThe contribution of the term second order in magnon variables to the occupation number can be calculated analogous\nand reads\n˙n(2)\nq=2π\nℏ\u0010∆\nSN\u00112X\nkνν′σ,q′n\n(nq+ 1)nq′\u0010\n(1−fFD(εk−q+q′νσ, Te))fFD(εkν′σ, Te)δ(ℏωq−ℏωq′+εk−q+q′νσ−εkν′σ)\u0011\n−\u0010\nq↔q′\u0011o\n(32)\n=2π\nℏ\u0010∆\nSN\u00112X\nkνν′σ,q′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te)\u0010\nfFD(εk−q+q′νσ, Te)−fFD(εkν′σ, Te)\u0011\n×\nδ(ℏωq−ℏωq′+εk−q+q′νσ−εkν′σ)−\u0010\nq↔q′\u0011o(33)\n≈2π\nℏ\u0010∆\nSN\u00112X\nkνν′σ,q′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te)(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te)−\u0010\nq↔q′\u0011o\n(34)\n=2π\nℏ\u0010∆\nSN\u00112X\nq′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te) +\u0000\nq↔q′\u0001oX\nkνν′σ(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te)\n(35)\n=2π\nℏ\u0010∆\nSN\u00112X\nq′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te) +\u0000\nq↔q′\u0001oX\nkνν′σ(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te)\n(36)\n=X\nq′n\n(nq+ 1)nq′nBE(ωq−ωq′, Te) +\u0000\nq↔q′\u0001o\nΓqq′(Te)(37)\nwith\nΓqq′(Te) =2π\nℏ\u0010∆\nSN\u00112\n(ℏωq−ℏωq′)X\nσIσσ(Te)X\nkνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ). (38)10\nB. Derivation of the three temperature model\nIn what follows, it is demonstrated that the three temperature model (3TM) can be obtained from the (N+2)-\ntemperature model derived in the main text,\n˙nq=h\nnBE(ωq, Te)−nqi\nγq+X\nq′h\n(nq+ 1)nq′nBE(ωq−ωq′, Te) + (q↔q′)i\nΓqq′,(39)\n˙Te=1\nCeh\n−X\nqℏωq˙nq+Gep(Tp−Te) +P(t)i\n, (40)\n˙Tp=−Gep\nCp(Tp−Te), (41)\nby assuming instantaneous relaxation of the magnon occupation numbers to the Bose-Einstein distribution with a\nsingle magnon temperature Tm, i.e., nq=nBE(ωq, Tm). For the sake of readability we rewrite nBE(ωq, Tm) =nq(Tm).\nWe start with the first order scattering term:\n˙n(1)\nq= [nq(Te)−nq(Tm)]γq≈(Te−Tm)∂nq(T)\n∂T\f\f\f\f\nT=Tmγq(Te) = (Te−Tm)Cqγq\nℏωq. (42)\nHere we have introduced the mode-dependent magnon heat capacity Cq=ℏωq∂nq(Tm)\n∂T.\nIn order to calculate the scattering term second order in the magnon variables, we first introduce the following\nrelation\n\u0000\nnq′(Tm) + 1\u0001\nnq(Tm) =\u0002\nnq′(Tm)−nq(Tm)\u0003\nnq−q′(Tm). (43)\nNow we calculate\n˙n(2)\nq=X\nq′\u0010\n(nq(Tm) + 1) nq′(Tm)nq−q′(Te) + (q↔q′)\u0011\nΓqq′ (44)\n=X\nq′\u0010\nnq′−q(Tm)nq−q′(Te)−(q↔q′)\u0011\n×\u0000\nnq(Tm)−nq′(Tm)\u0001\nΓqq′ (45)\n=X\nq′1\n2\u0012\ncoth\u0012ℏ(ωq′−ωq)\n2kBTe\u0013\n−coth\u0012ℏ(ωq′−ωq)\n2kBTm\u0013\u0013\u0000\nnq(Tm)−nq′(Tm)\u0001\nΓqq′ (46)\n≈X\nq′nq(Tm)−nq′(Tm)\nℏ(ωq′−ωq)kB(Te−Tm)Γqq′ (47)\n≈X\nq′∂nq(Tm)\n∂(ℏωq)kB(Tm−Te)Γqq′ (48)\n=X\nq′∂nq(T)\n∂T\f\f\f\f\nT=TmkBTm\nℏωq(Te−Tm)Γqq′ (49)\n=X\nq′CqkBTm\n(ℏωq)2(Te−Tm)Γqq′. (50)\nUsing the expressions for ˙ n(1)\nqand ˙n(2)\nq, the change in total energy of the magnons can then be calculated as\n∂Em\n∂t=∂Em\n∂Tm∂Tm\n∂t=X\nqℏωq∂nq(T)\n∂T|T=Tm\n| {z }\nCm∂Tm\n∂t= (Te−Tm)X\nqCq\u0010\nγq+X\nq′kBTm\nℏωqΓqq′\u0011\n.\n| {z }\nGem(51)\nWith that, the (N+2)TM transforms into the 3TM (in the absence of magnon-phonon coupling), which is given by\nCm˙Tm=Gem(Te−Tm),\nCe˙Te=Gem(Tm−Te) +Gep(Tp−Te) +P(t),\nCp˙Tp=Gep(Te−Tp).(52)11\nC.Ab initio calculations\nTo obtain a full solution of the (N+2)TM, it is necessary to compute the material specific quantities ∆, γq, Γqq′\nandIσσ(Te). For this purpose, we use the full-potential linear augmented plane wave code ELK [85].\nAs a first step, we determine the coupling parameter ∆ of the sp−dmodel, which sets the general scale of the\nelectron-magnon scattering. As shown in the main text, the first term (zeroth order in magnon variables) in the\nelectron-magnon scattering Hamiltonian reads ˆH(0)\nem=−∆P\nkν(ˆc†\nkν↑ˆckν↑−ˆc†\nkν↓ˆckν↓), with ν∈ {s, p}. Based on this,\n∆ can be estimated from the projected density of states (DOS), since it is one half of the spin-dependent energy\nsplitting of the s- and p-bands. In general, this splitting may vary for different electronic states. This is not accounted\nfor in the model used here, where instead a single parameter is used to model the spin splitting. We find, however,\nthat for bcc iron this is justified, since the shift in both s- and p-bands around the Fermi energy – the relevant\nregion for electron-magnon scattering – between spin up and down states is approximately constant with a value of\n∆≈0.75 eV, see left panel of Figure 5.\nNow we calculate the first and second order scattering rates using the formulas derived above,\nγq=4π∆2\nSNωqI↑↓(Te)X\nkνν′δ(εF−εk−qν↑)δ(εF−εkν′↓), (53)\nΓqq′=2π∆2\nS2N2(ωq−ωq′)X\nσIσσ(Te)X\nkνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ). (54)\nThe calculation of both quantities requires a spin-dependent summation over the Fermi surface, analogous to what\nwas done in Ref. [103] for the evaluation of the spin-dependent Eliashberg function for electron-phonon scattering.\nAs in Ref. [103] we use a Gaussian broadening of the Dirac delta distributions by 0 .03 eV. Also, since we only include\nthe contribution of s- and p-states (indicated by ν, ν′) to the scattering, we have to project the Kohn-Sham states\n(indicated by n, n′) onto the spherical harmonics Ym\nlvia\nδ(εF−εkνσ)δ(εF−εk′ν′σ′) =X\nnn′Pnν\nkσPn′ν′\nk′σ′δ(εF−εknσ)δ(εF−εk′n′σ′), (55)\nwith Pnν\nkσbeing projector functions.\nThe functions Iσσ′(Te) describe corrections to the scattering rate at high electron temperatures and are given by\nIσσ′(Te) =Z\ndε(−1)∂fFD(ε, Te)\n∂εgσ(ε)g′\nσ(ε)\ngσ(εF)g′σ(εF), (56)\nwith gσ(ε) =P\nkνδ(ε−εkνσ) =P\nkνP\nnPnν\nkσδ(ε−εknσ) being the cumulative DOS of both s- and p-states. We\nfind that they increase monotonously with the electron temperature (see right panel of Figure 5). However, even\nfor temperature up to 2000 K, the Iσσ′(Te) functions are below two. Hence, we concluded that the approximation\nIσσ′= 1 is reasonable for the laser fluences – heating the electrons up to around 700 K – considered in the main text.\nFigure 6 depicts the numerically calculated scattering rates using Iσσ′= 1 and ∆ = 0 .75 eV as obtained above. In\nthe left panel, we show the scattering rate γqthat is first order in the magnon variables through color code on the\nmagnon dispersion. It is strictly positive and tends to increase with magnon frequency. The right panel shows the\neffective scattering rate γ(2)\nq=P\nq′Γqq′due to the scattering term second order in magnon variables. Notably, this\nquantity is negative for low frequencies and positive for high frequencies, indicating that it leads to a depopulation\nof magnons at low energies due a scattering from low to high energies (the total magnon number is kept constant).\nIn general, the values of the effective second order scattering rate are comparable to the one first order in magnon\nvariables. They are, however, distributed differently: e.g., for magnons close to the Γ point the second order scattering\nrate is by far the dominating one. This is the reason why, as demonstrated in the main text, a laser pulse can in fact\nlead to a cooling of low energy magnons, i.e., to a decrease of their occupation numbers.\nLastly, we show in Figure 7 the ab initio computed mode-dependent Gilbert damping, αq=ωq/γq. Interestingly,\nthe Gilbert damping αqis large ( ∼0.01) at the BZ center and at the high-symmetry points H, N and P at the\nBZ edge. There is also a noticeable directional anisotropy in the Gilbert damping for modes along Γ −H and Γ −P.\nWe emphasize that the Gilbert damping is here due to the electron-magnon scattering term that is first order in\nthe magnon variables. Other scattering mechanisms as phonon-magnon scattering could contribute further to the\nmode-specific Gilbert damping.12\n−10−5 0 5 10\nenergyε−εF(eV)−0.04−0.020.000.020.040.06projected DOS (eV−1)s\np\n500 1000 1500 2000\nelectron temperature Te(K)1.01.21.41.61.8thermal correction factorI↑↓\nI↑↑\nI↓↓\nFigure 5. Left: Projected spin-polarized DOS for bcc iron. Spin-minority density is shown by positive values, spin-majority\ndensity by negative values. The exchange splitting is 2∆ ≈1.5 eV in a large interval around the Fermi energy and for both s-\nandp-states. Right : Thermal correction factors Iσσ′versus electron temperature Tecalculated from the projected DOS.\nΓ H N Γ P H020406080100120magnon frequency (THz)\n12345\nscattering rate γq(THz)\nΓ H N Γ P H020406080100120magnon frequency (THz)\n−4−3−2−101\nscattering rate γ(2)\nq(THz)\nFigure 6. Magnon dispersion of bcc iron along high-symmetry lines of the Brillouin zone. The color coding describes ( left)\nthe scattering rates γqdue to the electron-magnon scattering term first order in magnon variables γqand ( right) the effective\nscattering rate γ(2)\nq=P\nq′Γqq′due to the term second order in magnon variables, calculated with Iσσ′= 1 and ∆ = 0 .75 eV.\nΓ H N Γ P H020406080100120magnon frequency (THz)\n0.0020.0040.0060.0080.010\nGilbert damping αq\nFigure 7. Calculated mode-specific Gilbert damping αq=ωq/γq, depicted by the color code on the magnon dispersion of bcc\niron. The mode-specific Gilbert damping αqis due to the electron-magnon scattering term first order in magnon variables.13\nACKNOWLEDGMENTS\nThe authors thank K. Carva for valuable discussions.\nThis work has been supported by the Swedish Re-\nsearch Council (VR), the German Research Foundation\n(Deutsche Forschungsgemeinschaft) through CRC/TRR\n227 “Ultrafast Spin Dynamics” (project MF, project-ID:\n328545488), and the K. and A. Wallenberg Foundation\n(Grant No. 2022.0079). 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Nikoli´ c∗\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\n(Dated: December 15, 2023)\nUnderstanding spin wave (SW) damping, and how to control it to the point of being able to\namplify SW-mediated signals, is one of the key requirements to bring the envisaged magnonic tech-\nnologies to fruition. Even widely used magnetic insulators with low magnetization damping in their\nbulk, such as yttrium iron garnet, exhibit 100-fold increase in SW damping due to inevitable con-\ntact with metallic layers in magnonic circuits, as observed in very recent experiments [I. Bertelli\net al. , Adv. Quantum Technol. 4, 2100094 (2021)] mapping SW damping in spatially-resolved\nfashion. Here, we provide microscopic and rigorous understanding of wavevector-dependent SW\ndamping using extended Landau-Lifshitz-Gilbert equation with nonlocal damping tensor , instead\nof conventional local scalar Gilbert damping, as derived from Schwinger-Keldysh nonequilibrium\nquantum field theory. In this picture, the origin of nonlocal magnetization damping and thereby in-\nduced wavevector-dependent SW damping is interaction of localized magnetic moments of magnetic\ninsulator with conduction electrons from the examined three different types of metallic overlayers—\nnormal, heavy, and altermagnetic. Due to spin-split energy-momentum dispersion of conduction\nelectrons in the latter two cases, the nonlocal damping is anisotropic in spin and space, and it can\nbe dramatically reduced by changing the relative orientation of the two layers when compared to\nthe usage of normal metal overlayer.\nIntroduction. —Spin wave (SW) or magnon damping\nis a problem of great interest to both basic and ap-\nplied research. For basic research, its measurements [1–4]\ncan reveal microscopic details of boson-boson or boson-\nfermion quasiparticle interactions in solids, such as:\nmagnon-magnon interactions (as described by second-\nquantized Hamiltonians containing products of three or\nmore bosonic operators [5, 6]), which are frequently en-\ncountered in antiferromagnets [4, 5] and quantum spin\nliquids [7], wherein they play a much more important\nrole [8] than boson-boson interactions in other condensed\nphases, like anharmonic crystalline lattices or superflu-\nids [5]; magnon-phonon interactions [3], especially rel-\nevant for recently discovered two-dimensional magnetic\nmaterials [2]; and magnon-electron interactions in mag-\nnetic metals [1, 9–12]. For the envisaged magnon-\nbased digital and analog computing technologies [13–\n17], understanding magnon damping makes it possible\nto develop schemes to suppress [18] it, and, further-\nmore, achieve amplification of nonequilibrium fluxes of\nmagnons [19–22]. In fact, overcoming damping and\nachieving amplification is the keyto enable complex\nmagnon circuits where, e.g., a logic gate output must\nbe able to drive the input of multiple follow-up gates.\nLet us recall that the concept of SW was introduced by\nBloch [23] as a wave-like disturbance in the local mag-\nnetic ordering of a magnetic material. The quanta [6] of\nenergy of SWs of frequency ωbehave as quasiparticles\ntermed magnons, each of which carries energy ℏωand\nspin ℏ. As regards terminology, we note that in magnon-\nics [13] SW is often used for excitations driven by an-\ntennas [24–27] and/or described by the classical Landau-\nLifshitz-Gilbert (LLG) equation [9, 10, 28, 29], whereas\nmagnon is used for the quantized version of the same ex-\ne\nee\nee\neFIG. 1. (a) Schematic view of bilayers where a metallic over-\nlayer covers the top surface of magnetic insulator, as often\nencountered in spintronics and magnonics [13, 30]. Three\ndifferent energy-momentum dispersion of conduction elec-\ntrons at the interface are considered, with their Fermi sur-\nfaces shown in panel (b)—normal metal (NM); heavy metal\n(HM) with the Rashba SOC [31, 32], and altermagnetic metal\n(AM) [33, 34]—with the latter two being spin-split. The rel-\native alignment of the layers is labeled by an angle θ[33, 34],\nmeaning that the wavevector qof SWs within FI is at an an-\ngleθaway from the kx-axis.\ncitation [5], or these two terms are used interchangingly.\nIn particular, experiments focused on SW damp-\ning in metallic ferromagnets have observed [1] its de-\npendence on the wavevector qwhich cannot be ex-\nplained by using the standard LLG equation [28,\n29],∂tMn=−Mn×Beff\nn+αGMn×∂tMn(where ∂t≡\n∂/∂t), describing dynamics of localized magnetic mo-\nments (LMMs) Mnat site nof crystalline lattice (also\nused in atomistic spin dynamics [28]) viewed as classi-\ncal vectors of unit length. This is because αG, as the\nGilbert damping parameter [35, 36], is a local scalar (i.e.,\nposition-independent constant). Instead, various forms\nof spatially nonuniform (i.e., coordinate-dependent) and\nnonlocal (i.e., magnetization-texture-dependent) damp-\ning due to conduction electrons have been proposed [9,arXiv:2312.09140v1  [cond-mat.mes-hall]  14 Dec 20232\n10, 37–39], or extracted from first-principles calcula-\ntions [40], to account for observed wavevector-dependent\ndamping of SWs, such as ∝q2(q=|q|) measured in\nRef. [1]. The nonlocal damping terms require neither\nspin-orbit coupling (SOC) nor magnetic disorder scatter-\ning, in contrast to αGwhich is considered to vanish [41]\nin their absence.\nThus, in magnonics, it has been considered [30] that\nusage of magnetic insulators, such as yttrium iron gar-\nnet (YIG) exhibiting ultralow αG≃10−4(achieved on\na proper substrate [42]), is critical to evade much larger\nand/or nonlocal damping of SWs found in ferromagnetic\nmetals. However, very recent experiments [24–27] have\nobserved 100-fold increase of SW damping in the segment\nof YIG thin film that was covered by a metallic overlayer.\nSuch spatially-resolved measurement [24] of SW damp-\ning was made possible by the advent of quantum sensing\nbased on nitrogen vacancy (NV) centers in diamond [43],\nand it was also subsequently confirmed by other meth-\nods [25–27]. Since excitation, control, and detection of\nSWs requires to couple YIG to metallic electrodes [13],\nunderstanding the origin and means to control/suppress\nlarge increase in SW damping underneath metallic over-\nlayer is crucial for realizing magnonic technologies. To\nexplain their experiments, Refs. [24–27] have employed\nthe LLG equation with ad hoc intuitively-justified terms\n(such as, effective magnetic field due to SW induced eddy\ncurrents within metallic overlayer [24]) that can fit the\nexperimental data, which is nonuniversal and unsatisfac-\ntory (many other examples of similar phenomenological\nstrategy exist [1, 44]).\nIn contrast, in this Letter we employ recently derived\n∂tMn=−Mn×Beff\nn+Mn×X\nn′(αGδnn′+λR)·∂tMn′,\n(1)\nextended LLG equation with all terms obtained [45]\nmicroscopically from Schwinger-Keldysh nonequilibrium\nquantum field theory [46] and confirmed [45] via exact\nquantum-classical numerics [47–50]. It includes nonlo-\ncal damping as the third term on the right-hand side\n(RHS), where its nonlocality is signified by dependence\nonR=rn−rn′, where rnis the position vector of lat-\ntice site n. Equation (1) is applied to a setup depicted in\nFig. 1 where conduction electron spins from three differ-\nent choices for metallic overlayer are assumed to interact\nwith LMMs of ferromagnetic insulator (FI) at the inter-\nface via sdexchange interaction of strength Jsd, as well\nas possibly underneath the top surface of FI because of\nelectronic evanescent wavefunction penetrating into it.\nNote that FI/normal metal (NM) bilayer directly mod-\nels recent experiments [24] where FI was a thin film of\nYIG and NM was Au, and SW damping within FI was\nquantified using quantum magnetometry via NV centers\nin diamond. Next, the FI/heavy metal (HM) bilayer,\nsuch as YIG/Pt [18, 27], is frequently encountered in\n0 2 4\nK/J0.51.0q (1/a)\n kF= 0\n.92kF= 0\n.99kF= 1\n.08kF= 1\n.15kF= 1\n.22kF= 1\n.30kF= 1\n.38(a)\n1.0 1.2 1.4\nkF0.81.01.21.4qmax(1/a)\n∝kF(b)FIG. 2. (a) Wavevector qof SW generated by injecting spin-\npolarized current in TDNEGF+LLG simulations of NM over-\nlayer on the top of 1D FI [Fig. 1(a)] as a function of anisotropy\nK[Eq. (3)] for different electronic Fermi wavevectors kF. (b)\nMaximum wavevector qmaxof SWs that can be generated by\ncurrent injection [21, 57] before wavevector-dependent SW\ndamping becomes operative, as signified by the drop around\nkFin curves plotted in panel (a).\nvarious spintronics and magnonics phenomena [13, 30].\nFinally, due to recent explosion of interest in altermag-\nnets [33, 34], the FI/altermagnetic metal (AM) bilay-\ners, such as YIG/RuO 2, have been explored experimen-\ntally to characterize RuO 2as a spin-to-charge conver-\nsion medium [51]. The Schwinger-Keldysh field theory\n(SKFT), commonly used in high energy physics and cos-\nmology [52–54], allows one to “integrate out” unobserved\ndegrees of freedom, such as the conduction electrons in\nthe setup of Fig. 1, leaving behind a time-retarded dis-\nsipation kernel [48, 55, 56] that encompasses electronic\neffects on the remaining degrees of freedom. This ap-\nproach then rigorously yields the effective equation for\nLMMs only , such as Eq. (1) [45, 56] which bypasses the\nneed for adding [1, 24, 44] phenomenological wavevector-\ndependent terms into the standard LLG equation. In\nour approach, the nonlocal damping is extracted from\nthe time-retarded dissipation kernel [45].\nSKFT-based theory of SW damping in FI/metal bilay-\ners.—The nonlocal damping [45] λRin the third term\non the RHS of extended LLG Eq. (1) stems from back-\naction of conduction electrons responding nonadiabati-\ncally [48, 59]—i.e., with electronic spin expectation value\n⟨ˆsn⟩being always somewhat behind LMM which gener-\nates spin torque [60] ∝ ⟨ˆsn⟩×Mn—to dynamics of LMMs.\nIt is, in general, a nearly symmetric 3 ×3 tensor whose\ncomponents are given by [45]\nλαβ\nR=−J2\nsd\n2πZ\ndε∂f\n∂εTr\u0002\nσαAnn′σβAn′n\u0003\n. (2)\nHere, f(ε) is the Fermi function; α, β =x, y, z ;σα\nis the Pauli matrix; and A(ε) = i\u0002\nGR(ε)−GA(ε)\u0003\nis\nthe spectral function in the position representation ob-\ntained from the retarded/advanced Green’s functions\n(GFs) GR/A(ε) =\u0000\nε−H±iη\u0001−1. Thus, the calcula-\ntion of λRrequires only an electronic Hamiltonian H\nas input, which makes theory fully microscopic (i.e.,3\n−5 0 5\nX−505Y\nλNM\nR(a)NM\n−1 0 1\nλα\nR\n−5 0 5\nX−505Y\nλxx\nR(b)HM t SOC= 0.3t0\n−5 0 5\nX−505Y\nλzz\nR(c)\n−5 0 5\nX−505Y\nλ⊥\nR(d)AM t AM= 0.5t0\n0.0 0.5 1.0 1.5\nq (1/a)0123Γq≡Im(ωq) (J/¯ h)×10−1\nEq.(6)(e)\nη= 0.1\nη= 0\n−5 0 5\nX−505Y\nλyy\nR(f)\n−5 0 5\nX−505Y\nλxy\nR(g)\n−5 0 5\nX−505Y\nλ/bardbl\nR(h)\nFIG. 3. (a)–(d) and (f)–(h) Elements of SKFT-derived nonlocal damping tensor in 2D FI, λRwhere R= (X, Y, Z ) is the\nrelative vector between two sites within FI, covered by NM [Eq. (5)], HM [Eqs. (8)] or AM [Eqs. (9)] metallic overlayer. (e)\nWavevector-dependent damping Γ qof SWs due to NM overlayer, where the gray line is based on Eq. (6) in the continuous\nlimit [58] and the other two lines are numerical solutions of extended LLG Eq. (1) for discrete lattices of LMMs within FI. The\ndotted line in (e) is obtained in the absence of nonlocal damping ( η= 0), which is flat at small q.\nHamiltonian-based). Although the SKFT-based deriva-\ntion [45] yields an additional antisymmetric term, not\ndisplayed in Eq. (2), such term vanishes if the system\nhas inversion symmetry. Even when this symmetry is\nbroken, like in the presence of SOC, the antisymmet-\nric component is often orders of magnitude smaller [56],\ntherefore, we neglect it. The first term on the RHS of ex-\ntended LLG Eq. (1) is the usual one [28, 29], describing\nprecession of LMMs in the effective magnetic field, Beff\nn,\nwhich is the sum of both internal and external ( Bextez)\nfields. It is obtained as Beff\nn=−∂H/∂Mnwhere His\nthe classical Hamiltonian of LMMs\nH=−JX\n⟨nn′⟩Mn·Mn′+K\n2X\nn(Mz\nn)2−BextX\nnMz\nn.(3)\nHere we use g= 1 for gyromagnetic ratio, which sim-\nplifies Eq. (1); Jis the Heisenberg exchange coupling\nbetween the nearest-neighbors (NN) sites; and Kis the\nmagnetic anisotropy.\nWhen nonlocal damping tensor, λRis proportional\nto 3×3 identity matrix, I3, a closed formula for the\nSW dispersion can be obtained via hydrodynamic the-\nory [58]. In this theory, the localized spins in Eq. (1),\nMn= (Re ϕn,Imϕn,1−m)T, are expressed using com-\nplex field ϕnand uniform spin density m≪1. Then,\nusing the SW ansatz ϕn(t) =P\nqUqei(q·rn−ωqt), we ob-\ntain the dispersion relation for the SWs\nωq= (Jq2+K−B)\u0002\n1 +i(αG+˜λq)\u0003\n, (4)\nwhere qis the wavevector and ωis their frequency. Thedamping of the SW is then given by the imaginary part\nof the dispersion in Eq. (4), Γ q≡Imωq. It is comprised\nby contributions from the local scalar Gilbert damping\nαGand the Fourier transform of the nonlocal damping\ntensor, ˜λq=R\ndrnλrneiq·rn.\nResults for FI/NM bilayer. —We warm up by extract-\ning Γ qfor the simplest of the three cases in Fig. 1, a\none-dimensional (1D) FI chain under a 1D NM over-\nlayer with spin-degenerate quadratic electronic energy-\nmomentum dispersion, ϵkσ=t0k2\nx, where t0=ℏ2/2m.\nThe GFs and spectral functions in Eq. (2), can be\ncalculated in the momentum representation, yielding\nλ1D\nR=2J2\nsd\nπv2\nFcos2(kFR)I3, where vFis the Fermi velocity,\nR≡ |R|, and kFis the Fermi wavevector. Moreover, its\nFourier transform, ˜λq=2J2\nsd\nv2\nF[δ(q) +δ(q−2kF)/2], dic-\ntates additional damping to SWs of wavevector q=\n0,±2kF. Although the Dirac delta function in this ex-\npression is unbounded, this unphysical feature is an ar-\ntifact of the small amplitude, m≪1, approximation\nwithin the hydrodynamic approach [58]. The features of\nsuch wavevector-dependent damping in 1D can be cor-\nroborated via TDNEGF+LLG numerically exact simu-\nlations [47–50] of a finite-size nanowire, similar to the\nsetup depicted in Fig. 1(a) but sandwiched between two\nNM semi-infinite leads. For example, by exciting SWs\nvia injection of spin-polarized current into the metallic\noverlayer of such a system, as pursued experimentally in\nspintronics and magnonics [21, 57], we find in Fig. 2(a)\nthat wavevector qof thereby excited coherent SW in-4\ncreases with increasing anisotropy K. However, the max-\nimum wavevector qmaxis limited by kF[Fig. 2(b)]. This\nmeans that SWs with q≳kFare subjected to additional\ndamping, inhibiting their generation. Although our an-\nalytical results predict extra damping at q= 2kF, finite\nsize effects and the inclusion of semi-infinite leads in TD-\nNEGF+LLG simulations lower this cutoff to kF.\nSince SW experiments are conducted on higher-\ndimensional systems, we also investigate damping\non SWs in a two-dimensional (2D) FI/NM bilayer.\nThe electronic energy-momentum dispersion is then\nϵkσ=t0(k2\nx+k2\ny), and the nonlocal damping and its\nFourier transform are given by\nλNM\nR=k2\nFJ2\nsd\n2πv2\nFJ2\n0(kFR)I3, (5)\n˜λNM\nq=kFJ2\nsdΘ(2kF−q)\n2πv2\nFqp\n1−(q/2kF)2, (6)\nwhere Jn(x) is the n-th Bessel function of the first kind,\nand Θ( x) is the Heaviside step function. The nonlo-\ncal damping in Eqs. (5) and (6) is plotted in Fig. 3(a),\nshowing realistic decay with increasing R, in contrast to\nunphysical infinite range found in 1D case. Addition-ally, SW damping in Eq. (6) is operative for wavectors\n0≤q≤2kF, again diverging for q= 0,2kFdue to arti-\nfacts of hydrodynamic theory [58]. Therefore, unphysical\ndivergence can be removed by going back to discrete lat-\ntice, such as solid curves in Fig. 3(e) obtained for n=1–\n100 LMMs by solving numerically a system of coupled\nLLG Eq. (1) where λRin 2D is used [45]. In this numer-\nical treatment, we use kF= 0.5a−1where ais the lattice\nspacing; k2\nFJ2\nsd/2πv2\nF=η= 0.1;K= 0; Bext= 0.1J;\nandαG= 0.1.\nResults for FI/HM bilayer. —Heavy metals (such as of-\nten employed Pt, W, Ta) exhibit strong SOC effects due\nto their large atomic number. We mimic their presence at\nthe FI/HM interface [31] by using 2D energy-momentum\ndispersion ϵk=t0(k2\nx+k2\ny) +tSOC(σxky−σykx), which\nincludes spin-splitting due to the Rashba SOC [31, 32].\nUsing this dispersion, Eq. (2) yields\nλHM\nR=\nλxx\nRλxy\nR0\nλxy\nRλyy\nR0\n0 0 λzz\nR\n, (7)\nfor the nonlocal damping tensor. Its components are, in\ngeneral, different from each other\nλxx\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n+ cos(2 θ)\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8a)\nλyy\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n−cos(2 θ)\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8b)\nλzz\nR=J2\nsd\n4π\u0014\u0012kF↑\nvF↑J0(kF↑R) +kF↓\nvF↓J0(kF↓R)\u00132\n−\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\u0015\n, (8c)\nλxy\nR=−J2\nsdsin(2θ)\n4π\u0012kF↑\nvF↑J1(kF↑R)−kF↓\nvF↓J1(kF↓R)\u00132\n, (8d)\nwhere kF↑andkF↓are the spin-split Fermi wavevec-\ntors [Fig. 1(b)], and θis the relative orientation angle\n[Fig. 1(b)] between the SW wavevector qand the kxdi-\nrection. Thus, the nonlocal damping tensor in Eq. (7)\ngenerated by HM overlayer is anisotropic in spin due to\nits different diagonal elements, as well as nonzero off-\ndiagonal elements. It is also anisotropic in space due to\nits dependence on the angle θ. Its elements [Eqs. (8)]\nare plotted in Figs. 3(b), 3(c), 3(f), and 3(g) using\ntSOC= 0.3t0. They may become negative, signifying the\npossibility of antidamping torque [21] exerted by conduc-\ntion electrons. However, the dominant effect of nearby\nLMMs and the presence of local scalar αGensures that\nLMM dynamics is damped overall. Although there is no\nclosed expression for the SW dispersion in the presence of\nanisotropic λHM\nR, we can still extract SW damping Γ qin-duced by an HM overlayer from the exponential decay of\nthe SW amplitude in numerical integration of extended\nLLG Eq. (1) using SW initial conditions with varying q.\nFor an HM overlayer with realistic [31, 32] tSOC= 0.1t0\nthe results in Fig. (4)(a) are very similar to those ob-\ntained for NM overlayer with the same Fermi energy.\nAlso, the spatial anisotropy of λHM\nRdid not translate into\nθ-dependence of the SW damping.\nResults for FI/AM bilayer. —Altermagnets [33, 34] are\na novel class of antiferromagnets with spin-split elec-\ntronic energy-momentum dispersion despite zero net\nmagnetization or lack of SOC. They are currently in-\ntensely explored as a new resource for spintronics [51,\n61, 62] and magnonics [63, 64]. A simple model for\nan AM overlayer employs energy-momentum dispersion\nϵkσ=t0(k2\nx+k2\ny)−tAMσ(k2\nx−k2\ny) [33, 34], where tAMis5\n0.0 0.5 1.0 1.5\nq (1/a)1.01.52.02.5Γq≡Im(ωq) (J/¯ h)×10−1\n(a)NM\nHM\n0.0 0.5 1.0 1.5\nq (1/a)1234Γq≡Im(ωq) (J/¯ h)×10−1\n(b)NM\nAM :θ= 45◦\nAM :θ= 0◦\nFIG. 4. (a) Wavevector-dependent damping Γ qof SWs under\nNM or HM overlayer with the Rashba SOC of strength tSOC=\n0.1t0. (b) Γ qof SWs under AM overlayer with tAM= 0.8t0\nand for different relative orientations of FI and AM layers\nmeasured by angle θ[Fig. 1]. All calculations employ η= 0.1\nand Fermi energy εF= 0.25t0.\nthe parameter characterizing anisotropy in the AM. The\ncorresponding λAM\nR= diag( λ⊥\nR, λ⊥\nR, λ∥\nR) tensor has three\ncomponents, which we derive from Eq. (2) as\nλ⊥\nR=J2\nsd\n4πA+A−\u0014\nJ2\n0\u0012rϵF\nt0R+\u0013\n+J2\n0\u0012rϵF\nt0R−\u0013\u0015\n,\n(9a)\nλ∥\nR=J2\nsd\n2πA+A−J0\u0012rϵF\nt0R+\u0013\nJ0\u0012rϵF\nt0R−\u0013\n, (9b)\nwhere A±=t0±tAMandR2\n±=X2/A±+Y2/A∓is\nthe anisotropically rescaled norm of R. They are plot-\nted in Figs. 3(d) and 3(h), demonstrating that λAM\nRis\nhighly anisotropic in space and spin due to the impor-\ntance of angle θ[61, 65, 66]. Its components can also\ntake negative values, akin to the case of λHM\nR. It is inter-\nesting to note that along the direction of θ= 45◦[gray\ndashed line in Figs. 3(d) and 3(h)], λ⊥\nR=λ∥\nRso that\nnonlocal damping tensor is isotropic in spin. The SW\ndamping Γ qinduced by an AM overlayer is extracted\nfrom numerical integration of extended LLG Eq. (1) and\nplotted in Fig. (4)(b). Using a relatively large, but real-\nistic [33], AM parameter tAM= 0.8t0, the SW damping\nfor experimentally relevant small wavevectors is reduced\nwhen compared to the one due to NM overlayer by up to\n65% for θ= 0◦[Fig. 4(b)]. Additional nontrivial features\nare observed at higher |q|, such as being operative for a\ngreater range of wavevectors and with maxima around\n|q|= 2p\nϵF/t0and|q|= 3p\nϵF/t0. Remarkably, these\npeaks vanish for wavevectors along the isotropic direction\nθ= 45◦[Fig. 4(b)].\nConclusions. —In conclusion, using SKFT-derived non-\nlocal damping tensor [45], we demonstrated a rigorous\npath to obtain wavevector damping of SWs in magnetic\ninsulator due to interaction with conduction electrons\nof metallic overlayer, as a setup often encountered in\nmagnonics [13–17, 30] where such SW damping was di-\nrectly measured in very recent experiments [24–27]. Ouranalytical expressions [Eqs. (5), (7), and (9)] for nonlo-\ncal damping tensor—using simple models of NM, HM,\nand AM overlayers as an input—can be directly plugged\ninto atomistic spin dynamics simulations [28]. For more\ncomplicated band structures of metallic overlayers, one\ncan compute λRnumerically via Eq. (2), including com-\nbination with first-principles calculations [40]. 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B 108, 054511 (2023)."    },    {        "title": "2305.00814v2.Coherent_and_incoherent_magnons_induced_by_strong_ultrafast_demagnetization_in_thin_permalloy_films.pdf",        "content": "Coherent and incoherent magnons induced by strong ultrafast demagnetization in\nthin permalloy films\nAnulekha De,1,∗Akira Lentfert,1Laura Scheuer,1Benjamin Stadtmüller,1, 2\nGeorg von Freymann,1, 3Martin Aeschlimann,1and Philipp Pirro1,†\n1Department of Physics and Research Center OPTIMAS,\nRheinland-Pfälzische Technische Universität Kaiserslautern-Landau, 67663 Kaiserslautern, Germany\n2Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany\n3Fraunhofer Institute for Industrial Mathematics, 67663 Kaiserslautern, Germany\n(Dated: August 14, 2023)\nUnderstanding spin dynamics on femto- and picosecond timescales offers new opportunities for\nfaster and more efficient spintronic devices. Here, we experimentally investigate the coherent spin\ndynamics after ultrashort laser excitation by time-resolved magneto optical Kerr effect (TR-MOKE)\nin thin Ni80Fe20films. We provide a detailed study of the magnetic field and pump fluence depen-\ndenceofthecoherentprecessionaldynamics. Weshowthatthecoherentprecessionlifetimeincreases\nwith the applied external magnetic field which cannot be understood by viscous Gilbert damping\nof the coherent magnons. Instead, it can be explained by nonlinear magnon interactions and by the\nchange in the fraction of incoherent magnons. This interpretation is in agreement with the observed\ntrends of the coherent magnon amplitude and lifetime as a function of the exciting laser fluence.\nOur results provide a new insight into the magnetization relaxation processes in ferromagnetic thin\nfilms, which is of great importance for further spintronic applications.\nI. INTRODUCTION\nThe microscopic mechanism of laser-induced magneti-\nzation dynamics in femto-, pico- and nanosecond time-\nscales remains still a challenge in condensed matter\nphysics. Using the time-resolved magneto-optical Kerr\neffect (TR-MOKE), one can directly address the pro-\ncesses responsible for the excitation and relaxation of\na magnetic system on their characteristic timescales [1–\n4]. The pioneering work of Beaurepaire et al. in 1996\non femtosecond laser-induced ultrafast demagnetization\nopened up a new avenue for ultrafast manipulation of\nthe magnetization in magnetic materials [5]. Only a few\nyears later, experiments could show that the ultrafast\nlaser pulses also generate coherent magnons, manifested\nas precessional dynamics on the nanosecond time scale.\nThe precessional region in magnetic materials is a key\nparameter for encoding and transferring information in\nspintronic devices. This regime allows the study of the\nmagnetic anisotropy, damping, and precession frequency\nof different dynamic modes in continuous thin films and\npatterned nanostructures [2–4, 6–10]. However, the tran-\nsition between the two regimes of the magnetization dy-\nnamics, i.e., the ultrafast demagnetization and the coher-\nent precessional motion on the other side, is very crucial\nand raises intriguing questions. Many efforts have been\nmade to understand the leading mechanisms as well as\nto explore the characteristic time scales. For example,\nvan Kampen, et al. [2] demonstrated using TR-MOKE\nthat an optical pump pulse can induce coherent uniform\nspin precession in a ferromagnet. The excitation of these\n∗ade@rptu.de\n†ppirro@rptu.demagnons was explained by a transient change in mag-\nnetic anisotropy. Later, several works were concerned\nwith exploring the precessional relaxation mechanisms in\nferromagnets after ultrafast excitation [11–15]. A recent\nwork showed the excitation of the precession dynamics in\nferromagnets with two non-collinear optical pulses, that\ncan affect precessional relaxation mechanisms and damp-\ning [16].\nIn this article, we take a closer look at the relaxation\nof the coherent precession induced by ultrafast demag-\nnetization and study in detail the decay time of the\nmeasured coherent oscillations as a function of magnetic\nfield and pump fluence. We use femtosecond amplified\nlaser pulses to excite and detect magnetization dynamics\nin thin permalloy films, including ultrafast demagneti-\nzation, fast and slow remagnetization and precessional\ndynamics. We observe an unusual magnetic field depen-\ndence of the precessional relaxation time which is not\nin accordance with the expectations from the Gilbert\nmodel. Extrinsic contributions such as two-magnon scat-\ntering, magnetic anisotropy and spin pumping [11, 17–\n22] could also affect the magnetization relaxation process\nand the damping, but we show that these effects can be\nneglected in out case. Instead, we can relate our observa-\ntions to nonlinear magnon interactions and the variable\ncontribution of the incoherent magnon background after\nexcitation by a femtosecond optical pulse. Our findings\nare of general importance for the interpretation of the\ncoherent dynamics measured in similar experiments af-\nter ultrafast stimuli.\nII. MATERIALS AND METHODS\nPermalloy ( Ni80Fe20, Py) films of 5 nm and 2.8 nm\nthickness were deposited on MgO substrates by molecu-2\nlar beam epitaxy (MBE) technique in an ultrahigh vac-\nuum chamber. The samples were capped with a 3 nm\nthick layer of Al2O3to protect them from environmen-\ntal degradation, oxidation and laser ablation during the\npump-probe experiment using femtosecond laser pulses.\nFIG. 1. (a) Schematic diagram of the measurement geometry.\n(b) Typical experimental TR-MOKE data showing different\ntemporal regimes of the magnetization dynamics for 5 nm Py\nsample measured at µ0H= 113 mT and F= 4.6 mJ cm−2.\nThe magnetization dynamics were measured using a\nTR-MOKE setup based on a two-color, non-collinear op-\ntical pump-probe technique. A schematic diagram of the\nexperimental geometry is shown in Fig. 1(a). In this\nexperiment, we use the fundamental output of an ampli-\nfied femtosecond laser system with wavelength λ= 800\nnm, repetition rate of 1 kHz, and pulse width of ∼35\nfs (Libra, Coherent Inc.) as the pump pulse, while its\nsecond harmonic with λ= 400 nm is used to probe the\ndynamics. The probe is normally incident on the sam-\nple, while the pump is incident obliquely ( ∼30°) with\nrespect to the surface normal. During the measurements,\nwe have applied a magnetic field inclined at a small an-\ngle of ∼15°to the sample plane. The inclination of the\nmagnetization provides a finite out-of-plane (OOP) de-\nmagnetization field which is transiently modified by the\npump pulse, inducing a coherent precession of the mag-\nnetization[1,2]. Thissymmetrybreakingisimportantto\nobtainthesamestartingphaseoftheprecessionalmotion\nforeachlaserpumppulsetoavoidthatthelossofthepre-\ncession signal in a TR-MOKE experiment averaged over\nthousands of pump-probe cycles. For a completely in-\nplane configuration of the external magnetic field, we ob-\nserve a complete reduction of the precessional signal [see\nSupplemental Materials]. To obtain the intrinsic mag-\nnetic response, we performed the measurements for two\nopposite magnetization directions of the sample and ex-\ntracted the pure magnetic response from the difference\nof the two resulting Kerr signals. This is done to elim-\ninate any nonmagnetic signal, i.e., any signal that does\nnot depend of the direction of the sample magnetization\n[23]. We have used a specially designed photodetector\nconnected to a lock-in amplifier to measure the dynamic\nKerr rotation signal. All measurements have been per-\nformed under ambient condition and room temperature.III. RESULTS AND DISCUSSION\nSeveral processes occur when a femtosecond laser pulse\ninteracts with a ferromagnetic thin film in its saturation\ncondition. First of all, the magnetization of the sys-\ntem is partially or completely lost within hundreds of\nfemtoseconds, which is known as ultrafast demagnetiza-\ntion [5]. This is generally followed by a fast recovery of\nthe magnetization within sub-picoseconds to a few pi-\ncoseconds and a slower recovery within hundreds of pi-\ncoseconds, known as the fast and slow remagnetization.\nThe slower recovery is accompanied by a precession of\nthe magnetization. On a much longer time scale of a\nfew nanoseconds, the magnetization returns to its ini-\ntial equilibrium, which can be described phenomenolog-\nically by the Gilbert damping [24]. Figure 1(b) shows\nthe representative Kerr rotation data of the 5 nm Py\nsample for pump fluence F= 4.6 mJ cm−2and��0H=\n113mT consisting of three temporal regions of the mag-\nnetization dynamics, i.e., the ultrafast demagnetization,\nthe fast remagnetization followed by the slow remagne-\ntization superposed with the damped precession within\nthe time window of 900 ps. The slow remagnetization is\nmainly due to heat diffusion from the lattice to the sub-\nstrate and the surroundings. Region I is characterized by\nthe demagnetization time τM, region II is characterized\nby the fast remagnetization time τE. For region III, we\ncharacterize the dynamics by the precession frequency f\nand the precessional relaxation time τd.\nFIG. 2. Ultrafast demagnetization traces at different pump\nfluences for 5 nm Py sample in (a) polar (b) longitudinal\nMOKE geometry. (c) Demagnetization times ( τM) vs. pump\nfluence (d) Fast remagnetization times ( τE) vs. pump fluence\nand (e) quenching vs. pump fluence measured in both po-\nlar and longitudinal geometry. Solid circles and open circles\nrepresent the data corresponding to polar and longitudinal\nMOKE respectively.\nWe start our discussion with the ultrafast demagne-\ntization and fast remagnetization processes of our sys-\ntem to study their dependence on the pump laser fluence3\nand the external magnetic field. In general, the ultrafast\ndemagnetization studies are performed in the longitudi-\nnal MOKE geometry. However, to measure the coherent\nprecessionofthemagnetization, weneedtoperformmea-\nsurementsinapolarMOKEgeometrywithaslightlyout-\nof-planeinclinedexternalmagneticfield[1,2]. Therefore,\nwe first check whether there is a significant difference in\nthe laser-induced ultrafast demagnetization in the two\nMOKE geometries. Figure 2(a) and (b) show the ultra-\nfast demagnetization traces obtained for 5 nm Py in the\npolar and longitudinal MOKE geometries, respectively.\nThe pump fluence is varied between 4.6 - 8.4 mJ cm−2\nby varying the power of the pump pulse. We restrict\nourselves to the low pump fluence regime to avoid sam-\nple damage, and the probe fluence is kept constant at\na very low value ( ∼fewµW) to avoid any additional\ncontribution to the spin dynamics by probe excitation.\nThe ultrafast demagnetization traces for both geometries\nshow qualitatively similar trends. However, minor dis-\ncrepancies between thequantitative values obtained from\ntwo different geometries arise due to slight differences in\nthe pump spot sizes. The amplitude of the maximum\nquenching of the Kerr rotation signal increases almost\nlinearly with the laser fluence. Closer inspection of the\ntraces also reveals an increase in the τMandτMwith\nincreasing fluence. To quantify this increase, we fit our\ndemagnetization traces with a phenomenological thermo-\ndynamic model, the so-called three-temperature model\n(3TM) [25], which is obtained by solving the energy rate\nequation between three different degrees of freedom, e.g.\nelectron, spin and lattice, under low pump fluence con-\nditions [see Supplemental Materials].\nFIG. 3. (a) Ultrafast demagnetization traces for 5 nm Py\nsample measured at different values of magnetic field and for\nfixedF= 4.6 mJ cm−2. (b) upper panel: Demagnetization\ntimes (τM) vs. magnetic field and lower panel: Fast remagne-\ntization times ( τE) vs. magnetic field.\nThe fluence-dependent behavior of τM,τEand quench-\ning in both MOKE geometries, as shown in Fig. 2(c),\n(d) and (e), respectively, is an indication of the spin-\nflip process-dominated ultrafast demagnetization in our\nsystems [26–28]. The values of τMextracted from our\nexperiments are on the same time scale as previous re-\nports [29], and are too large to represent superdiffusive\ntransport driven demagnetization [30]. These values areslightly larger in the longitudinal geometry compared to\nthe polar geometry. However, they do not vary signifi-\ncantly with the applied external magnetic field (as shown\nin Fig. 3), indicating that, as expected, the compara-\ntively small variations in Zeeman energy as well as the\nsmall change in magnetization direction associated with\nthe change in magnetic field strength do not affect the\nultrafast magnetization dynamics.\nAfter quantifying the ultrafast demagnetization and\nfast remagnetization dynamics (region I and II), we turn\nto the region III, which is characterized by the coher-\nent precessional magnetization dynamics induced by the\npump laser pulse. These dynamics in the GHz- range\naregenerallydescribedbythephenomenologicalLandau-\nLifshitz-Gilbert (LLG) equation [24],\nd⃗M\ndt=−γ⃗M×/bracketleftigg\nµ0⃗Heff−α\nγMSd⃗M\ndt/bracketrightigg\n(1)\nwhereγisthegyromagneticratio, MSisthesaturation\nmagnetization, αis the Gilbert damping constant, and\n⃗Heffis the effective magnetic field consisting of several\nfield components. The first term on the right side of\nEq. (1) accounts for the precession of the magnetization\nvector (⃗M) around⃗Heff. The second term with the first-\norder time derivative of ⃗M, is the Gilbert damping term\n[24], which models the transfer of energy and angular\nmomentum of ⃗Mto the surrounding degrees of freedom\n(relaxation of ⃗Mtowards⃗Heff). Figure 4(a) shows the\nbackground subtracted time-resolved Kerr rotation data\n(precessional part) for two different values of the applied\nmagnetic field, fitted with a damped sinusoidal function,\nM(t) =M(0)e−t/τdsin(2πft) (2)\nHere,M(0)istheinitialamplitudeoftheprecession, τd\nis the relaxation time of the coherent precession obtained\nas a fitting parameter, and fis the precession frequency,\nwhichcanalsobeextracteddirectlyfromtheFastFourier\nTransform (FFT) of the precessional oscillation. Due to\nthe size of the laser spot (D ∼500µm), our measure-\nment basically detects only magnon wavevectors up to\napproximately k ∼π/500 rad/µm, thus essentially only\nthe ferromagnetic resonance (FMR). The effective mag-\nnetization ( Meff), which includes the saturation magneti-\nzation and potential additional out-of-plane anisotropies,\nis calculated from the magnetic field dependence of the\nprecession frequencies (Fig. 4(b)) and fitting the data\npoints with the Kittel formula [31],\nf=1\n2π/radicalbig\nωH(ωH+ωM) (3)\nwhereωH=γµ0H,ωM=γµ0MeffandHis the ex-\nternally applied magnetic field and γ= 1.83×1011rad\ns−1T−1for Py. Strictly speaking, Eq. 3 is only valid4\nfor a completely in-plane magnetized film, but we have\nverified using micromagnetic simulations that it approxi-\nmates our experimental situation very well. From the fit,\nMeffis obtained to be ∼700±25kA m−1for 5 nm Py\n[and∼670±20kA m−1for 2.8 nm Py, see Supplemental\nMaterials] measured at F= 4.6 mJ cm−2.\nFIG. 4. Magnetic field dependent dynamics: (a) Background-\nsubtracted time-resolved Kerr rotation data for 5 nm Py sam-\nple measured at two different magnetic fields and at F= 4.6\nmJ cm−2. Solid lines are fitting lines. (b) Magnetic field\ndependence of precession frequency. Solid line represent the\nKittel fit to the data points. (c) Magnetic field dependence\nof quenching (green) and of precessional amplitude (red). (d)\nMagnetic field dependence of the precessional relaxation time\n(τd) measured after ultrafast demagnetization in TR-MOKE\n(red), usingmicrowave spectroscopy (blue) andanalytical cal-\nculation (black).\nWhenstudyingtheexcitationofthecoherentmagnons,\npoints of interest are the dependence of the precession\nfrequency, amplitude and lifetime on the external con-\nditions. Concerning the amplitude, Fig. 4(c) shows the\ndependence of quenching and precessional amplitude on\nthe magnetic field for a fixed pump fluence ( F= 4.6\nmJ cm−2). As expected, the quenching, which is a mea-\nsure of the energy initially introduced into the magnetic\nsystem, is independent of the applied magnetic field.\nSurprisingly at first glance, the precessional amplitude,\nwhich is usually considered as a measure for the energy of\nthe coherent oscillations, is strongly dependent on mag-\nnetic field. We interpret the observed increase of the\nprecession amplitude with increasing magnetic field as an\nincrease of the part of the coherent precession that has\na well-defined and constant phase relation to the pump\npulse. This can be explained by the fact that the static\nout-of-plane component of the magnetization, which is\nrequired for TR-MOKE to measure the coherent preces-\nsion [1, 2], increases with the strength of the applied bias\nmagnetic field.Interestingly, we also observe that the precessional re-\nlaxation time ( τd) as obtained from TR-MOKE mea-\nsurements, increases with increasing magnetic field (red\ndots in Fig. 4(d)). This is unexpected if one assumes\nthat Gilbert damping, described by a material parameter\n(Gilbert damping constant α) is responsible for the de-\ncayoftheprecession. SinceGilbertdampingisviscous, it\npredictsadecreaseofthemagnonlifetimewithincreasing\nfrequency, which is equivalent to an increase of the mag-\nnetic field strength in the presented geometry. Analyt-\nical calculations of the magnon characteristics based on\nthe Gilbert model [32–35] show that the Gilbert-induced\nlifetime decreases with increasing magnetic field (black\nsolid line in Fig. 4(d)). For the analytical calculations,\nwe have assumed an in-plane magnetic field, but we have\nverified again with micromagnetic simulations that this\napproximation is well justified. The lifetime ( τd) of the\nhomogeneousFMRmodeforanin-planemagnetized(the\napplied in-plane magnetic field is assumed to be H) thin\nfilm is calculated using the following expression [35]:\n1\nτd=α/parenleftig\nωH+ωM\n2/parenrightig\n(4)\nA possible interpretation for this intriguing discrep-\nancy between the lifetimes measured with different ex-\ncitation mechanisms could be the contribution of non-\nGilbert damping mechanisms. One of these is the\nso-called two-magnon scattering mechanism, where the\nmagnon energy is redistributed from the FMR ( k= 0)\ntoothershort-wavelengthmagnons( k>0)duetodefect-\ninduced scattering. Arias and Mills [17] developed a the-\nory describing the contributions of two-magnon scatter-\ning to the FMR linewidth. The two-magnon process is\nlinear in magnon amplitude, thus the broadening of the\nFMR linewidth is independent of the magnon amplitude.\nWoltersdorf et al., showed by TR-MOKE experiments\nthat different capping layers can affect spin relaxation\nand damping of Fe films in different ways [19]. They\nalso showed the increase of relaxation time or decrease of\ndamping with increasing magnetic field for Cu capped Fe\nfilms and interpreted it by two-magnon scattering. Liu\net al. [21] showed that the effective damping constant\ndecreases with the increasing magnetic field, suggesting\na contribution of magnetic anisotropy to the enhanced\ndamping. Some other reports have also discussed the en-\nhancement of damping with decreasing magnetic fields\ndue to two-magnon scattering, magnetic anisotropy or\nspin pumping effects [11, 20, 22].\nTo clarify a possible contribution of two-magnon scat-\ntering from defects at surfaces and interfaces [17, 18]\nto the measured decay, we performed additional inde-\npendent measurements of the FMR lifetime using induc-\ntive microwave spectroscopy measured with a vector net-\nwork analyzer (VNA), as shown by the blue dots in Fig.\n4(d). Thistechniqueisknowntobesensitivetolinewidth\nbroadeninginducedbytwo-magnonscattering[36]. How-\never, the measurements show a decrease of lifetime in ac-\ncordance with the Gilbert model. In addition, a theoreti-5\ncal work [18] predicts that a two-magnon contribution to\nthe linewidth should increase with resonance frequency\nand magnetic field in our experimental case (small angles\nof less than 15 °to the films plane). Thus, we can con-\nclude that the defect induced two-magnon scattering is\nnot responsible for the increase in lifetime ( τd) observed\nin our TR-MOKE measurements. Instead, we interpret\nthe change in lifetime due to coherent precession as fol-\nlows. Both incoherent and coherent magnons are excited\nwhenthesampleishitwithafslaserpulseinTR-MOKE.\nThe ratioof coherentto incoherent magnonsis influenced\nby the out-of-plane component of the static magnetiza-\ntion, which breaks the symmetry of the system. As in the\ncase of the precession amplitude, a higher external field\nstrength increases the static out-of-plane component and\nthus the relative proportion of coherent magnons with\na defined and constant phase relationship to the laser\npulses. Due to the reduced excitation of the incoher-\nent magnons at higher fields, the dephasing of the pre-\ncession signal is weaker, leading to a longer lifetime of\nthe measured coherent precession signal. As the external\nmagnetic field decreases, the relative proportion of inco-\nherent magnons to coherent magnons increases, leading\nto a lower lifetime. This is in contrast to microwave spec-\ntroscopymeasurementswhereonlycoherentmagnonsare\nexcited and only a negligible amount of thermally ex-\ncited incoherent magnons are present. This explains the\ndifferent trends in the lifetime of magnons excited and\nmeasured by microwave spectroscopy and TR-MOKE.\nFIG. 5. Pump fluence dependent dynamics: (a) Background-\nsubtracted time-resolved Kerr rotation data for 5 nm Py sam-\nple measured at µ0H= 113 mT and different pump fluences.\nSolid lines are fitting lines. (b) Pump fluence dependence of\nprecessional frequencies at µ0H= 113 mT. (c) Pump fluence\ndependence of quenching (olive) and precessional amplitude\n(brown). Pump fluence dependent quenching plotted here\n(olive) is the same as plotted by solid circles in Fig. 2(e). (d)\nPump fluence dependence of precessional relaxation time ( τd)\nmeasured after ultrafast demagnetization in TR-MOKE.\nAnother interesting parameter to study is the influenceof the excitation intensity, which in our case is given by\nthe pump fluence, on the coherent dynamics. The back-\nground subtracted time-resolved Kerr rotation data for\nthe 5 nm Py sample measured at different pump fluences\n(F) are shown in Fig. 5(a). The energy deposited by\nthe pump pulse, in the form of heat within the probed\nvolume, plays a very crucial role in the modification of\nthe local magnetic properties, i.e. the magnetic moment,\nanisotropy, coercivity, magnetic susceptibility, etc., as\nwell as the precession frequency can experience a vari-\nation with the pump fluence[12, 37]. However, we do not\nobserve any significant frequency shift within our exper-\nimental fluence range as shown in Fig. 5(b). This also\nindicates that the temperature of the sample is not sig-\nnificantly increased on longer timescales after the initial,\nfastremagnetization. Consequently, Meffcalculatedfrom\nEqn. (3) shows no significant dependence on Fwithin\nour experimental fluence range. Thus, we conclude that\nas the pump fluence increases, there is no further change\nin the anisotropy field that can modify the effective mag-\nnetization of the system to this extent [21]. Figure 5(c)\nshows that the quenching increases with pump fluence as\nexpected. However, although the initial quenching is a\nmeasure of the absorbed energy and thus the source of\ncoherent precession, the precession amplitude decreases\nwith fluence. The precessional relaxation time ( τd) also\nshows a decrease with pump fluence (Fig. 5(d)). The\ndecrease of both the precession amplitude and the pre-\ncessionrelaxationtimecanbeexplainedbythedephasing\nof the magnons due to nonlinear magnon-magnon inter-\nactions. These interactions generally increase with an\nincrease in the total magnon population (incoherent and\ncoherent),whichwebelieveisproportionaltothequench-\ning. In addition, the probability of excitation of incoher-\nent magnons relative to coherent magnons increases with\nthe increase of disorder in the system and thus with the\nquenching / pump fluence.\nIV. SUMMARY\nIn summary, the spin dynamics on different time scales\nin thin Ni80Fe20films have been studied using an all-\noptical TR-MOKE technique. We study the preces-\nsion dynamics in the GHz range after femtosecond laser-\ninduced ultrafast demagnetization. The demagnetiza-\ntion time, fast remagnetization time, and magnetization\nquenching studied in both longitudinal and polar geome-\ntry show an increasing trend with excitation fluence, con-\nsistent with a spin-flip scattering-dominated demagneti-\nzation process. On a longer timescale of several hundreds\nof picoseconds, we observe an increase of the coherent\nprecessional relaxation time with magnetic field / reso-\nnance frequency, which cannot be explained by viscous\nGilbert damping. Using standard FMR techniques, we\nconclude that two-magnon scattering is not responsible\nfor this behavior. Instead, we can consistently explain all\nobserved trends by considering the different relative con-6\ntributions of coherent and incoherent magnons produced\nin the ultrafast demagnetization process and the nonlin-\near interaction between them. This interpretation also\nexplains the dependence of the coherent magnon ampli-\ntude and relaxation time on the excitation fluence. We\nexpectthatourresultswillpavethewayforfutureexper-\nimental and theoretical investigations towards a deeper\nunderstanding of the photon-to-magnon conversion in ul-trafast demagnetization processes.\nV. ACKNOWLEDGMENTS\nThe authors thank Eva Prinz, Jonas Hoefer, and Mar-\ntin Stiehl for technical assistance. The work was funded\nby the Deutsche Forschungsgemeinschaft (DFG, Ger-\nman Research Foundation) under granz No. TRR 173-\n268565370 Spin+X: spin in its collective environment\n(project B11 and B03).\n[1] G. Ju, A. V. Nurmikko, R. F. C. Farrow, R. F. Marks,\nM. J. Carey, and B. A. Gurney, Phys. Rev. Lett. 82, 3705\n(1999).\n[2] M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair,\nL. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys.\nRev. Lett. 88, 227201 (2002).\n[3] B. Koopmans, J. J. M. Ruigrok, F. D. Longa, and\nW. J. M. de Jonge, Phys. Rev. Lett. 95, 267207 (2005).\n[4] J. Walowski, G. Müller, M. Djordjevic, M. Münzenberg,\nM. Kläui, C. A. F. Vaz, and J. A. C. Bland, Phys. Rev.\nLett.101, 237401 (2008).\n[5] E.Beaurepaire, J.-C.Merle, A.Daunois,andJ.-Y.Bigot,\nPhys. Rev. Lett. 76, 4250 (1996).\n[6] M. Vomir, L. H. F. Andrade, L. Guidoni, E. Beaurepaire,\nand J.-Y. 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Exp. 1, 121301\n(2008).Supplemental Materials  \nCoherent and incoherent magnons induced by strong ultrafast demagnetization in thin \npermalloy films  \nAnulekha De,1,∗ Akira Lentfert,1 Laura Scheuer,1 Benjamin Stadtmüller,1,2 Georg von Freymann,1,3 Martin \nAeschlimann,1 and Philipp Pirro1,† \n1Department of Physics and Research Center OPTIMAS, Rheinland -Pfälzische Technische Universität \nKaiserslautern -Landau, 67663 Kaiserslautern, Germany  \n2Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany  \n3Fraunhofer Institute for Industrial Mathematics, 67663 Kaiserslautern, Germany  \n \nS1. Three Temperature Model : \nThe dynamics of the spin fluctuations after excitation by  ultrafast laser pulses can be described by a \nphenomenological  thermodynamic model, the so -called three -temperature model  (3TM), which pictures \nhow energy is redistributed among  electrons, spins, and the lattice after the absorption of the laser  power by \nthe electronic system. The energy flow ultimately  leads to an increase in the spin temperat ure, thereby \nreducing  the magnetization. The expression is given by,  \n−∆𝑀\n𝑀= {[𝐴1\n(1+𝑡𝜏0⁄)12⁄−𝐴2𝜏𝐸−𝐴1𝜏𝑀\n𝜏𝐸−𝜏𝑀𝑒−𝑡𝜏𝑀⁄−𝜏𝐸(𝐴1−𝐴2)\n𝜏𝐸−𝜏𝑀𝑒−𝑡𝜏𝐸⁄]𝐻(𝑡)+𝐴3𝛿(𝑡)}𝐺(𝑡)   \nHere, τ M and τ E are ultrafast demagnetization and first remagnetization  times respectively. A 1 represents the \namplitude  of magnetization after fast relaxation, A 2 is proportional to  the maximum electron temperature \nrise, and A 3 represents  the state filling effects during pump -probe temporal overlap.  HS(t) is the Heaviside \nstep function, δ(t) is the Dirac delta  distribution, and G(t) is a Gaussian function corresponding to  the laser \npulse. τ 0 represents the cooling time through heat  diffusion. This model is very useful in analysing \nexperimental  data and extracting quantitative information on the timescales  of the different processes taking \nplace during the laser induced  ultrafast demagnetization.  \nS2. Precession dynamics for 2.8 nm Py sample : \nFigure S1(a) shows the background subtracted time -resolved  Kerr rotation data ( precessional part) measured \nat different  values of magnetic fields, fitted with a damped sinusoidal  function (Eqn. 2 of the main article) \nfor 2.8 nm Py sample.  All measurements are done at particular pump fluence of F =  4.6 mJ cm−2. Figure \nS1(b) shows magn etic field dependence  of the precession frequencies obtained from the fast Fourier  \ntransform (FFT) of the precessional oscillation, from which  we calculate the effective magnetization (M eff) \nusing the Kittel  formula (Eqn. (3) of the main article). The valu e of M eff obtained from fit is 670 ± 20 kA \nm−1, which is slightly  less than the thicker (5 nm Py ) film. Figure S1(c) shows the  variation of precessional \nrelaxation time (τ d) with magnetic  as obtained after ultrafast demagnetization in TR -MOKE  measurements \n(red), microwave spectroscopy measurements  (blue) and analytical calculations (black). We observe that  τd \nincreases with magnetic field for TR -MOKE measurements which is in contrast to  both the microwave \nspectroscopy measurements and analytical  calculation s. Similar results are obtained from 5 nm Py sample  \nand are thoroughly discussed in the main article.  \n \n \n  \nS3. Effect of in -plane magnetic field : \nFor the in -plane configuration of the magnetic field, we  observe a reduction in precessional amplitude \nleading to a  poor signal -to-noise ratio (as shown in S2(a)). As the magnetic  field is tilted slightly ( ∼ 15°) in \nthe out -of-plane direction,  the precessional amplitude increases and we observe a clear  time-resolve d Kerr \nrotation trace resulting clear fast Fourier  transformed magnon modes. In the in -plane configuration the  \ndominance of nonmagnetic noise due to two -magnon scattering has suppressed the features of magnetic \npeaks and only a  few spurious peaks are prese nt in the spectra. A comparison  between the FFT powers for \nin-plane and tilted magnetic fields  are shown in Fig. S2(c), where the power is negligibly small in  in-plane \nconfiguration as compared to the tilted configuration  of magnetic field.    \nFig. S1:  Magnetic field dependent dynamics of 2.8 nm Py sample: (a) Background -subtracted time -resolved Kerr rotation data \nmeasured at different values of magnetic field and at F = 4.6 mJ cm−2. Solid lines are fitting lines. (b) Magnetic field dependence \nof precession frequency. Solid line represent the Kittel fit to the data points. (c) Magnetic field dependence of precessiona l \nrelaxation time (τ d) measured after ultrafast demagnetization in TR -MOKE (red), microwave spectroscopy (blue) and analytical \ncalculation (black).  \n \nFig. S 2: (a) Precessional dynamics at two different values of in -plane magnetic field. (b) The corresponding FFT power spectra \n(c) The comparison between FFT spectra at in -plane and tilted magnetic fields. The values of magnetic fields are mentioned in \nthe respective graphs.  \n"    },    {        "title": "0811.3472v1.Spin_Transfer_Torque_as_a_Non_Conservative_Pseudo_Field.pdf",        "content": "1 Spin Transfer Torque as a Non-Conservative Pseudo-Field  \nSayeef Salahuddin*, Deepanjan Datta and Supriyo Datta  \nSchool of Electrical and Computer Engineering and NSF Center for Computational \nNanotechnology (NCN), Purdue University, West Lafayette, IN 47906  \n*Present address: Electrical Engineering and Computer Science, UC Berkeley, CA-94720 \nAbstract: \nIn this paper we show that the spin transfer torque can be described by a pseudo magnetic field, \nproportional to the magnetic moment of the itinerant electrons that enters the Landau-Lifshitz-\nGilbert equation in the same way as other external or internal magnetic fields. However, unlike \nan ordinary magnetic field, which is always conservative in nature, the spin torque induced \n‘pseudo field’ may have both conservative and non-conservative components. We further show \nthat the magnetic moment of itinerant electrons develops an out-of-plane component only at non-\nequilibrium and this component is responsible for the ‘Slonczewski’ type switching that acts \nagainst the damping and is always non-conservative. On the other hand, the in-plane components \nof the pseudo field exist both at equilibrium and out-of-equilibrium, and are responsible for the \n‘field like’ term. For tunnel based devices, this term results in lower switching current for anti-\nparallel (AP) to parallel (P) switching compared to P to AP, even when the torque magnitudes \nare completely symmetric with voltage. \n \n \n \n \n 2 1. Introduction \nSpin torque devices [1, 2]  that switch the magnetization of small magnets with spin polarized \ncurrents without any external magnetic field, have stirred tremendous interest due to their \npotential application as non volatile memory and also as nanoscale microwave oscillators. \nAlthough the concept of spin transfer torque has been demonstrated by a number of experiments \n[3, 4], quantitative measurement of spin transfer torque has been achieved only very recently [5, \n6, 7]. All these measurements show a significant ‘field-like’ or out-of-plane torque in addition to \nthe original in-plane torque predicted by Slonczewski [1]. This is very different from metallic \nchannel based devices where the field like term is minimal. Recent theoretical studies have also \nshown the field like term to be significant in tunnel based devices [8, 9, 10, 11]. However, the \ndetails of how this field-like torque can affect the switching behavior is yet to be understood \nproperly [5, 6, 7, 12, 13].  \n \nIn this paper we first show that spin torque can be described by a pseudo magnetic field \nproportional to the net magnetic moment  of the itinerant electrons, (normalized to the Bohr \nmagneton ) providing a natural relationship between Slonczewski and field like terms: \n \n                                                                                      (1)                      \nEqn. (1) is the central result of this paper and is derived in Section 2, starting from the Gilbert \nform of the LLG equation and introducing the spin-torque in terms of  obtained from non-\nequilibrium Green function (NEGF) formalism for the conduction electrons. Note that so \nthat the pseudo field  is in the same direction as  and enters Eqn. (1) just like other 3 magnetic fields included in. This may seem surprising; since it is well-known that spin-\ntorque leads to phenomena like coherent precession that do not arise from ordinary magnetic \nfields. We show in section 3 that such phenomena can also be understood in terms of Eqn. (1) \nonce we note that the pseudo-field  representing the spin-torque has both a conservative \ncomponent like the conventional magnetic fields included in  and also a non-conservative \ncomponent that makes the curl of overall  to be non-zero: ; (Note that   \n ) .                                                                                                                        \nWe show that the out-of-plane component of  is responsible for the Slonczewski term and \nis always non-conservative. On the other hand, the in plane components give the field like term \nand can introduce asymmetry in switching currents for opposite polarity in the voltage bias. \nSpecifically, we shall show that for tunnel based devices, this field like torque can result in a \nlower switching voltage for AP to P switching compared to P to AP, even when the torque \nmagnitudes are completely symmetric with voltage. This can be understood by noting that for \ntunneling devices, the in plane component of the pseudo field (responsible for the field like term) \nremains conservative even away from equilibrium, and thus acting like an ordinary magnetic \nfield parallel to the direction of the fixed magnet that helps switching from AP to P while \nhindering P to AP transition. \n \n2. Spin-Torque as a Pseudo Field \nA typical spin torque device is shown schematically in Fig. 1. Left contact is the fixed \nferromagnet having magnetization along . Right contact is soft layer and its magnetization \npoints along  which is free to rotate in easy (z-x) plane. An insulating layer separates the 4 ferromagnetic contacts. Following Gilbert’s prescription, we write, the rate of change of the \ndirection  of the magnetization as \n                                                                                                (2) \nwhere the spin torque  is obtained by integrating the divergence of the spin current  carried \nby the conduction electrons over the volume of the magnet. Below we will use the non-\nequilibrium Green’s function formalism to show that  \n                                                                                                  (3)        \nwhere is the magnetic moment of the conduction electrons (normalized to )  and is the \nenergy splitting of the conduction electrons due to the exchange interaction with the localized \nspins that comprise the magnet. Combining Eqns (2) and (3) we obtain our central result stated \nearlier in Eqn. (1) with .  \n \nProof of Eqn. 3: We start from the expression for the (2x2) operator representing  at site  \nin a discrete representation (see Eqn. (8.6.3), page 317, [15]) for the conduction electrons\n.  is the (2x2) correlation matrix at site  and the \nHamiltonian [ ] is given by  , where  is the spin-independent part and \n is the spin-dependent part arising from the exchange interaction with the magnet \npointing along , with  being a 2x2 identity matrix and  representing the Pauli spin \nmatrices. \nThe divergence of the spin-current is obtained from the  operator  \n           (4) \nand substituting for , we get (note:  is the Levi-Civita antisymmetric tensor)  5  (5) \nso that the spin-torque is given by         \n                                                                                                                        (6) \n \nDefining  as the magnetic moment (normalized to ) of the \nconduction electrons, we obtain  which is the same as \n as stated above in Eqn. (3). This completes our proof of Eqn. (1). Note \nthat this expression for torque is consistent with previous studies [9,10,11].  \n \n3. Relation to the standard form \nIt is shown in the Appendix A  that if the conduction electrons are in equilibrium then the spin \ndensity  can be written in the form,  \n                 (7) \nbut away from equilibrium, the spin density remarkably develops an additional out-of plane \ncomponent that is perpendicular to the magnetization of both magnets: \n                                                                                                     (8) \nso that from Eqn. (3) the spin-torque comes out as . \nwhich has the same form as the standard torque equations used extensively in literature [5, 6, 7].  \n \nOur formulation leads to a simple criterion for coherent precession  which is considered one of \nthe hallmarks of spin-torque. To see this we note that one can write Eqn. (1) in the following \nform  6                     \n                                                                                                                                                       (9)  \n \nNoting that coherent precession arises when the second term is zero, we obtain \n                                                                        (10)  \nso that Eqn. (9) reduces to yielding  as the precession \nfrequency. Since the \" \" term is zero under equilibrium conditions (see Appendix A ), coherent \nprecession is possible only under non-equilibrium conditions, as one would expect. \n \nNature of the pseudo field: \nNow that we have established the relationship between our concept of pseudo field and the \nstandard form of torque having a Slonczewski and a field like term, let us try to examine the \npseudo field more deeply. The first term in Eqn. (8) is in the same direction as the magnet. \nHence this does not contribute anything to the torque and may be ignored. As for the second \nterm, we see that if the coefficient  were independent of and , . This means \nthat for the case when is independent of and , the second term acts as a conservative field. \nAs for the third term in Eqn. (8) we show in Appendix B , that independent of the angular \ndependence of c, the third term always constitutes a non-conservative field. To summarize, the \npseudo field that gives the spin transfer torque has two terms, one of which is in-plane with the \nmagnets and may or may not be a conservative field. On the other hand, the second term is out-\nof-plane, is always non-conservative and can only appear at out-of-equilibrium.  \n 7 4. Switching behavior in tunneling barrier based spin torque devices: \nLet us now consider switching in tunneling barrier based spin torque devices. Our formulation is \nbased on the coupled NEGF-LLG methodology described above. The details of NEGF \nimplementation of transport for tunneling barrier based spin torque devices have been discussed \nin [16]. Here we shall skip the details and only present results. In brief, our formulation is based \non effective mass description. We sum over the transverse modes assuming that the inter-mode \ncoupling is negligible. Also, we only take the torque at the surface of the soft magnet. We have \nshown [16] that this methodology gives reasonable agreement with both the current and the \ntunneling magneto resistance (TMR) as a function of voltage by using effective mass and barrier \nheight as fitting parameters. In this case, we shall use similar parameters as used in [16, 17]. A \ntypical bias and angular dependence of and and the torque components are shown in Fig. 2. \nNote that the bias and angular dependence of the torque components show the same qualitative \ndependence as in the recent ab-initio study [10].  The bias dependence of and can be \napproximately written as .  \n \nAlso, from the Fig. 2, it is evident that both  and are completely independent of and  for \nthe tunneling device as we have considered here. This means that the pseudo field will have a \nconservative part due to ( ), where  is symmetric with voltage. Fig. 3 shows the switching of \nmagnetization with applied voltage. One would see that it takes less time to go from AP to P \nconfiguration compared to P to AP for the same magnitude of voltage. This means that it would \ntake more voltage to switch from P to AP for a particular width of the voltage pulse. This result \nis surprising considering that both the torque magnitudes shown in Fig. 2 are completely \nsymmetric with voltage. However the reason would be clear if we look at the pseudo field. As 8 mentioned above, ( ) is conservative and does not change polarity with voltage. This means \nthat ( ) acts as if an external magnetic field was applied in the direction of  irrespective of \nthe voltage polarity. As a result, it directly changes the potential energy of the system helping the \nAP to P transition while acting against the P to AP switching.  \n \nAn important thing to note is the fact that , the equilibrium component of , would also \nintroduce an asymmetry in switching voltage and it manifests itself as an exchange field in the \nequilibrium R-H loops. However, the significance of  being independent of angular position is \nthat even if we compensate for this exchange field by making the equilibrium hysteresis loop \ncompletely symmetric, for example, by applying an external magnetic field, there will still be an \nasymmetry in the switching current due to .  \n \nNotice that this asymmetry in the switching voltage is not dependent on the symmetric nature of \n shown in Fig. 2. As long as  is not purely anti-symmetric, the effect remains. This \nasymmetry is also in addition to that arising from any voltage asymmetry in the magnitude of , \ni.e., the in-plane torque component. It is worth mentioning, however, that two [6,7] of the three \ntorque measurement experiments done so far have found  to be anti-symmetric (making its \nmagnitude symmetric) at least in the low voltage region in agreement with ab-initio calculation \n[10]. Our own calculations also support the anti-symmetric nature of . This suggests that the \ndominant reason for the asymmetry in switching voltages for tunnel based devices [18] may arise \nfrom field like terms. This is surprising considering the fact that the field like term was minimal \nand was normally ignored in the earlier devices based on metallic channels. \n 9 5. Conclusion: \nBy formulating spin transfer torque as a pseudo field proportional to the spin resolved electron \ndensity, we have been able to show how the field like torque can introduce a voltage symmetric \nconservative torque on the magnet and thereby cause an asymmetry in the switching voltages for \ntunneling barrier based spin torque devices. It will be interesting to explore if this effect can be \nutilized to reduce the switching voltage by appropriate device design.  Our results also suggest \nthat that one should consider maximizing the electron density while exploring novel device \ndesigns [19, 20, 21] involving spin transfer torque. Furthermore, the ability to change the \npotential energy of a system (by virtue of a voltage induced conservative field [22]) may also \nhave important implications for voltage induced energy conversion and phase transition. \n \n \n \n \n \n \n \n \n \n \n \n \n \n 10 Appendix A: Proof of Eqn(s) (7), (8) \n \nLet us assume that the fixed magnet  and the soft magnet are both in the  plane (see \nFig. 1.) so that the Hamiltonian  (see Eqn. (6)) completely real, assuming the vector potential \n to be zero. It can then be shown that the Green’s function is symmetric (Chapter 3, [15]):  \n. We shall use this symmetry property of the Green’s function to understand the form of\n which is defined in terms of correlation function ,  with \ngiven by [14]  \n \n                                                                                                                                                    (A1)  \n \nwhere,  and  are the partial spectral functions due to contact 1 and 2 respectively\n. Now, both are Hermitian, but not symmetric, since , so that\n. However, the total spectral function can be \nwritten as  and hence symmetric: . This means that is purely real and can be \nexpressed as  \n                                                                                                                                                    (A2) \nWhile                                                                                                                                         \nAt equilibrium, only the  term in Eqn. (A1) is non-zero, so that the magnetization can \nbe written as stated in Eqn. (7) while under non-equilibrium condition, it has the more general \nform stated in Eqn. (8):                                           \n \n \n \n 11 Appendix B: Non-Conservative Nature of Pseudo-field \n \nIn Appendix A  we showed that the pseudo-field  lies entirely in-plane at equilibrium, but can \nhave an out-of-plane component away from equilibrium. We will now show that at equilibrium it \nis conservative, but away from equilibrium, the out-of-plane component makes it non-\nconservative. \n \nAssume that the fixed magnet  points along  (Fig.1 ) and the soft magnet points along \nwhere defined in a spherical co-ordinate system. The other \nunit vectors can be written as and . \nWe can write the curl of the pseudo-field as \n               \n                                                         (B.1) \n \nwhere we have dropped terms involving ,  since we assume ,  to be fixed \nand only consider changes in the direction  of the magnetization of the soft magnet relative to \nthe fixed magnet ( ). \n \nWe write the pseudo-field as  so that we obtain (with\n),  \n                                         (B.2a) \n                   (B.2b) \n                     (B.2c) \n \nNow, if we change the  of the soft magnet, its angle with the fixed magnet changes and in \nresponse the pseudo field could in general change arbitrarily making both terms in Eqn.(B.1) \nnon-zero. But the component  contributes nothing to the actual torque, and we could \narbitrarily define it to be a constant so that the only non-zero curl arises from the first term: 12                                                                                                                                                    (B.4)    \n \nThis means the curl is non-zero unless  and it does not change as \nthe  of the soft magnet is rotated. This can happen only if  is identically zero, which is \nexactly what happens under equilibrium conditions (see Appendix A ): the pseudo-field only has \nin-plane components, which means that , making . Hence the pseudo-field is in-\nplane and conservative in equilibrium, but away from equilibrium it can have an out-of-plane \ncomponent that will make it non-conservative. \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n 13 References:  \n1. J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). \n2. L. Berger, Phys. Rev. B 54, 9353 (1996). \n \n3. I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman, \nScience, 307, 228 (2005). \n4. G. D. Fuchs, J. A. Katine, S. I. Kiselev, D. Mauri, K. S. Wooley, D. C. Ralph, and R. A. \nBuhrman, Phys. Rev. Lett. 96, 186603 (2006). \n5. Kubota et. al.,  Nature Phys., 4, 37 (2008). \n6. Sankey J. K. et. al.,  Nature Phys, 4, 67 (2008). \n7. Deac, A. M. et. al.,  Nature Phys., 4, 803 (2008). \n8. Slonczewski J. C. and Sun, J. Z., J. Magn. Magn. Mater. 310, 169 (2007). \n9. Theodonis, I et. al.,  Phys. Rev. Lett, 97, 237205 (2006). \n10. Heiliger, C. and Stiles, M.D., Phys Rev. Lett., 100, 186805 (2008). \n11. P.M Haney et. al.,  J. Magn. Magn. Mater. 320, 1300 (2008). \n12. Sun, J.Z. and Ralph, D.C., J. Magn. Magn. Mater. 320, 1227 (2008). \n13. Ito, K. et al.,  Appl. Phys Lett., 89, 252509 (2006).               \n14. S. Datta, Quantum Transport: Atom to transistor , Cambridge University Press (2005). \n15. S. Datta. Electronic Transport in Mesoscopic Systems , Cambridge University Press (1995). \n16.  S. Salahuddin et. al.,  Technical Digest of IEDM, 121 (2007). \n17. Recent theoretical studies have successfully achieved quantitative agreement for torque with \nexperimental measurement [10] based on ab-initio band structure. However, since we are only \ninterested in the qualitative nature of the torque, we believe that our effective mass treatment \nshould suffice. \n18. M. Hosomi et al.,  Technical Digest of IEDM, 473, (2005). \n19.  Huai et. al.,  Appl. Phys. Lett. 87, 222510, (2005).  \n \n20. Meng et al.,  Appl. Phys. Lett. 88,082504, (2006).  \n 14 21. Fuchs G. D. et. al ., Appl. Phys. Lett. 86, 152509 (2005). \n22. Di Ventra M. et. al ., Phys. Rev. Lett, 92, 176803 (2004). \n 15 Figure Captions: \nFig. 1.  Schematic of tri-layer device. The left contact is the pinned ferromagnet having magnetization \nalong the z-axis.  The right contact is the free layer and the channel material is an oxide.  is the easy \naxis and  is the easy plane. Transport occurs in y-direction.  The device region is modeled using \nappropriate Hamiltonian, , and electrostatic potential  and the contacts are taken into account by self \nenergy matrices and , whose anti-Hermitian components                            are broadening matrices \ndue to contacts 1 and 2 respectively [14]. \nFig. 2. (a)  Typical variation of  and  as a function of voltage for tunnel based spin torque devices.  \nshows symmetric and  shows anti-symmetric voltage dependence. (b) Bias dependence of in-plane and \nout-of-plane components of Torque for tunnel based spin torque devices. (c), (d)  The variation of  and \n as a function of  at a fixed  and as a function of  at a fixed  respectively at a fixed voltage for a \ntunnel based spin torque device. We see that   and  at a fixed voltage are independent of both  and\n. (e) Typical variation of differential torque (w.r.t. voltage) as a function of the relative angle  \nbetween the magnetizations of the ferromagnetic electrodes.  \nFig. 3. The switching dynamics with same voltages with opposite polarity: positive voltage for AP to P \nand negative voltage for P to AP. For clarity, we have only marked the z component with bold blue color. \nThe dashed curve shows AP to P and the solid curve shows P to AP transitions. (a) For the same voltage \namplitude, the AP to P transition is faster than P to AP. Note the dashed line where the AP to P transition \nis almost complete while the P-to-AP transition is just around its half-way mark. (b) To get a symmetric \nswitching time, it takes almost 30% more voltage (V-) for P-to AP compared to the AP-to-P \ntransition. No external magnetic field has been assumed.  \n \n \n 16  \nFig. 1 17  \n \n \n \n \n \n \n \n  \n  \n  \n  \n  \n  \n  \n \n Fig. 2 (a) Fig. 2 (b) \nFig. 2 (e) \nFig. 2 (c) Fig. 2 (d) 18 \nFig. 3 (a) Fig. 3 (b) "    },    {        "title": "1701.03083v2.The_Cauchy_problem_for_the_Landau_Lifshitz_Gilbert_equation_in_BMO_and_self_similar_solutions.pdf",        "content": "The Cauchy problem for the Landau–Lifshitz–Gilbert equation\nin BMO and self-similar solutions\nSusana Gutiérrez1and André de Laire2\nAbstract\nWe prove a global well-posedness result for the Landau–Lifshitz equation with Gilbert\ndamping provided that the BMO semi-norm of the initial data is small. As a consequence,\nwe deduce the existence of self-similar solutions in any dimension. In the one-dimensional\ncase, we characterize the self-similar solutions associated with an initial data given by some\n(S2-valued) step function and establish their stability. We also show the existence of multiple\nsolutions if the damping is strong enough.\nOur arguments rely on the study of a dissipative quasilinear Schrödinger equation ob-\ntained via the stereographic projection and techniques introduced by Koch and Tataru.\nKeywords and phrases: Landau–Lifshitz–Gilbert equation, global well-posedness, discontin-\nuous initial data, stability, self-similar solutions, dissipative Schrödinger equation, complex\nGinzburg–Landau equation, ferromagnetic spin chain, heat-flow for harmonic maps.\n2010Mathematics Subject Classification: 35R05, 35Q60, 35A01, 35C06, 35B35, 35Q55,\n35Q56, 35A02, 53C44.\nContents\n1 Introduction and main results 2\n2 The Cauchy problem 6\n2.1 The Cauchy problem for a dissipative quasilinear Schrödinger equation . . . . . . 6\n2.2 The Cauchy problem for the LLG equation . . . . . . . . . . . . . . . . . . . . . 15\n3 Applications 22\n3.1 Existence of self-similar solutions in RN. . . . . . . . . . . . . . . . . . . . . . . 22\n3.2 The Cauchy problem for the one-dimensional LLG equation with a jump initial\ndata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23\n3.2.1 Existence, uniqueness and stability. Proof of Theorem 1.2 . . . . . . . . . 24\n3.2.2 Multiplicity of solutions. Proof of Theorem 1.3 . . . . . . . . . . . . . . . 28\n3.3 A singular solution for a nonlocal Schrödinger equation . . . . . . . . . . . . . . . 30\n4 Appendix 34\n1School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom.\nE-mail: s.gutierrez@bham.ac.uk\n2Univ. Lille, CNRS, Inria, UMR 8524, Laboratoire Paul Painlevé, F-59000 Lille, France.\nE-mail: andre.de-laire@univ-lille.fr\n1arXiv:1701.03083v2  [math.AP]  20 Mar 20191 Introduction and main results\nWe consider the Landau–Lifshitz–Gilbert (LLG) equation\n@tm=\fm\u0002\u0001m\u0000\u000bm\u0002(m\u0002\u0001m);onRN\u0002R+; (LLG\u000b)\nwhere m= (m1;m2;m3) :RN\u0002R+\u0000!S2is the spin vector, \f\u00150,\u000b\u00150;\u0002denotes the usual\ncross-product in R3, and S2is the unit sphere in R3. This model introduced by Landau and\nLifshitz describes the dynamics for the spin in ferromagnetic materials [26, 16] and constitutes a\nfundamental equation in the magnetic recording industry [36]. The parameters \f\u00150and\u000b\u00150\nare respectively the so-called exchange constant and Gilbert damping, and take into account the\nexchange of energy in the system and the effect of damping on the spin chain. Note that, by\nperforming a time-scaling, we assume w.l.o.g. that\n\u000b2[0;1]and\f=p\n1\u0000\u000b2:\nThe Landau–Lifshitz family of equations includes as special cases the well-known heat-flow for\nharmonic maps and the Schrödinger map equation onto the 2-sphere. In the limit case \f= 0\n(and so\u000b= 1) the LLG equation reduces to the heat-flow equation for harmonic maps\n@tm\u0000\u0001m=jrmj2m;onRN\u0002R+: (HFHM)\nThe case when \u000b= 0(i.e. no dissipation/damping) corresponds to the Schrödinger map equation\n@tm=m\u0002\u0001m;onRN\u0002R+: (SM)\nIn the one-dimensional case N= 1, we established in [17] the existence and asymptotics of the\nfamilyfmc;\u000bgc>0of self-similar solutions of (LLG \u000b) for any fixed \u000b2[0;1], extending the results\nin Gutiérrez, Rivas and Vega [18] in the setting of the Schrödinger map equation and related\nbinormal flow equation. The motivation for the results presented in this paper first originated\nfrom the desire to study further properties of the self-similar solutions found in [17], and in\nparticular their stability. In the case \u000b= 0, the stability of the self-similar solutions of the\nSchrödinger map has been considered in the series of papers by Banica and Vega [5, 6, 7], but\nno stability result is known for these solutions in the presence of damping, i.e. \u000b > 0. One of\nthe key ingredients in the analysis given by Banica and Vega is the reversibility in time of the\nequation in the absence of damping. However, since (LLG \u000b) is a dissipative equation for \u000b>0,\nthis property is no longer available and a new approach is needed.\nIn the one-dimensional case and for fixed \u000b2[0;1], the self-similar solutions of (LLG \u000b)\nconstitute a uniparametric family fmc;\u000bgc>0where mc;\u000bis defined by\nmc;\u000b(x;t) =f\u0012xp\nt\u0013\n;\nfor some profile f:R\u0000!S2, and is associated with an initial condition given by a step function\n(at least when cis small) of the form\nm0\nc;\u000b:=A+\nc;\u000b\u001fR++A\u0000\nc;\u000b\u001fR\u0000; (1.1)\nwhere A\u0006\nc;\u000bare certain unitary vectors and \u001fEdenotes the characteristic function of a set E. In\nparticular, when \u000b>0, the Dirichlet energy associated with the solutions mc;\u000bgiven by\nkrmc;\u000b(\u0001;t)k2\nL2=c2\u00102\u0019\n\u000bt\u00111=2\n; t> 0; (1.2)\n2diverges as t!0+3.\nA first natural question in the study of the stability properties of the family of solutions\nfmc;\u000bgc>0is whether or not it is possible to develop a well-posedness theory for the Cauchy\nproblem for (LLG \u000b) in a functional framework that allows us to handle initial conditions of the\ntype (1.1). In view of (1.1) and (1.2), such a framework should allow some “rough” functions\n(i.e. function spaces beyond the “classical” energy ones) and step functions.\nA few remarks about previously known results in this setting are in order. In the case \u000b>0,\nglobal well-posedness results for (LLG \u000b) have been established in N\u00152by Melcher [31] and by\nLin, Lai and Wang [30] for initial conditions with a smallness condition on the gradient in the\nLN(RN)and the Morrey M2;2(RN)norm4, respectively. Therefore these results do not apply\nto the initial condition m0\nc;\u000b. When\u000b= 1, global well-posedness results for the heat flow for\nharmonic maps (HFHM) have been obtained by Koch and Lamm [22] for an initial condition\nL1-close to a point and improved to an initial data with small BMO semi-norm by Wang [35].\nThe ideas used in [22] and [35] rely on techniques introduced by Koch and Tataru [23] for the\nNavier–Stokes equation. Since m0\nc;\u000bhas a small BMO semi-norm if cis small, the results in [35]\napply to the case \u000b= 1.\nTherearetwomainpurposesinthispaper. Thefirstoneistoadaptandextendthetechniques\ndeveloped in [22, 23, 35] to prove a global well-posedness result for (LLG \u000b) with\u000b2(0;1]for\ndatam0inL1(RN;S2)with small BMO semi-norm. The second one is to apply this result to\nestablish the stability of the family of self-similar solutions fmc;\u000bgc>0found in [17] and derive\nfurther properties for these solutions. In particular, a further understanding of the properties of\nthe functions mc;\u000bwill allow us to prove the existence of multiple smooth solutions of (LLG \u000b)\nassociated with the same initial condition, provided that \u000bis close to one.\nIn order to state the first of our results, we introduce the function space Xas follows:\nX=fv:Rn\u0002R+!R3:v;rv2L1\nloc(RN\u0002R+)andkvkX:=supt>0kv(t)kL1+ [v]X<1g\nwhere\n[v]X:=supt>0p\ntkrvkL1+ sup\nx2RN\nr>0 \n1\nrN\u0002\nBr(x)\u0002[0;r2]jrv(y;t)j2dtdy!1\n2\n;\nandBr(x)denotes the ball with center xand radius r >0inRN. Let us remark that the first\nterm in the definition of [v]Xallows to capture a blow-up rate of 1=p\ntforkrv(t)kL1, ast!0+.\nThis is exactly the blow-up rate for the self-similar solutions (see (3.1) and (3.12)). The integral\nterm in [v]Xis associated with the space BMO as explained in Subsection 2.1, and it is also well\nadapted to the self-similar solutions (see Proposition 3.4 and its proof).\nWe can now state the following (global) well-posedness result for the Cauchy problem for the\nLLG equation:\nTheorem 1.1. Let\u000b2(0;1]. There exist constants M1;M2;M3>0, depending only on \u000band\nNsuch that the following holds. For any m02L1(RN;S2),Q2S2,\u000e2(0;2]and\"0>0such\nthat\"0\u0014M1\u000e6,\ninf\nRNjm0\u0000Qj2\u00152\u000eand [m0]BMO\u0014\"0; (1.3)\nthere exists a unique solution m2X(RN\u0002R+;S2)of(LLG\u000b)with initial condition m0such\nthat\ninf\nx2RN\nt>0jm(x;t)\u0000Qj2\u00154\n1 +M2\n2(M3\u000e4+\u000e\u00001)2and [m]X\u00144M2(M3\u000e4+ 8\u000e\u00002\"0):(1.4)\n3We refer the reader to Theorem A.5 in the Appendix and to [17] for precise statements of these results.\n4See footnote in Section 3.3 for the definition of the Morrey space M2;2(RN).\n3In addition, mis a smooth function belonging to C1(RN\u0002R+;S2). Furthermore, assume that\nnis a solution to (LLG\u000b)fulfilling (1.4), with initial condition n0satisfying (1.3). Then\nkm\u0000nkX\u0014120M2\n\u000e2km0\u0000n0kL1: (1.5)\nAs we will see in Section 2, the proof of Theorem 1.1 relies on the use of the stereographic pro-\njection to reduce Theorem 1.1 to establish a well-posedness result for the associated dissipative\n(quasilinear) Schrödinger equation (see Theorem 2.1). In order to be able to apply Theorem 1.1\nto the study of both the initial value problem related to the LLG equation with a jump initial\ncondition, and the stability of the self-similar solutions found in [17], we will need a more quanti-\ntative version of this result. A more refined version of Theorem 1.1 will be stated in Theorem 2.9\nin Subsection 2.2.\nTheorem 1.1 (or more precisely Theorem 2.9) has two important consequences for the Cauchy\nproblem related to (LLG \u000b) in one dimension:\n8\n<\n:@tm=\fm\u0002@xxm\u0000\u000bm\u0002(m\u0002@xxm);onR\u0002R+;\nm0\nA\u0006:=A+\u001fR++A\u0000\u001fR\u0000;(1.6)\nwhere A\u0006are two given unitary vectors such that the angle between A+andA\u0000is sufficiently\nsmall:\n(a)From the uniqueness statement in Theorem 1.1, we can deduce that the solution to (1.6)\nprovided by Theorem 1.1 is a rotation of a self-similar solution mc;\u000bfor an appropriate\nvalue ofc(see Theorem 3.3 for a precise statement).\n(b)(Stability) From the dependence of the solution with respect to the initial data established\nin (1.5) and the analysis of the 1d-self-similar solutions mc;\u000bcarried out in [17], we obtain\nthe following stability result: For any given m02S2satisfying (1.3) and close enough to\nm0\nA\u0006, the solution mof (LLG\u000b) associated with m0given by Theorem 1.1 must remain\nclose to a rotation of a self-similar solution mc;\u000b, for somec>0. In particular, mremains\nclose to a self-similar solution.\nThe precise statement is provided in the following theorem.\nTheorem 1.2. Let\u000b2(0;1]. There exist constants L1;L2;L3>0,\u000e\u00032(\u00001;0),#\u0003>0such\nthat the following holds. Let A+,A\u00002S2with angle#between them. If\n0<#\u0014#\u0003;\nthen there is c>0such that for every m0satisfying\nkm0\u0000m0\nA\u0006kL1\u0014cp\u0019\n2p\u000b;\nthere existsR2SO(3), depending only on A+,A\u0000,\u000bandc, such that there is a unique global\nsmooth solution mof(LLG\u000b)with initial condition m0that satisfies\ninf\nx2R\nt>0(Rm)3(x;t)\u0015\u000e\u0003and [m]X\u0014L1+L2c: (1.7)\nMoreover,\nkm\u0000Rmc;\u000bkX\u0014L3km0\u0000m0\nA\u0006kL1:\n4In particular,\nk@xm\u0000@xRmc;\u000bkL1\u0014L3p\ntkm0\u0000m0\nA\u0006kL1;\nfor allt>0.\nNotice that Theorem 1.2 provides the existence of a unique solution in the set defined by the\nconditions (1.7), and hence it does not exclude the possibility of the existence of other solutions\nnot satisfying these conditions. In fact, as we will see in Theorem 1.3 below, one can prove the\nexistence of multiple solutions of the initial value problem (1.6), at least in the case when \u000bis\nclose to 1.\nWe point out that our results are valid only for \u000b>0. If we let\u000b!0, then the constants M1\nandM3in Theorem 1.1 go to 0 and M2blows up. Indeed, we use that the kernel associated with\nthe Ginzburg–Landau semigroup e(\u000b+i\f)t\u0001belongs toL1and its exponential decay. Therefore\nour techniques cannot be generalized (in a simple way) to cover the critical case \u000b= 0. In\nparticular, we cannot recover the stability results for the self-similar solutions in the case of\nSchrödinger maps proved by Banica and Vega in [5, 6, 7].\nAs mentioned before, in [30] and [31] some global well-posedness results for (LLG \u000b) with\n\u000b2(0;1]were proved for initial conditions with small gradient in LN(RN)andM2;2(RN),\nrespectively (see footnote in Subsection 3.3 for the definition of the space M2;2(RN)). In view of\nthe embeddings\nLN(RN)\u001aM2;2(RN)\u001aBMO\u00001(RN);\nforN\u00152, Theorem 1.1 can be seen as generalization of these results since it covers the case\nof less regular initial conditions. The arguments in [30, 31] are based on the method of moving\nframes that produces a covariant complex Ginzburg–Landau equation. In Subsection 3.3 we give\nmore details and discuss the corresponding equation in the one-dimensional case and provide\nsome properties related to the self-similar solutions.\nOur existence and uniqueness result given by Theorem 1.1 requires the initial condition to be\nsmall in the BMO semi-norm. Without this condition, the solution could develop a singularity\nin finite time. In fact, in dimensions N= 3;4, Ding and Wang [13] have proved that for some\nsmooth initial conditions with small (Dirichlet) energy, the associated solutions of (LLG \u000b) blow\nup in finite time.\nIn the context of the initial value problem (1.6), the smallness condition in the BMO semi-\nnorm is equivalent to the smallness of the angle between A+andA\u0000. As discussed in [17], in the\none dimensional case N= 1for fixed\u000b2(0;1]there is some numerical evidence that indicates\nthe existence of multiple (self-similar) solutions associated with the same initial condition of the\ntype in (1.6) (see Figures 2 and 3 in [17]). This suggests that the Cauchy problem for (LLG \u000b)\nwith initial condition (1.6) is ill-posed for general A+andA\u0000unitary vectors.\nThe following result states that in the case when \u000bis close to 1, one can actually prove the\nexistenceofmultiplesmoothsolutionsassociatedwiththesameinitialcondition m0\nA\u0006. Moreover,\ngiven any angle #2(0;\u0019)between two vectors A+andA\u00002S2, one can generate any number\nof distinct solutions by considering values of \u000bsufficiently close to 1.\nTheorem 1.3. Letk2N,A+,A\u00002S2and let#be the angle between A+andA\u0000. If\n#2(0;\u0019), then there exists \u000bk2(0;1)such that for every \u000b2[\u000bk;1]there are at least kdistinct\nsmooth self-similar solutions fmjgk\nj=1inX(R\u0002R+;S2)of(LLG\u000b)with initial condition m0\nA\u0006.\nThese solutions are characterized by a strictly increasing sequence of values fcjgk\nj=1, withck!1\nask!1, such that\nmj=Rjmcj;\u000b; (1.8)\n5whereRj2SO(3). In particular\np\ntk@xmj(\u0001;t)kL1=cj;for allt>0: (1.9)\nFurthermore, if \u000b= 1and#2[0;\u0019], then there is an infinite number of distinct smooth self-\nsimilar solutions fmjgj\u00151inX(R\u0002R+;S2)of(LLG\u000b)with initial condition m0\nA\u0006. These\nsolutions are also characterized by a sequence fcjg1\nj=1such that (1.8)and(1.9)are satisfied.\nThis sequence is explicitly given by\nc2`+1=`p\u0019\u0000#\n2p\u0019; c 2`=`p\u0019+#\n2p\u0019;for`\u00150: (1.10)\nIt is important to remark that in particular Theorem 1.3 asserts that when \u000b= 1, given\nA+;A\u00002S2such that A+=A\u0000, there exists an infinite number of distinct solutions fmjgj\u00151\ninX(R\u0002R+;S2)of (LLG\u000b) with initial condition m0\nA\u0006such that [m0\nA\u0006]BMO = 0. This\nparticular case shows that a condition on the size of X-norm of the solution as that given in\n(1.4) in Theorem 1.1 is necessary for the uniqueness of solution. We recall that for finite energy\nsolutions of (HFHM) there are several nonuniqueness results based on Coron’s technique [11] in\ndimensionN= 3. Alouges and Soyeur [2] successfully adapted this idea to prove the existence of\nmultiple solutions of the (LLG \u000b), with\u000b>0, for maps m: \n\u0000!S2, with \na bounded regular\ndomain of R3. In our case, since fcjgk\nj=1is strictly increasing, we have at least kgenuinely\ndifferent smoothsolutions. Notice also that the identity (1.9) implies that the X-norm of the\nsolution is large as j!1.\nStructure of the paper. This paper is organized as follows: in Section 2 we use the ste-\nreographic projection to reduce matters to the study the initial value problem for the resulting\ndissipative Schrödinger equation, prove its global well-posedness in well-adapted normed spaces,\nand use this result to establish Theorem 2.9 (a more quantitative version of Theorem 1.1). In\nSection 3 we focus on the self-similar solutions and we prove Theorems 1.2 and 1.3. In Section 3.3\nwe discuss some implications of the existence of explicit self-similar solutions for the Schrödinger\nequation obtained by means of the Hasimoto transformation. Finally, and for the convenience of\nthe reader, we have included some regularity results for the complex Ginzburg–Landau equation\nand some properties of the self-similar solutions mc;\u000bin the Appendix.\nNotations. We write R+= (0;1). Throughout this paper we will assume that \u000b2(0;1]and\nthe constants can depend on \u000b. In the proofs A.Bstands forA\u0014CBfor some constant\nC > 0depending only on \u000bandN. We denote in bold the vector-valued variables.\nSince we are interested in S2-valued functions, with a slightly abuse of notation, we denote\nbyL1(RN;S2)(resp.X(RN;S2)) the space of function in L1(RN;R3)(resp.X(RN;R3)) such\nthatjmj=1 a.e. on RN.\n2 The Cauchy problem\n2.1 The Cauchy problem for a dissipative quasilinear Schrödinger equation\nOur approach to study the Cauchy problem for (LLG \u000b) consists in analyzing the Cauchy prob-\nlem for the associated dissipative quasilinear Schrödinger equation through the stereographic\nprojection, and then “transferring” the results back to the original equation. To this end, we\nintroduce the stereographic projection from the South Pole P:S2nf(0;0;\u00001)g!Cdefined for\nby\nP(m) =m1+im2\n1 +m3:\n6Letmbe a smooth solution of (LLG \u000b) withm3>\u00001, then its stereographic projection u=\nP(m)satisfies the quasilinear dissipative Schrödinger equation (see e.g. [25] for details)\niut+ (\f\u0000i\u000b)\u0001u= 2(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2: (DNLS)\nAt least formally, the Duhamel formula gives the integral equation:\nu(x;t) =S\u000b(t)u0+\u0002t\n0S\u000b(t\u0000s)g(u)(s)ds; (IDNLS)\nwhereu0=u(\u0001;0)corresponds to the initial condition,\ng(u) =\u00002i(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2\nandS\u000b(t)is the dissipative Schrödinger semigroup (also called the complex Ginzburg–Landau\nsemigroup) given by S\u000b(t)\u001e=e(\u000b+i\f)t\u0001\u001e, i.e.\n(S\u000b(t)\u001e)(x) =\u0002\nRNG\u000b(x\u0000y;t)\u001e(y)dy;withG\u000b(x;t) =e\u0000jxj2\n4(\u000b+i\f)t\n(4\u0019(\u000b+i\f)t)N=2:(2.1)\nOne difficulty in studying (IDNLS) is to handle the term g(u). Taking into account that\nj\f\u0000i\u000bj= 1anda\n1 +a2\u00141\n2;for alla\u00150; (2.2)\nwe see that\njg(u)j\u0014jruj2; (2.3)\nso we need to control jruj2. Koch and Taratu dealt with a similar problem when studying the\nwell-posedness for the Navier–Stokes equation in [23]. Their approach was to introduce some new\nspaces related to BMO and BMO\u00001. Later, Koch and Lamm [22] and Wang [35] have adapted\nthese spaces to study some geometric flows. Following these ideas, we define the Banach spaces\nX(RN\u0002R+;F) =fv:RN\u0002R+!F:v;rv2L1\nloc(RN\u0002R+);kvkX<1gand\nY(RN\u0002R+;F) =fv:RN\u0002R+!F:v2L1\nloc(RN\u0002R+);kvkY<1g;\nwhere\nkvkX:= sup\nt>0kvkL1+ [v]X;with\n[v]X:= sup\nt>0p\ntkrvkL1+ sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jrv(y;t)j2dtdy!1\n2\n;and\nkvkY= sup\nt>0tkvkL1+ sup\nx2RN\nr>01\nrN\u0002\nQr(x)jv(y;t)jdtdy:\nHereQr(x)denotes the parabolic ball Qr(x) =Br(x)\u0002[0;r2]andFis either CorR3. The\nabsolute value stands for the complex absolute value if F=Cand for the euclidean norm if\nF=R3. We denote with the same symbol the absolute value in FandF3. Here and in the\nsequel we will omit the domain in the norms and semi-norms when they are taken in the whole\nspace, for example k\u0001kLpstands fork\u0001kLp(RN), forp2[1;1].\n7The spaces XandYare related to the spaces BMO (RN)and BMO\u00001(RN)and are well-\nadapted to study problems involving the heat semigroup S1(t) =et\u0001. In order to establish\nthe properties of the semigroup S\u000b(t)with\u000b2(0;1], we introduce the spaces BMO \u000b(RN)and\nBMO\u00001\n\u000b(RN)as the space of distributions f2S0(RN;F)such that the semi-norm and norm\ngiven respectively by\n[f]BMO\u000b:= sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jrS\u000b(t)fj2dtdy!1\n2\n;and\nkfkBMO\u00001\n\u000b:= sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jS\u000b(t)fj2dtdy!1\n2\n;\nare finite.\nOntheonehand, theCarlesonmeasurecharacterizationofBMOfunctions(see[34, Chapter4]\nand [27, Chapter 10]) yields that for fixed \u000b2(0;1],BMO\u000b(RN)coincides with the classical\nBMO (RN)space5, that is for all \u000b2(0;1]there exists a constant \u0003>0depending only on \u000b\nandNsuch that\n\u0003[f]BMO\u0014[f]BMO\u000b\u0014\u0003\u00001[f]BMO: (2.4)\nOn the other hand, Koch and Tataru proved in [23] that BMO\u00001(or equivalently BMO\u00001\n1,\nusing our notation) can be characterized as the space of derivatives of functions in BMO. A\nstraightforward generalization of their argument shows that the same result holds for BMO\u00001\n\u000b\n(see Theorem A.1). Hence, using the Carleson measure characterization theorem, we conclude\nthat BMO\u00001\n\u000bcoincides with the space BMO\u00001and that there exists a constant ~\u0003>0, depending\nonly on\u000bandN, such that\n~\u0003kfkBMO\u00001\u0014kfkBMO\u00001\n\u000b\u0014~\u0003\u00001kfkBMO\u00001: (2.5)\nThe above remarks allows us to use several of the estimates proved in [22, 23, 35] in the case\n\u000b= 1to study the integral equation (IDNLS) by using a fixed-point approach.\nOur first result concerns the global well-posedness of the Cauchy problem for (IDNLS) with\nsmall initial data in BMO (RN).\nTheorem 2.1. Let\u000b2(0;1]. There exist constants C;K\u00151such that for every L\u00150,\">0,\nand\u001a>0satisfying\n8C(\u001a+\")2\u0014\u001a; (2.6)\nifu02L1(RN;C), with\nku0kL1\u0014Land [u0]BMO\u0014\"; (2.7)\nthen there exists a unique solution u2X(RN\u0002R+;C)to(IDNLS) such that\n[u]X\u0014K(\u001a+\"): (2.8)\nMoreover,\n5\nBMO (RN) =ff:RN\u0002[0;1)!F:f2L1\nloc(RN);[f]BMO <1g;\nwith the semi-norm\n[f]BMO = sup\nx2RN\nr>0 \nBr(x)jf(y)\u0000fx;rjdy;\nwhere fx;ris the average\nfx;r= \nBr(x)f(y)dy=1\njBr(x)j\u0002\nBr(x)f(y)dy:\n8(i)supt>0kukL1\u0014K(\u001a+L).\n(ii)u2C1(RN\u0002R+)and(DNLS) holds pointwise.\n(iii) lim\nt!0+u(\u0001;t) =u0as tempered distributions. Moreover, for every '2S(RN), we have\nk(u(\u0001;t)\u0000u0)'kL1!0;ast!0+: (2.9)\n(iv) (Dependence on the initial data) Assume that uandvare respectively solutions to (IDNLS)\nfulfilling (2.8)with initial conditions u0andv0satisfying (2.7). Then\nku\u0000vkX\u00146Kku0\u0000v0kL1: (2.10)\nAlthough condition (2.6) appears naturally from the fixed-point used in the proof, it may be\nno so clear at first glance. To better understand it, let us define for C > 0\nS(C) =f(\u001a;\")2R+\u0002R+:C(\u001a+\")2\u0014\u001ag: (2.11)\nWe see that if (\u001a;\")2S(C), then\u001a;\"> 0and\n\"\u0014p\u001ap\nC\u0000\u001a: (2.12)\nTherefore the set S(C)is non-empty and bounded. The shape of this set is depicted in Figure 1.\nIn particular, we infer from (2.12) that if (\u001a;\")2S(C), then\n1\n4C\n1\n4C1\nCρε\nFigure 1: The shape of the set S(C).\n\u001a\u00141\nCand\"\u00141\n4C: (2.13)\nIn addition, if ~C\u0015C, then\nS(~C)\u0012S(C): (2.14)\nMoreover, taking for instance \u001a= 1=(32C), Theorem 2.1 asserts that for fixed \u000b2(0;1], we can\ntake for instance \"= 1=(32C)(that depends on \u000bandN, but not on the L1-norm of the initial\ndata) such that for any given initial condition u02L1(RN)with [u0]BMO\u0014\", there exists a\nglobal (smooth) solution u2X(RN\u0002R+;C)of (DNLS). Notice that u0is allowed to have a\nlargeL1-norm as long as [u0]BMOis sufficiently small; this is a weaker requirement that asking\nfor theL1-norm ofu0to be sufficiently small, since\n[f]BMO\u00142kfkL1;for allf2L1(RN): (2.15)\n9Remark 2.2. The smallness condition in (2.8) is necessary for the uniqueness of the solution.\nAs we will see in Subsection 3.2.2, at least in dimension one, it is possible to construct multiple\nsolutions of (IDNLS) in X(RN\u0002R+;C), if\u000bis close enough to 1.\nThe aim of this section is to prove Theorem 2.1 using a fixed-point technique. To this pursuit\nwe write (IDNLS) as\nu(t) =Tu0(u)(t); (2.16)\nwhere\nTu0(u)(t) =S\u000b(t)u0+T(g(u))(t)andT(f)(t) =\u0002t\n0S\u000b(t\u0000s)f(s)ds: (2.17)\nIn the next lemmas we study the semigroup S\u000band the operator Tto establish that the appli-\ncationTu0is a contraction on the ball\nB\u001a(u0) =fu2X(RN\u0002R+;C) :ku\u0000S\u000b(t)u0kX\u0014\u001ag;\nfor some\u001a>0depending on the size of the initial data.\nLemma 2.3. There exists C0>0such that for all f2BMO\u00001\n\u000b(RN),\nsup\nt>0p\ntkS\u000b(t)fkL1(RN)\u0014C0kfkBMO\u00001\n\u000b: (2.18)\nProof.The proof in the case \u000b= 1is done in [27, Lemma 16.1]. For \u000b2(0;1), decomposing\nS\u000b(t) =S\u000b(t\u0000s)S\u000b(s)and using the decay properties of the kernel G\u000bassociated with the\noperatorsS\u000b(t)(see (2.1)), we can check that the same proof still applies.\nLemma 2.4. There exists C1\u00151such that for all f2Y(RN\u0002R+;C),\nkT(f)kX\u0014C1kfkY: (2.19)\nProof.Estimate (2.19) can be proved using the arguments given in [23] or [35]. For the conve-\nnience of the reader, we sketch the proof following the lines in [35, Lemma 3.1]. By scaling and\ntranslation, it suffices to show that\njT(f)(0;1)j+jrT(f)(0;1)j+ \u0002\nQ1(0)jrT(f)j2!1=2\n.kfkY: (2.20)\nLetBr=Br(0). SettingW=T(f), we have\nW(0;1) =\u00021\n0\u0002\nRNG\u000b(\u0000y;1\u0000s)f(y;s)dyds\n= \u00021\n1=2\u0002\nRN+\u00021=2\n0\u0002\nB2+\u00021=2\n0\u0002\nRNnB2!\nG\u000b(\u0000y;1\u0000s)f(y;s)dyds\n:=I1+I2+I3:\nSincejG\u000b(y;1\u0000s)j=e\u0000\u000bjyj2\n4(1\u0000s)\n(4\u0019(1\u0000s))N=2;we obtain\njI1j\u0014\u00021\n1=2\u0002\nRNjG\u000b(\u0000y;1\u0000s)jjf(y;s)jdyds\n\u0014sup\n1\n2\u0014s\u00141kf(s)kL1 \u00021\n1\n2\u0002\nRnjG\u000b(\u0000y;1\u0000s)jdyds!\n.kfkY;\n10jI2j\u0014\u00021=2\n0\u0002\nB2jG\u000b(\u0000y;1\u0000s)jjf(y;s)jdyds\n.sup\n0\u0014s\u00141\n2kG\u000b(\u0001;1\u0000s)kL1(RN)\u0002\nB2\u0002[0;1\n2]jf(y; s)jdyds.kfkY\nand\njI3j\u0014\u00021=2\n0\u0002\nRNnB2jG\u000b(\u0000y;1\u0000s)jjf(y;s)jdyds\n\u0014C\u00021\n2\n0\u0002\nRNnB2e\u0000\u000bjyj2\n4jf(y; s)jdyds\n\u0014C 1X\nk=2kn\u00001e\u0000\u000bk2\n4! \nsup\ny2RN\u0002\nQ1(y)jf(y; s)jdyds!\n.kfkY:\nThe quantityjrT(f)(0;1)jcan be bounded in a similar way. The last term in the l.h.s. of (2.20)\ncan be controlled using an energy estimate. Indeed, Wsatisfies the equation\ni@tW+ (\f\u0000i\u000b)\u0001W=if (2.21)\nwith initial condition W(\u0001;0) = 0. Let\u00112C1\n0(B2)be a real-valued cut-off function such that\n0\u0014\u0011\u00141onRNand\u0011= 1onB1. By multiplying (2.21) by \u0000i\u00112W, integrating and taking\nreal part, we get\n1\n2@t\u0002\nRN\u00112jWj2+\u000b\u0002\nRN\u00112jrWj2+ 2 Re\u0012\n(\u000b+i\f)\u0002\nRN\u0011r\u0011WrW\u0013\n=\u0002\nRN\u00112Re(fW):\nUsing thatj\u000b+i\fj= 1and integrating in time between 0and1, it follows that\n1\n2\u0002\nRN\u00112jW(x;1)j2+\u000b\u0002\nRN\u0002[0;1]\u00112jrWj2\u0014\u0002\nRN\u0002[0;1](2\u0011jr\u0011jjWjjrWj+\u00112jfjjWj):\nFrom the inequality ab\u0014\"a2+b2=(4\");witha=\u0011jrWj,b= 2jr\u0011jjWjand\"=\u000b=2, we deduce\nthat\u000b\n2\u0002\nRN\u0002[0;1]\u00112jrWj2\u0014\u0002\nRN\u0002[0;1]\u00002\n\u000bjr\u0011j2jWj2+\u00112jfjjWj\u0001\n:\nBy the definition of \u0011, this implies that\nkrWk2\nL2(B1\u0002[0;1]).kWk2\nL1(B2\u0002[0;1])+kWkL1(B2\u0002[0;1])kfkL1(B2\u0002[0;1]):(2.22)\nFrom the first part of the proof, we have\nkWkL1(B2\u0002[0;1])\u0014CkfkY:\nUsing also that\nkfkL1(B2\u0002[0;1]).kfkY;\nwe conclude from (2.22) that\nkrWkL2(B1\u0002[0;1]).kfkY;\nwhich finishes the proof.\n11Lemma 2.5. Let\u000b2(0;1]and\u001a;\";L> 0. There exists C2\u00151, depending on \u000bandN, such\nthat for all u02L1(RN)\nkS\u000b(t)u0kX\u0014C2(ku0kL1+ [u0]BMO ): (2.23)\nIf in additionku0kL1\u0014Land[u0]BMO\u0014\", then for all u2B\u001a(u0)we have\nsup\nt>0kukL1\u0014C2(\u001a+L)and [u]X\u0014C2(\u001a+\"): (2.24)\nProof.We first controlkS\u000b(t)u0kX. On the one hand, using the definition of G\u000band the relation\n\u000b2+\f2= 1, we obtain\nkS\u000b(t)u0kL1=kG\u000b\u0003u0kL1\u0014kG\u000bkL1ku0kL1=\u000b\u0000N\n2ku0kL1;8t>0:\nThus\nsup\nt>0kS\u000b(t)u0kL1\u0014\u000b\u0000N\n2ku0kL1: (2.25)\nOn the other hand, using Lemma 2.3, Theorem A.1 and (2.4),\n[S\u000b(t)u0]X= sup\nt>0p\ntkrS\u000b(t)u0kL1+ sup\nx2RN\nr>0 \n1\nrN\u0002\nQr(x)jrS\u000b(t)u0j2dtdy!1\n2\n.kru0kBMO\u00001\n\u000b+ [u0]BMO\u000b\n.[u0]BMO\u000b\n.[u0]BMO:(2.26)\nThe estimate in (2.23) follows from (2.25) and (2.26), and we w.l.o.g. can choose C2\u00151.\nFinally, using (2.25), given u0such thatku0kL1\u0014Land[u0]BMO\u0014\", for allu2B\u001a(u0)we\nhave\nkukL1\u0014ku\u0000S\u000b(t)u0kL1+kS\u000b(t)u0kL1\u0014ku\u0000S\u000b(t)u0kX+kS\u000b(t)u0kL1\u0014C2(\u001a+L);\nand, using (2.26),\n[u]X\u0014[u\u0000S\u000b(t)u0]X+ [S\u000b(t)u0]X\u0014ku\u0000S\u000b(t)u0kX+ [S\u000b(t)u0]X\u0014C2(\u001a+\");\nwhich finishes the proof of (2.24).\nNow we proceed to bound the nonlinear term\ng(u) =\u00002i(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2:\nLemma 2.6. For allu2X(RN\u0002R+;C), we have\nkg(u)kY\u0014[u]2\nX:\nProof.Letu2X(RN\u0002R+;C). Using (2.3) and the definitions of the norms in YandX, it\nfollows that\nkg(u)kY\u0014\u0012\nsup\nt>0p\ntkrukL1\u00132\n+ sup\nx2RN\nr>01\nrN\u0002\nQr(x)jruj2dtdy\u0014[u]2\nX:\n12Now we have all the estimates to prove that Tu0is a contraction on B\u001a(u0).\nProposition 2.7. Let\u000b2(0;1]and\u001a;\"> 0. Given any u02L1(RN)with [u0]BMO\u0014\", the\noperatorTu0given in (2.17)defines a contraction on B\u001a(u0), whenever \u001aand\"satisfy\n8C1C2\n2(\u001a+\")2\u0014\u001a: (2.27)\nMoreover, for all u;v2X(RN\u0002R+;C),\nkT(g(u))\u0000T(g(v))kX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX: (2.28)\nHere,C1\u00151andC2\u00151are the constants in Lemmas 2.4 and 2.5, respectively.\nRemark 2.8. Using the notation introduced in (2.11), the hypothesis (2.27) means that (\u001a;\")2\nS(8C1C2\n2). Therefore, by (2.13),\n\u001a\u00141\n8C1C2\n2;and\"\u00141\n32C1C2\n2; (2.29)\nso\u001aand\"are actually small. Since C1;C2\u00151, we have\nC2(\u001a+\")\u00145\n32: (2.30)\nProof.Letu02L1(RN)withku0kL1\u0014Land[u0]BMO\u0014\", andu2B\u001a(u0). Using Lemma 2.4,\nLemma 2.5 and Lemma 2.6, we have\nkTu0(u)\u0000S\u000b(t)u0kX=kT(g(u))kX\u0014C1kg(u)kY\u0014C1[u]2\nX\u0014C1C2\n2(\u001a+\")2:\nThereforeTu0mapsB\u001a(u0)into itself provided that\nC1C2\n2(\u001a+\")2\u0014\u001a: (2.31)\nNotice that by (2.14), the condition (2.27) implies that (2.31) is satisfied.\nTo prove (2.28), we use the decomposition\ng(u)\u0000g(v) =\u00002i(\f\u0000i\u000b)\u0014\u0012\u0016u\n1 +juj2\u0000\u0016v\n1 +jvj2\u0013\n(ru)2+\u0016v\n1 +jvj2((ru)2\u0000(rv)2))\u0015\n:\nSince \f\f\f\f\u0016u\n1 +juj2\u0000\u0016v\n1 +jvj2\f\f\f\f\u0014ju\u0000vj1 +jujjvj\n(1 +juj2)(1 +jvj2)\u0014ju\u0000vj;\nand using (2.2), we obtain\njg(u)\u0000g(v)j\u00142ju\u0000vjjruj2+jru\u0000rvj(jruj+jrvj):\nTherefore\nkg(u)\u0000g(v)kY\u00142kju\u0000vjjruj2kY+kjru\u0000rvj(jruj+jrvj)kY:=I1+I2:(2.32)\nForI1, it is immediate that\nI1\u00142 sup\nt>0ku\u0000vkL12\n64\u0010\nsup\nt>0p\ntkrukL1\u00112\n+ sup\nx2RN\nr>01\nrN\u0002\nQr(x)jruj2dtdy3\n75\u00142ku\u0000vkX[u]2\nX:\n(2.33)\n13Similarly, using the Cauchy–Schwarz inequality,\nI2\u0014\u0012\nsup\nt>0p\ntkru\u0000rvkL1\u0013\u0012\nsup\nt>0p\nt(krukL1+krvkL1)\u0013\n+ sup\nx2RN\nr>01\nrN\u0000\nkru\u0000rvkL2(Qr(x))\u0001\u0000\nkrukL2(Qr(x))+krvkL2(Qr(x))\u0001\n\u0014ku\u0000vkX([u]X+ [v]X):(2.34)\nUsing Lemma 2.4, (2.32), (2.33) and (2.34), we conclude that\nkT(g(u))\u0000T(g(v))kX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX: (2.35)\nLetu;v2B\u001a(u0), by Lemma 2.5 and (2.30)\n[u]X\u0014C2(\u001a+\")\u00145\n32; (2.36)\nso that\n2[u]2\nX+ [u]X+ [v]X\u001437\n16C2(\u001a+\")<3C2(\u001a+\"): (2.37)\nThen (2.35) implies that\nkTu0(u)\u0000Tu0(v)kX\u00143C1C2(\u001a+\")ku\u0000vkX: (2.38)\nFrom (2.29), we conclude that\n3C1C2(\u001a+\")\u001415\n32\u00141\n2; (2.39)\nand then (2.38) yields that the operator Tu0defined in (2.17) is a contraction on B\u001a(u0). This\nconcludes the proof of the proposition.\nProof of Theorem 2.1. Let us setC=C1C2\n2andK=C2, whereC1andC2are the constants\nin Lemma 2.4 and Lemma 2.5 respectively. Since \u001asatisfies (2.6), Proposition 2.7 implies that\nthere exists a solution uof equation (2.16) in the ball B\u001a(u0), and in particular from Lemma 2.5\nsup\nt>0kukL1\u0014K(\u001a+L)and [u]X\u0014K(\u001a+\"):\nTo prove the uniqueness part of the theorem, let us assume that uandvare solutions of (IDNLS)\ninX(RN\u0002R+;C)such that\n[u]X;[v]X\u0014K(\u001a+\"); (2.40)\nwith the same initial condition u0. By the definitions of CandK, (2.6) and (2.40), the estimates\nin (2.29) and (2.30) hold. It follows that (2.36), (2.37) and (2.39) are satisfied. Then, using\n(2.28),\nku\u0000vkX=kT(g(u))\u0000T(g(v))kX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX\n\u00141\n2ku\u0000vkX:\nFrom which it follows that u=v.\nTo prove the dependence of the solution with respect to the initial data (part (iv)), consider\nuandvsolutions of (IDNLS) satisfying (2.40) with initial conditions u0andv0. Then, by\ndefinition,u=Tu0(u),v=Tv0(v)and\nku\u0000vkX=kTu0(u)\u0000Tv0(v)kX\u0014kS\u000b(u0\u0000v0)kX+kT(g(u))\u0000T(g(v))kX:\n14Using (2.15), (2.23) and (2.28) and arguing as above, we have\nku\u0000vkX\u0014C2(ku0\u0000v0kL1+ [u0\u0000v0]BMO ) +C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX\n\u00143C2ku0\u0000v0kL1+1\n2ku\u0000vkX:\nThis yields (2.10), since K=C2.\nThe assertions in (ii)and(iii)follow from Theorem A.3.\n2.2 The Cauchy problem for the LLG equation\nBy using the inverse of the stereographic projection P\u00001:C!S2nf0;0;\u00001g, that is explicitly\ngiven by m= (m1;m2;m3) =P\u00001(u), with\nm1=2 Reu\n1 +juj2; m 2=2 Imu\n1 +juj2; m 3=1\u0000juj2\n1 +juj2; (2.41)\nwe will be able to establish the following global well-posedness result for (LLG \u000b).\nTheorem 2.9. Let\u000b2(0;1]. There exist constants C\u00151andK\u00154, such that for any\n\u000e2(0;2],\"0>0and\u001a>0such that\n8K4C\u000e\u00004(\u001a+ 8\u000e\u00002\"0)2\u0014\u001a; (2.42)\nifm0= (m0\n1;m0\n2;m0\n3)2L1(RN;S2)satisfies\ninf\nRNm0\n3\u0015\u00001 +\u000eand [m0]BMO\u0014\"0; (2.43)\nthen there exists a unique solution m= (m1;m2;m3)2X(RN\u0002R+;S2)of(LLG\u000b)such that\ninf\nx2RN\nt>0m3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+\u000e\u00001)2and [m]X\u00144K(\u001a+ 8\u000e\u00002\"0):(2.44)\nMoreover, we have the following properties.\ni)m2C1(RN\u0002R+;S2).\nii)jm(\u0001;t)\u0000m0j\u0000! 0inS0(RN)ast\u0000!0+.\niii) Assume that mandnare respectively smooth solutions to (IDNLS) satisfying (2.44)with\ninitial conditions m0andn0satisfying (2.43). Then\nkm\u0000nkX\u0014120K\u000e\u00002km0\u0000n0kL1: (2.45)\nRemark 2.10. The restriction (2.42) on the parameters is similar to (2.27), but we need to\ninclude\u000e. To better understand the role of \u000e, we can proceed as before. Indeed, setting for\na;\u000e> 0,\nS\u000e(a) =f(\u001a;\"0)2R+\u0002R+:a\u000e\u00004(\u001a+ 8\u000e\u00002\"0)2\u0014\u001ag;\nwe see that its shape is similar to the one in Figure 1. It is simple to verify that for any\n(\u001a;\"0)2S\u000e(a), we have the bounds\n\u001a\u0014\u000e4\naand\"0\u0014\u000e6\n32a; (2.46)\nand the maximum value \"\u0003\n0=\u000e6\n32ais attained at \u001a\u0003=\u000e4\n4a. Also, the sets are well ordered, i.e. if\n~a\u0015a>0, thenS\u000e(~a)\u0012S\u000e(a).\n15We emphasize that the first condition in (2.43) is rather technical. Indeed, we need the\nessential range of m0to be far from the South Pole in order to use the stereographic projection.\nIn the case\u000b= 1, Wang [35] proved the global well-posedness using only the second restriction in\n(2.43). It is an open problem to determinate if this condition is necessary in the case \u000b2(0;1).\nThe choice of the South Pole is of course arbitrary. By using the invariance of (LLG \u000b) under\nrotations, we have the existence of solutions provided that the essential range of the initial\ncondition m0is far from an arbitrary point Q2S2. Precisely,\nCorollary 2.11. Let\u000b2(0;1],Q2S2,\u000e2(0;2], and\"0;\u001a> 0such that (2.42)holds. Given\nm0= (m0\n1;m0\n2;m0\n3)2L1(RN;S2)satisfying\ninf\nRNjm0\u0000Qj2\u00152\u000eand [m0]BMO\u0014\"0;\nthere exists a unique smooth solution m2X(RN\u0002R+;S2)of(LLG\u000b)with initial condition m0\nsuch that\ninf\nx2RN\nt>0jm(x;t)\u0000Qj2\u00154\n1 +K2(\u001a+\u000e\u00001)2and [m]X\u00144K(\u001a+ 8\u000e\u00002\"0):(2.47)\nFor the sake of clarity, before proving Theorem 2.9, we provide a precise meaning of what we\nrefer to as a weak and smooth global solution of the (LLG \u000b) equation. The definition below is\nmotivated by the following vector identities for a smooth function mwithjmj= 1:\nm\u0002\u0001m= div( m\u0002rm);\n\u0000m\u0002(m\u0002\u0001m) = \u0001m+jrmj2m:\nDefinition 2.12. LetT2(0;1]andm02L1(RN;S2). We say that\nm2L1\nloc((0;T);H1\nloc(RN;S2))\nis a weak solution of (LLG\u000b)in(0;T)with initial condition m0if\n\u0000hm;@t'i=\fhm\u0002rm;r'i\u0000\u000bhrm;r'i+\u000bhjrmj2m;'i;\nand\nk(m(t)\u0000m0)'kL1!0;ast!0+;for all'2C1\n0(RN\u0002(0;T)): (2.48)\nIfT=1, and in addition m2C1(RN\u0002R+), we say that mis a smooth global solution of\n(LLG\u000b)inRN\u0002R+with initial condition m0. Hereh\u0001;\u0001istands for\nhf1;f2i=\u00021\n0\u0002\nRNf1\u0001f2dxdt:\nWith this definition, we see the following: Assume that mis a smooth global solution of\n(LLG\u000b) with initial condition m0and consider its stereographic projection P(m). IfP(m)and\nP(m0)are well-defined, then P(m)2C1(RN\u0002R+;C)satisfies (DNLS) pointwise, and\nlim\nt!0+P(m) =P(m0)inS0(RN):\nTherefore,ifinaddition P(m)2X(RN\u0002R;C),thenP(m)isasmoothglobalsolutionof (DNLS)\nwith initial condition P(m0). Reciprocally, suppose that u2X(RN\u0002R+;C)\\C1(RN\u0002R+)\nis a solution of (IDNLS) with initial condition u02L1(RN)such that (2.9) holds. If P\u00001(u)\nandP\u00001(u0)are in appropriate spaces, then P\u00001(u)is a global smooth solution of (LLG \u000b) with\ninitial conditionP\u00001(u0). The above (formal) argument allows us to obtain Theorem 2.9 from\nTheorem 2.1 once we have established good estimates for the mappings PandP\u00001. In this\ncontext, we have the following\n16Lemma 2.13. Letu;v2C1(RN;C),m= (m1;m2;m3);n= (n1;n2;n3)2C1(RN;S2).\na) Assume that inf\nRNm3\u0015\u00001+\u000eandinf\nRNn3\u0015\u00001+\u000efor some constant \u000e2(0;2]. Ifu=P(m)\nandv=P(n), then\nju(x)\u0000v(x)j\u00144\n\u000e2jm(x)\u0000n(x)j; (2.49)\n[u]BMO\u00148\n\u000e2[m]BMO; (2.50)\njru(x)j\u00144\n\u000e2jrm(x)j; (2.51)\nfor allx2RN.\nb) Assume thatkukL1\u0014M,kvkL1\u0014M, for some constant M\u00150. Ifm=P\u00001(u)and\nn=P\u00001(v), then\ninf\nRNm3\u0015\u00001 +2\n1 +M2; (2.52)\njm(x)\u0000n(x)j\u00143ju(x)\u0000v(x)j; (2.53)\njrm(x)j\u00144jru(x)j; (2.54)\njrm(x)\u0000rn(x)j\u00144jru(x)\u0000rv(x)j+ 12ju(x)\u0000v(x)j(jru(x)j+jrv(x)j):(2.55)\nProof.In the proof we will use the notation \u0014m:=m1+im2. To establish (2.49), we write\nu(x)\u0000v(x) =\u0014m(x)\u0000\u0014n(x)\n1 +m3(x)+\u0014n(x)(n3(x)\u0000m3(x))\n(1 +m3(x))(1 +n3(x)):\nHence, sincej\u0014nj\u00141,m3(x) + 1\u0015\u000eandn3(x) + 1\u0015\u000e,8x2RN,\nju(x)\u0000v(x)j\u0014j\u0014m(x)\u0000\u0014n(x)j\n\u000e+jn3(x)\u0000m3(x)j\n\u000e2:\nUsing that\nj\u0014m\u0000\u0014nj\u0014jm\u0000nj (2.56)\nand that\nmax\u001a1\na;1\na2\u001b\n\u00142\na2;for alla2(0;2];\nwe obtain (2.49). The same argument also shows that\nju(y)\u0000u(z)j\u00144\n\u000e2jm(y)\u0000m(z)j;for ally;z2RN: (2.57)\nTo verify (2.50), we recall the following inequalities in BMO (see [10]):\n[f]BMO\u0014sup\nx2RN \nBr(x) \nBr(x)jf(y)\u0000f(z)jdydz\u00142[f]BMO: (2.58)\nEstimate (2.50) is an immediate consequence of this inequality and (2.57). To prove (2.51) it is\nenough to remark that\njruj\u00142\n\u000e2(jrm1j+jrm2j+jrm3j)\u00144\n\u000e2jrmj:\n17We turn into (b). Using the explicit formula for P\u00001in (2.41), we can write\nm3=\u00001 +2\n1 +juj2:\nSincekukL1\u0014M, we obtain (2.52).\nTo show (2.53), we compute\n\u0014m\u0000\u0014n=2u\n1 +juj2\u00002v\n1 +jvj2=2(u\u0000v) + 2uv(\u0016v\u0000\u0016u)\n(1 +juj2)(1 +jvj2); (2.59)\nm3\u0000n3=1\u0000juj2\n1 +juj2\u00001\u0000jvj2\n1 +jvj2=2(jvj2\u0000juj2)\n(1 +juj2)(1 +jvj2): (2.60)\nUsing the inequalities\na\n1 +a2\u00141\n2;1 +ab\n(1 +a2)(1 +b2)\u00141;anda+b\n(1 +a2)(1 +b2)\u00141;for alla;b\u00150;(2.61)\nfrom (2.59) and (2.60) we deduce that\nj\u0014m\u0000\u0014nj\u00142ju\u0000vjandjm3\u0000n3j\u00142ju\u0000vj: (2.62)\nHence\njm\u0000nj=p\nj\u0014m\u0000\u0014nj2+jm3\u0000n3j2\u0014p\n8ju\u0000vj\u00143ju\u0000vj:\nTo estimate the gradient, we compute\nr\u0014m=2ru\n1 +juj2\u00004uRe(\u0016uru)\n(1 +juj2)2; (2.63)\nfrom which it follows that\njr\u0014mj\u0014jruj\u00122\n1 +juj2+4juj2\n(1 +juj2)2\u0013\n\u00143jruj;\nsince4a\n(1+a)2\u00141;for alla\u00150. Form3, we have\nrm3=\u00002 Re(\u0016uru)\n1 +juj2\u00002 Re(\u0016uru)(1\u0000juj2)\n(1 +juj2)2=\u00004 Re(\u0016uru)\n(1 +juj2)2;\nand thereforejrm3j\u00142jruj, since\na\n(1 +a2)2\u00141\n2;for alla\u00150: (2.64)\nHence\njrmj=p\njrm1j2+jrm2j2+jrm3j2\u0014p\n13jruj\u00144jruj;\nwhich gives (2.54).\nIn order to prove (2.55), we start differentiating (2.59)\nr\u0014m\u0000r\u0014n=2r(u\u0000v) +r(uv)(\u0016v\u0000\u0016u) +uvr(\u0016v\u0000\u0016u)\n(1 +juj2)(1 +jvj2)\n\u00004((u\u0000v) +uv(\u0016v\u0000\u0016u))(Re(\u0016uru)(1 +jvj2) + Re(\u0016vrv)(1 +juj2))\n(1 +juj2)2(1 +jvj2)2;\n18Hence, setting R= maxfjru(x)j;jrv(x)jg,\njr\u0014m\u0000r\u0014nj\u00142jru\u0000rvj\u00121 +jujjvj\n(1 +juj2)(1 +jvj2)\u0013\n+ 2Rju\u0000vj\u0012juj+jvj\n(1 +juj2)(1 +jvj2)\u0013\n+ 4Rju\u0000vj\u0012juj(1 +jujjvj)\n(1 +juj2)2(1 +jvj2)+jvj(1 +jujjvj)\n(1 +juj2)(1 +jvj2)2\u0013\n:\nUsing again (2.61), we get\njuj(1 +jujjvj)\n(1 +juj2)2(1 +jvj2)\u0014juj\n(1 +juj2)\u00141\n2:\nBy symmetry, the same estimate holds interchanging ubyv. Therefore, invoking again (2.61),\nwe obtain\njr\u0014m\u0000r\u0014nj\u00142jru\u0000rvj+ 6Rju\u0000vj: (2.65)\nSimilarly, writing juj2\u0000jvj2= (u\u0000v)\u0016u+ (\u0016u\u0000\u0016v)v, from (2.60) we have\njrm3\u0000rn3j\u00142jru\u0000rvj+ 6Rju\u0000vj: (2.66)\nTherefore, sincep\na2+b2\u0014a+b;8a;b\u00150;\ninequalities (2.65) and (2.66) yield (2.55).\nNow we have all the elements to establish Theorem 2.9.\nProof of Theorem 2.9. We continue to use the constants CandKdefined in Theorem 2.1. We\nrecall that they are given by C=C1C2\n2andK=C2, whereC1\u00151andC2\u00151are the constants\nin Lemmas 2.4 and 2.5, respectively. In addition, w.l.o.g. we assume that\nK=C2\u00154; (2.67)\nin order to simplify our computations.\nFirst we notice that by Remark 2.10, any \u001aand\"0fulfilling the condition (2.42), also satisfy\n8C(\u001a+ 8\u000e\u00002\"0)2\u0014\u001a; (2.68)\nsince\u000e4=K4\u00141(notice that K\u00154and\u000e2(0;2]).\nLetm0as in the statement of the theorem and set u0=P(m0). Using (2.50) in Lemma 2.13,\nwe have\nku0kL1\u0014\r\r\r1\n1 +m0\n3\r\r\r\nL1\u00141\n\u000eand [u0]BMO\u00148\"0\n\u000e2:\nTherefore, bearing in mind (2.68), we can apply Theorem 2.1 with\nL:=1\n\u000eand\":= 8\u000e\u00002\"0;\nto obtain a smooth solution u2X(RN\u0002R+;C)to (IDNLS) with initial condition u0. In\nparticularusatisfies\nsup\nt>0kukL1\u0014K(\u001a+\u000e\u00001)and [u]X\u0014K(\u001a+ 8\u000e\u00002\"0): (2.69)\n19Defining m=P\u00001(u), we infer that mis a smooth solution to (LLG \u000b) and, using the fact that\nk(u(\u0001;t)\u0000u0)'kL1!0(see (2.9)) and (2.53),\njm(\u0001;t)\u0000m0j\u0000! 0inS0(RN);ast!0+:\nNotice also that applying Lemma 2.13 we obtain\ninf\nx2RN\nt>0m3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+\u000e\u00001)2and [m]X\u00144[u]X\u00144K(\u001a+ 8\u000e\u00002\"0);\nwhich yields (2.44).\nLet us now prove the uniqueness. Let nbe a another smooth solution of (LLG \u000b) with initial\nconditionu0satisfying\ninf\nx2RN\nt>0n3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+\u000e\u00001)2and [n]X\u00144K(\u001a+ 8\u000e\u00002\"0); (2.70)\nand letv=P(n)be its stereographic projection. Then by (2.51),\n[v]X\u0014\u0010\n1 +K2(\u001a+\u000e\u00001)2\u00112\n[n]X: (2.71)\nWe continue to control the upper bounds for [v]Xand[u]Xin terms of \u000eand the constants\nC1\u00151andC2\u00154. Notice that since \u001aand\"0satisfy (2.42), from (2.46) with a= 8K4C, it\nfollows that\n\u001a\u0014\u000e4\n8K4Cand\"0\u0014\u000e6\n28K4C;\nor equivalently (recall that K=C2andC=C1C2\n2)\n\u001a\u0014\u000e4\n8C1C6\n2and8\"0\n\u000e2\u0014\u000e4\n32C1C6\n2: (2.72)\nHence\nK(\u001a+ 8\u000e\u00002\"0)\u00145\u000e4\n32C1C5\n2: (2.73)\nAlso, using (2.72), we have\n1 +K2(\u001a+\u000e\u00001)2=1 +C2\n2\n\u000e2(\u001a\u000e+ 1)2=C2\n2\n\u000e2\u0012\u000e2\nC2\n2+ (\u001a\u000e+ 1)2\u0013\n\u0014C2\n2\n\u000e2 \n\u000e2\nC2\n2+\u0012\u000e5\n8C1C6\n2+ 1\u00132!\n\u00142C2\n2\n\u000e2;(2.74)\nsinceC1\u00151,C2\u00154and\u000e\u00142.\nFrom the bounds in (2.73) and (2.74), combined with (2.69), (2.70) and (2.71), we obtain\n[u]X\u0014K(\u001a+ 8\u000e\u00002\"0)\u00145\u000e4\n32C1C5\n2\u00145\n211C1\nand\n[v]X\u0014(1+K2(\u001a+\u000e\u00001)2)2[n]X\u0014(1+K2(\u001a+\u000e\u00001)2)24K(\u001a+8\u000e\u00002\"0)\u0014\u0012\n2C2\n2\n\u000e2\u0013220\u000e4\n32C1C5\n2\u00145\n8C1;\n20since\u000e\u00142andC2\u00154. Finally, since uandvare solutions to (IDNLS) with initial condition\nu0, (2.28) and the above inequalities for [u]Xand[v]Xyield\nku\u0000vkX\u0014C1(2[u]2\nX+ [u]X+ [v]X)ku\u0000vkX\n\u0014C1 \n2\u00125\n211C1\u00132\n+5\n211C1+5\n8C1!\nku\u0000vkX;\nwhich implies that u=v, bearing in mind that the constant on the r.h.s. of the above inequality\nis strictly less that one. This completes the proof of the uniqueness.\nIt remains to establish (2.45). Let mandntwo smooth solutions of (LLG \u000b) satisfying (2.44).\nAs a consequence of the uniqueness, we see that mandnare the inverse stereographic projection\nof some functions uandvthat are solutions of (IDNLS) with initial condition u0=P(m0)and\nv0=P(n0), respectively. In particular, uandvsatisfy the estimates in (2.69). Using also (2.53)\nand (2.55), we deduce that\nkm\u0000nkX\u00143 sup\nt>0ku\u0000vkL1+ 4[u\u0000v]X+ 12 sup\nt>0ku\u0000vkL1([u]X+ [v]X])\n\u00144ku\u0000vkX+ 24C2(\u001a+ 8\u000e\u00002\"0)ku\u0000vkX;\n\u00145ku\u0000vkX;\nwhere we have used (2.73) in obtaining the last inequality. Finally, using also (2.43) and (2.49),\nand applying (2.10) in Theorem 2.1,\nkm\u0000nkX\u001430Kku0\u0000v0kL1\n\u0014120K\u000e\u00002km0\u0000n0kL1;\nwhich yields (2.45).\nProof of Corollary 2.11. LetR2SO(3)such thatRQ= (0;0;\u00001), i.e.Ris the rotation that\nmapsQto the South Pole. Let us set m0\nR=Rm0. Then\njm0\u0000Qj2=jR(m0\u0000Q)j2=jm0\nR\u0000(0;0;\u00001)j2= 2(1 +m0\n3;R):\nHence,\ninf\nx2RNm0\n3;R\u0015\u00001 +\u000e\nand\n[m0\nR]BMO = [m0]BMO\u0014\"0:\nTherefore, Theorem2.9providestheexistenceofauniquesmoothsolution mR2X(RN\u0002R+;S2)\nof (LLG\u000b) satisfying (2.44). Using the invariance of (LLG \u000b) and setting m=R\u00001mRwe obtain\nthe existence of the desired solution. To establish the uniqueness, it suffices to observe that if n\nis another smooth solution of (LLG \u000b) satisfying (2.47), then nR:=Rnis a solution of (LLG \u000b)\nwith initial condition m0\nRand it fulfills (2.44). Therefore, from the uniqueness of solution in\nTheorem 2.9, it follows that mR=nRand then m=n.\nProof of Theorem 1.1. In Theorem 2.9 and Corollary 2.11, the constants are given by C=C1C2\n2\nandK=C2. As discussed in Remark 2.10, the value\n\u001a\u0003=\u000e4\n32C1C2\n2\n21maximizes the range for \"0in (2.27) and this inequality is satisfied for any \"0>0such that\n\"0\u0014\u000e6\n256C1C2\n2:\nTaking\nM1=1\n256C1C2\n2; M 2=C2andM3=1\n32C1C2\n2;\nso that\u001a\u0003=M3\u000e4, the conclusion follows from Theorem 2.9 and Corollary 2.11.\nRemark2.14. Wefinallyremarkthatispossibletostatelocal(intime)versionsofTheorems2.1\nand 2.9 as it was done in [23, 22, 35]. In our context, the local well-posedness would concern\nsolutions with initial condition m02VMO (RN), i.e. such that\nlim\nr!0+sup\nx2RN \nBr(x)jm0(y)\u0000m0\nx;rjdy= 0: (2.75)\nMoreover, some uniqueness results have been established for solutions with this kind of initial\ndata by Miura [32] for the Navier–Stokes equation, and adapted by Lin [29] to (HFHM). It is\nalso possible to do this for (LLG \u000b), for\u000b > 0. We do not pursuit here these types of results\nbecause they do not apply to the self-similar solutions mc;\u000b. This is due to the facts that the\nfunction m0\nA\u0006does not belong to VMO (R)and that\nlim\nT!0+sup\n0<t<Tp\ntk@xmc;\u000bkL16= 0:\n3 Applications\n3.1 Existence of self-similar solutions in RN\nThe LLG equation is invariant under the scaling (x;t)!(\u0015x;\u00152t), for\u0015 > 0, that is if m\nsatisfies (LLG \u000b), then so does the function\nm\u0015(x;t) =m(\u0015x;\u00152t); \u0015> 0:\nTherefore is natural to study the existence of self-similar solutions (of expander type), i.e. a\nsolution msatisfying\nm(x;t) =m(\u0015x;\u00152t);8\u0015>0; (3.1)\nor, equivalently,\nm(x;t) =f\u0012xp\nt\u0013\n;\nfor some f:RN\u0000!S2profile of m. In particular we have the relation f(y) =m(y;1), for all\ny2RN. From (3.1) we see that, at least formally, a necessary condition for the existence of a\nself-similar solution is that initial condition m0be homogeneous of degree 0, i.e.\nm0(\u0015x) =m0(x);8\u0015>0:\nSince the norm in X(RN\u0002R+;R3)is invariant under this scaling, i.e.\nkm\u0015kX=kmkX;8\u0015>0;\nwhere m\u0015is defined by (3.1), Theorem 2.9 yields the following result concerning the existence\nof self-similar solutions.\n22Corollary 3.1. With the same notations and hypotheses as in Theorem 2.9, assume also that\nm0is homogeneous of degree zero. Then the solution mof(LLG\u000b)provided by Theorem 2.9 is\nself-similar. In particular there exists a smooth profile f:RN!S2such that\nm(x;t) =f\u0012xp\nt\u0013\n;\nfor allx2RNandt>0, andfsatisfies the equation\n\u00001\n2y\u0001rf(y) =\ff(y)\u0002\u0001f(y)\u0000\u000bf(y)\u0002(f(y)\u0002\u0001f(y));\nfor ally2RN. Herey\u0001rf(y) = (y\u0001rf1(y);:::;y\u0001rfN(y)).\nRemark 3.2. Analogously, Theorem 2.1 leads to the existence of self-similar solutions for\n(DNLS), provided that u0is a homogeneous function of degree zero.\nFor instance, in dimensions N\u00152, Corollary 3.1 applies to the initial condition\nm0(x) =H\u0012x\njxj\u0013\n;\nwithHa Lipschitz map from SN\u00001toS2\\f(x1;x2;x3) :x3\u0015\u00001=2g, provided that the Lipschitz\nconstant is small enough. Indeed, using (2.58), we have\n[m0]BMO\u00144kHkLip;\nso that taking\n\u000e= 1=2; \u001a =\u000e4\n32K4C; \" 0=\u000e6\n256K4CandkHkLip\u0014\"0;\nthe condition (2.42) is satisfied and we can invoke Corollary 3.1.\nOther authors have considered self-similar solutions for the harmonic map flow (i.e. (LLG \u000b)\nwith\u000b= 1) in different settings. Actually, equation (HFHM) can be generalized for maps\nm:M\u0002R+!N, withMandNRiemannian manifolds. Biernat and Bizoń [8] established\nresults whenM=N=Sdand3\u0014d\u00146:Also, Germain and Rupflin [15] have investigated the\ncaseM=RdandN=Sd, ind\u00153. In both works the analysis is done only for equivariant\nsolutions and does not cover the case M=RNandN=S2.\n3.2 The Cauchy problem for the one-dimensional LLG equation with a jump\ninitial data\nThis section is devoted to prove Theorems 1.2 and 1.3 in the introduction. These two results con-\ncern the question of well-posedness/ill-posedness of the Cauchy problem for the one-dimensional\nLLG equation associated with a step function initial condition of the form\nm0\nA\u0006:=A+\u001fR++A\u0000\u001fR\u0000; (3.2)\nwhere A+andA\u0000are two given unitary vectors in S2.\n233.2.1 Existence, uniqueness and stability. Proof of Theorem 1.2\nAs mentioned in the introduction, in [17] we proved the existence of the uniparametric smooth\nfamily of self-similar solutions fmc;\u000bgc>0of (LLG\u000b) for all\u000b2[0;1]with initial condition of the\ntype (3.2) given by\nm0\nc;\u000b:=A+\nc;\u000b\u001fR++A\u0000\nc;\u000b\u001fR\u0000; (3.3)\nwhere A\u0006\nc;\u000b2S2are given by Theorem A.5. For the convenience of the reader, we collect some\nof the results proved in [17] in the Appendix. The results in this section rely on a further\nunderstanding of the properties of the self-similar solutions mc;\u000b.\nIn Proposition 3.4 we show that\nmc;\u000b= (m1;c;\u000b;m2;c;\u000b;m3;c;\u000b)2X(R\u0002R+;S2);\nthatm3;c;\u000bis far from the South Pole and that [mc;\u000b]Xis small, if cis small enough. This\nwill yield that mc;\u000bcorresponds (up to a rotation) to the solution given by Corollary 3.1. More\nprecisely, using the invariance under rotations of (LLG \u000b), we can prove that, if the angle between\nA+andA\u0000is small enough, then the solution given by Corollary 3.1 with initial condition m0\nA\u0006\ncoincides (modulo a rotation) with mc;\u000b, for somec. We have the following:\nTheorem 3.3. Let\u000b2(0;1]. There exist L1;L2>0,\u000e\u00032(\u00001;0)and#\u0003>0such that the\nfollowing holds. Let A+,A\u00002S2and let#be the angle between them. If\n0<#\u0014#\u0003; (3.4)\nthen there exists a solution mof(LLG\u000b)with initial condition m0\nA\u0006. Moreover, there exists\n0< c <p\u000b\n2p\u0019, such that mcoincides up to a rotation with the self-similar solution mc;\u000b, i.e.\nthere existsR2SO(3), depending only on A+,A\u0000,\u000bandc, such that\nm=Rmc;\u000b; (3.5)\nandmis the unique solution satisfying\ninf\nx2R\nt>0m3(x;t)\u0015\u000e\u0003and [m]X\u0014L1+L2c: (3.6)\nIn order to prove Theorem 3.3, we need some preliminary estimates for mc;\u000bin terms of c\nand\u000b. To obtain them, we use some properties of the profile profile fc;\u000b= (f1;c;\u000b;f2;c;\u000b;f3;c;\u000b)\nconstructed in [17] using the Serret–Frenet equations with initial conditions\nf1;c;\u000b(0) = 1; f 2;c;\u000b(0) =f3;c;\u000b(0) = 0:\nAlso,\njf0\nj;c;\u000b(s)j\u0014ce\u0000\u000bs2=4;for alls2R;\nforj2f1;2;3gand\nmc;\u000b(x;t) =fc;\u000b\u0012xp\nt\u0013\n;for all (x;t)2R\u0002R+: (3.7)\nHence, for any x2R,\njf3;c\u000b(x)j=jf3;c\u000b(x)\u0000f3;c\u000b(0)j\u0014\u0002jxj\n0ce\u0000\u000b\u001b2=4d\u001b\u0014cp\u0019p\u000b:\n24Since the same estimate holds for f2;c;\u000b, we conclude that\njm2;c;\u000b(x;t)j\u0014cp\u0019p\u000b;andjm3;c;\u000b(x;t)j\u0014cp\u0019p\u000bfor all (x;t)2R\u0002R+:(3.8)\nMoreover, since\nA\u0006\nc;\u000b= lim\nx!\u00061fc;\u000b(x);\nwe also get\njA\u0006\nj;c;\u000bj\u0014cp\u0019p\u000b;forj2f2;3g: (3.9)\nWe now provide some further properties of the self-similar solutions.\nProposition 3.4. For\u000b2(0;1]andc>0, we have\nkm0\n2;c;\u000bkL1\u0014cp\u0019p\u000b;km0\n3;c;\u000bkL1\u0014cp\u0019p\u000b;sup\nt>0km3;c;\u000bkL1\u0014cp\u0019p\u000b;(3.10)\n[m0\nc;\u000b]BMO\u00142cp\n2\u0019p\u000b; (3.11)\np\ntk@xmc;\u000bk1=c;for allt>0; (3.12)\nsup\nx2R\nr>01\nr\u0002\nQr(x)j@ymc;\u000b(y;t)j2dtdy\u00142p\n2\u0019c2\np\u000b: (3.13)\nIn particular, mc;\u000b2X(R\u0002R+;S2)and\n[mc;\u000b]X\u00144c\n\u000b1\n4: (3.14)\nProof of Proposition 3.4. The estimates in (3.10) follow from (3.8) and (3.9). To prove (3.11),\nwe use (2.58), (3.3), (3.10) and the fact that\nA\u0000\nc;\u000b= (A+\n1;c;\u000b;\u0000A+\n2;c;\u000b;\u0000A+\n3;c;\u000b); (3.15)\n(see Theorem A.5) to get\n[m0\nc;\u000b]BMO\u0014sup\nx2RN \nBr(x) \nBr(x)jm0\nc;\u000b(y)\u0000m0\nc;\u000b(z)jdydz\n\u00142q\n(A+\n2;c;\u000b)2+ (A+\n3;c;\u000b)2sup\nx2RN \nBr(x) \nBr(x)dydz\n\u00142cp\n2\u0019p\u000b:\nFrom (A.12) we obtain the equality in (3.12) and also\nIr;x:=1\nr\u0002\nQr(x)j@ymc;\u000b(y;t)j2dtdy =c2\nr\u0002x+r\nx\u0000r\u0002r2\n0e\u0000\u000by2\n2t\ntdtdy: (3.16)\nPerforming the change of variables z= (\u000by2)=(2t), we see that\n\u0002r2\n0e\u0000\u000by2\n2t\ntdt=E1\u0012\u000by2\n2r2\u0013\n; (3.17)\n25whereE1is the exponential integral function\nE1(y) =\u00021\nye\u0000z\nzdz:\nThis function satisfies that limy!0+E1(y) =1andlimy!1E1(y) = 0(see e.g. [1, Chapter 5]).\nMoreover, taking \u000f>0and integrating by parts,\n\u00021\n\u000fE1(y2)dy=yE1(y2)\f\f1\n\u000f+ 2\u00021\n\u000fe\u0000y2dy; (3.18)\nso L’Hôpital’s rule shows that the first term in the r.h.s. of (3.18) vanishes as \u000f!0+. Therefore,\nthe Lebesgue’s monotone convergence theorem allows to conclude that E1(y2)2L1(R+)and\n\u00021\n0E1(y2) =p\u0019: (3.19)\nBy using (3.16), (3.17), (3.19), and making the change of variables z=p\u000by=(rp\n2), we obtain\nIr;x=c2\nr\u0002x+r\nx\u0000rE1\u0012\u000by2\n2r2\u0013\ndy=p\n2c2\np\u000b\u0002p\u000bp\n2(x\nr+1)\np\u000bp\n2(x\nr\u00001)E1(z2)dz\u0014p\n2c2\np\u000b\u00012p\u0019; (3.20)\nwhich leads to (3.13). Finally, the bound in (3.14) easily follows from those in (3.12) and (3.13)\nand the elementary inequality\n\u0010\n1 +\u00102p\n2\u0019p\u000b\u00111=2\u0011\n\u00141\n\u000b1\n4\u0000\n1 + (2p\n2\u0019)1=2\u0001\n\u00144\n\u000b1\n4; \u000b2(0;1]:\nProof of Theorem 3.3. First, we consider the case when A+=A+\nc;\u000bandA\u0000=A\u0000\nc;\u000b(i.e. when\nm0\nA\u0006=m0\nc;\u000b) for some c >0. We will continue to show that the solution provided by Theo-\nrem 2.9 is exactly mc;\u000b, forcsmall. Indeed, bearing in mind the estimates in Proposition 3.4,\nwe consider\nc\u0014p\u000b\n2p\u0019;\nso that\ninf\nx2Rm0\n3;c;\u000b(x)\u0015\u00001\n2: (3.21)\nIn view of (3.11), (3.21) and Remark 2.10, we set\n\"0:= 4cp\u0019p\u000b; \u000e :=1\n2; \u001a :=\u000e4\n8K4C=1\n27K4C; (3.22)\nwhereC;K\u00151are the constants given by Theorem 2.9. In this manner, from (3.11), (3.21) and\n(3.22), we have\ninf\nRm0\n3\u0015\u00001 +\u000eand [m0]BMO\u0014\"0;\nand the condition (2.42) is fulfilled if\n\"0\u0014\u000e6\n256K4C;\nor equivalently, if c\u0014~c, with\n~c:=p\u000b\n216K4Cp\u0019:\n26Observe that in particular ~c<p\u000b\n2p\u0019.\nFor fixed 0<c< ~c, we can apply Theorem 2.9 to deduce the existence and uniqueness of a\nsolution mof (LLG\u000b) satisfying\ninf\nx2R\nt>0m3(x;t)\u0015\u00001 +2\n1 +K2(\u001a+ 2)2and [m]X\u00144K\u001a+29Kcp\u0019p\u000b:(3.23)\nNow by Proposition 3.4, for fixed 0<c\u0014~c, we have the following estimates for mc;\u000b\n[mc;\u000b]X\u00144c=\u000b1\n4and inf\nx2R\nt>0m3;c;\u000b(x;t)\u0015\u00001\n2;\nso in particular mc;\u000bsatisfies (3.23). Thus the uniqueness of solution implies that m=mc;\u000b,\nprovided that c\u0014~c. Defining the constants L1,L2and\u000e\u0003by\nL1= 4K\u001a; L 2=29Kp\u0019p\u000band\u000e\u0003=\u00001 +2\n1 +K2(\u001a+ 2)2;(3.24)\nthe theorem is proved in the case A\u0006=A\u0006\nc;\u000b.\nFor the general case, we would like to understand which angles can be reached by varying the\nparametercin the range (0;~c]. To this end, for fixed 0< c\u0014~c, let#c;\u000bbe the angle between\nA+\nc;\u000bandA\u0000\nc;\u000b. From Lemma A.6,\n#c;\u000b\u0015arccos\u0012\n1\u0000c2\u0019+ 32c3p\u0019\n\u000b2\u0013\n;for allc2\u0010\n0;\u000b2p\u0019\n32i\n:\nNow, it is easy to see that the function F(c) = arccos\u0010\n1\u0000c2\u0019+ 32c3p\u0019\n\u000b2\u0011\nis strictly increasing\non the interval [0;\u000b2p\u0019\n48]so that\nF(c)>F(0) = 0;for allc2\u0010\n0;\u000b2p\u0019\n48i\n: (3.25)\nLetc\u0003= min(~c;\u000b2p\u0019\n48)and consider the map T\u000b:c\u0000!#c;\u000bon[0;c\u0003]. By Lemma A.6, T\u000bis\ncontinuous on [0;c\u0003],T\u000b(0) = lim c!0+T\u000b(c) = 0and, bearing in mind (3.25), T(c\u0003) =#c\u0003;\u000b>0.\nThus, from the intermediate value theorem we infer that for any #2(0;#c\u0003;\u000b), there exists\nc2(0;c\u0003)such that\n#=T\u000b(c) =#c;\u000b:\nWe can now complete the proof for any A+,A\u00002S2. Let#be the angle between A+and\nA\u0000. From the previous lines, we know that there exists #\u0003:=#c\u0003;\u000bsuch that if #2(0;#\u0003),\nthere exists c2(0;c\u0003)such that#=#c;\u000b. For this value of c, consider the initial value problem\nassociated with m0\nc;\u000band the constants defined in (3.24). We have already seen the existence\nof a unique solution mc;\u000bof the LLG equation associated with this initial condition satisfying\n(3.6). LetR2SO(3)be the rotation on R3such that A+=RA+\nc;\u000bandA\u0000=RA\u0000\nc;\u000b. Then\nm:=Rmc;\u000bsolves (LLG \u000b) with initial condition m0\nA\u0006. Finally, recalling the above definition\nofL1,L2and\u000e\u0003, using the invariance of the norms under rotations and the fact that mc;\u000bis the\nunique solution satisfying (3.23), it follows that mis the unique solution satisfying the conditions\nin the statement of the theorem.\nWe are now in position to give the proof of Theorem 1.2, the second of our main results\nin this paper. In fact, we will see that Theorem 1.2 easily follows from Theorem 3.3 and the\nwell-posedness for the LLG equation stated in Theorem 2.9.\n27Proof of Theorem 1.2. Let#\u0003,\u000e\u0003,L1andL2betheconstantsdefinedintheproofofTheorem3.3.\nGiven A+andA\u0000such that 0<#<#\u0003, Theorem 3.3 asserts the existence of\n0<c<p\u000b\n2p\u0019(3.26)\nandR2SO(3)such thatRmc;\u000bis the unique solution of (LLG \u000b) with initial condition m0\nA\u0006\nsatisfying (3.6), and in particular m0\nA\u0006=Rm0\nc;\u000b. By hypothesis m0satisfies\nkm0\u0000m0\nA\u0006kL1\u0014cp\u0019\n2p\u000b: (3.27)\nHence, defining m0\nR=R\u00001m0, recalling that [f]BMO\u00142kfkL1and using the invariance of the\nnorms under rotations, we deduce from (3.27) that\nkm0\nRkL1\u0014km0\nc;\u000bkL1+cp\u0019\n2p\u000band [m0\nR]BMO\u0014[m0\nc;\u000b]BMO +cp\u0019p\u000b:\nThen, by Proposition 3.4,\nkm0\n3;RkL1\u00142cp\u0019p\u000band [m0\nR]BMO\u00144cp\u0019p\u000b: (3.28)\nFrom (3.26) and (3.28), it follows that\nm0\n3;R(x)\u0015\u00001=2;for allx2R:\nTherefore, as in the proof of Theorem 3.3, we can apply Theorem 2.9 with the values of \"0,\u000e\nand\u001agiven in (3.22) to deduce the existence of a unique (smooth) solution mRof (LLG\u000b) with\ninitial condition m0\nRsatisfying\ninf\nx2R\nt>0m3;R(x;t)\u0015\u00001 +2\n1 +K2(\u001a+ 2)2=\u000e\u0003and [mR]X\u00144K\u001a+29Kcp\u0019p\u000b=L1+L2c:\nSince we have taken the values for \"0,\u000eand\u001aas in the proof Theorem 3.3, Theorem 2.9 also\nimplies that\nkmR\u0000mc;\u000bkX\u0014480Kkm0\nR\u0000m0\nc;\u000bkL1:\nThe conclusion of the theorem follows defining m=RmRandL3= 480K, and using once\nagain the invariance of the norm under rotations.\n3.2.2 Multiplicity of solutions. Proof of Theorem 1.3\nAs proved in [17], when \u000b= 1, the self-similar solutions are explicitly given by\nmc;1(x;t) = (cos(cErf(x=p\nt));sin(cErf(x=p\nt));0);for all (x;t)2R\u0002R+;(3.29)\nfor everyc>0, where Erf(\u0001)is the non-normalized error function\nErf(s) =\u0002s\n0e\u0000\u001b2=4d\u001b:\nIn particular,\n~A\u0006\nc;1= (cos(cp\u0019);\u0006sin(cp\u0019);0)\n28#c;1\n\u0019\nc\nFigure 2: The angle #c;\u000bas a function of cfor\u000b= 1.\nand the angle between A+\nc;1andA\u0000\nc;1is given by\n#c;1= arccos(cos(2 cp\u0019)): (3.30)\nFormula (3.30) and Figure 2 show that there are infinite values of cthat allow to reach any\nangle in [0;\u0019]. Therefore, using the invariance of (LLG \u000b) under rotations, in the case when\n\u000b= 1, one can easily prove the existence of multiple solutions associated with a given initial\ndata of the form m0\nA\u0006for any given vectors A\u00062S2(see argument in the proof included below).\nIn the case that \u000bis close enough to 1, we can use a continuity argument to prove that we still\nhave multiple solutions. More precisely, Theorem 1.3 asserts that for any given initial data of\nthe form m0\nA\u0006with angle between A+andA\u0000in the interval (0;\u0019), if\u000bis sufficiently close\nto one, then there exist at leastk-distinct solutions of (LLG \u000b) associated with the same initial\ncondition, for any given k2N.\nThe rest of this section is devoted to the proof of Theorem 1.3.\nProof of Theorem 1.3. Letk2N,A\u00062S2and#2(0;\u0019)be the angle between A+andA\u0000.\nUsing the invariance of (LLG \u000b) under rotations, it suffices to prove the existence of \u000bk2(0;1)\nsuch that for every \u000b2[\u000bk;1]there exist 0< c1<\u0001\u0001\u0001< cksuch that the angle #cj;\u000bbetween\nA+\ncj;\u000bandA\u0000\ncj;\u000b, satisfies\n#cj;\u000b=#;for allj2f1;:::;kg: (3.31)\nIn what follows, and since we want to show the existence of at least k-distinct solutions, we will\nassume without loss of generality that kis large enough.\nFirst observe that, since A\u0000\nc;\u000b= (A+\n1;c;\u000b;\u0000A+\n2;c;\u000b;\u0000A+\n3;c;\u000b), we have the explicit formula\ncos(#c;\u000b) = 2(A+\n1;c;\u000b)2\u00001;\nand using Lemma A.8 in the Appendix, we get\njcos(#c;\u000b)\u0000cos(#c;1)j=j2((A+\n1;c;\u000b)2\u0000(A+\n1;c;1)2)j\u00144jA+\n1;c;\u000b\u0000A+\n1;c;1j\u00144h(c)p\n1\u0000\u000b;(3.32)\nfor all\u000b2[1=2;1], withh:R+\u0000!R+an increasing function satisfying lims!1h(s) =1.\nForj2N, we setaj= (2j+ 1)p\u0019=2andbj= (2j+ 2)p\u0019=2, so that (3.30) and (3.32) yield\ncos(#aj;\u000b)\u0014\u00001 + 4h(aj)p\n1\u0000\u000band cos(#bj;\u000b)\u00151\u00004h(bj)p\n1\u0000\u000b;8\u000b2[1=2;1]:\n(3.33)\n29Definel= cos(#)and\n\u000bk= max\u0012\n1\u0000\u00101\u0000l\n8h(bk)\u00112\n;1\u0000\u00101 +l\n8h(bk)\u00112\u0013\n:\nNotice that, since #2(0;\u0019), we have\u00001< l < 1and thus\u000bk<1. Also, since hdiverges to\n1, we can assume without loss of generality that \u000bk2[1=2;1), and from the definition of \u000bkwe\nhave\n0<p1\u0000\u000bk<min\u00121\u0000l\n8h(bk);1 +l\n8h(bk)\u0013\n:\nTherefore, from (3.33) and h(aj)< h(bj)\u0014h(bk)(sincehis a strictly increasing function), we\nget\ncos(#aj;\u000b)\u0014\u00001 +l\n2and1 +l\n2\u0014cos(#bj;\u000b);8j2f1;:::;kg;8\u000b2[\u000bk;1];\nand thus\ncos(#aj;\u000b)<l< cos(#bj;\u000b);8j2f1;:::;kg;8\u000b2[\u000bk;1]; (3.34)\nsincel2(\u00001;1).\nLet us fix\u000b2[\u000bk;1]andj2f1;:::;kg. By Lemma A.8, c!cos(#c;\u000b)is a continuous\nfunction on c. Therefore (3.34) and the intermediate value theorem yield the existence of cj2\n[aj;bj]such that\ncos(#cj;\u000b) =l= cos(#);\nor equivalently, such that #cj;\u000b=#.\nFinally, for each j2f1;:::;kg, letRj2SO(3)be such that m0\nA\u0006=Rjm0\ncj;\u000b, and define mj\nbymj=Rjmcj;\u000b. Then mjsolves (LLG \u000b) with initial data m0\nA\u0006and the control of @xmjin\n(1.9) follows from the definition of mjin terms of mcj;\u000band the analogous property established\nin Theorem A.5 for the self-similar solution mcj;\u000b(see (A.12)).\nIn the case when \u000b= 1and#2[0;\u0019], formula (3.30) shows that sequence fcjgj\u00151in (1.10)\nsatisfies#cj;1=#for allj2N\u0003. The result follows by considering the sequence of solutions\nfmjgj\u00151described above.\nRemark 3.5. Notice that the proof given above also shows how close \u000bkneeds to be to 1 in\nterms of the (fixed) angle #2(0;\u0019), and in particular \u000bk!1ask!1, and\u000bk!1as#!0\n(i.e. whenl!1).\nRemark 3.6. For\u000b= 1andc>0, the function\nu(x;t) =P(mc;1) = exp\u0000\nicErf(x=p\nt)\u0001\nis a solution of (DNLS) with initial condition\nu0=eicp\u0019\u001fR++e\u0000icp\u0019\u001fR\u0000:\nTherefore there is also a multiplicity phenomenon for the equation (DNLS).\n3.3 A singular solution for a nonlocal Schrödinger equation\nWe have used the stereographic projection to establish a well-posedness result for (LLG \u000b).\nMelcher [31] showed a global well-posedness result, provided that\nkrm0kLN\u0014\";m0\u0000Q2H1(RN)\\W1;N(RN); \u000b> 0; N\u00153;\n30for some Q2S2and\">0small. Later, Lin, Lan and Wang [30] improved this result and proved\nglobal well-posedness under the conditions\nkrm0kM2;2\u0014\";m0\u0000Q2L2(RN); \u000b> 0; N\u00152;\nfor some Q2S2and\">0small.6In the context of Theorem 1.1 and using the characterization\nofBMO\u00001in Theorem A.1, the second condition in (1.3) says that krm0kBMO\u00001is small. In\nview of the embeddings\nLN(RN)\u001aM2;2(RN)\u001aBMO\u00001(RN);\nforN\u00152, we deduce that Theorem 1.1 includes initial conditions with less regularity, as long\nas their essential range is not S2. The argument in [30, 31] is based on the method of moving\nframes that produces a covariant complex Ginzburg–Landau equation. One of the aims of this\nsubsection is to compare their approach in the context of the self-similar solutions mc;\u000b, and in\nparticular to draw attention to a possible difficulty in using it to study these solutions.\nIn the sequel we consider the one-dimensional case N= 1and\u000b2[0;1]. Then the moving\nframes technique can be recast as a Hasimoto transformation as follows. Assume that mis the\ntangent vector of a curve in R3, i.e.m=@xX, for some curve X(x;t)2R3parametrized by the\narc-length. It can be shown (see [12]) that if mevolves under (LLG \u000b), then the torsion \u001cand\nthe curvature cofXsatisfy\n@t\u001c=\f\u0012\nc@xc +@x\u0010@xxc\u0000c\u001c2\nc\u0011\u0013\n+\u000b\u0012\nc2\u001c+@x\u0010@x(c\u001c) +\u001c@xc\nc\u0011\u0013\n;\n@tc =\f(\u0000@x(c\u001c)\u0000\u001c@xc) +\u000b\u0000\n@xc\u0000c\u001c2\u0001\n:\nHence, defining the Hasimoto transformation [19] (also called filament function)\nv(x;t) = c(x;t)ei\u0001x\n0\u001c(\u001b;t)d\u001b; (3.35)\nwe verify that vsolves the following dissipative Schrödinger (or complex Ginzburg–Landau)\nequation\ni@tv+ (\f\u0000i\u000b)@xxv+v\n2\u0012\n\fjvj2+ 2\u000b\u0002x\n0Im(\u0016v@xv)\u0000A(t)\u0013\n= 0; (3.36)\nwhere\f=p\n1\u0000\u000b2and\nA(t) =\u0012\n\f\u0012\nc2+2(@xxc\u0000c\u001c2)\nc\u0013\n+ 2\u000b\u0012@x(c\u001c) +\u001c@xc\nc\u0013\u0013\n(0;t):\nThe curvature and torsion associated with the self-similar solutions mc;\u000bare (see [17]):\ncc;\u000b(x;t) =cp\nte\u0000\u000bx2\n4tand\u001cc;\u000b(x;t) =\fx\n2p\nt: (3.37)\nTherefore in this case\nA(t) =\fc2\nt(3.38)\n6We recall that v2M2;2(RN)ifv2L2\nloc(RN)and\nkvkM2;2:= sup\nx2RN\nr>01\nr(N\u00002)=2kvkL2(Br(x))<1:\n31and the Hasimoto transformation of mc;\u000bis\nvc;\u000b(x;t) =cp\nte(\u0000\u000b+i\f)x2\n4t:\nIn particular vc;\u000bis a solution of (3.36) with A(t)as in (3.38), for all \u000b2[0;1]andc > 0.\nMoreover, the Fourier transform of this function (w.r.t. the space variable) is\nbvc;\u000b(\u0018;t) = 2cp\n\u0019(\u000b+i\f)e\u0000(\u000b+i\f)\u00182t;\nso thatvc;\u000bis a solution of (3.36) with a Dirac delta as initial condition:\nvc;\u000b(\u0001;0) = 2cp\n\u0019(\u000b+i\f)\u000e:\nHere\u000edenotes the delta distribution at the point x= 0andpzdenotes the square root of a\ncomplex number zsuch that Im(pz)>0.\nIn the limit cases \u000b= 0and\u000b= 1, the first three terms in equation (3.36) lead to a cubic\nSchrödinger equation and to a linear heat equation, respectively. The Cauchy problem with\na Dirac delta for these kind of equations associated with a power type non-linearity has been\nstudied by several authors (see e.g. [4] and the reference therein). We recall two classical results.\nTheorem 3.7 ([9]).Letp\u00152andu2Lp\nloc(R\u0002R+)be a solution in the sense of distributions\nof\n@tu\u0000@xxu+jujpu= 0onR\u0002R+: (3.39)\nAssume that\nlim\nt!0+\u0002\nRu(x;t)'(x)dx= 0;for all'2C0(Rnf0g); (3.40)\nwhereC0(Rnf0g)denotes the space of continuous functions with compact support in Rnf0g.\nThenu2C2;1(R\u0002[0;1))andu(x;0) = 0for allx2R. In particular there is no solution of\n(3.39)such that\nlim\nt!0+\u0002\nRu(x;t)'(x)dx='(0);for all'2C0(RN):\nIn[9]itisalsoprovedthatif 1<p< 2, equation(3.39)hasaglobalsolutionwithaDiracdelta\nas initial condition, as in the case of the linear parabolic equation. Concerning the Schrödinger\nequation, we have the following ill-posedness result due to Kenig, Ponce and Vega [21].\nTheorem 3.8 ([21]).Letp\u00152. Either there is no solution in the sense of distributions of\ni@tu+@xxu+jujpu= 0onR\u0002R+; (3.41)\nwith\nlim\nt!0+u(\u0001;t) =\u000einS0(R);\nin the class u;jujpu2L1(R+;S0(R)), or there is more than one.\nAfter performing an appropriate change of variables, equation (3.36) leads to the following\nequation\ni@tu+ (\f\u0000i\u000b)@xxu+u\n2\u0012\n\fjuj2+ 2\u000b\u0002x\n0Im(\u0016u@yu)dy\u0013\n= 0: (3.42)\nSince\u000b2[0;1], the above equation can be seen as an intermediate model between (3.39) and\n(3.41). Therefore one could expect that when \u000b2(0;1], the solutions of (3.42) share similar\nproperties to those established in Theorem 3.7 for the equation (3.39). We will continue to\nobserve that this is not necessarily the case. To this end, we need the following:\n32Proposition 3.9. For all\u000b2[0;1]and for all c2Cnf0gthe function wc;\u000b:R\u0002R+!Cgiven\nby\nwc;\u000b(x;t) =cp\ntexp\u0012i\fjcj2\n2ln(t) + (i\f\u0000\u000b)x2\n4t\u0013\nis a solution of (3.42). In addition,\ni) If\u000b2[0;1), thenwc;\u000b(\u0001;t)does not converge in S0(R)ast!0+.\nii) If\u000b2(0;1], then\nlim\nt!0+\u0002\nRwc;\u000b(x;t)'(x)dx= 0;for all'2C0(Rnf0g): (3.43)\nProof.A straightforward computation shows that wc;\u000bsatisfies (3.42). The proof that wc;\u000b(\u0001;t)\ndoes not converge in S0(R)ast!0+ifc6= 0is the same as in [21]. Indeed, for '2S(R), by\nParseval’s theorem,\u0002\nR\u0016wc;\u000b(x;t)'(x)dx=1\n2\u0019\u0002\nRbwc;\u000b(\u0018;t)b'(\u0018)d\u0018\n=\u0016ce\u0000i\fjcj2ln(t)=2\np\u0019p\n\u000b+i\f\u0002\nRe\u0000(\u000b+i\f)\u00182tb'(\u0018)d\u0018:\nBy the dominated convergence theorem, the last integral converges:\nlim\nt!0+\u0002\nRe\u0000(\u000b+i\f)\u00182tb'(\u0018)d\u0018=\u0002\nRb'(\u0018)d\u0018= 2\u0019'(0):\nSince\f6= 0,e\u0000i\fjcj2ln(t)=2does not admit a limit at 0inS0(R). We conclude that wc;\u000b(\u0001;t)does\nnot converge in S0(R)ast!0+.\nIt remains to prove (3.43). Since now '2C0(Rnf0g), we cannot proceed as before. However,\nusing the change of variables x=p\nty, we have\nlim\nt!0+\u0002\nRwc;\u000b(x;t)'(x)dx=cei\fjcj2ln(t)=2\u0002\nRe(\u0000\u000b+i\f)y2=4'(p\nty)dy:\nTherefore, since \u000b>0and'(0) = 0, the dominated convergence theorem implies that\nlim\nt!0+\u0002\nRe(\u0000\u000b+i\f)y2=4'(p\nty)dy='(0)\u0002\nRe(\u0000\u000b+i\f)y2=4dy= 0:\nSincejei\fjcj2ln(t)=2j= 1, we obtain (3.43).\nThe results in Proposition 3.9 lead to the following remarks:\n1. Observe that if \u000b2(0;1),wc;\u000bprovides a solution to the dissipative equation (3.42).\nMoreover, form part (ii)in Proposition 3.9, wc;\u000bsatisfies the condition (3.40). However,\nnotice that wc;\u000bcannot be extended to C2;1(R\u0002[0;1))due to the presence of a logarithmic\noscillation. This is in contrast with the properties for solutions of the cubic heat equation\n(3.39) established in Theorem 3.7.\n2. In the case \u000b= 0, equation (3.42) corresponds to (3.41) with p= 2, i.e. to the equation\ncubic NLS equation that is invariant under the Galilean transformation. The proof of the\nill-posedness result given in Theorem 3.8 relies on this invariance and part (i)of Proposi-\ntion 3.9 with \u000b= 0. Although when \u000b > 0, equation (3.42) is no longer invariant under\nthe Galilean transformation, part (i)of Proposition 3.9 could be an indicator that that the\nCauchy problem (3.42) with a delta as initial condition is still ill-posed. This question rests\nopen for the moment and it seems that the use of (3.36) (or (3.42)) can be more difficult\nto formulate a Cauchy theory for (LLG \u000b) including self-similar solutions.\n334 Appendix\nThe characterization of BMO\u00001\n1(RN)as sum of derivatives of functions in BMO was proved by\nKoch and Tataru in [23]. A straightforward generalization of their proof leads to the following\ncharacterization of BMO\u00001\n\u000b(RN).\nTheorem A.1. Let\u000b2(0;1]andf2S0(RN). Thenf2BMO\u00001\n\u000b(RN)if and only if there exist\nf1;:::;fN2BMO\u000b(RN)such thatf=PN\nj=1@jfj. In addition, if such a decomposing holds,\nthen\nkfkBMO\u00001\n\u000b.NX\nj=1[fj]BMO\u000b:\nThe next results provide the equivalence between the weak solutions and the Duhamel for-\nmulation. We first need to introduce for T >0the spaceL1\nuloc(RN\u0002(0;T))defined as the space\nof measurable functions on RN\u0002(0;T)such that the norm\nkfkuloc;T:= sup\nx02RN\u0002\nB(x0;1)\u0002T\n0jf(y;t)jdtdy\nis finite. We refer the reader to Lemarié–Rieusset’s book [27] for more details about these kinds\nof spaces. In particular, we recall the following result corresponding to Lemma 11.3 in [27] in\nthe case\u000b= 1. It is straightforward to check that the same proof still applies if \u000b2(0;1).\nLemma A.2. Let\u000b2(0;1],T2(0;1)andw2L1\nuloc(RN\u0002(0;T)). Then the function\nW(x;t) :=\u0002t\n0S\u000b(t\u0000s)w(x;s)ds\nis well defined and belongs to L1\nuloc(RN\u0002(0;T)). Moreover,\ni@tW+ (\f\u0000i\u000b)\u0001W=winD0(RN\u0002R+);\nand the application\n[0;T]!R\nt7! kW(\u0001;t)kL1(B1(x0))\nis continuous for any x02R, withkW(\u0001;t)kL1(B1(x0))!0, ast!0+, uniformly in x0.\nFollowing the ideas in [27], we can establish now the equivalence between the notions of\nsolutions as well as the regularity.\nTheorem A.3. Let\u000b2(0;1]andu2X(RN\u0002R+;C). Then the following assertions are\nequivalent:\ni) The function usatisfies\niut+ (\f\u0000i\u000b)\u0001u= 2(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2inD0(RN\u0002R+): (A.1)\nii) There exists u02S0(RN)such thatusatisfies\nu(t) =S\u000b(t)u0\u00002(\f\u0000i\u000b)\u0002t\n0S\u000b(t\u0000s)\u0016u(ru)2\n1 +juj2ds:\n34Moreover, if (ii) holds, then u2C1(RN\u0002R+)and\nk(u(t)\u0000u0)'kL1(RN)!0;ast!0+; (A.2)\nfor any'2S(RN).\nProof.In view of Lemma A.2, we need to prove that the function\ng(u) =\u00002(\f\u0000i\u000b)\u0016u(ru)2\n1 +juj2\nbelongs toL1\nuloc(RN\u0002(0;T)), for allT >0. Indeed, by (2.3) we have\nkg(u)kuloc;T\u0014kjruj2kuloc;T: (A.3)\nIfT\u00141, then\nkjruj2kuloc;T\u0014sup\nx02RN\u0002\nQ1(x0)jru(y;t)j2dtdy\u0014kuk2\nX: (A.4)\nIfT\u00151, using that\njruj\u0014[u]Xp\nt;for anyt>0;\nwe get\nkjruj2kT;uloc\u0014sup\nx02RN\u0002\nQ1(x0)jru(y;t)j2dtdy + sup\nx02RN\u0002T\n1\u0002\nB1(x0)jruj2dydt\n\u0014kuk2\nX+ [u]2\nXjB1(0)j\u0002T\n11\ntdt\n\u0014kuk2\nX(1 +jB1(0)jln(T)):(A.5)\nIn conclusion, we deduce from (A.3), (A.4) and (A.5) that g(u)2L1\nuloc(RN\u0002(0;T))and then\nit follows from Lemma A.2 that (ii) implies (i). The other implication can be established as in\n[27, Theorem 11.2]. Moreover, we deduce that the function\nW(x;t) :=T(g(u))(x;t) =\u0002t\n0S\u000b(t\u0000s)g(u)ds\nsatisfieskW(\u0001;t)kL1(B1(x0))!0, ast!0+, uniformly in x02RN. Let us take '2S(RN)and\na constantC'>0such thatj'(x)j\u0014C'(2 +jxj)\u0000N\u00001. Then\n\u0002\nRNj'(y)W(y;t)jdy\u0014X\nk2ZN\u0002\nB1(k)C'\n(2 +jxj)N+1jW(y;t)jdy\n\u0014sup\nx02RNkW(\u0001;t)kL1(B1(x0))X\nk2ZNC'\n(1 +jkj)N+1;\nso thatk'W(\u0001;t)kL1(RN)!0ast!0+, i.e.\nk(u(t)\u0000S\u000b(t)u0)'kL1(RN)!0;ast!0+: (A.6)\nOn the other hand, since u02L1(RN),\nkS\u000b(t)u0\u0000u0kL1(Br(0))!0;ast!0+; (A.7)\n35for anyr>0(see e.g. [3, Corollary 2.4]). Given \u000f>0, we fixr\u000f>0such that\n2ku0k1k'kL1(Bcr\u000f(0))\u0014\u000f:\nUsing (A.7), we obtain\nlim\nt!0+k(S\u000b(t)u0\u0000u0)'kL1(Br\u000f(0))= 0:\nThen, passing to limit in the inequality\nk(S\u000b(t)u0\u0000u0)'kL1(RN)\u0014k(S\u000b(t)u0\u0000u0)'kL1(Br\u000f(0))+ 2ku0kL1(RN)k'kL1(Bcr\u000f(0));(A.8)\nwe obtain\nlim sup\nt!0+k(S\u000b(t)u0\u0000u0)'kL1(RN)\u0014\u000f: (A.9)\nTherefore\nlim\nt!0+k(S\u000b(t)u0\u0000u0)'kL1(RN)= 0:\nCombining with (A.6), we conclude the proof of (A.2).\nIt remains to prove that uis smooth for t >0. Sinceu2X(RN\u0002R+;C), we get that\nu;ru2L1\nloc(RN\u0002R+). Theng(u)2L2\nloc(RN\u0002R+)so theLp-regularity theory for parabolic\nequations implies that a function usatisfying (A.1) belongs to u2H2;1\nloc(RN\u0002R+)(see [28, 24]\nand [33, Remark 48.3] for notations and more details). Since the space Hk\\L1is stable under\nmultiplication (see e.g. [20, Chapter 6]), we can use a bootstrap argument to conclude that\nu2C1(RN\u0002R+).\nRemark A.4. Several authors have studied further properties of the solutions found by Koch\nand Tataru for the Navier–Stokes equations. For instance, analyticity, decay rates of the higher-\norder derivatives in space and time have been investigated by Miura and Sawada [32], Germain,\nPavlović and Staffilani [14], among others. A similar analysis for the solution uof (DNLS) is\nbeyond the scope of this paper, but it can probably be performed using the same arguments\ngiven in [32, 14].\nWe end this appendix with some properties of the self-similar found in [17].\nTheorem A.5 ([17]).LetN= 1. For every \u000b2[0;1]andc >0, there exists a profile fc;\u000b2\nC1(R;S2)such that\nmc;\u000b(x;t) =fc;\u000b\u0012xp\nt\u0013\n;for all (x;t)2R\u0002R+;\nis a smooth solution of (LLG\u000b)onR\u0002R+. Moreover,\n(i) There exist unitary vectors A\u0006\nc;\u000b= (A\u0006\nj;c;\u000b)3\nj=12S2such that the following pointwise con-\nvergence holds when tgoes to zero:\nlim\nt!0+mc;\u000b(x;t) =8\n<\n:A+\nc;\u000b;ifx>0;\nA\u0000\nc;\u000b;ifx<0;(A.10)\nandA\u0000\nc;\u000b= (A+\n1;c;\u000b;\u0000A+\n2;c;\u000b;\u0000A+\n3;c;\u000b).\n(ii) There exists a constant C(c;\u000b;p ), depending only on c,\u000bandpsuch that for all t>0\nkmc;\u000b(\u0001;t)\u0000A+\nc;\u000b\u001f(0;1)(\u0001)\u0000A\u0000\nc;\u000b\u001f(\u00001;0)(\u0001)kLp(R)\u0014C(c;\u000b;p )t1\n2p; (A.11)\nfor allp2(1;1). In addition, if \u000b>0,(A.11)also holds for p= 1.\n36(iii) Fort>0andx2R, the derivative in space satisfies\nj@xmc;\u000b(x;t)j=cp\nte\u0000\u000bx2\n4t: (A.12)\n(iv) Let\u000b2[0;1]. Then A+\nc;\u000b!(1;0;0)asc!0+.\nLemma A.6. Letc >0,\u000b2(0;1],A+\nc;\u000b;A\u0000\nc;\u000bbe the unit vectors given in Theorem A.5 and\n#c;\u000bthe angle between A+\nc;\u000bandA\u0000\nc;\u000b. Then, for fixed \u000b2(0;1],#c;\u000bis a continuous function\ninc. Also, for 0<c<\u000b2p\u0019=32,\n#c;\u000b\u0015arccos\u0012\n1\u0000c2\u0019+32c3p\u0019\n\u000b2\u0013\n: (A.13)\nRemark A.7. If\u000b2(0;1]andc2(0;\u000b2p\u0019=32), then 1\u0000c2\u0019+32c3p\u0019\n\u000b22(0;1), so that its\narccosis well-defined.\nProof.The continuity was proved in [17]. To show the estimate (A.13), we use Theorem 1.3 in\n[17], to get\n\f\f\f\f\fA+\n2;c;\u000b\u0000cp\n\u0019(1 +\u000b)p\n2\f\f\f\f\f\u0014c2\u0019\n4+c2\u0019\n\u000bp\n2 \n1 +c2\u0019\n8+cp\n\u0019(1 +\u000b)\n2p\n2!\n+\u0012c2\u0019\n2p\n2\u000b\u00132\n:\nSince\u000b2(0;1], we have for any c2(0;1],\n\u0019\n4+\u0019\n\u000bp\n2 \n1 +c2\u0019\n8+cp\n\u0019(1 +\u000b)\n2p\n2!\n+c2\u00192\n8\u000b2\u00141\n\u000b2\u0012\u0019\n4+\u0019p\n2\u0012\n1 +\u0019\n8+p\u0019\n2\u0013\n+\u00192\n8\u0013\n\u00148\n\u000b2:\nWe deduce that for all \u000b;c2(0;1],\nA+\n2;c;\u000b\u0015cp\n\u0019(1 +\u000b)p\n2\u00008c2\n\u000b2\u0015cp\u0019p\n2\u00008c2\n\u000b2:\nIn particular A+\n2;c;\u000b\u00150ifc\u0014\u000b2p\u0019=(8p\n2). Thus\n(A+\n2;c;\u000b)2\u0015c2\u0019\n2\u000016c3p\u0019p\n2\u000b2+64c4\n\u000b4\u0015c2\u0019\n2\u000016c3p\u0019\n\u000b2;\nso that\ncos(#c;\u000b) = 1\u00002((A+\n2;c;\u000b)2+ (A+\n3;c;\u000b)2)\u00141\u0000c2\u0019+32c3p\u0019\n\u000b2;\nwhich implies (A.13).\nThe following lemma is a slightly refinement of Theorem 1.4 in [17].\nLemma A.8 ([17]).Letc >0,\u000b2[0;1]andA+\nc;\u000bbe the unit vector given in Theorem A.5.\nThenA+\nc;\u000bis a continuous function of \u000bin[0;1]and\njA+\nc;\u000b\u0000A+\nc;1j\u0014h(c)p\n1\u0000\u000b;for all\u000b2[1=2;1]; (A.14)\nwhereh:R+!R+is a strictly increasing function satisfying\nlim\ns!1h(s) =1:\n37Proof.Inviewof[17, Theorem1.4], weonlyneedtoprovethattheconstant C(c)inthestatement\nof the Theorem 1.4 (notice that c0in [17] corresponds to cin our notation) is polynomial in c\nwith nonnegative coefficients. Looking at the proof of [17, Theorem 1.4], we see that the constant\nC(c)behaves like the constant in inequality (3.108) in [17]. In view of (3.17), the estimate (3.23)\nin [17] can be written as\njf(s)j\u0014p\n2andjf0(s)j\u0014c\n2e\u0000\u000bs2=4;\nand then (3.18) can be recast as\njgj\u0014 \nc\n4+c2p\n2\n8!\u0012s\n\fe\u0000\u000bs2=4+s2e\u0000\u000bs2=2\u0013\n:\nThen, it can be easily checked that the function his a polynomial with nonnegative coefficients.\nAcknowledgments. A.deLairewaspartiallysupportedbytheLabexCEMPI(ANR-11-LABX-\n0007-01) and the MathAmSud program. S. Gutierrez was partially supported by the EPSRC\ngrant EP/J01155X/1 and the ERCEA Advanced Grant 2014 669689 - HADE.\nReferences\n[1] M.AbramowitzandI.A.Stegun. Handbook of mathematical functions with formulas, graphs,\nand mathematical tables , volume 55 of National Bureau of Standards Applied Mathematics\nSeries. For sale by the Superintendent of Documents, U.S. Government Printing Office,\nWashington, D.C., 1964.\n[2] F.AlougesandA.Soyeur. OnglobalweaksolutionsforLandau-Lifshitzequations: existence\nand nonuniqueness. 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Germain and M. Rupflin. Selfsimilar expanders of the harmonic map flow. Ann. Inst. H.\nPoincaré Anal. Non Linéaire , 28(5):743–773, 2011.\n[16] T. L. Gilbert. A lagrangian formulation of the gyromagnetic equation of the magnetization\nfield.Phys. Rev. , 100:1243, 1955.\n[17] S. Gutiérrez and A. de Laire. Self-similar solutions of the one-dimensional Landau–Lifshitz–\nGilbert equation. Nonlinearity , 28(5):1307, 2015.\n[18] S. Gutiérrez, J. Rivas, and L. Vega. Formation of singularities and self-similar vortex motion\nunder the localized induction approximation. Comm. Partial Differential Equations , 28(5-\n6):927–968, 2003.\n[19] H. Hasimoto. A soliton on a vortex filament. J. Fluid Mech , 51(3):477–485, 1972.\n[20] L. Hörmander. Lectures on nonlinear hyperbolic differential equations , volume 26 of Math-\nématiques & Applications (Berlin) [Mathematics & Applications] . Springer-Verlag, Berlin,\n1997.\n[21] C. E. Kenig, G. Ponce, and L. Vega. On the ill-posedness of some canonical dispersive\nequations. Duke Math. J. , 106(3):617–633, 2001.\n[22] H. Koch and T. Lamm. Geometric flows with rough initial data. Asian J. Math. , 16(2):209–\n235, 2012.\n[23] H. Koch and D. Tataru. Well-posedness for the Navier-Stokes equations. Adv. Math. ,\n157(1):22–35, 2001.\n[24] O. Ladyzhenskaya, V. Solonnikov, and N. Ural’tseva. Linear and quasi-linear equations of\nparabolic type . Amer. Math. Soc., Transl. Math. Monographs. Providence, R.I., 1968.\n[25] M. Lakshmanan and K. Nakamura. Landau-Lifshitz equation of ferromagnetism: Exact\ntreatment of the Gilbert damping. Phys. Rev. Lett. , 53:2497–2499, 1984.\n[26] L. Landau and E. Lifshitz. On the theory of the dispersion of magnetic permeability in\nferromagnetic bodies. Phys. Z. Sowjetunion , 8:153–169, 1935.\n[27] P. G. Lemarié-Rieusset. Recent developments in the Navier-Stokes problem , volume 431\nofChapman & Hall/CRC Research Notes in Mathematics . Chapman & Hall/CRC, Boca\nRaton, FL, 2002.\n39[28] G. M. Lieberman. Second order parabolic differential equations . World Scientific Publishing\nCo., Inc., River Edge, NJ, 1996.\n[29] J. Lin. Uniqueness of harmonic map heat flows and liquid crystal flows. Discrete Contin.\nDyn. Syst. , 33(2):739–755, 2013.\n[30] J. Lin, B. Lai, and C. Wang. Global well-posedness of the Landau-Lifshitz-Gilbert equation\nfor initial data in Morrey spaces. Calc. Var. Partial Differential Equations , 54(1):665–692,\n2015.\n[31] C. Melcher. Global solvability of the Cauchy problem for the Landau-Lifshitz-Gilbert equa-\ntion in higher dimensions. Indiana Univ. Math. J. , 61(3):1175–1200, 2012.\n[32] H. Miura and O. Sawada. On the regularizing rate estimates of Koch-Tataru’s solution to\nthe Navier-Stokes equations. Asymptot. Anal. , 49(1-2):1–15, 2006.\n[33] P. Quittner and P. Souplet. Superlinear parabolic problems . Birkhäuser Advanced Texts:\nBasler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag,\nBasel, 2007. Blow-up, global existence and steady states.\n[34] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory inte-\ngrals, volume 43 of Princeton Mathematical Series . Princeton University Press, Princeton,\nNJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis,\nIII.\n[35] C. Wang. Well-posedness for the heat flow of harmonic maps and the liquid crystal flow\nwith rough initial data. Arch. Ration. Mech. Anal. , 200(1):1–19, 2011.\n[36] D. Wei. Micromagnetics and Recording Materials . SpringerBriefs in Applied Sciences and\nTechnology. Springer Berlin Heidelberg, 2012.\n40"    },    {        "title": "1804.09242v1.Generalisation_of_Gilbert_damping_and_magnetic_inertia_parameter_as_a_series_of_higher_order_relativistic_terms.pdf",        "content": "Generalisation of Gilbert damping and magnetic\ninertia parameter as a series of higher-order\nrelativistic terms\nRitwik Mondalz, Marco Berritta and Peter M. Oppeneer\nDepartment of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-751 20\nUppsala, Sweden\nE-mail: ritwik.mondal@physics.uu.se\nAbstract. The phenomenological Landau-Lifshitz-Gilbert (LLG) equation of motion\nremains as the cornerstone of contemporary magnetisation dynamics studies, wherein\nthe Gilbert damping parameter has been attributed to \frst-order relativistic e\u000bects.\nTo include magnetic inertial e\u000bects the LLG equation has previously been extended\nwith a supplemental inertia term and the arising inertial dynamics has been related\nto second-order relativistic e\u000bects. Here we start from the relativistic Dirac equation\nand, performing a Foldy-Wouthuysen transformation, derive a generalised Pauli spin\nHamiltonian that contains relativistic correction terms to any higher order. Using the\nHeisenberg equation of spin motion we derive general relativistic expressions for the\ntensorial Gilbert damping and magnetic inertia parameters, and show that these ten-\nsors can be expressed as series of higher-order relativistic correction terms. We further\nshow that, in the case of a harmonic external driving \feld, these series can be summed\nand we provide closed analytical expressions for the Gilbert and inertial parameters\nthat are functions of the frequency of the driving \feld.\n1. Introduction\nSpin dynamics in magnetic systems has often been described by the phenomenological\nLandau-Lifshitz (LL) equation of motion of the following form [1]\n@M\n@t=\u0000\rM\u0002He\u000b\u0000\u0015M\u0002[M\u0002He\u000b]; (1)\nwhere\ris the gyromagnetic ratio, He\u000bis the e\u000bective magnetic \feld, and \u0015is an\nisotropic damping parameter. The \frst term describes the precession of the local,\nclassical magnetisation vector M(r;t) around the e\u000bective \feld He\u000b. The second term\ndescribes the magnetisation relaxation such that the magnetisation vector relaxes to the\ndirection of the e\u000bective \feld until \fnally it is aligned with the e\u000bective \feld. To include\nzPresent address: Department of Physics, University of Konstanz, D -78457 Konstanz, GermanyarXiv:1804.09242v1  [cond-mat.other]  3 Apr 20182\nlarge damping, the relaxation term in the LL equation was reformulated by Gilbert [2, 3]\nto give the Landau-Lifshitz-Gilbert (LLG) equation,\n@M\n@t=\u0000\rM\u0002He\u000b+\u000bM\u0002@M\n@t; (2)\nwhere\u000bis the Gilbert damping constant. Note that both damping parameters \u000band\u0015\nare here scalars, which corresponds to the assumption of an isotropic medium. Both the\nLL and LLG equations preserve the length of the magnetisation during the dynamics and\nare mathematically equivalent (see, e.g. [4]). Recently, there have also been attempts\nMHeff\nPrecession\nNutationDamping\nFigure 1. Sketch of extended LLG magnetisation dynamics. The green arrow denotes\nthe classical magnetisation vector which precesses around an e\u000bective \feld. The red\nsolid and dotted lines depict the precession and damping. The yellow path signi\fes\nthe nutation, or inertial damping, of the magnetisation vector.\nto investigate the magnetic inertial dynamics which is essentially an extension to the\nLLG equation with an additional term [5{7]. Phenomenologically this additional term of\nmagnetic inertial dynamics, M\u0002I@2M=@t2, can be seen as a torque due to second-order\ntime derivative of the magnetisation [8{11]. The essence of the terms in the extended\nLLG equation is described pictorially in Fig. 1. Note that in the LLG dynamics the\nmagnetisation is described as a classical vector \feld and not as a quantum spin vector.\nIn their original work, Landau and Lifshitz attributed the damping constant \u0015to\nrelativistic origins [1]; later on, it has been more speci\fcally attributed to spin-orbit\ncoupling [12{15]. In the last few decades, several explanations have been proposed\ntowards the origin of damping mechanisms, e.g., the breathing Fermi surface model\n[16, 17], torque-torque correlation model [18], scattering theory formulation [19], e\u000bective\n\feld theories [20] etc. On the other hand, the origin of magnetic inertia is less discussed\nin the literature, although it's application to ultrafast spin dynamics and switching\ncould potentially be rich [9]. To account for the magnetic inertia, the breathing Fermi\nsurface model has been extended [11, 21] and the inertia parameter has been associated\nwith the magnetic susceptibility [22]. However, the microscopic origins of both Gilbert3\ndamping and magnetic inertia are still under debate and pose a fundamental question\nthat requires to be further investigated.\nIn two recent works [23, 24], we have shown that both quantities are of relativistic\norigin. In particular, we derived the Gilbert damping dynamics from the relativistic\nspin-orbit coupling and showed that the damping parameter is not a scalar quantity\nbut rather a tensor that involves two main contributions: electronic and magnetic\nones [23]. The electronic contribution is calculated as an electronic states' expectation\nvalue of the product of di\u000berent components of position and momentum operators;\nhowever, the magnetic contribution is given by the imaginary part of the susceptibility\ntensor. In an another work, we have derived the magnetic inertial dynamics from a\nhigher-order (1 =c4) spin-orbit coupling and showed that the corresponding parameter\nis also a tensor which depends on the real part of the susceptibility [24]. Both these\ninvestigations used a semirelativistic expansion of the Dirac Hamiltonian employing the\nFoldy-Wouthuysen transformation to obtain an extended Pauli Hamiltonian including\nthe relativistic corrections [25, 26]. The thus-obtained semirelativistic Hamiltonian was\nthen used to calculate the magnetisation dynamics, especially for the derivation of the\nLLG equation and magnetic inertial dynamics.\nIn this article we use an extended approach towards a derivation of the\ngeneralisation of those two (Gilbert damping and magnetic inertia) parameters from\nthe relativistic Dirac Hamiltonian, developing a series to fully include the occurring\nhigher-order relativistic terms. To this end we start from the Dirac Hamiltonian in\nthe presence of an external electromagnetic \feld and derive a semirelativistic expansion\nof it. By doing so, we consider the direct \feld-spin coupling terms and show that\nthese terms can be written as a series of higher-order relativistic contributions. Using\nthe latter Hamiltonian, we derive the corresponding spin dynamics. Our results show\nthat the Gilbert damping parameter and inertia parameter can be expressed as a\nconvergent series of higher-order relativistic terms and we derive closed expressions\nfor both quantities. At the lowest order, we \fnd exactly the same tensorial quantities\nthat have been found in earlier works.\n2. Relativistic Hamiltonian Formulation\nTo describe a relativistic particle, we start with a Dirac particle [27] inside a material,\nand, in the presence of an external \feld, for which one can write the Dirac equation\nasi~@ (r;t)\n@t=H (r;t) for a Dirac bi-spinor  . Adopting furthermore the relativistic\ndensity functional theory (DFT) framework we write the corresponding Hamiltonian as\n[23{25]\nH=c\u000b\u0001(p\u0000eA) + (\f\u0000 1)mc2+V 1\n=O+ (\f\u0000 1)mc2+E; (3)\nwhereVis the e\u000bective unpolarised Kohn-Sham potential created by the ion-ion, ion-\nelectron and electron-electron interactions. Generally, to describe magnetic systems, an4\nadditional spin-polarised energy (exchange energy) term is required. However, we have\ntreated e\u000bects of the exchange \feld previously, and since it doesn't contribute to the\ndamping terms we do not consider it explicitly here (for details of the calculations\ninvolving the exchange potential, see Ref. [23, 25]). The e\u000bect of the external\nelectromagnetic \feld has been accounted through the vector potential, A(r;t),cde\fnes\nthe speed of light, mis particle's mass and 1is the 4\u00024 unit matrix. \u000band\fare the\nDirac matrices which have the form\n\u000b= \n0\u001b\n\u001b0!\n; \f = \n10\n0\u00001!\n;\nwhere\u001b= (\u001bx;\u001by;\u001bz) are the Pauli spin matrix vectors and 1is 2\u00022 unit matrix.\nNote that the Dirac matrices form the diagonal and o\u000b-diagonal matrix elements of\nthe Hamiltonian in Eq. (3). For example, the o\u000b-diagonal elements can be denoted as\nO=c\u000b\u0001(p\u0000eA), and the diagonal matrix elements can be written as E=V 1.\nIn the nonrelativistic limit, the Dirac Hamiltonian equals the Pauli Hamiltonian,\nsee e.g. [28]. In this respect, one has to consider that the Dirac bi-spinor can be written\nas\n (r;t) = \n\u001e(r;t)\n\u0011(r;t)!\n;\nwhere the upper \u001eand lower\u0011components have to be considered as \\large\" and \\small\"\ncomponents, respectively. This nonrelativistic limit is only valid for the case when the\nparticle's momentum is much smaller than the rest mass energy, otherwise it gives\nan unsatisfactory result [26]. Therefore, the issue of separating the wave functions of\nparticles from those of antiparticles is not clear for any given momentum. This is mainly\nbecause the o\u000b-diagonal Hamiltonian elements link the particle and antiparticle. The\nFoldy-Wouthuysen (FW) transformation [29] has been a very successful attempt to \fnd\na representation where the o\u000b-diagonal elements have been reduced in every step of the\ntransformation. Thereafter, neglecting the higher-order o\u000b-diagonal elements, one \fnds\nthe correct Hamiltonian that describes the particles e\u000eciently. The FW transformation\nis an unitary transformation obtained by suitably choosing the FW operator [29],\nUFW=\u0000i\n2mc2\fO: (4)\nThe minus sign in front of the operator is because of the property that \fandO\nanticommute with each other. With the FW operator, the FW transformation of the\nwave function adopts the form  0(r;t) =eiUFW (r;t) such that the probability density\nremains the same, j j2=j 0j2. In this way, the time-dependent FW transformed\nHamiltonian can be expressed as [26, 28, 30]\nHFW=eiUFW\u0012\nH\u0000i~@\n@t\u0013\ne\u0000iUFW+i~@\n@t: (5)5\nAccording to the Baker-Campbell-Hausdor\u000b formula, the above transformed Hamilto-\nnian can be written as a series of commutators, and the \fnally transformed Hamiltonian\nreads\nHFW=H+i\u0014\nUFW;H\u0000i~@\n@t\u0015\n+i2\n2!\u0014\nUFW;\u0014\nUFW;H\u0000i~@\n@t\u0015\u0015\n+i3\n3!\u0014\nUFW;\u0014\nUFW;\u0014\nUFW;H\u0000i~@\n@t\u0015\u0015\u0015\n+:::: : (6)\nIn general, for a time-independent FW transformation, one has to work with@UFW\n@t= 0.\nHowever, this is only valid if the odd operator does not contain any time dependency. In\nour case, a time-dependent transformation is needed as the vector potential is notably\ntime-varying. In this regard, we notice that the even operators and the term i~@=@t\ntransform in a similar way. Therefore, we de\fne a term Fsuch thatF=E\u0000i~@=@t.\nThe main theme of the FW transformation is to make the odd terms smaller in every\nstep of the transformation. After a fourth transformation and neglecting the higher\norder terms, the Hamiltonian with only the even terms can be shown to have the form\nas [26, 30{33]\nH000\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6+O6\n16m5c10\u0013\n+E\u00001\n8m2c4[O;[O;F]]\n\u0000\f\n8m3c6[O;F]2+3\n64m4c8\b\nO2;[O;[O;F]]\t\n+5\n128m4c8\u0002\nO2;\u0002\nO2;F\u0003\u0003\n:(7)\nHere, for any two operators AandBthe commutator is de\fned as [ A;B] and the\nanticommutator as fA;Bg. As already pointed out, the original FW transformation\ncan only produce correct and expected higher-order terms up to \frst order i.e., 1 =c4\n[26, 30, 33]. In fact, in their original work Foldy and Wouthuysen derived only the\nterms up to 1 =c4, i.e., only the terms in the \frst line of Eq. (7), however, notably\nwith the exception of the fourth term [29]. The higher-order terms in the original FW\ntransformation are of doubtful value [32, 34, 35]. Therefore, the Hamiltonian in Eq. (7)\nis not trustable and corrections are needed to achieve the expected higher-order terms.\nThe main problem with the original FW transformation is that the unitary operators in\ntwo preceding transformations do not commute with each other. For example, for the\nexponential operators eiUFWandeiU0\nFW, the commutator [ UFW;U0\nFW]6= 0. Moreover, as\nthe unitary operators are odd, this commutator produces even terms that have not been\nconsidered in the original FW transformation [26, 30, 33]. Taking into account those\nterms, the correction of the FW transformation generates the Hamiltonian as [33]\nHcorr:\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6+O6\n16m5c10\u0013\n+E\u00001\n8m2c4[O;[O;F]]\n+\f\n16m3c6fO;[[O;F];F]g+3\n64m4c8\b\nO2;[O;[O;F]]\t\n+1\n128m4c8\u0002\nO2;\u0002\nO2;F\u0003\u0003\n\u00001\n32m4c8[O;[[[O;F];F];F]]: (8)6\nNote the di\u000berence between two Hamiltonians in Eq. (7) and Eq. (8) that are observed\nin the second and consequent lines in both the equations, however, the terms in the\n\frst line are the same. Eq. (8) provides the correct higher-order terms of the FW\ntransformation. In this regard, we mention that an another approach towards the correct\nFW transformation has been employed by Eriksen; this is a single step approach that\nproduces the expected FW transformed higher-order terms [34]. Once the transformed\nHamiltonian has been obtained as a function of odd and even terms, the \fnal form\nis achieved by substituting the correct form of odd terms Oand even termsEin the\nexpression of Eq. (8) and calculating term by term.\nSince we perform here the time-dependent FW transformation, we note that the\ncommutator [O;F] can be evaluated as [ O;F] =i~@O=@t. Therefore, following the\nde\fnition of the odd operator, the time-varying \felds are taken into account through\nthis term. We evaluate each of the terms in Eq. (8) separately and obtain that the\nparticles can be described by the following extended Pauli Hamiltonian [24, 26, 36]\nHcorr:\nFW=(p\u0000eA)2\n2m+V\u0000e~\n2m\u001b\u0001B\u0000(p\u0000eA)4\n8m3c2+(p\u0000eA)6\n16m5c4\n\u0000\u0012e~\n2m\u00132B2\n2mc2+e~\n4m2c2(\n(p\u0000eA)2\n2m;\u001b\u0001B)\n\u0000e~2\n8m2c2r\u0001Etot\u0000e~\n8m2c2\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000e~2\n16m3c4\u001a\n(p\u0000eA);@Etot\n@t\u001b\n\u0000ie~2\n16m3c4\u001b\u0001\u0014@Etot\n@t\u0002(p\u0000eA) + (p\u0000eA)\u0002@Etot\n@t\u0015\n+3e~\n64m4c4n\n(p\u0000eA)2\u0000e~\u001b\u0001B;~r\u0001Etot+\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]o\n+e~4\n32m4c6r\u0001@2Etot\n@t2+e~3\n32m4c6\u001b\u0001\u0014@2Etot\n@t2\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002@2Etot\n@t2\u0015\n:\n(9)\nThe \felds in the last Hamiltonian (9) are de\fned as B=r\u0002A, the external magnetic\n\feld,Etot=Eint+Eextare the electric \felds where Eint=\u00001\nerVis the internal \feld\nthat exists even without any perturbation and Eext=\u0000@A\n@tis the external \feld (only\nthe temporal part is retained here because of the Coulomb gauge). It is clear that as the\ninternal \feld is time-independent, it does not contribute to the fourth and sixth lines\nof Eq. (9). However, the external \feld does contribute to the above terms wherever it\nappears in the Hamiltonian.\nThe above-derived Hamiltonian can be split in two parts: (1) a spin-independent\nHamiltonian and (2) a spin-dependent Hamiltonian that involves the Pauli spin matrices.\nThe spin-dependent Hamiltonian, furthermore, has two types of coupling terms. The\ndirect \feld-spin coupling terms are those which directly couples the \felds with the\nmagnetic moments e.g., the third term in the \frst line, the second term in the third\nline of Eq. (9) etc. On the other hand, there are relativistic terms that do not directly\ncouple the spins to the electromagnetic \feld - indirect \feld-spin coupling terms. These7\nterms include e.g., the second term of the second line, the \ffth line of Eq. (9) etc. The\ndirect \feld-spin interaction terms are most important because these govern the directly\nmanipulation of the spins in a system with an electromagnetic \feld. For the external\nelectric \feld, these terms can be written together as a function of electric and magnetic\n\feld. These terms are taken into account and discussed in the next section. The indirect\ncoupling terms are often not taken into consideration and not included in the discussion\n(see Ref. [36, 37] for details). In this context, we reiterate that our current approach of\nderiving relativistic terms does not include the exchange and correlation e\u000bect. A similar\nFW transformed Hamiltonian has previously been derived, however, with a general\nKohn-Sham exchange \feld [23, 25, 26]. As mentioned before, in this article we do not\nintend to include the exchange-correlation e\u000bect, while mostly focussing on the magnetic\nrelaxation and magnetic inertial dynamics.\n2.1. The spin Hamiltonian\nThe aim of this work is to formulate the spin dynamics on the basis of the Hamiltonian\nin Eq. (9). The direct \feld-spin interaction terms can be written together as electric or\nmagnetic contributions. These two contributions can be expressed as a series up to an\norder of 1=m5[36]\nHS\nmagnetic =\u0000e\nmS\u0001\"\nB+1\n2X\nn=1;2;3;4\u00121\n2i!c\u0013n@nB\n@tn#\n+O\u00121\nm6\u0013\n; (10)\nHS\nelectric =\u0000e\nmS\u0001\"\n1\n2mc2X\nn=0;2\u0012i\n2!c\u0013n@nE\n@tn\u0002(p\u0000eA)#\n+O\u00121\nm6\u0013\n; (11)\nwhere the Compton wavelength and pulsation have been expressed by the usual\nde\fnitions \u0015c=h=mc and!c= 2\u0019c=\u0015cwith Plank's constant h. We also have used\nthe spin angular momentum operator as S= (~=2)\u001b. Note that we have dropped\nthe notion of total electric \feld because the the involved \felds ( B,E,A) are external\nonly, the internal \felds are considered as time-independent. The involved terms in the\nabove two spin-dependent Hamiltonians can readily be explained. The \frst term in the\nmagnetic contribution in Eq. (10) explains the Zeeman coupling of spins to the external\nmagnetic \feld. The rest of the terms in both the Hamiltonians in Eqs. (11) and (10)\nrepresent the spin-orbit coupling and its higher-order corrections. We note that these\ntwo spin Hamiltonians are individually not Hermitian, however, it can be shown that\ntogether they form a Hermitian Hamiltonian [38]. As these Hamiltonians describe a\nsemirelativistic Dirac particle, it is possible to derive from them the spin dynamics of\na single Dirac particle [24]. The e\u000bect of the indirect \feld-spin terms is not yet well\nunderstood, but they could become important too in magnetism [36, 37], however, those\nterms are not of our interest here.\nThe electric Hamiltonian can be written in terms of magnetic contributions with\nthe choice of a gauge A=B\u0002r=2. The justi\fcation of the gauge lies in the fact8\nthat the magnetic \feld inside the system being studied is uniform [26]. The transverse\nelectric \feld in the Hamiltonian (10) can be written as\nE=1\n2\u0012\nr\u0002@B\n@t\u0013\n: (12)\nReplacing this expression in the electric spin Hamiltonian in Eq. (11), one can obtain a\ngeneralised expression of the total spin-dependent Hamiltonian as\nHS(t) =\u0000e\nmS\u0001h\nB+1\n21X\nn=1;2;:::\u00121\n2i!c\u0013n@nB\n@tn\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u0012\nr\u0002@n+1B\n@tn+1\u0013\n\u0002(p\u0000eA)i\n: (13)\nIt is important to stress that the above spin-Hamiltonian is a generalisation of the two\nHamiltonians in Eqs. (10) and (11). We have already evaluated the Hamiltonian forms\nforn= 1;2;3;4 and assume that the higher-order terms will have the same form [36].\nThis Hamiltonian consists of the direct \feld-spin interaction terms that are linear and/or\nquadratic in the \felds. In the following we consider only the linear interaction terms,\nthat is we neglect the eAterm in Eq. (13). Here, we mention that the quadratic terms\ncould provide an explanation towards the previously unknown origin of spin-photon\ncoupling or optical spin-orbit torque and angular magneto-electric coupling [38{40].\nThe linear direct \feld-spin Hamiltonian can then be recast as\nHS(t) =\u0000e\nmS\u0001h\nB+1\n21X\nn=1;2;:::\u00121\n2i!c\u0013n@nB\n@tn\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u001a@n+1B\n@tn+1(r\u0001p)\u0000r\u0012@n+1B\n@tn+1\u0001p\u0013\u001bi\n: (14)\nThis is \fnal form of the Hamiltonian and we are interested to describe to evaluate its\ncontribution to the spin dynamics.\n3. Spin dynamics\nOnce we have the explicit form of the spin Hamiltonian in Eq. (14), we can proceed to\nderive the corresponding classical magnetisation dynamics. Following similar procedures\nof previous work [23, 24], and introducing a magnetisation element M(r;t), the\nmagnetisation dynamics can be calculated by the following equation of motion\n@M\n@t=X\njg\u0016B\n\n1\ni~D\u0002\nSj;HS(t)\u0003E\n; (15)\nwhere\u0016Bis the Bohr magneton, gis the Land\u0013 e g-factor that takes a value \u00192 for electron\nspins and \n is a suitably chosen volume element. Having the spin Hamiltonian in Eq.9\n(14), we evaluate the corresponding commutators. As the spin Hamiltonian involves the\nmagnetic \felds, one can classify the magnetisation dynamics into two situations: (a) the\nsystem is driven by a harmonic \feld, (b) the system is driven by a non-harmonic \feld.\nHowever, in the below we continue the derivation of magnetisation dynamics with the\nharmonic driven \felds. The magnetisation dynamics driven by the non-harmonic \felds\nhas been discussed in the context of Gilbert damping and inertial dynamics where it was\nshown that an additional torque contribution (the \feld-derivative torque) is expected\nto play a crucial role [23, 24, 26].\nThe magnetisation dynamics due to the very \frst term of the Hamiltonian in Eq.\n(14) is derived as [24]\n@M(1)\n@t=\u0000\rM\u0002B; (16)\nwith the gyromagnetic ratio \r=gjej=2m. Here the commutators between two spin\noperators have been evaluated using [ Sj;Sk] =i~Sl\u000fjkl, where\u000fjklis the Levi-Civita\ntensor. This dynamics actually produces the precession of magnetisation vector around\nan e\u000bective \feld. To get the usual form of Landau-Lifshitz precessional dynamics, one\nhas to use a linear relationship of magnetisation and magnetic \feld as B=\u00160(M+H).\nWith the latter relation, the precessional dynamics becomes \u0000\r0M\u0002H, where\r0=\r\u00160\nde\fnes the e\u000bective gyromagnetic ratio. We point out that the there are relativistic\ncontributions to the precession dynamics as well, e.g., from the spin-orbit coupling due\nto the time-independent \feld Eint[23]. Moreover, the contributions to the magnetisation\nprecession due to exchange \feld appear here, but are not explicitly considered in this\narticle as they are not in the focus of the current investigations (see Ref. [23] for details).\nThe rest of the terms in the spin Hamiltonian in Eq. (14) is of much importance\nbecause they involve the time-variation of the magnetic induction. As it has been shown\nin an earlier work [23] that for the external \felds and speci\fcally the terms with n= 1\nin the second terms and n= 0 in the third terms of Eq. (14), these terms together\nare Hermitian. These terms contribute to the magnetisation dynamics as the Gilbert\nrelaxation within the LLG equation of motion,\n@M(2)\n@t=M\u0002\u0012\nA\u0001@M\n@t\u0013\n; (17)\nwhere the Gilbert damping parameter Ahas been derived to be a tensor that has mainly\ntwo contributions: electronic and magnetic. The damping parameter Ahas the form\n[23, 24]\nAij=\u0000e\u00160\n8m2c2X\n`;k\u0002\nhripk+pkrii\u0000hr`p`+p`r`i\u000eik\u0003\n\u0002\u0000\n1+\u001f\u00001\u0001\nkj; (18)\nwhere 1is the 3\u00023 unit matrix and \u001fis the magnetic susceptibility tensor that can be\nintroduced only if the system is driven by a \feld which is single harmonic [26]. Note\nthat the electronic contributions to the Gilbert damping parameter are given by the10\nexpectation value hripkiand the magnetic contributions by the susceptibility. We also\nmention that the tensorial Gilbert damping tensor has been shown to contain a scalar,\nisotropic Heisenberg-like contribution, an anisotropic Ising-like tensorial contribution\nand a chiral Dzyaloshinskii-Moriya-like contribution [23].\nIn an another work, we took into account the terms with n= 2 in the second term\nof Eq. (14) and it has been shown that those containing the second-order time variation\nof the magnetic induction result in the magnetic inertial dynamics. Note that these\nterms provide a contribution to the higher-order relativistic e\u000bects. The corresponding\nmagnetisation dynamics can be written as [24]\n@M(3)\n@t=M\u0002\u0012\nC\u0001@M\n@t+D\u0001@2M\n@t2\u0013\n; (19)\nwith a higher-order Gilbert damping tensor Cijand inertia parameter Dijthat have the\nfollowing expressions Cij=\r0~2\n8m2c4@\n@t( 1+\u001f\u00001)ijandDij=\r0~2\n8m2c4( 1+\u001f\u00001)ij. We note\nthat Eq. (19) contains two fundamentally di\u000berent dynamics { the \frst term on the\nright-hand side has the exact form of Gilbert damping dynamics whereas the second\nterm has the form of magnetic inertial dynamics [24].\nThe main aim of this article is to formulate a general magnetisation dynamics\nequation and an extension of the traditional LLG equation to include higher-order\nrelativistic e\u000bects. The calculated magnetisation dynamics due to the second and third\nterms of Eq. (14) can be expressed as\n@M\n@t=e\nmM\u0002h1\n21X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n+1B\n@tn+1\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u001a@n+1B\n@tn+1hr\u0001pi\u0000D\nr\u0012@n+1B\n@tn+1\u0001p\u0013E\u001bi\n: (20)\nNote the di\u000berence in the summation of \frst terms from the Hamiltonian in Eq. (14).\nTo obtain explicit expressions for the Gilbert damping dynamics, we employ a general\nlinear relationship between magnetisation and magnetic induction, B=\u00160(H+M).\nThe time-derivative of the magnetic induction can then be replaced by magnetisation\nand magnetic susceptibility. For the n-th order time-derivative of the magnetic induction\nwe \fnd\n@nB\n@tn=\u00160\u0012@nH\n@tn+@nM\n@tn\u0013\n: (21)\nNote that this equation is valid for the case when the magnetisation is time-dependent.\nSubstituting this expression into the Eq. (20), one can derive the general LLG equation\nand its extensions. Moreover, as we work out the derivation in the case of harmonic\ndriving \felds, the di\u000berential susceptibility can be introduced as \u001f=@M=@H. The\n\frst term ( n-th derivative of the magnetic \feld) can consequently be written by the11\nfollowing Leibniz formula as\n@nH\n@tn=n\u00001X\nk=0(n\u00001)!\nk!(n\u0000k\u00001)!@n\u0000k\u00001(\u001f\u00001)\n@tn\u0000k\u00001\u0001@k\n@tk\u0012@M\n@t\u0013\n; (22)\nwhere the magnetic susceptibility \u001f\u00001is a time-dependent tensorial quantity and\nharmonic. Using this relation, the \frst term and second terms in Eq. (20) assume\nthe form\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=0;1;:::\u00121\n2i!c\u0013n+1nX\nk=0n!\nk!(n\u0000k)!@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\n;\n(23)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u0002\n1X\nn=0;2;:::\u00121\n2i!c\u0013nnX\nk=0n!\nk!(n\u0000k)!h@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\u001b\n\u0001p\u0013Ei\n:(24)\nThese two equations already provide a generalisation of the higher-order magnetisation\ndynamics including the Gilbert damping (i.e., the terms with k= 0) and the inertial\ndynamics (the terms with k= 1) and so on.\n4. Discussion\n4.1. Gilbert damping parameter\nIt is obvious that, as Gilbert damping dynamics involves the \frst-order time derivative of\nthe magnetisation and a torque due to it, kmust take the value k= 0 in the equations\n(23) and (24). Therefore, the Gilbert damping dynamics can be achieved from the\nfollowing equations:\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t; (25)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u00021X\nn=0;2;:::\u00121\n2i!c\u0013nh\u0012@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t\u001b\n\u0001p\u0013Ei\n: (26)\nNote that these equations can be written in the usual form of Gilbert damping as\nM\u0002\u0000\nG\u0001@M\n@t\u0001\n, where the Gilbert damping parameter Gis notably a tensor [2, 23]. The12\ngeneral expression for the tensor can be given by a series of higher-order relativistic\nterms as follows\nGij=e\u00160\n2m1X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n( 1+\u001f\u00001)ij\n@tn\n+e\u00160\n4m2c21X\nn=0;2;:::\u00121\n2i!c\u0013nh@n( 1+\u001f\u00001)ij\n@tn(hrlpli\u0000hrlpii)i\n: (27)\nHere we have used the Einstein summation convention on the index l. Note that there\nare two series: the \frst series runs over even and odd numbers ( n= 0;1;2;3;\u0001\u0001\u0001),\nhowever, the second series runs only over the even numbers ( n= 0;2;4;\u0001\u0001\u0001). Eq. (27)\nrepresents a general relativistic expression for the Gilbert damping tensor, given as a\nseries of higher-order terms. This equation is one of the central results of this article. It\nis important to observe that this expression provides the correct Gilbert tensor at the\nlowest relativistic order, i.e., putting n= 0 the expression for the tensor is found to be\nexactly the same as Eq. (18).\nThe analytic summation of the above series of higher-order relativistic contributions\ncan be carried out when the susceptibility depends on the frequency of the harmonic\ndriving \feld. This is in general true for ferromagnets where a di\u000berential susceptibility\nis introduced because there exists a spontaneous magnetisation in ferromagnets even\nwithout application of a harmonic external \feld. However, if the system is driven by a\nnonharmonic \feld, the introduction of the susceptibility is not valid anymore. In general\nthe magnetic susceptibility is a function of wave vector and frequency in reciprocal space,\ni.e.,\u001f=\u001f(q;!). Therefore, for the single harmonic applied \feld, we use \u001f\u00001/ei!tand\nthen-th order derivative will follow @n=@tn(\u001f\u00001)/(i!)n\u001f\u00001. With these arguments,\none can express the damping parameter of Eq. (27) as (see Appendix A for detailed\ncalculations)\nGij=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n(2!!c+!2)~\ni+!2(hrlpli\u0000hrlpii)\n4!2\nc\u0000!2#\n\u001f\u00001\nij: (28)\nHere, the \frst term in the last expression is exactly the same as the one that has been\nderived in our earlier investigation [23]. As the expression of the expectation value\nhripjiis imaginary, the real Gilbert damping parameter will be given by the imaginary\npart of the susceptibility tensor. This holds consistently for the higher-order terms\nas well. The second term in Eq. (28) stems essentially from an in\fnite series which\ncontain higher-order relativistic contributions to the Gilbert damping parameter. As\n!cscales with c, these higher-order terms will scale with c\u00004or more and thus their\ncontributions will be smaller than the \frst term. Note that the higher-order terms will\ndiverge when != 2!c\u00191021sec\u00001, which means that the theory breaks down at the\nlimit!!2!c. In this limit, the original FW transformation is not de\fned any more\nbecause the particles and antiparticles cannot be separated at this energy limit.13\n4.2. Magnetic inertia parameter\nMagnetic inertial dynamics, in contrast, involves a torque due to the second-order time-\nderivative of the magnetisation. In this case, kmust adopt the value k= 1 in the\nafore-derived two equations (23) and (24). However, if k= 1, the constraint n\u0000k\u00150\ndictates that n\u00151. Therefore, the magnetic inertial dynamics can be described with\nthe following equations:\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=1;2;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!@n\u00001( 1+\u001f\u00001)\n@tn\u00001\u0001@2M\n@t2; (29)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u00021X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!h\u0012@n\u00001( 1+\u001f\u00001)\n@tn\u00001\u0001@2M\n@t2\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@2M\n@t2\u001b\n\u0001p\u0013Ei\n: (30)\nSimilar to the Gilbert damping dynamics, these dynamical terms can be expressed\nasM\u0002\u0010\nI\u0001@2M\n@t2\u0011\nwhich is the magnetic inertial dynamics [8]. The corresponding\nparameter has the following expression\nIij=e\u00160\n2m1X\nn=1;2;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001\n+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!h@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001(hrlpli\u0000hripli)i\n: (31)\nNote that as ncannot adopt the value n= 0, the starting values of nare di\u000berent in\nthe two terms. Importantly, if n= 1 we recover the expression for the lowest order\nmagnetic inertia parameter Dij, as given in the equation (19) [24].\nUsing similar arguments as in the case of the generalised Gilbert damping\nparameter, when we consider a single harmonic \feld as driving \feld, the inertia\nparameter can be rewritten as follows (see Appendix A for detailed calculations)\nIij=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c4\u0012\u0000!2+ 4!!c\n(2!c\u0000!)2\u0013\n\u001f\u00001\nij\n+e\u00160\n8m3c4~\ni(hrlpli\u0000hripli)\u001216!!3\nc\n(4!2\nc\u0000!2)2\u0013\n\u001f\u00001\nij: (32)\nThe \frst term here is exactly the same as the one that was obtained in our earlier\ninvestigation [24]. However, there are now two extra terms which depend on the\nfrequency of the driving \feld and that vanish for !!0. Again, in the limit !!2!c,\nthese two terms diverge and hence this expression is not valid anymore. The inertia\nparameter will consistently be given by the real part of the susceptibility.14\n5. Summary\nWe have developed a generalised LLG equation of motion starting from fundamental\nquantum relativistic theory. Our approach leads to higher-order relativistic correction\nterms in the equation of spin dynamics of Landau and Lifshitz. To achieve this, we have\nstarted from the foundational Dirac equation under the presence of an electromagnetic\n\feld (e.g., external driving \felds or THz excitations) and have employed the FW\ntransformation to separate out the particles from the antiparticles in the Dirac equation.\nIn this way, we derive an extended Pauli Hamiltonian which e\u000eciently describes the\ninteractions between the quantum spin-half particles and the applied \feld. The thus-\nderived direct \feld-spin interaction Hamiltonian can be generalised for any higher-order\nrelativistic corrections and has been expressed as a series. To derive the dynamical\nequation, we have used this generalised spin Hamiltonian to calculate the corresponding\nspin dynamics using the Heisenberg equation of motion. The obtained spin dynamical\nequation provides a generalisation of the phenomenological LLG equation of motion\nand moreover, puts the LLG equation on a rigorous foundational footing. The equation\nincludes all the torque terms of higher-order time-derivatives of the magnetisation (apart\nfrom the Gilbert damping and magnetic inertial dynamics). Speci\fcally, however, we\nhave focussed on deriving an analytic expression for the generalised Gilbert damping\nand for the magnetic inertial parameter. Our results show that both these parameters\ncan be expressed as a series of higher-order relativistic contributions and that they\nare tensors. These series can be summed up for the case of a harmonic driving \feld,\nleading to closed analytic expressions. We have further shown that the imaginary part\nof the susceptibility contributes to the Gilbert damping parameter while the real part\ncontributes to the magnetic inertia parameter. Lastly, with respect to the applicability\nlimits of the derived expressions we have pointed out that when the frequency of the\ndriving \feld becomes comparable to the Compton pulsation, our theory will not be valid\nanymore because of the spontaneous particle-antiparticle pair-production.\n6. Acknowledgments\nWe thank P-A. Hervieux for valuable discussions. This work has been supported\nby the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation\n(Contract No. 2015.0060), the European Union's Horizon2020 Research and\nInnovation Programme under grant agreement No. 737709 (FEMTOTERABYTE,\nhttp://www.physics.gu.se/femtoterabyte).15\nAppendix A. Detailed calculations of the parameters for a harmonic \feld\nIn the following we provide the calculational details of the summation towards the results\ngiven in Eqs. (28) and (32).\nAppendix A.1. Gilbert damping parameter\nEq. (27) can be expanded as follows\nGij=e\u00160\n2m1\n2i!c( 1+\u001f\u00001)ij+e\u00160\n4m2c2(hrlpli\u0000hrlpii) ( 1+\u001f\u00001)ij\n+e\u00160\n2m1X\nn=1;2;:::\u00121\n2i!c\u0013n+1\n(i!)n\u001f\u00001\nij+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013n\n(hrlpli\u0000hrlpii) (i!)n\u001f\u00001\nij\n=e\u00160\n2m1\n2i!c( 1+\u001f\u00001)ij+e\u00160\n4m2c2(hrlpli\u0000hrlpii) ( 1+\u001f\u00001)ij\n+e\u00160\n2m1\n2i!c1X\nn=1;2;:::\u0012!\n2!c\u0013n\n\u001f\u00001\nij+e\u00160\n4m2c21X\nn=2;4;:::\u0012!\n2!c\u0013n\n(hrlpli\u0000hrlpii)\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n~\ni1X\nn=1;2;:::\u0012!\n2!c\u0013n\n+ (hrlpli\u0000hrlpii)1X\nn=2;4;:::\u0012!\n2!c\u0013n#\n\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\u0014~\ni!\n2!c\u0000!+ (hrlpli\u0000hrlpii)!2\n4!2\nc\u0000!2\u0015\n\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n(2!!c+!2)~\ni+!2(hrlpli\u0000hrlpii)\n4!2\nc\u0000!2#\n\u001f\u00001\nij: (A.1)\nWe have used the fact that!\n!c<1 and the summation formula\n1 +x+x2+x3+:::=1\n1\u0000x;\u00001<x< 1: (A.2)REFERENCES 16\nAppendix A.2. Magnetic inertia parameter\nEq. (31) can be expanded as follows\nIij=e\u00160\n2m\u00121\n2i!c\u00132\n( 1+\u001f\u00001)ij+e\u00160\n2m1X\nn=2;3;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001\n+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!h@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001(hrlpli\u0000hripli)i\n=e\u00160\n2m\u00121\n2i!c\u00132\n( 1+\u001f\u00001)ij+1X\nn=2;3;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!(i!)n\u00001\u001f\u00001\nij\n+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!(hrlpli\u0000hripli) (i!)n\u00001\u001f\u00001\nij\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij+e\u00160\n2m\u00121\n2i!c\u001321X\nn=2;3;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!\u001f\u00001\nij\n+e\u00160\n4m2c2\u00121\n2i!c\u00131X\nn=2;4;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!(hrlpli\u0000hripli)\u001f\u00001\nij\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c41X\nn=2;3;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!\u001f\u00001\nij\n+e\u00160\n8m3c4~\ni1X\nn=2;4;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!(hrlpli\u0000hripli)\u001f\u00001\nij\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c4\u001f\u00001\nij\"\n2\u0012!\n2!c\u0013\n+ 3\u0012!\n2!c\u00132\n+ 4\u0012!\n2!c\u00133\n+:::#\n+e\u00160\n8m3c4~\ni(hrlpli\u0000hripli)\u001f\u00001\nij\"\n2\u0012!\n2!c\u0013\n+ 4\u0012!\n2!c\u00133\n+ 6\u0012!\n2!c\u00135\n+:::#\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c4\u0012\u0000!2+ 4!!c\n(2!c\u0000!)2\u0013\n\u001f\u00001\nij\n+e\u00160\n8m3c4~\ni(hrlpli\u0000hripli)\u001216!!3\nc\n(4!2\nc\u0000!2)2\u0013\n\u001f\u00001\nij: (A.3)\nHere we have used the formula\n1 + 2x+ 3x2+ 4x3+ 5x4+:::=1\n(1\u0000x)2;\u00001<x< 1: (A.4)\nReferences\n[1] Landau L D and Lifshitz E M 1935 Phys. 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Matter 29\n194002 URL http://stacks.iop.org/0953-8984/29/i=19/a=194002\n[40] Paillard C, Mondal R, Berritta M, Dkhil B, Singh S, Oppeneer P M and Bellaiche\nL 2016 Proc. SPIE 9931 99312E{16 URL http://dx.doi.org/10.1117/12.\n2238196"    },    {        "title": "1810.11016v2.Time_retarded_damping_and_magnetic_inertia_in_the_Landau_Lifshitz_Gilbert_equation_self_consistently_coupled_to_electronic_time_dependent_nonequilibrium_Green_functions.pdf",        "content": "Time-retarded damping and magnetic inertia in the Landau-Lifshitz-Gilbert equation\nself-consistently coupled to electronic time-dependent nonequilibrium Green functions\nUtkarsh Bajpai and Branislav K. Nikoli´ c∗\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\nThe conventional Landau-Lifshitz-Gilbert (LLG) equation is a widely used tool to describe dy-\nnamics of local magnetic moments, viewed as classical vectors of fixed length, with their change\nassumed to take place simultaneously with the cause. Here we demonstrate that recently devel-\noped [M. D. Petrovi´ c et al. , Phys. Rev. Applied 10, 054038 (2018)] self-consistent coupling of\nthe LLG equation to time-dependent quantum-mechanical description of electrons—where nonequi-\nlibrium spin density from time-dependent nonequilibrium Green function (TDNEGF) calculations\nis inserted within a torque term into the LLG equation while local magnetic moments evolved by\nthe LLG equation introduce time-dependent potential in the quantum Hamiltonian of electrons—\nmicroscopically generates time-retarded damping in the LLG equation described by a memory kernel\nwhich is also spatially dependent. For sufficiently slow dynamics of local magnetic moments on the\nmemory time scale, the kernel can be expanded into power series to extract the Gilbert damping\n(proportional to first time derivative of magnetization) and magnetic inertia (proportional to second\ntime derivative of magnetization) terms whose parameters, however, are time-dependent in contrast\nto time-independent parameters used in the conventional LLG equation. We use examples of single\nor multiple local magnetic moments precessing in an external magnetic field, as well as field-driven\nmotion of a magnetic domain wall (DW), to quantify the difference in their time evolution computed\nfrom conventional LLG equation vs. TDNEGF+LLG quantum-classical hybrid approach. The faster\nDW motion predicted by TDNEGF+LLG approach reveals that important quantum effects, stem-\nming essentially from a finite amount of time which it takes for conduction electron spin to react\nto the motion of classical local magnetic moments, are missing from conventional classical micro-\nmagnetics simulations. We also demonstrate large discrepancy between TDNEGF+LLG-computed\nnumerically exact and, therefore, nonperturbative result for charge current pumped by a moving\nDW and the same quantity computed by perturbative spin motive force formula combined with the\nconventional LLG equation.\nI. INTRODUCTION\nThe conventional Landau-Lifshitz-Gilbert (LLG)\nequation [1–3] is the cornerstone of numerical micro-\nmagnetics [4] and atomistic spin dynamics [5] where one\nsimulates the classical time evolution of many magnetic\nunits coupled by exchange or magnetostatic interactions.\nThe LLG equation\n∂m(r,t)\n∂t=−gm(r,t)×Beff(r,t)+λGm(r,t)×∂m(r,t)\n∂t,\n(1)\ndescribes time evolution of m(r,t) as the unit vector\n|m|= 1 of constant length representing the direction\nof the local magnetization. Here gis the gyromag-\nnetic ratio and Beffis the sum of an external mag-\nnetic field and effective magnetic fields due to magnetic\nanisotropy and exchange coupling (additional stochastic\nmagnetic field can contribute to Beffto take into ac-\ncount finite temperature effects [6]). The second term\non the right-hand side of Eq. (1) is introduced phe-\nnomenologically to break the time-inversion symmetry,\nthereby generating a damping mechanism. The conven-\ntional intrinsic Gilbert damping λGis assumed to be ma-\nterials specific and, therefore, time-independent parame-\n∗bnikolic@udel.eduter. It is typically computed using the so-called breath-\ning Fermi surface [7] or torque-torque correlation formu-\nlas [8] within single-particle quantum-mechanical frame-\nwork (additional many-body processes have to be taken\ninto account to make λGfinite in the clean limit at low\ntemperatures [9]). In the original form, Eq. (1) is written\nfor a bulk material as a highly nonlinear partial differen-\ntial equation. It can also be re-written for a macrospin or\na lattice of atomic spins leading to a system of nonlinear\nordinary differential equations [5].\nIn the case of conducting ferromagnets, LLG equation\nhas to be extended by including additional terms, such as:\n(i) spin-transfer torque [10] T∝/angbracketleftˆs/angbracketright×mdue to injected\nelectrons generating nonequilibrium spin density /angbracketleftˆs/angbracketrightthat\nis noncollinear to local magnetization; ( ii) additional\nGilbert damping, ( g↑↓/4π)m×∂m/∂t, due to pumping\nof spin currents by the dynamics of m(t) whereg↑↓is the\nso-called spin-mixing conductance [11]; ( iii) additional\nnonlocal Gilbert damping [12–18], m×(D·∂m/∂t), due\nto spin pumping by noncollinear magnetic textures where\nDαβ=η/summationtext\ni(m×∂im)α(m×∂im)βis the 3×3 damping\ntensor,∂i=∂/∂iandα,β,i∈{x,y,z}; and ( iv) mag-\nnetic inertia [19–25], Im×∂2m/∂t2, of relevance to ul-\ntrafast magnetization dynamics. Like the original Gilbert\ndamping parameter λGin Eq. (1), T,g↑↓,DandIrequire\nmicroscopic quantum-mechanical calculations which are\noften combined [8, 9, 26–31] with first-principles Hamil-\ntonians of realistic materials.\nFurthermore, generalizations of LLG equation havearXiv:1810.11016v2  [cond-mat.mes-hall]  6 Dec 20182\nbeen considered to take into account the retardation ef-\nfects [32, 33]\n∂m(r,t)\n∂t=\u0002t\n0dt/prime\u0002\nd3r/primeΓ(r,t;r/prime,t/prime)m(r/prime,t/prime)\n×/bracketleftbigg\n−gBeff(r/prime,t/prime) +λG∂m(r/prime,t/prime)\n∂t/prime/bracketrightbigg\n,(2)\nby introducing a memory kernel Γ( r,t;r/prime,t/prime). The mem-\nory kernel models space-time correlation between local\nmagnetic moments, i.e., the fact that the cause for the\nchange of local magnetization occurs at time t−t/primeand\nat position r−r/primewhile the effect at position ris vis-\nible at the later time t. It has been specified phe-\nnomenologically, such as the sum of an instantaneous\nand time-dependent part which exponentially decays on\na characteristic time scale defining the strength of mem-\nory [32, 33]. It is also often simplified [33] by considering\ntime-retardation only, Γ( r,t;r/prime,t/prime)→Γ(t,t/prime), so that any\nspace-retardation effects are included only through the\neffective field Beff(r,t/prime).\nThe time-retardation described by Γ( t,t/prime) is a damping\nmechanism in addition to well-established mechanisms—\nthe combined effects of spin-orbit coupling and electron-\nphonon interaction [7, 8]—which govern λGin Eq. (1).\nHowever, the magnitude of Γ( t,t/prime) cannot be deduced\nfrom purely phenomenological considerations [32, 33].\nInstead, the introduction of the memory kernel Γ( t,t/prime)\ncan be justified microscopically [6, 22, 34–37] by us-\ningquantum-classical hybrid approaches, where time-\ndependent quantum formalism is used to compute /angbracketleftˆs/angbracketright(t)\nwhich is then fed into the LLG equation, while in turn, lo-\ncal magnetization from the LLG equation generates time-\ndependent field in the quantum Hamiltonian of electrons.\nAlthough electron dynamics is assumed to be much faster\nthan that of local magnetic moments, it still takes fi-\nnite time for electron spin to react to new position of\nm(r,t). This is the fundamental reason for time-retarded\ndamping effects encoded by Eq. (2), which are present\neven if the intrinsic Gilbert damping in Eq. (1) is van-\nishingly small due to small spin-orbit coupling (nonzero\nλGrequires spin-orbit coupling [7–9] and scales quadrati-\ncally with it [38]). Since classical micromagnetics simula-\ntions typically use only the conventional intrinsic Gilbert\ndamping term in Eq. (1), while not considering explicitly\nthe flow of conduction electrons in the presence of mag-\nnetization dynamics, the question arises about the mag-\nnitude of neglected effects like time-retarded damping in\nstandard simulations of magnetic-field- or current-driven\ndynamics of noncollinear magnetic textures such as mag-\nnetic domain walls (DWs) [39–47] and skyrmions [48, 49].\nAlthough quantum-classical approaches which auto-\nmatically include time-retardation effects have been dis-\ncussed previously [6, 22, 34–37], they have been focused\non the simples examples where one or two local magnetic\nmoments (pertinent to, e.g., magnetic molecules) inter-\nact with either closed electronic quantum system [22, 35]\n(i.e., not attached to macroscopic reservoirs to allow elec-\ntron spin and charge currents to flow into and from an\nFIG. 1. Schematic view of two-terminal devices where an infi-\nnite 1D TB chain, describing electrons quantum-mechanically,\nis attached to two macroscopic reservoirs while its middle part\nhosts: (a) single local magnetic moment, initially oriented in\nthe +x-direction, placed in an external magnetic field point-\ning along the + z-direction; (b) 11 local magnetic moments\n(illustration shows 7 of them), initially oriented in the + x-\ndirection, placed in an external magnetic field pointing along\nthe +z-direction; (c) three-site-wide head-to-head magnetic\nDW whose motion is driven by an external magnetic field\npointing in the + x-direction. Electrons within 1D TB chain\nand classical local magnetic moments interact via the s-dex-\nchange coupling of strength Jsd, and classical local magnetic\nmoments within the DW in (c) additionally interact with each\nother via the Heisenberg exchange coupling of strength J.\nexternal circuit), or open electronic quantum system but\nemploying approximations [6, 34, 36, 37] to obtain analyt-\nical solution. Thus, these approaches are not suitable for\nsimulations of spintronic devices containing large number\nof noncollinear local magnetic moments.\nHere we employ recently developed [50] numerically\nexact and, therefore, nonperturbative algorithm combin-\ning time-dependent nonequilibrium Green function for-\nmalism [51, 52] with the conventional LLG Eq. (1) (TD-\nNEGF+LLG) to demonstrate how it effectively generates\ntime-retardation effects, whose memory kernel can be ex-\nplicitly extracted in terms of TDNEGFs only in some lim-\nits (such as weak electron-spin/local-magnetic-moment\ninteraction and weak coupling of the active region to\nmacroscopic reservoirs). The paper is organized as fol-\nlows. Section II A introduces model quantum Hamilto-\nnian for electronic subsystem and classical Hamiltonian\nfor the subsystem comprised of local magnetic moments.\nIn Sec. II B, we show how the nonequilibrium expectation3\nvalue of spin density\n/angbracketleftˆs/angbracketrighti(t) =~\n2Tr [(\u001aneq(t)−\u001aeq)|i/angbracketright/angbracketlefti|⊗\u001b], (3)\ninserted into the LLG Eq. (1) generates a memory ker-\nnel because of the structure of the nonequilibrium time-\ndependent density matrix \u001aneq(t) obtained from TD-\nNEGF calculations. Here \u001aeqis the grand canonical equi-\nlibrium density matrix; \u001b= (ˆσx,ˆσy,ˆσz) is the vector of\nthe Pauli matrices; |i/angbracketrightelectron orbital centered on site i;\nand the operator |i/angbracketright/angbracketlefti|⊗\u001bacts in the composite Hilbert\nspaceH=Horb⊗Hspinof electronic orbital and spin\ndegrees of freedom. In this Section, we also discuss how\nin the limit of slow magnetization dynamics the memory\nkernel can be expanded in a Taylor series in order to ex-\ntract conventional Gilbert damping and magnetic inertia\nterms, but with time- and spatially-dependent parame-\ntersλD\ni(t) andID\ni(t). In Secs. III A–III C we compare the\ndynamics of local magnetic moments driven by an exter-\nnal magnetic field as computed by TDNEGF+LLG vs.\nconventional LLG simulations for three one-dimensional\n(1D) examples depicted in Fig. 1(a)–(c), respectively.\nSec. III C also compares pumped charge current due to\nthe DW motion as computed by TDNEGF+LLG vs. the\nwidely-used spin motive force (SMF) theory [12, 53] com-\nbined [54–56] with the conventional LLG equation. We\nconclude in Sec. IV.\nII. MODELS AND METHODS\nA. Coupled quantum and classical Hamiltonians\nThe conduction electron subsystem is modeled by a\nquantum Hamiltonian\nH(t) =−γ/summationdisplay\n/angbracketleftij/angbracketrightˆc†\niˆci−Jsd/summationdisplay\niˆc†\ni\u001b·Mi(t)ˆci, (4)\nwhich is (assumed to be 1D for simplicity) tight-binding\n(TB) model where electron interacts with magnetic mo-\nments localized at sites iand described by the classical\nvector Mi(t) of unit length. Here ˆ c†\ni= (ˆc†\ni↑,ˆc†\ni↓) is a row\nvector containing operators ˆ c†\niσwhich create an electron\nof spinσ=↑,↓at sitei; ˆciis a column vector that con-\ntains the corresponding annihilation operators; γ= 1 eV\nis the nearest neighbor hopping; and Jsdis thes-dex-\nchange coupling parameter between conduction electrons\nand local magnetic moments. The active region of de-\nvices depicted in Fig. 1(a)–(c) consists of 1, 11 and 21\nTB sites, respectively. These are attached to the left (L)\nand right (R) semi-infinite ideal leads modeled by the\nsame Hamiltonian in Eq. (4) but with Jsd= 0 eV. The\nleads are assumed to terminate into macroscopic reser-\nvoirs kept at the same chemical potential since we do not\napply any bias voltage to the devices in Fig. 1(a)–(c).The classical Hamiltonian describing the local mag-\nnetic moments is given by\nH=−J/summationdisplay\nijMi·Mj−µM/summationdisplay\niMi·Bi\next−K/summationdisplay\ni(Mx\ni)2\n−Jsd/summationdisplay\ni/angbracketleft^s/angbracketrighti·Mi,(5)\nwhereJis the Heisenberg exchange coupling parame-\nter;Bi\nextis the applied external magnetic field; Kis the\nmagnetic anisotropy (in the x-direction) and/angbracketleftˆs/angbracketrightiis the\nnonequilibrium electronic spin density computed from\nEq. (3).\nB. Time-retarded damping and magnetic inertia in\nthe LLG equation self-consistently coupled to\nTDNEGF\nThe quantum equation of motion for the nonequilib-\nrium density matrix of electrons [57, 58]\ni~∂\u001aneq(t)\n∂t= [H(t),\u001aneq(t)] +/summationdisplay\np=L,Ri[Πp(t) +Π†\np(t)].\n(6)\nis an example of a master equation for an open (i.e., con-\nnected to macroscopic reservoirs) quantum system [59]\ndue to the presence of the second term on the right hand\nside, in addition to standard terms of the von Neumann\nequation. This term and the density matrix itself can be\nexpressed using TDNEGF formalism [51, 52] as\n\u001aneq(t) =1\niG<(t,t/prime)|t=t/prime, (7)\nΠp(t/prime) =\u0002t/prime\n−∞dt1[G>(t/prime,t1)Σ<\np(t1,t/prime)−\nG<(t/prime,t1)Σ>\np(t1,t/prime)].(8)\nThe central quantities of the TDNEGF formalism are\nthe retarded Gr,σσ/prime\nii/prime(t,t/prime) =−iΘ(t−t/prime)/angbracketleft{ˆciσ(t),ˆci/primeσ/prime(t)}/angbracketright\nand the lesser G<,σσ/prime\nii/prime(t,t/prime) =i/angbracketleftˆc†\ni/primeσ/prime(t/prime)ˆciσ(t)/angbracketrightGreen\nfunctions (GFs) which describe the available density of\nstates and how electrons occupy those states, respec-\ntively. In addition, it is also useful to introduce the\ngreater GF, G>(t,t/prime) = [G<(t/prime,t)]†, and the advanced\nGF,Ga(t,t/prime) = [Gr(t,t/prime)]†. The current matrices Πp(t)\nmake it possible to compute directly [57, 58] charge cur-\nrent\nIp(t) =e\n~Tr[Πp(t)], (9)\nand spin current\nISα\np(t) =e\n~Tr[ˆσαΠp(t)], (10)4\nin the L and R semi-infinite leads. The equation of mo-\ntion for the lesser and greater GFs is given by\ni~∂G>,<(t,t1)\n∂t=H(t)G>,<(t,t1)+\n+∞\u0002\n−∞dt2/bracketleftbigg\nΣr\ntot(t,t2)G>,<(t2,t) +Σ>,<\ntot(t,t2)Ga(t2,t)/bracketrightbigg\n,\n(11)\nwhere Σr,>,<\ntot (t,t2) =/summationtext\np=L,RΣr,>,<\np (t,t2) and\nΣr,>,<\np (t,t2) are the lead self-energy matrices [52, 57, 58].\nThe classical equation of motion for the magnetic mo-\nment localized at site iis the Landau-Lifshitz equation\n∂Mi(t)\n∂t=−gMi(t)×Beff\ni(t), (12)\nwhere the effective magnetic field is\nBeff\ni(t) =−1\nµM∂H/∂MiandµMis the magnitude\nof the magnetic moment [5].\nThe full TDNEGF+LLG framework [50], which we\nalso denote as TDNEGF \u001cLLG, consists of self-\nconsistent combination of Eq.(6) and (12) where one first\nsolves for the nonequilibrium electronic spin density in\nEq.(3), which is then fed into Eq. (12) to propagate local\nmagnetic moments Mi(t) in the next time step. Evolving\n\u001aneq(t) via Eq. (6) requires time step δt= 0.1 fs for nu-\nmerical stability, and we use the same time step to evolve\nLLG or Landau-Lifshitz equations for Mi(t). These up-\ndated local magnetic moments are fed back into the quan-\ntum Hamiltonian of conduction electron subsystem in\nEq. (6). Thus obtained solutions for Mi(t),/angbracketleftˆs/angbracketrighti(t),Ip(t)\nandISαp(t) are numerically exact. For testing the im-\nportance of the self-consistent feedback loop, we also use\nTDNEGF←LLG where TDNEGF is utilized to obtain\nIp(t) andISαp(t) while the local magnetic moments are\nevolved solely by the conventional LLG Eq. (1), i.e., by\nusingJsd≡0 in Eq. (5) but Jsd/negationslash= 0 is used in Eq. (4).\nIn the weak-coupling limit [34, 60] (i.e., small Jsd)\nfor electron-spin/local-magnetic-moment interaction it is\npossible to extract explicitly the generalized LLG equa-\ntion with a memory kernel. For this purpose we use the\nfollowing expansions in the powers of small Jsd\n\u001aneq(t) =∞/summationdisplay\nn=0\u001an(t)Jn\nsd, (13)\nΠp(t/prime) =∞/summationdisplay\nn=0Π(n)\np(t/prime)Jn\nsd, (14)\nGr,a,>,<(t/prime,t1) =∞/summationdisplay\nn=0Gr,a,>,<\nn (t/prime,t1)Jn\nsd. (15)In Appendix A, we show how to combine Eqs.(6), (11),\n(13), (14) and (15) to obtain the perturbative equation\n∂Mi(t)\n∂t=−g/bracketleftbigg\nMi(t)×Beff,0\ni(t)+\nJ2\nsd\nµM/summationdisplay\np=L,RMi(t)×+∞\u0002\n−∞dt/prime/primeMi(t/prime/prime){Kp\ni(t/prime/prime,t)+Kp∗\ni(t/prime/prime,t)}/bracketrightbigg\n,\n(16)\nfor the dynamics of each local magnetic moment at site i,\nby retaining only the terms linear in Jsdin Eqs. (13)–(15).\nHere Beff,0\ni≡−1\nµM∂H0/∂Mi,H0is the classical Hamil-\ntonian in Eq. (5) with Jsd≡0 and Kp\ni(t/prime/prime,t) is defined in\nAppendix A. The physical origin [35] of time-retardation\neffects described by the second term in Eq. (16) is that,\neven though electron dynamics is much faster than the\ndynamics of local magnetic moments, the nonequilibrium\nspin density in Eq. (3) is always behind Mi(t) and, there-\nfore, never parallel to it which introduces spin torque\nterm into the Landau-Lifshitz Eq. (12). In other words it\ntakes finite amount of time for conduction electron spin\nto react to the motion of classical local magnetic mo-\nments, so that nonequilibrium electrons effectively me-\ndiate interaction of Mi(t) with the same local magnetic\nmoment at time t/prime< t. In the full TDNEGF+LLG,\nsuch retardation effects are mediated by the nonequilib-\nrium electrons starting at site iat timet/primeand returning\nback to the same site at time t > t/prime, while in the per-\nturbative limit the same effect is captured by the second\nterm in Eq. (16). The perturbative formula Eq. (16) is\nexpected [35] to breakdown after propagation over time\nt∼~/Jsd.\nFurther approximation to Eq. (16) can be made by\nconsidering sufficiently slow dynamics of local magnetic\nmoments so that higher order terms in the Taylor series\nMi(t/prime/prime)≈Mi(t)+∂Mi(t)\n∂t(t/prime/prime−t)+1\n2∂2Mi(t)\n∂t2(t/prime/prime−t)2+...,\n(17)\ncan be neglected. By defining the following quantities\nλD\np,i(t)≡+∞\u0002\n−∞dt/prime/prime(t/prime/prime−t)[Kp\ni(t/prime/prime,t) + K∗p\ni(t/prime/prime,t)],(18)\nand\nID\np,i(t)≡1\n2+∞\u0002\n−∞dt/prime/prime(t/prime/prime−t)2[Kp\ni(t/prime/prime,t) + K∗p\ni(t/prime/prime,t)],(19)\nand by retaining terms up to the second order in Eq. (17)5\nwe obtain the conventionally looking LLG equation\n∂Mi(t)\n∂t=−g/bracketleftbigg\nMi(t)×Beff,0\ni(t)+\nJ2\nsd\nµM/braceleftbigg/summationdisplay\np=L,RλD\np,n(t)/bracerightbigg\nMi(t)×∂Mi(t)\n∂t+\nJ2\nsd\nµM/braceleftbigg/summationdisplay\np=L,RID\np,i(t)/bracerightbigg\nMi(t)×∂2Mi(t)\n∂t2/bracketrightbigg\n.(20)\nHowever, the Gilbert damping term prefactor\nλD\ni(t) =J2\nsd\nµM/summationdisplay\np=L,RλD\np,i(t), (21)\nand the magnetic inertia term prefactor\nID\ni(t) =J2\nsd\nµM/summationdisplay\np=L,RID\np,i(t), (22)\nin Eq. (20) are now time- and position-dependent. This\nis in sharp contrast to conventional LLG Eq. (1) em-\nployed in classical micromagnetics where Gilbert damp-\ning and magnetic inertia prefactors are material specific\nconstants.\nIII. RESULTS AND DISCUSSION\nA. Single local magnetic moment in an external\nmagnetic field\nTo compare the dynamics of local magnetic moments\nin full TDNEGF+LLG quantum-classical simulations vs.\nconventional LLG classical simulations, we first consider\na well-known example [5] for which the conventional LLG\nequation can be analytically solved—a single local mag-\nnetic moment which at t= 0 points along the + x-\ndirection and then starts to precesses due to an external\nmagnetic field pointing in the + z-direction. Its trajectory\nis given by [5]\nMx(t) = sech/parenleftbigggλGB\n1 +λGt/parenrightbigg\ncos/parenleftbigggB\n1 +λ2\nGt/parenrightbigg\n,(23a)\nMy(t) = sech/parenleftbigggλGB\n1 +λGt/parenrightbigg\nsin/parenleftbigggB\n1 +λ2\nGt/parenrightbigg\n,(23b)\nMz(t) = tanh/parenleftbigggλGB\n1 +λGt/parenrightbigg\n, (23c)\nwhere B= (0,0,B) is the applied external mag-\nnetic field. Thus, if the conventional intrinsic Gilbert\ndamping parameter is set to zero, λG= 0, then\nthe local magnetic moment precesses steadily around\nthez-axis with Mz≡0. On the other hand,\nfor nonzero λG>0, the local magnetic moment\nFIG. 2. (a) Time dependence of tanh−1(Mz) for a sin-\ngle local magnetic moment in Fig. 1(a) obtained from TD-\nNEGF+LLG simulations. Colors red to blue indicate in-\ncreasings-dexchange coupling in steps of 0 .1 eV, ranging\nfromJsd= 0 eV toJsd= 1.9 eV. (b) The dynamical Gilbert\ndamping parameter in Eq. (21) extracted from panel (a) as\na function of Jsd. (c) Time dependence of Mzcomponent\nfor a single local magnetic moment in Fig. 1(a) at large\nJsd= 2.0 eV exhibits nutation as a signature of magnetic in-\nertia. To generate fast magnetization dynamics and reduce\nsimulation time, we use an unrealistically large external mag-\nnetic field of strength B= 1000 T. The conventional intrinsic\nGilbert damping parameter is set to zero, λG= 0, and the\nFermi energy is EF= 0 eV.\nwill relax towards the direction of magnetic field,\ni.e., lim\nt→∞(Mx(t),My(t),Mz(t)) = (0,0,1). Thus, such\ndamped dynamics is signified by a linear tanh−1(Mz)\nvs. time dependence. Figure 2 plots results of TD-\nNEGF+LLG simulations for the same problem. Even\nthough we set conventional intrinsic Gilbert damping\nto zero,λG= 0, Fig. 2(a) shows linear tanh−1(Mz) vs.\ntime, independently of the strength of s-dexchange cou-\npling as long as Jsd.2 eV. This means that the lo-\ncal magnetic moment is experiencing (time-independent)\ndynamical Gilbert damping λD∝J2\nsd, in accord with\nEq. (21) and as shown in Fig. 2(b), which is generated\nsolely by the TDNEGF part of the self-consistent loop\nwithin the full TDNEGF+LLG scheme.\nForJsd&2 eV, the dynamics of the local magnetic mo-\nment also exhibits nutation [35], as shown in Fig. 2(c),\nwhich is the signature of the magnetic inertia [19–24]\nterm∝Mi×∂2Mi/∂t2in Eq. (20). Thus, nutation be-\ncomes conspicuous when the dynamics of the local mag-\nnetic moments is sufficiently fast, so that ∂2Mi/∂t2is\nlarge, as well as when the interaction between the itiner-\nant and localized spins is sufficiently large.6\nFIG. 3. TDNEGF+LLG-computed trajectories\n(Mx(t),My(t),Mz(t)) on the Bloch sphere of local magnetic\nmoment in the setup of Fig. 1(b) at: (a) site 1; and (c) site\n6. The total number of local magnetic moments is N= 11,\nand they do not interact with each other via exchange\ncoupling [i.e., J= 0 eV in Eq. (5)]. Panels (b) and (d) show\nthe corresponding time dependence of Mzcomponent from\npanels (a) and (c), respectively. The external magnetic\nfield isB= 1000 T, and the s-dexchange coupling strength\nJsd= 0.1 eV is nonperturbative in this setup, therefore,\nnotallowing us to extract explicitly the dynamical Gilbert\ndamping parameter from Eq. (21). The conventional intrinsic\nGilbert damping parameter is set to zero, λG= 0, and the\nFermi energy is EF= 0 eV.\nB. Multiple exchange-uncoupled local magnetic\nmoments in an external magnetic field\nIn order to examine possible spatial dependence of the\ndynamical Gilbert damping parameter or emergence of\ndynamical exchange coupling [61, 62] between local mag-\nnetic moments, we consider a chain of N= 11 magnetic\nmoments which do not interact with each other ( J= 0)\nbut interact with conduction electron spin ( Jsd/negationslash= 0), as\nillustrated in Fig. 1(b). At t= 0, all magnetic moments\npoint in the + x-direction while the external magnetic\nfield is in the + zdirection, and the conventional intrin-\nsic Gilbert damping is set to zero, λG= 0.\nFigures 3(a) and 3(c) show the trajectory of selected\nlocal magnetic moments ( i= 1 and 6) on the Bloch\nsphere forJsd= 0.1 eV. In contrast to single local mag-\nnetic moment in Fig. 2(a), for which tanh−1(Mz) vs.\ntime is linear using Jsd= 0.1 eV, we find that in case of\nmultiple exchange-uncoupled magnetic moments this is\nno longer the case, as demonstrated by Figs. 3(b) and\n3(d). Hence, the trajectory followed by these local mag-\nnetic moments cannot be described by Eq. (23) so that\nFIG. 4. (a) TDNEGF+LLG-computed time dependence of\nMzcomponent of local magnetic moment on sites 1, 3 and 6 in\nthe setup of Fig. 1(b) with a total of N= 11 moments. (b) Po-\nsition dependence of the dynamical Gilbert damping param-\neter in Eq. (21). The external magnetic field is B= 1000 T,\nand thes-dexchange coupling strength Jsd= 0.01 eV is per-\nturbative in this setup, therefore, allowing us to extract the\ndynamical Gilbert damping explicitly from Eq. (21). The con-\nventional intrinsic Gilbert damping parameter is set to zero,\nλG= 0, and the Fermi energy is EF= 0 eV.\nthe conventional-like Gilbert damping parameter cannot\nbe extracted anymore. Thus, such a nonstandard damp-\ning of the dynamics of local magnetic moments originates\nfrom time-dependence of the dynamical damping param-\neterλD\niin Eq. (21).\nFigure 4(a) shows tanh−1(Mz) vs. time for selected\nlocal magnetic moments ( i= 1,3 and 6) and smaller\nJsd= 0.01 eV. Although all local magnetic moments fol-\nlow linear tanh−1(Mz) vs. time, as predicted by the\nsolution in Eq. (23c) of the conventional LLG equation,\nthe dynamical Gilbert damping extracted from Eq. (23)\nchanges from site to site as shown in Fig. 4(b). Further-\nmore, the linear tanh−1(Mz) vs. time relation breaks\ndown for times t&50 ps at specific sites, which then pre-\nvents extracting time-independent λD\niat those sites.\nC. Magnetic field-driven motion of a domain wall\ncomposed of multiple exchange-coupled local\nmagnetic moments\nIn order to examine difference in predicted dynam-\nics of exchange-coupled local magnetic moments by TD-\nNEGF+LLG framework vs. conventional LLG equa-\ntion, we consider the simplest example of 1D head-to-\nhead magnetic DW depicted in Fig. 1(c). Its motion is\ndriven by applying an external magnetic field in the + x-\ndirection. Some type of damping mechanism is crucial\nfor the DW to move, as demonstrated by solid lines in\nFig. 5(e)–(h), obtained by solving the conventional LLG\nequation with λG= 0, which show how local magnetic\nmoments precess around the magnetic field but without\nnet displacement of the center of the DW.\nOn the other hand, even though we set λG= 0 in TD-\nNEGF+LLG simulations in Fig. 5(a)–(d), the center of\nthe DW moves to the right due to dynamically generated7\nFIG. 5. (a)–(d) TDNEGF+LLG-computed snapshots of head-to-head DW in the setup of Fig. 1(c) driven by an external\nmagnetic field of strength B= 100 T pointing in the + x-direction, in the absence ( λG= 0) or presence ( λG= 0.01) of the\nconventional intrinsic Gilbert damping. Panels (e)–(h) show the corresponding snapshots computed solely by the conventional\nLLG Eq. (1) where in the absence ( λG= 0) of the conventional intrinsic Gilbert damping the DW does not move at all. The\nHeisenberg exchange coupling between local magnetic moments is J= 0.01 eV;s-dexchange coupling between electrons and\nlocal magnetic moments is Jsd= 0.1 eV; magnetic anisotropy (in the x-direction) is K= 0.01 eV; and the Fermi energy of\nelectrons is EF=−1.9 eV. The magnetic field is applied at t= 2 ps, while prior to that we evolve the conduction electron\nsubsystem with TDNEGF until it reaches the thermodynamic equilibrium where all transient spin and charge currents have\ndecayed to zero.\ntime-retarded damping encoded by the memory kernel\nin Eq. (16). Including the conventional intrinsic Gilbert\ndamping,λG= 0.01 as often used in micromagnetic sim-\nulations of DW along magnetic nanowires [43, 44, 63],\nchanges only slightly the result of TDNEGF+LLG sim-\nulations which demonstrates that the effective dynam-\nical Gilbert damping (which is also time-dependent) is\nabout an order of magnitude larger than λG. This is also\nreflected in the DW velocity being much larger in TD-\nNEGF+LLG simulations with λG= 0 in Fig. 5(a)–(d)\nthan in the conventional LLG equation simulations with\nλG= 0.01 in Fig. 5(e)–(h).\nIt has been predicted theoretically [12, 53, 64–68] and\nconfirmed experimentally [69] that a moving DW will\npump charge current even in the absence of any applied\nbias voltage. The corresponding open circuit pumping\nvoltage in the so-called spin motive force (SMF) the-\nory [12, 53] is given by\nVSMF=1\nG0\u0002\njxdx, (24a)\njα(r) =Pσ0~\n2e[∂tm(r,t)×∂αm(r,t)]·m(r,t),(24b)\nwherejxis the pumped local charge current along the\nx-axis. Here σ0=σ↑+σ↓is the total conductivity;\nP= (σ↑−σ↓)/(σ↑+σ↓) is the spin polarization of the\nferromagnet; and ∂t=∂/∂t. Equation (24) is typicallycombined [54–56] with classical micromagnetics which\nsupplies Mi(t) that is then plugged into the discretized\nversion [50]\njx(i)∝1\na[∂tMi(t)×(Mi+1(t)−Mi(t))]·Mi(t)\n∝1\na[∂Mi(t)×Mi+1(t)]·Mi(t). (25)\nof Eq. (24b). We denote this approach as SMF ←LLG,\nwhich is perturbative in nature [67, 70] since it considers\nonly the lowest temporal and spatial derivatives.\nOn the other hand, the same pumping voltage can be\ncomputed nonperturbatively\nVTDNEGF =Ip(t)\nG(t), (26)\nusing TDNEGF expression for charge current in lead p\nin Eq. 9, where TDNEGF calculations are coupled to\nLLG calculations either self-consistently (i.e., by using\nTDNEGF\u001cLLG) or non-self-consistently (i.e., by us-\ning TDNEGF←LLG). Here, G(t) is the conductance\ncomputed using the Landauer formula applied to two-\nterminal devices with a frozen at time ttexture of local\nmagnetic moments.\nFigures 6(a) and 6(b) plot the pumping voltage cal-\nculated by TDNEGF \u001cLLG for DW motion shown in\nFig. 5(a)–(d) in the absence or presence of conventional8\nFIG. 6. Time dependence of pumping voltage generated by\nthe DW motion depicted in Fig. 5(a)–(d) for: (a) λG= 0;\n(b)λG= 0.01. In panels (a) and (b) local magnetic mo-\nments evolve in time by the full TDNEGF+LLG framework\nwhere the arrows indicate how TDNEGF sends nonequilib-\nrium electronic spin density into the LLG equation which, in\nturn, sends trajectories of local magnetic moments into TD-\nNEGF. Time dependence of pumping voltage generated by\nDW motion depicted in Fig. 5(e)–(h) for: (c) λG= 0; (d)\nλG= 0.01. In panels (c) and (d) local magnetic moments\nevolve in time using the conventional LLG equation which\nsends their trajectories into either TDNEGF (green) or SMF\nformulas (blue) in Eq. (26) or Eq. (24), respectively, to obtain\nthe corresponding pumping voltage.\nGilbert damping, respectively. The two cases are virtu-\nally identical due to an order of magnitude larger dynam-\nical Gilbert damping that is automatically generated by\nTDNEGF\u001cLLG in both Figs. 6(a) and 6(b). The\nnonperturbative results in Figs. 6(a) and Fig. 6(b) are\nquite different from SMF ←LLG predictions in Figs. 6(c)\nand Fig. 6(d), respectively. This is due to both failure\nof Eqs. (24) and (25) to describe noncoplanar and non-\ncollinear magnetic textures with neighboring local mag-\nnetic moments tilted by more than 10◦[50] and lack\nof dynamical Gilbert damping in SMF ←LLG simula-\ntions [54–56]. The latter effect is also emphasized by the\ninability of TDNEGF ←LLG in Figs. 6(c) and Fig. 6(d)\nto reproduce the results of self-consistent TDNEGF \u001c\nLLG in Figs. 6(a) and Fig. 6(b), respectively.\nIV. CONCLUSIONS\nIn conclusion, we delineated a hierarchy of theoret-\nical descriptions of a nonequilibrium quantum many-\nbody system in which conduction electron spins inter-\nact with local magnetic moments within a ferromagneticlayer sandwiched between normal metal electrodes. On\nthe top of the hierarchy is a fully quantum approach,\nfor both electrons and local magnetic moments, whose\ncomputational complexity (using either original spin op-\nerators [71, 72] for local magnetic moments, or their\nmapping to bosonic operators in order to enable ap-\nplication of many-body perturbation theory within the\nNEGF formalism [73]) makes it impractical for systems\ncontaining large number of local magnetic moments.\nThe next approach in the hierarchy is computation-\nally much less expensive quantum-classical hybrid [74]\nbased on self-consistent coupling [50] of TDNEGF (which\ncan be implemented using algorithms that scale linearly\nwith both system size and simulation time [52, 58, 75])\nwith classical LLG equation for local magnetic moments.\nSuch TDNEGF+LLG approach is numerically exact and,\ntherefore, nonperturbative in the strength of electron-\nspin/local-magnetic-moment interaction, speed of local\nmagnetic moment dynamics and degree of noncollinearity\nbetween them. Even though electron dynamics is much\nfaster than localized spin dynamics, the most general sit-\nuation cannot be handled by integrating out [6, 34] the\nconduction electron degrees of freedom and by focusing\nonly on the LLG-type equation where a much larger time\nstep can be used to propagate spins only.\nNevertheless, in the limit [34, 60] of weak electron-\nspin/local-magnetic-moment interaction [i.e., small Jsd\nin Eqs. (4) and (5)] one can derive analytically a type\nof generalized LLG equation [34–37] for each local mag-\nnetic moment which is next approach in the hierarchy\nthat sheds light onto different effects included in the nu-\nmerically exact TDNEGF+LLG scheme. Instead of the\nconventional Gilbert damping term in Eq. (1), the gen-\neralized LLG equation we derive as Eq. (16) contains\na microscopically determined memory kernel which de-\nscribes time-retardation effects generated by the coupling\nto TDNEGF. Fundamentally, the memory kernel is due\nto the fact that electron spin can never follow instanta-\nneously change in the orientation of the local magnetic\nmoments [35]. In the limit of slow dynamics of local\nmagnetic moments, one can further expand the memory\nkernel into a Taylor series to obtain the final approach\nwithin the hierarchy whose LLG Eq. (20) is akin to the\nconventional one, but which contains both Gilbert damp-\ning (proportional to first time derivative of local mag-\nnetization) and magnetic inertia terms (proportional to\nsecond time derivative of local magnetization) with time-\ndependent parameters instead of usually assumed mate-\nrials specific constants.\nUsing three simple examples—single or multiple local\nmagnetic moments precessing in an external magnetic\nfield or magnetic-field-driven magnetic DW motion—\nwe demonstrate the importance of dynamically induced\ndamping which operates even if conventional static\nGilbert damping is set to zero. In the case of field-\ndriven magnetic DW motion, we can estimate that the\nstrength of dynamical damping is effectively an order of\nmagnitude larger than typically assumed [43, 44, 63] con-9\nventional static Gilbert damping λG/similarequal0.01 in classical\nmicromagnetic simulations of magnetic nanowires. In ad-\ndition, we show that charge pumping by the dynamics of\nnoncoplanar and noncollinear magnetic textures, which\nis outside of the scope of pure micromagnetic simulations\nbut it is often described by combining [54–56] them with\nthe SMF theory formula [12, 53], requires to take into ac-\ncount both the dynamical Gilbert damping and possiblylarge angle between neighboring local magnetic moments\nin order to reproduce numerically exact results of TD-\nNEGF+LLG scheme.\nACKNOWLEDGMENTS\nThis work was supported by NSF Grant No. ECCS\n150909.\nAppendix A: Derivation of Memory Kernel in LLG equation self-consistently coupled to TDNEGF\nIn this Appendix, we provide a detailed derivation of the memory kernel in Eq. (16). To obtain the perturbative\nequation of motion for local magnetic moments we start from Landau-Lifshitz Eq. (12) where the effective magnetic\nfield can be written as\nBeff\ni(t) =Beff,0\ni(t) +Jsd/angbracketleftˆs/angbracketrighti(t). (A.1)\nThe nonequilibrium spin density is expanded up to terms linear in Jsdusing Eq. (13)\n/angbracketleftˆs/angbracketrighti(t) =~\n2Tr[\u001aneq(t)|i/angbracketright/angbracketlefti|⊗\u001b]−/angbracketleftˆs/angbracketrighti\neq≈~\n2Tr/bracketleftbigg\n{\u001a0(t)+Jsd\u001a1(t)}|i/angbracketright/angbracketlefti|⊗\u001b/bracketrightbigg\n−/angbracketleftˆs/angbracketrighti\neq=Jsd~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗\u001b]−/angbracketleftˆs/angbracketrighti\neq.\n(A.2)\nHere/angbracketleftˆs/angbracketrighti\neqis the equilibrium electronic spin density i.e., /angbracketleftˆs/angbracketrighti\neq= (~/2) Tr [ \u001aeq|i/angbracketright/angbracketlefti|⊗\u001b]. Furthermore, the electronic\nspin density in the zeroth order must vanish, i.e., Tr [ \u001a0(t)|i/angbracketright/angbracketlefti|⊗\u001b] = 0 since for Jsd= 0 electrons are not spin-\npolarized. Hence, we can write Eq. (12) as\n∂Mi(t)\n∂t=−gMi(t)×/bracketleftbigg\nBeff,0\ni(t) +J2\nsd~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗\u001b]−Jsd/angbracketleftˆs/angbracketrighti\neq/bracketrightbigg\n. (A.3)\nTo obtain analytical results, we assume that the equilibrium spin density follows the direction of local magnetic\nmoments, so that Mi(t)×/angbracketleftˆs/angbracketrighti\neq= 0. By expanding Eq. (6) we obtain\ni~∂\u001a0(t)\n∂t= [H0(t),\u001a0(t)] +/summationdisplay\np=L,Ri[Π(0)\np(t) +Π(0)†\np(t)], (A.4)\nand\ni~∂\u001a1(t)\n∂t= [H1(t),\u001a0(t)] +i/summationdisplay\np=L,R[Π(1)\np(t) +Π(1)†\np(t)], (A.5)\nwhere H1(t) =−/summationtext\ni|i/angbracketright/angbracketlefti|⊗\u001b·Mi(t). One can formally integrate Eq. (A.5) which leads to\n~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗\u001b] =/summationdisplay\np=L,R1\n2t\u0002\n−∞dt/primeTr/bracketleftbigg\n{Π(1)\np(t/prime) +Π(1)†\np(t/prime)}|i/angbracketright/angbracketlefti|⊗\u001b/bracketrightbigg\n. (A.6)\nwhich requires to find an expression for Π(1)\np(t/prime). Using Eq. (8) and the fact that lead self-energy matrices do not\ndepend onJsdleads to\nΠ(1)\np(t/prime) =t/prime\u0002\n−∞dt1[G>\n1(t/prime,t1)Σ<\np(t1,t/prime)−G<\n1(t/prime,t1)Σ>\np(t1,t/prime)]. (A.7)\nEquations (11) and (15) can be formally integrated to yield lesser and greater GFs in Eq. (A.7)\nG>,<\n1(t/prime,t1) =1\ni~/parenleftbiggt/prime\u0002\n−∞dt/prime/primeH1(t/prime/prime)G>,<\n0(t/prime/prime,t1) +t/prime\u0002\n−∞dt/prime/prime+∞\u0002\n−∞dt2/bracketleftbigg\nΣr\ntot(t/prime/prime,t2)G>,<\n1(t2,t/prime/prime) +Σ>,<\ntot(t/prime/prime,t2)Ga\n1(t2,t/prime/prime)/bracketrightbigg/parenrightbigg\n.\n(A.8)10\nWe further assume that the active region in Fig. 1 is weakly coupled with semi-infinite leads and, therefore, macroscopic\nreservoirs into which they terminate. This means that after we substitute Eq. (A.8) into Eq. (A.7) we can keep only\nthose terms that are linear in the self-energy\nΠ(1)\np(t/prime) =1\n2it/prime\u0002\n−∞dt/prime/primeH1(t/prime/prime)t/prime\u0002\n−∞dt1/bracketleftbigg\nG>\n0(t/prime/prime,t1)Σ<\np(t1,t/prime)−G<\n0(t/prime/prime,t1)Σ>\np(t1,t/prime)/bracketrightbigg\n(A.9)\n=i\n2/summationdisplay\nit/prime\u0002\n−∞dt/prime/prime|i/angbracketright/angbracketlefti|⊗\u001b·Mi(t/prime/prime)t/prime\u0002\n−∞dt1/bracketleftbigg\nG>\n0(t/prime/prime,t1)Σ<\np(t1,t/prime)−G<\n0(t/prime/prime,t1)Σ>\np(t1,t/prime)/bracketrightbigg\n(A.10)\n=i/summationdisplay\nit/prime\u0002\n−∞dt/prime/prime|i/angbracketright/angbracketlefti|⊗\u001b·Mi(t/prime/prime)A0\np(t/prime/prime,t/prime), (A.11)\nwhere A0\np(t/prime/prime,t/prime) is an operator constructed out of the zeroth order terms in the expansion of GFs shown in Eq. (15)\nA0\np(t/prime/prime,t/prime)≡i\n2t/prime\u0002\n−∞dt1/bracketleftbigg\nG>\n0(t/prime/prime,t1)Σ<\np(t1,t/prime)−G<\n0(t/prime/prime,t1)Σ>\np(t1,t/prime)/bracketrightbigg\n. (A.12)\nBy plugging in Eqs. (A.11) and (A.12) into Eq. (A.6) we obtain\n~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗ˆσµ] =/summationdisplay\np=L,R/summationdisplay\nj/summationdisplay\nνt\u0002\n−∞dt/primet/prime\u0002\n−∞dt/prime/primeMν\nj(t/prime/prime) Tr/bracketleftbigg\n|j/angbracketright/angbracketleftj|⊗ˆσν{A0\np(t/prime/prime,t/prime)+A0†\np(t/prime/prime,t/prime)}|i/angbracketright/angbracketlefti|⊗σµ/bracketrightbigg\n.(A.13)\nSince A0\np(t/prime/prime,t/prime) is an operator constructed from the zeroth order GFs, it can be written in the followin form\nA0\np(t/prime/prime,t/prime) =1\n2/summationdisplay\nmnAp\nmn(t/prime/prime,t/prime)|m/angbracketright/angbracketleftn|⊗12, (A.14)\nwhere 12is a 2×2 identity matrix. 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B 93, 134506\n(2016)."    },    {        "title": "1708.02402v2.Spin_orbit_torque_driven_magnetoimpedance_in_Pt_layer_magnetic_ribbon_heterostructures.pdf",        "content": "1 \n Spin-orbit -torque driven magnetoimpedance in  \nPt-layer/magnetic -ribbon heterostructures  \nM. R. Hajiali1, †, S. Morteza Mohseni2, †, &, L. Jamilpanah2, M. Hamdi2,  \nS. E. Roozmeh1, S. Majid. Mohseni 2, * \n1Department of Physics, University of Kashan, 87317 Kashan, Iran  \n2Faculty of Physics, Shahid Beheshti University, Evin, 19839 Tehran, Iran  \n \nWhen a flow of electron passes through a paramagnetic layer with strong spin -orbit -coupling  such as  \nplatinum  (Pt), a net spin current is produced via spin Hall effect (SHE) . This spin current can exert a \ntorque on the magnetization of an adjacent ferromagnetic layer  which can be probed via magnetization \ndynamic response , e.g. spin-torque ferromagnetic resonance  (ST-FMR) . Nevertheless,  that effect in \nlower frequency magnetization dynamic regime  (MHz)  where skin effect occurs in high permeability \nferromagnetic conductors namely the magneto -impedance (MI) effect can be fundamentally important  \nwhich has  not been  studied  so far . Here, by u tilizing the MI effect in magnetic -ribbon/Pt heterostructure  \nwith high transvers magnetic permeability that allow s the ac current effectively confined  at the skin \ndepth of ~100 nm thickness , the effect of spin-orbit -torque (SOT)  induced by  the SHE  probed via  MI \nmeasurement  is investigated . We observe d a systematic MI  frequency shift that increase s by increasing \nthe applied current amplitude and thickness of the Pt layer (varying from 0 nm to 20 nm) . In addi tion, \nthe role of Pt layer in ribbon/Pt heterostructure  is evaluated with ferromagnetic resonance (FMR) effect \nrepresenting standard Gilbert damping increase as the result of presence of the SHE. Our results unveil \nthe role of SOT  in dynamic control of  the transverse magnetic permeability probed with impedance \nspectroscopy  as useful and valuable technique for detection of future SHE devices .  \n \n \n \n \n \n \n \n       \n \n*Corresponding author’s email address: m-mohseni@sbu.ac.ir , majidmohseni@gmail.com  \n† These authors contributed equally.  \n& Current affiliation: Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität  \nKaiserslautern, 67663 Kaiserslautern, Germany.  2 \n Spin-orbit torques (SOTs) generated by current injection in ferromagnet (FM)/ heavy -metal (HM) \nheterostructures  have attracted considerable attention as a method to effectively manipulate the \nmagnetization of thin FM films1–7. The spin Hall effect (SHE)8 is reported to be the dominant source of \nthe damping -like (DL) SOT  in such heterostructures  that is responsible for magnetization switching9,10, \ndomain  wall (DW)  motion11–13, skyrmion manipulation14,15 and high-frequency magnetization \ndynamics16–18. Rashba -Edelstein effect (REE)  as another source of SOT is also present in FM/HM \nbilayers and depends  on the interface structure of the se bilayer and the ir corresponding thicknesses19. \nThis mechanism can result in  the exertion of field -like (FL) torque on FM.  \nQuantif ication of  the SOTs in FM/HM heterostructures  are based on  spin-torque ferromagnetic \nresonance (ST -FMR)20, planar Hall effect21, low-frequency (~max imum  to few 100 Hz) harmonic Hall \nvoltage6,22, spin Hall magnetoresistance23–25, DW creep velocity26 and magneto -optical effect27. They \nrequires high frequency (few GHz) instruments  or need a complicated assessment process , hence, \ndemonstration of  SOT  materials and techniques  in low frequency regime (MHz) via easy experimental  \nprocess is desirable . \nThe studied heterostructure  in this letter is made of a FM Co68.15Fe4.35Si12.5B15 ribbon and a thin layer of \nplatinum  (Pt). Such ribbon among soft magnetic material is one of the most promising candidates  for \nthe MI effect28-30, primarily because of its application in low -cost and high sensitive magnetic \nsensors31,32 and MI magnetic random access memory (MRAM)33. This effect causes a change in \nelectrical impedance of a conducting FM  with high transverse magnetic permeability  (𝜇𝑡) in the \npresence of a static magnetic field34. By applying an external magnetic field, the skin depth ( 𝛿) changes \ndue to change in 𝜇𝑡, thus varying the impedance of the FM. In the case of the ribbon, with width 𝑙 and \nlength 𝐿, the impedance is approximately35  \n𝑍=(1−𝑖)𝜌𝐿\n2𝑙𝛿=(1−𝑖)𝐿\n2𝑙(𝜋𝜌𝑓𝜇𝑡)1\n2 \n  \n(1) \nwhere 𝜌 is electric resistivity, 𝑓 is frequency of the current and 𝑖 = imaginary unit. Therefor e, the \nimpedance of the ribbon is a function of frequency, driving current and the external dc magnetic field \n(Hdc) through 𝜇𝑡 and 𝛿. \nHere, we study SOT effect on the MI response of the Co-based amorphous ribbon  (~30 µm) /Pt (0-20 \nnm) heterostructure  by measuring its external magnetic field and frequency dependence impedance  \nresponse . It is however noted that , the MI will be studied in a system including a thick FM layer, but \nbased on the aforementioned skin effect , the current distributes at approximately 100 nm thickness close \nto the thin Pt layer  deposited at the interface. This  enables to uncover the dynamic of domain and DW  \nin the present of SOT within a thin region of skin depth . We observe that the impedance response is \nstrongly dependent on the thicknesses of the Pt layer and the applied current amplitude . Our  results \nreveal the possibility of SOT  detection in a FM using the impedance spectroscopy of FM in low \nfrequencies ~MHz  in high transvers magnetic permeability structures. Moreover, we have used 3 \n ferromagnetic resonance  (FMR ) (see section S3 of the supplementary material s) for a better \nunderstanding of the mechanism happening in this system to represent the role of SOT effect based on \nfundamental and standard measurement to confirm the validity of our technique.  \nAmorphous Co-based ribbons (2 mm width , 30 mm length and ~30 µm thickness) were prepared by a \nconventional melt -spinning technique. Before deposition of Pt layer, about 40 nm of ribbons surface \nwere sputter etched via Ar to have clean and oxygen free components. Pt thin layers with thickness o f \n10 nm and 20 nm were deposited on the soft surface (wheel side) of those ribbons in the Ar with gas \npressure of 5 mTorr, base pressure better than 5×10-8 Torr and growth rate of 3 nm/minute. (See \nsupplementary materials including MI measurement , X-ray diffraction (XRD)  analysis  and FMR \nmeasurements)  \nSchematic illustration of a FM/ HM heterostructure system and the definition of the Cartesian coordinate \nsystem in this work are presented in Fig. 1.  The high 𝜇𝑡 of these ribbons allows the skin effect to occur \nin the MHz frequency range with thickness  <100 nm . As shown schematically in Fig. 1(a), an ac charge \ncurrent in the HM layer generates a pure spin -current, oscillating at the same frequency, perpendicular \nto the charge curre nt direction thanks to the SHE. This oscillating spin current flows into the adjacent \nFM layer, exerts two different types of oscillating SOT s5,6,3 6. \n \n \nFIG 1:  (a) Schematic illustration of a FM /HM heterostructure  system. An in -plane charge current Iac generates a perpendicular \nspin current, which in turn generates SOTs acting on ferromagnetic moments.  Oscillations of the magnetization due to ( b) \ndamping -like SOT (T AD) and ( c) the field -like SOT and Oersted field (T FL + T Oe) induced by an ac current.  We should note \nhere that this scenario happen s at the  skin depth 𝛿 of the ribbon.  \nThey are field-like (FL) torque TFL ∼ m × y and damping -like (DL) torque TDL ∼ m × (y × m), where \nm is the magnetization unit vector and y is the in -plane axis perpendicular to current flow direction x \n(Fig. 1 (b, c)). TDL originates from the SHE in the adjacent HM layer. The magnitude of this TDL depends \n4 \n on the transmission of spin current across the FM/HM interface36,37. TFL can be originated  from the REE  \nat the FM/ HM interface due to the structural inversion asymmetry or from the spin current through  HM \nvia the SHE19. When the magnetization lies in -the-plane  of the bilayer sample , the action of TFL is \nequivalent to an in -plane field hFL ∼ y, and that of TDL establishes an out -of-plane field hDL ∼ m × y.  \nAlthough we have  a thick FM layer, a t the studied frequency range ( 1-25 MHz ) in our sample s, the \ncurrent passes through the skin depth 𝛿 of ribbon  (~few nm to 100 nm) , therefore the above mentioned \nscenario is valid in the MI measurement that we introduce here . The magnitude  of TFL varies  \nsignificantly  with thickness of FM  layer5, the type of FM and HM38,39 and the direction of magnetization \nin the FM37. Two origins of TFL are known generally, one due to REE  and the other due to SHE. It is \nadmitted that TFL due to REE, reveals in FM/ HM heterostructures  with 1 -nm-thick FM41,42 and TFL due \nto SHE remains very weak in metallic systems43. Also it is shown that  very large  TFL occur s in magnetic \ntunnel junctions44 and HM/ nonmagnetic /FM/oxide heterostructures45. Therefor e, in our studied \nheterostructure  the contributions of TFL can be neglected  because of the metallic  nature of  layers  and \nlarge thicknesses of  FM layer . From now on, we consider both TDL that comes primarily from the SHE \nand Oersted  torque (TOe) due to Oersted field  generated from the charge current  that depends upon the \nconductivity of each layer and skin depth  𝛿. As the thickness of FM layer is much larger than its skin \ndepth, the Oersted field from FM layer can be important20. In order to detect the effect of these  torque s \nwe carry out  impedance measurement by applying an ac charge current with frequency 𝑓 to the samples , \nand investigate  how the impedance 𝑍 of the bilayer changes as a function of frequency  and field .  \nComparison of frequency sweep MI measurement for 0, 10 and 20 nm Pt is shown in Fig 2(a) where an \nexternal field of 120 Oe was applied to saturate the sample in the plane and an ac current with peak to \npeak amplitude of 66 mA was used to excite the sample.  According to equation 1 and based on \nliteratures arguments the impedance depends on current frequency 𝑓 and transverse magnetic \npermeability  𝜇𝑡(𝑓) with decreas ing trend  at high frequencies46. Therefore, with increasing 𝑓, the \nimpedance of sample increases up to some frequency and further increase of 𝑓 results in the reduction \nof impedance where strong reduction of 𝜇𝑡(𝑓) occurs.   \nWhen an ac current flows through a FM layer the magnetization oscillates about its equilibriu m position , \ny direction , due to Oersted field. Because of the presence of Pt layer, generated spin currents due to the \nSHE  from Pt layer consequences into T DL that derives the  magnetization oscillation in the z direction. \nIt can be seen that the frequency of the maximum impedance of the sample shifts towards high -\nfrequency values, increased from 17 MHz for 0 nm Pt to 18.5 and 19.5 MHz for 10 and 20 nm Pt \ndeposited ribbons, respectively. There is another confirmation for this fact (that will be discussed lat er) \nthat MI versus field shows reduced transverse anisotropy as the magnetization oscillation changed its \norientation toward z direction. We speculate that the angle of precession decreases from that transversal \norientation (without T DL) and the peak posit ion that represents the maximum 𝜇𝑡 goes to higher values 5 \n as shown in Fig. 2(b). The magnitude of h DL can be affected by changing the thickness of the HM and \nFM5 while we have varying thickness of the HM layer. The spin current in FM/HM heterostructures  \nobeys 𝐽𝑠(𝑡) ≈1−sech(𝑡𝐻𝑀𝜆𝑆𝑑⁄)20, where 𝑡𝐻𝑀 is the thickness of HM and 𝜆𝑆𝑑 is the spin -diffusion \nlength. In this relation, as the spin diffusion length of Pt layers is in the Co 75Fe25/Pt bilayer film was \nestimated to be 2.1 ± 0.2 nm47, therefore 𝐽𝑠(𝑡) does not have to change for 10 and 20 nm thickness of \nPt contrary to the frequency shift observed from Pt (10  nm) to that for Pt (20  nm), represented in Fig. \n2(b). However, this effect can be explained based on the resistivity of FM and HM layers. The resistivity  \nof Pt layer is 𝜌 =20 𝜇Ωcm and that for the ribbon is 𝜌 =130 𝜇Ωcm which might pinpoint as a fact that \nat the skin depth region the current in the thicker Pt deposited layer is more than when Pt thickness is \n10 nm. This implies a fact that the current effect and therefore the SHE effect is more significant for \nsample d eposited with 20 nm Pt with enhanced T DL that results in frequency shift.  \n16000000 18000000 200000000.960.981.00\n  \n 0 nm\n 10 nm\n 20 nmZnorm\nFrequency (MHz)(a)\n20 18 16\n \n0 5 10 15 201617181920\n  Frequency Max (MHz)\nPt Thickness (nm)(b)  \nFIG. 2. (a) Frequency sweep of impedance  measurement  of 0, 10 and 20 nm Pt  deposited  ribbon  normalized to maximum \nshowing a shift towards higher f. (b) The maximum frequency vs Pt thickness obtained from (a)  showing higher shift for higher \nthickness of Pt . \nWe measured the frequency sweep  of the MI response with different amplitude  of current app lied to \nthe samples to better elucidate the origin of the frequency shift. Considering the relation between the \nspin current 𝐽𝑠 and the charge current ( 𝐽𝑐), increasing the applied current amplitude results in higher \nspin current generation and higher h DL magnitude (h DL ∝𝐽𝑠)43,48. Frequency sweep impedance \nmeasurement against  ac current with peak to peak amplitude of 33, 66 and 99 mA are shown in Fig. 3 \n(a-c) for 0, 10 and 20 nm Pt deposited ribbons while the ribbons were saturated at 120 Oe. It is cle ar \nfrom Fig. 3(a) that increasing the magnitude of the applied current for 0 nm Pt does not affect the  peak \nposition of the impedance . Whereas for 10 and 20 nm Pt deposited ribbons, increasing the amplitude of \nthe applied current results in a shift in the  maximum impedance frequency. A comparison  between the \nmaximum impedance frequency shift and the applied current for all samples is shown in Fig. 3(d). As \ncan be seen, the role of current for 20 nm Pt is more pronounced with larger frequency -current slop.  6 \n \n15.0M 20.0M0.960.981.00\n18.0M 21.0M0.960.981.0015.0M 20.0M0.960.981.00\n40 60 80 10016182022\n  Znorm \nFrequency (MHz)(b) 10 nm   Znorm \nFrequency (MHz)(c) 20 nm\n  \n 33 mA\n 66 mA\n 99 mAZnorm\nFrequency (MHz)(a) 0 nm\n  Frequency Max  (MHz)\nI (mA) 0 nm \n 10 nm \n 20 nm  (d) \nFIG. 3. Frequency sweep  impedance measurement of (a) 0, (b) 10 and (c) 20 nm Pt at the presence of  an external field of 120  \nOe in different ac current peak to peak amplitude of 33, 66  and 99  mA with a higher frequency shift for higher driving currents . \n(d) Maximum frequency obtained from (a), (b) and (c) versus  ac current amplitude  indicating higher slope of increment for 20 \nnm Pt deposited  ribbon than that for 10 nm Pt deposited  one. \nMI response of a ribbon can give us detailed  information about magnetic anisotropy and transverse \nmagnetic permeability . Therefore, we measured field sweep impedance measurement in an arbitrary \nfrequency of 6 MHz for 0, 10 and 20 nm Pt samples with 66 mA c urrent applied to the samples, as \npresented in Fig. 4(a, b) . It is considered that based on equation S1 in supplement ary materials , the MI \ndecreases from 191% for bare sample to 169% for 10 nm Pt and 152% for 20 nm Pt deposited samples. \nThis behavior is consistent with the TDL tends to reduce the 𝜇𝑡 by exerting a torque perpendicular to \nequilibrium angle of magnetization. As can be seen in Fig. 4(b), the bare ribbon s hows a double peak \nbehavior and Pt deposited ribbons show a single peak behavior.  The observed single - or double -peak \nbehaviors are associated with the longitudinal or transverse magnetic anisotropy with respect to the \nexternal field direction49,50. The disappearance of the transverse anisotropy in ribbon/Pt heterostructures \ncould stem from the T DL which is perpendicular to the plane of the ribbon thus forces the magnetization \nfrom transverse alignment and reduces the transverse  magnetic  permeability.  Furthermore, as another \ntestifier,  we repeated the experiment for ribbon sample coated with 20 nm Copper (Cu) coated layer \nand observed double peak behavior and no frequency shift similar to the bare ribbon ( see FIG. S3 and \nS4 in  supplementary materials). Cu is a representative light metal with weak spin –orbit coupling51 and \nwe expect to see double peak behavior . 7 \n \n-120 -80 -40 040 801200100200\n  MI (%)  \nField (Oe) 0 nm \n 10 nm \n 20 nm  (a) \n-10 -5 0 5 10100125150175200\n  MI (%)\nField (Oe) 0 nm\n 10 nm\n 20 nm(b)  \n0 2 4 6 8 1080120160200\n  MI (%)\nFrequency (MHz) 0 nm \n 10 nm \n 20 nm  (c)  \nFIG. 4.  (a) Comparison of the MI response of the 0, 10 and 20 nm Pt  coated samples in an arb itrary frequency of 6  MHz  and \napplied current of 66  mA. (b) A zoom -in window of MI in the low fields  shows the disappearance of the double peak behavior \nfor Pt deposited  ribbons . (c) MI ratio of 0, 10 and 20 nm Pt deposited  ribbons versus frequency of the applied current with a \nreductive behavior for Pt deposited  ribbons.  \nThe maximum MI ratio of all samples versus frequency are plotted in Fig. 4(c) to better illustrate the \neffect of Pt layer. MI measurements were done at dif ferent frequencies ranging from 1 MHz to 10 MHz. \nIt is noted in Fig. 4(c) that for all investigated samples, with increasing frequency, the maximum MI \nratio first increases, reaches to a maximum at a particular frequency (6 MHz), and then decreases for \nhigher frequencies. This trend can be interpreted by considering the relative contributions of DW \nmotion and moment  rotation to the transverse  magnetic  permeability and hence to the MI52,53. Noted \nthat as frequency increases well above 100 kHz, the contribution of DW motion is damped due to \npresence of the eddy current and moment rotation becomes dominant46,53,54. Thus, t he MI ratio decreases \nat high frequencies. Here, the  𝜇𝑡 decreases, thus resulting in the observed drops of the MI ratio at all \nfrequencies46,52. It is known that DW motion speed increases in the present of SHE12,55,56. MI ratio \nfrequency peak is correlated to the DW relaxation and suggests how DW follows the ac current \nfrequency or correlated with DW speed. Such increase in frequency for 20 nm Pt has same fashion as \nDW does in the present of SHE, dictating another qualitative confirmation.  \nIn summery we have proposed that impedance spectroscopy ca n be used for detection of SOT resulting \nfrom  the SHE  in magnetic -ribbon  /Pt heterostructures . Tunable  impedance response correlated to SOT \ninduced moment re alignments within  FM can be detected . 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Hossain1,5,†\n1Department of Physics, Indian Institute of Technology, Kan pur 208016, India.\n2Institute of Physics, Bhubaneswar 751005, India.\n3Institute of Physics, Johannes Gutenberg-University Main z, 55099 Mainz, Germany.\n4Homi Bhabha National Institute, Anushakti Nagar, Mumbai 40 0085, India.\n5Institute of Low Temperature and Structure Research, 50-42 2 Wroclaw, Poland.\n(Dated: September 27, 2021)\nWe report strong damping enhancement in a 200 nm thick yttriu m iron garnet (YIG) film due\nto spin inhomogeneity at the interface. The growth-induced thin interfacial gadolinium iron garnet\n(GdIG) layer antiferromagnetically (AFM) exchange couple s with the rest of the YIG layer. The\nout-of-plane angular variation of ferromagnetic resonanc e (FMR) linewidth ∆ Hreflects a large in-\nhomogeneous distribution of effective magnetization ∆4 πMeffdue to the presence of an exchange\nspringlike moments arrangement in YIG. We probe the spin inh omogeneity at the YIG-GdIG inter-\nface by performing an in-plane angular variation of resonan ce fieldHr, leading to a unidirectional\nfeature. The large extrinsic ∆4 πMeffcontribution, apart from the inherent intrinsic Gilbert co n-\ntribution, manifests enhanced precessional damping in YIG film.\nI. INTRODUCTION\nThe viability of spintronics demands novel magnetic\nmaterials and YIG is a potential candidate as it ex-\nhibits ultra-low precessional damping, α∼3×10−5[1].\nThe magnetic properties of YIG thin films epitaxially\ngrown on top of Gd 3Ga5O12(GGG) vary significantly\ndue to growth tuning[ 2,3], film thickness[ 4], heavy met-\nals substitution[ 5–7] and coupling with thin metallic\nlayers[8–10]. The growth processes may also induce the\nformation of a thin interfacial-GdIG layer at the YIG-\nGGG interface[ 11–13]. The YIG-GdIG heterostructure\nderived out of monolithic YIG film growth on GGG ex-\nhibits interestingphenomenasuchasall-insulatingequiv-\nalent of a synthetic antiferromagnet[ 12] and hysteresis\nloop inversion governed by positive exchange-bias [ 13].\nThe radio frequency magnetization dynamics on YIG-\nGdIG heterostructure still remains unexplored and need\na detailed FMR study.\nThe relaxation of magnetic excitation towards equi-\nlibrium is governed by intrinsic and extrinsic mecha-\nnisms, leading to a finite ∆ H[14,15]. The former mech-\nanism dictates Gilbert type relaxation, a consequence of\ndirect energy transfer to the lattice governed by both\nspin-orbit coupling and exchange interaction in all mag-\nnetic materials[ 14,15]. Whereas, the latter mechanism\nis a non-Gilbert-type relaxation, divided mainly into two\ncategories[ 14,15]- (i) the magnetic inhomogeneity in-\nduced broadening: inhomogeneity in the internal static\nmagnetic field, and the crystallographic axis orienta-\ntion; (ii) two-magnon scattering: the energy dissipates in\nthe spin subsystem by virtue of magnon scattering with\nnonzero wave vector, k∝negationslash= 0, where, the uniform reso-\nnance mode couples with the degeneratespin waves. The\n∗ravindk@iitk.ac.in\n†zakir@iitk.ac.inangular variation of Hrprovides information about the\npresence of different magnetic anisotropies[ 4,6]. Most\nattention has been paid towards the angular dependence\nofHr[4,6], whereas, the angular variation of the ∆ H\nis sparsely investigated. The studies involving angular\ndependence of ∆ Hmay help to probe different contribu-\ntions to the precessional damping.\nInthispaper, theeffectsofintrinsicandextrinsicrelax-\nation mechanisms on precessionaldamping of YIG film is\nstudied extensively using FMR technique. An enhanced\nvalue of α∼1.2×10−3is realized, which is almost two\norders of magnitude higher than what is usually seen in\nYIGthinfilms, ∼6×10−5[1,2]. Theout-of-planeangular\nvariation of ∆ Hshows an unusual behaviour where spin\ninhomogeneity at the interface plays significant role in\ndefining the ∆ Hbroadening and enhanced α. In-plane\nangular variation showing a unidirectional feature, de-\nmandstheincorporationofanexchangeanisotropytothe\nfree energy density, evidence of the presence of an AFM\nexchangecoupling at the YIG-GdIG interface. The AFM\nexchange coupling leads to a Bloch domain-wall-like spi-\nralmoments arrangementin YIG and givesrise to a large\n∆4πMeff. This extrinsic ∆4 πMeffcontribution due to\nspin inhomogeneity at the interface adds up to the inher-\nent Gilbert contribution, which may lead to a significant\nenhancement in precessional damping.\nII. SAMPLE AND MEASUREMENT SETUPS\nWe deposit a ∼200 nm thick epitaxial YIG film on\nGGG(111)-substrate by employing a KrF Excimer laser\n(Lambda Physik COMPex Pro, λ= 248 nm) of 20 ns\npulse width. A solid state synthesized Y3Fe5O12target\nis ablated using an areal energy of 2.12 J.cm−2with a\nrepetition frequency of 10 Hz. The GGG(111) substrate\nis placed 50 mm away from the target. The film is grown\nat 800oC temperature and in-situpost annealed at the\nsame temperature for 60 minutes in pure oxygen envi-ronment. The θ−2θX-ray diffraction pattern shows epi-\ntaxial growth with trails of Laue oscillations (Fig. 3(a)\nof ref[3]). FMR measurements are performed using a\nBruker EMX EPR spectrometer and a broadband copla-\nnar waveguide (CPW) setup. The former technique uses\na cavity mode frequency f≈9.60 GHz, and enables us\nto perform FMR spectra for various θHandφHangu-\nlar variations. The latter technique enables us to mea-\nsure frequency dependent FMR spectra. We define the\nconfigurations Hparallel ( θH= 90o) and perpendicular\n(θH= 0o) to the film plane for rf frequency and angu-\nlar dependent measurements. The resultant spectra are\nobtained as the derivative of microwave absorption w.r.t.\nthe applied field H.\nIII. RESULTS AND DISCUSSION\nA. Broadband FMR\nFig.1(a) shows typical broadband FMR spectra in\na frequency frange of 1.5 to 13 GHz for 200 nm thick\nYIG film at temperature T= 300 K and θH= 90o.\nThe mode appearing at a lower field value is the main\nmode, whereas the one at higher field value represents\nsurface mode. We discuss all these features in detail in\nthe succeeding subsection IIIB. We determine the res-\nonance field Hrand linewidth (peak-to-peak linewidth)\n∆Hfrom the first derivative of the absorption spectra.\nFig.1(b) shows the rf frequency dependence of Hrat\nθH= 90oand 0o. We use the Kittel equation for fitting\nthe frequency vs. Hrdata from the resonance condi-\ntion expressed as[ 10],f=γ[Hr(Hr+4πMeff)]1/2/(2π)\nforθH= 90oandf=γ(Hr−4πMeff)/(2π) for\nθH= 0o. Where, γ=gµB/ℏis the gyromagnetic ratio,\n4πMeff= 4πMS−Haniis the effective magnetization\nconsisting of 4 πMSsaturation magnetization (calculated\nusing M(H)) and Hanianisotropy field parametrizing cu-\nbic and out-of-plane uniaxial anisotropies. The fitting\ngives 4πMeff≈2000 Oe, which is used to calculate the\nHani≈ −370 Oe.\nFig.1(c) shows the frequency dependence of ∆ Hat\nθH= 90o. The intrinsic and extrinsic damping contri-\nbutions are responsible for a finite width of the FMR\nsignal. The intrinsic damping ∆ Hintarises due to the\nGilbert damping of the precessing moments. Whereas,\nthe extrinsic damping ∆ Hextexists due to different non-\nGilbert-type relaxations such as inhomogeneity due to\nthe distribution of magnetic anisotropy ∆ Hinhom, or\ntwo-magnon scattering (TMS) ∆ HTMS. The intrinsic\nGilbert damping coefficient ( α) can be determined using\nthe Landau-Liftshitz-Gilbert equation expressed as[ 10],\n∆H= ∆Hin+ ∆Hinhom= (4πα/√\n3γ)f+ ∆Hinhom.\nConsidering the above equation where ∆ Hobeys lin-\nearfdependence, the slope determines the value of α,\nand ∆Hinhomcorresponds to the intercept on the ver-\ntical axis. We observe a very weak non-linearity in the\nfdependence of ∆ H, which is believed to be due to thecontribution of TMS to the linewidth ∆ HTMS. The non-\nlinearfdependence of ∆ Hin Fig.1(c) can be described\nin terms of TMS, assuming ∆ H= ∆Hin+ ∆Hinhom+\n∆HTMS. We put a factor of 1 /√\n3 to ∆Hdue to the\npeak-to-peak linewidth value extraction[ 14]. The TMS\ninduces non-linear slope at low frequencies, whereas a\nsaturation is expected at high frequencies. TMS is in-\nduced by scattering centers and surface defects in the\nsample. The defects with size comparable to the wave-\nlength of spin waves are supposed to act as scattering\ncentres. The TMS term at θH= 90ocan be expressed\nas[16]-\n∆HTMS(ω) = Γsin−1/radicalBigg/radicalbig\nω2+(ω0/2)2−ω0/2/radicalbig\nω2+(ω0/2)2+ω0/2,(1)\nwithω= 2πfandω0=γ4πMeff. The prefactor Γ\ndefines the strength of TMS. The extracted values are\nas follows: α= 1.2×10−3, ∆H0= 13 Oe and Γ = 2 .5\nOe. The Gilbert damping for even very thin YIG film\nis extremely low, ∼6×10−5. Whereas, the value we\nachieved is higher than the reported in the literature for\nYIG thin films[ 2]. Also, the value of Γ is insignificant,\nimplying negligible contribution to the damping.\nB. Cavity FMR\nFig.2(a) shows typical T= 300 K cavity-FMR\n(f≈9.6 GHz) spectra for YIG film performed at dif-\nferentθH. The FMR spectra exhibit some universal\nfeatures: (i) Spin-Wave resonance (SWR) spectrum for\nθH= 0o; (ii) rotating the Haway from the θH= 0o,\nthe SWR modes successively start diminishing, and at\ncertain critical angle θc(falls in a range of 30 −35o;\nshaded region in Fig. 2(b)), all the modes vanish except\na single mode (uniform FMR mode). Further rotation\nofHforθH> θc, the SWR modes start re-emerging.\nWe observe that the SWR mode appearing at the higher\nfield side for θH> θc, represents an exchange-dominated\nnon-propagating surface mode[ 17–19]. The above dis-\ncussed complexity in HrvsθHbehaviour has already\nbeen realized in some material systems[ 19], including a\nµ-thick YIG film[ 18]. The localized mode or surface\nspin-wave mode appears for H∝bardblbut not⊥to the film-\nplane[17–19]. WeassigntheSWR modesforthesequence\nn= 1,2,3,...., as it provides the best correspondence\ntoHex∝n2, where, Hex=Hr(n)−Hr(0) defines ex-\nchange field[ 20]. The exchange stiffness can be obtained\nby considering the modified Schreiber and Frait classical\napproach using the mode number n2dependence of res-\nonance field (inset Fig. 3(c))[ 20]. For a fixed frequency,\nthe exchange field Hexof thickness modes is determined\nby subtracting the highest field resonance mode ( n= 1)\nfrom the higher modes ( n∝negationslash= 1). In modified Schreiber\nand Frait equation, the Hexshows direct dependency on\nthe exchange stiffness D:µ0Hex=Dπ2\nd2n2(wheredis\n2/s40/s99/s41/s40/s98/s41/s40/s97/s41\nFIG. 1. Room temperature frequency dependent FMR measureme nts. (a) Representative FMR derivative spectra for differen t\nfrequencies at θH= 90o. (b) Resonance field vs. frequency data for θH= 90oandθH= 0oare represented using red and\nblue data points, respectively. The fitting to both the data a re shown using black lines. (c) Linewidth vs. frequency data at\nθH= 90o. The solid red circles represent experimental data, wherea s the solid black line represents ∆ Hfitting. Inhomogeneous\n(∆Hinhom), Gilbert (∆ Hα) and two-magnon scattering (∆ HTMS) contributions to ∆ Hare shown using dashed green, solid\nyellow and blue lines, respectively.\nthe film thickness). The linear fit of data shown in the\ninset of Fig. 2(b) gives D= 3.15×10−17T.m2. The ex-\nchange stiffness constant Acan be determined using the\nrelationA=D MS/2. The calculated value is A= 2.05\npJ.m−1, which is comparable to the value calculated for\nYIG,A= 3.7 pJ.m−1[20].\nYIG thin films with in-plane easy magnetization ex-\nhibit extrinsic uniaxial magnetic and intrinsic magne-\ntocrystallinecubic anisotropies[ 21]. The total free energy\ndensity for YIG(111) is given by[ 21,22]:\nF=−HMS/bracketleftbigg\nsinθHsinθMcos(φH−φM)\n+cosθHcosθM/bracketrightbigg\n+2πM2\nScos2θM−Kucos2θM\n+K1\n12/parenleftbigg7sin4θM−8sin2θM+4−\n4√\n2sin3θMcosθMcos3φM/parenrightbigg\n+K2\n108\n−24sin6θM+45sin4θM−24sin2θM+4\n−2√\n2sin3θMcosθM/parenleftbig\n5sin2θM−2/parenrightbig\ncos3φM\n+sin6θMcos6φM\n\n(2)\nThe Eq. 2consists of the following different energy\nterms; the first term is the Zeeman energy, the second\nterm is the demagnetization energy, the third term is\nthe out-of-plane uniaxial magnetocrystalline anisotropy\nenergyKu, and the last two terms are the first and sec-\nond order cubic magnetocrystalline anisotropy energies\n(K1andK2), respectively. The total free energy density\nequation is minimized by taking partial derivatives w.r.t.\ntoθMandφMtoobtaintheequilibriumorientationofthe\nmagnetization vector M(H), i.e.,∂F/∂θ M=∂F/∂φ M=\n0. Theresonancefrequencyofuniformprecessionatequi-\nlibrium condition is expressed as[ 21,23,24]:\nωres=γ\nMSsinθM/bracketleftBigg\n∂2F\n∂θ2\nM∂2F\n∂φ2\nM−/parenleftbigg∂2F\n∂θM∂φM/parenrightbigg2/bracketrightBigg1/2\n(3)\nMathematica is used to numerically solve the reso-nance condition described by Eq. 3for the energy den-\nsity given by Eq. 2. The solution for a fixed frequency\nis used to fit the angle dependent resonance data ( Hr\nvs.θH) shown in Fig. 2(b). The main mode data\nsimulation is shown using a black line. The parame-\nters obtained from the simulation are Ku=−1.45×104\nerg.cm−3,K1= 1.50×103erg.cm−3, andK2= 0.13×103\nerg.cm−3. The calculated uniaxial anisotropy field value\nisHu∼ −223 Oe.\nThe ∆Hmanifests the spin dynamics and related re-\nlaxation mechanisms in a magnetic system. The intrinsic\ncontribution to ∆ Harises due to Gilbert term ∆ Hint≈\n∆Hα, whereas, the extrinsic contribution ∆ Hextconsists\nof line broadening due to ∆ Hinhomand ∆HTMS. The\nterms representing the precessional damping due to in-\ntrinsic and extrinsic contributions can be expressed in\ndifferent phenomenologicalforms. Figure 2(c) shows∆ H\nas a function of θH. TheθHvariation of ∆ Hshows\ndistinct signatures due to different origins of magnetic\ndamping. We consider both ∆ Hintand ∆Hextmag-\nnetic damping contributions to the broadening of ∆ H,\n∆H= ∆Hα+∆Hinhom+∆HTMS. The first term can\nbe expressed as[ 14]-\n∆Hα=α\nMS/bracketleftbigg∂2F\n∂θ2\nM+1\nsin2θM∂2F\n∂φ2\nM/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂(2πf\nγ)\n∂Hr/vextendsingle/vextendsingle/vextendsingle/vextendsingle−1\n.(4)\nThe second term ∆ Hinhomhas a form[ 14]-\n∆Hinhom=/vextendsingle/vextendsingle/vextendsingle/vextendsingledHr\nd4πMeff/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆4πMeff+/vextendsingle/vextendsingle/vextendsingle/vextendsingledHr\ndθH/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆θH.(5)\nWhere, the dispersion of magnitude and direction of\nthe 4πMeffare represented by ∆4 πMeffand ∆θH, re-\nspectively. The ∆ Hinhomcontribution arises due to a\nsmall spread of the sample parameters such as thickness,\ninternal fields, or orientation of crystallites within the\nthin film. The third term ∆ HTMScan be written as[ 25]-\n3/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s49/s50/s51/s32/s72\n/s101/s120/s40/s107/s79/s101/s41\n/s110/s50/s67/s114/s105/s116/s105/s99/s97/s108/s32/s97/s110/s103/s108/s101/s54/s53/s52/s51/s72\n/s114/s32/s40/s32/s107/s79/s101/s32/s41/s50/s110/s32/s61/s32/s49\n/s72/s32/s40/s32/s68/s101/s103/s114/s101/s101/s32/s41/s50 /s51 /s52 /s53 /s54/s68/s101/s114/s105/s118/s97/s116/s105/s118/s101/s32/s97/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s72 /s32/s40/s107/s79/s101/s41/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48 /s55/s53 /s57/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s32/s32\n/s32/s69/s120/s112/s116/s46/s32/s68/s97/s116/s97\n/s32 /s72\n/s32 /s72\n/s32 /s77\n/s101/s102/s102\n/s32\n/s32 /s72\n/s84/s77/s83/s72 /s32/s40/s79/s101/s41/s32\n/s40/s100/s101/s103/s114/s101/s101 /s41/s40/s99/s41/s40/s98/s41/s40/s97/s41\nFIG. 2. Room temperature out-of-plane angular θHdependence of FMR. (a) Derivative FMR spectra shown for diffe rentθH\nperformed at ≈9.6 GHz. (b) θHvariation of uniform mode and SWR modes of resonance field Hr. Inset: Exchange field\n(Hex) vs mode number square ( n2). (c)θHvariation of the linewidth (∆ H), where, the experimental and simulated data are\nrepresented by solid yellow circles and black line, respect ively. The different contributions ∆ Hα, ∆4πMeff, ∆θHand ∆HTMS\nare represented by gray, purple, green and red lines, respec tively.\n∆HTMS=/summationtext\ni=1Γout\nifi(φH)\nµ0γΦsin−1/radicalbigg√\nω2+(ω0/2)2−ω0/2√\nω2+(ω0/2)2+ω0/2,\nΓout\ni= Γ0\niΦA(θ−π/4)dHr(θH)\ndω(θH)/slashbigg\ndHr(θH=0)\ndω(θH=0)\n(6)\nThe prefactor Γout\nidefines the TMS strength and has\naθHdependency in this case. The type and size of the\ndefects responsible for TMS is difficult to characterize\nwhich makes it non-trivial to express the exact form of\nΓout\ni. Although, it mayhaveasimplified expressiongiven\nin Eq.6, where, Γ0\niis a constant; A(θ−π/4), a step\nfunction which makes sure that the TMS is deactivated\nforθH< π/4; anddHr(θH)/dω(θH), a normalization\nfactor responsible for the θHdependence of the Γout\ni.\nIn fig.2(c)the solid dark yellow circles and black solid\nline represent the experimental and simulated ∆ HvsθH\ndata, respectively. We also plot contributions of different\nterms such as ∆ Hα(blue color line), ∆4 πMeff(purple\ncolorline), ∆ θH(greencolorline) and ∆ HTMS(red color\nline). The fitting provides following extracted parame-\nters,α= 1.3×10−3, ∆4πMeff= 58 Oe, ∆ θH= 0.29o\nand Γ0\ni= 1.3 Oe. The precessional damping calculated\nfromthe∆ Hvs.θHcorroboratewiththevalueextracted\nfrom the frequency dependence of ∆ Hdata (shown in\nFig.1(c));α= 1.2×10−3. The ∆ Hbroadening and\nthe overwhelmingly enhanced precessional damping are\nthedirectconsequenceofcontributionsfromintrinsicand\nextrinsic damping. Usually, the Gilbert term and the\ninhomogeneity due to sample quality contribute to the\nbroadening of ∆ Hand enhanced αin YIG thin films. If\nwe interpret the ∆ HvsθHdata, it is clear that damping\nenhancement in YIG is arising from the extrinsic mag-\nnetic inhomogeneity.The role of an interface in YIG coupled with metals\nor insulators leading to the increments in ∆ Handαhas\nbeen vastly explored. Wang et. al. [ 9] studied a variety\nof insulating spacers between YIG and Pt to probe the\neffect on spin pumping efficiency. Their results suggest\nthe generation of magnetic excitations in the adjacent\ninsulating layers due to the precessing magnetization in\nYIG at resonance. This happens either due to fluctu-\nating correlated moments or antiferromagnetic ordering,\nvia interfacial exchange coupling, leading to ∆ Hbroad-\nening and enhanced precessional damping of the YIG[ 9].\nTheimpurityrelaxationmechanismisalsoresponsiblefor\n∆HbroadeningandenhancedmagneticdampinginYIG,\nbut is prominent only at low temperatures[ 16]. Strong\nenhancement in magnetic damping of YIG capped with\nPt has been observed by Sun et. al. [ 8]. They suggest\nferromagneticorderingin an atomically thin Pt layerdue\nto proximity with YIG at the YIG-Pt interface, dynam-\nically exchange couples to the spins in YIG[ 8]. In recent\nyears, some research groups have reported the presence\nof a thin interfacial layer at the YIG-GGG interface[ 11–\n13]. The 200 nm film we used in this study is of high\nquality with a trails of sharp Laue oscillations [see Fig\n3(a) in ref.[ 3]]. Thus it is quite clear that the observed\n∆Hbroadening and enhanced αis not a consequence of\nsample inhomogeneity. The formation of an interfacial\nGdIG layer at the YIG-GGG interface, which exchange\ncouples with the YIG film may lead to ∆ Hbroadening\nand increased α. Considering the above experimental ev-\nidences leading to ∆ Hbroadening and enhanced Gilbert\ndampingdueto couplingwithmetals andinsulators[ 8,9],\nit is safe to assume that the interfacial GdIG layer at the\ninterface AFM exchange couples with the YIG[ 11–13],\nand responsible for enhanced ∆ Handα.\nFig.3shows in-plane φHangular variation of Hr. We\n4/s32/s68/s97/s116/s97\n/s32/s84/s111/s116/s97/s108\n/s32/s69/s120/s99/s104/s97/s110/s103/s101/s72\n/s114/s32/s40/s79/s101/s41/s50/s48/s48/s32/s110/s109/s40/s97/s41\n/s40/s98/s41\n/s49/s48/s48/s32/s110/s109\n/s72/s32/s40/s68/s101/s103/s114/s101/s101/s41\nFIG. 3. (a) In-plane angular φHvariation of Hr. The exper-\nimental data are represented by solid grey circles. Whereas ,\nthe simulated data for total and exchange (unidirectional)\nanisotropy are represented by black and red solid lines, re-\nspectively. (a) 200 nm thick YIG sample. (b) 100 nm thick\nYIG sample.\nsimulate the in-plane HrvsφHangular variation using\nthe free energy densities provided in ref. [ 26] and an\nadditional term, −KEA.sinθM.cosφM, representing the\nexchange anisotropy ( KEA). Even though φHvaria-\ntion ofHrshown in Fig. 3(a) is not so appreciable\nas the film is 200 nm thick, a very weak unidirectional\nanisotropy trend is visible, suggesting an AFM exchange\ncoupling between the interface and YIG. It has been\nshown that the large inhomogeneous 4 πMeffis a direct\nconsequence of the AFM exchange coupling at the inter-\nface of LSMO and a growth induced interfacial layer[ 27].\nThe YIG thin film system due to the presence of a hard\nferrimagnetic GdIG interfacial layer possesses AFM ex-\nchange coupling[ 11–13]. A Bloch domain-wall-like spiral\nmoments arrangement takes place due to the AFM ex-\nchange coupling acrossthe interfacial GdIG and top bulk\nYIG layer[ 11–13]. An exchange springlike characteris-\ntic is found in YIG film due to the spiral arrangement\nof the magnetic moments [ 11–13]. The FMR measure-\nment and the extracted value of ∆4 πMeffreflect inho-\nmogeneous distribution of 4 πMeffin YIG-GdIG bilayer\nsystem. The argument of Bloch domain-wall-like spiral\narrangement of moments is conceivable, as this arrange-\nment between the adjacent layers lowers the exchange\ninteraction energy[ 27]. To further substantiate the pres-ence of an interfacial AFM exchange coupling leading\nto spin inhomogeneity at YIG-GdIG interface, we per-\nformed in-plane φHvariation of Hron a relatively thin\nYIG film ( ∼100 nm with growth conditions leading to\nthe formation of a GdIG interfacial layer[ 13]). Fig.3(b)\nshows prominent feature of unidirectional anisotropydue\nto AFM exchange coupling in 100 nm thick film. It is\nevident that the interfacial layer exchange couples with\nthe rest of the YIG film and leads to a unidirectional\nanisotropy. We observethatthe interfacialexchangecou-\npling may cause ∆ Hbroadening and enhanced αdue to\nspin inhomogeneity at the YIG-GdIG interface, even in\na 200 nm thick YIG film.\nIV. CONCLUSIONS\nThe effects of spin inhomogeneity at the YIG and\ngrowth-induced GdIG interface on the magnetization dy-\nnamics of a 200 nm thick YIG film is studied extensively\nusing ferromagnetic resonance technique. The Gilbert\ndamping is almost two orders of magnitude larger\n(∼1.2×10−3) than usually reported in YIG thin films.\nThe out-of-plane angular dependence of ∆ Hshows\nan unusual behaviour which can only be justified after\nconsidering extrinsic mechanism in combination with the\nGilbert contribution. The extracted parameters from\nthe ∆HvsθHsimulation are, (i) α= 1.3×10−3from\nGilbert term; (ii) ∆4 πMeff= 58 Oe and ∆ θH= 0.29o\nfrom the inhomogeneity in effective magnetization and\nanisotropy axes, respectively; (iii) Γ0\ni= 1.3 Oe from\nTMS. The TMS strength Γ is not so appreciable,\nindicating high quality thin film with insignificant defect\nsites. The AFM exchange coupling between YIG and\nthe interfacial GdIG layer causes exchange springlike\nbehaviour of the magnetic moments in YIG, leading to a\nlarge ∆4 πMeff. The presence of large ∆4 πMeffimpels\nthe quick dragging of the precessional motion towards\nequilibrium. A unidirectional behaviour is observed in\nthe in-plane angular variation of resonance field due to\nthe presence of an exchange anisotropy. 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B 101, 024408 (2020) .\n6"    },    {        "title": "1606.06610v1.Torsion_Effects_and_LLG_Equation.pdf",        "content": "arXiv:1606.06610v1  [hep-th]  21 Jun 2016Torsion Effects and LLG Equation.\nCristine N. Ferreira†a, Cresus F. L. Godinho‡b, J. A. Helay¨ el Neto∗c\n†N´ ucleo de Estudos em F´ ısica, Instituto Federal de Educa¸ c ˜ ao, Ciˆ encia e Tecnologia\nFluminense, Rua Dr. Siqueira 273, Campos dos Goytacazes, 28 030-130 RJ, Brazil\n‡Grupo de F´ ısica Te´ orica, Departamento de F´ ısica, Univer sidade Federal Rural do Rio de\nJaneiro, BR 465-07, 23890-971, Serop´ edica, RJ, Brazil\n∗Centro Brasileiro de Pesquisas F´ ısicas (CBPF), Rua Dr. Xav ier Sigaud 150, Urca,\n22290-180, Rio de Janeiro, Brazil\nAbstract\nBased on the non-relativistic regime of the Dirac equation coupled to a torsion\npseudo-vector, we study the dynamics of magnetization and how it is affected by the\npresence of torsion. We consider that torsion interacting terms in Dirac equation\nappear in two ways one of these is thhrough the covariant derivativ e considering the\nspin connection and gauge magnetic field and the other is through a n on-minimal\nspin torsion coupling. We show within this framework, that it is possible to obtain\nthe most general Landau, Lifshitz and Gilbert (LLG) equation includ ing the torsion\neffects, where we refer to torsion as a geometric field playing an impo rtant role in the\nspin coupling process. We show that the torsion terms can give us tw o important\nlandscapes in the magnetization dynamics: one of them related with d amping and the\nother related with the screw dislocation that give us a global effect lik e a helix damping\nsharped. These terms are responsible for changes in the magnetiz ation precession\ndynamics.\nacrisnfer@iff.edu.br ,bcrgodinho@ufrrj.br ,chelayel@cbpf.br\n11 Introduction\nThe discovery of the graphene-like systems and topological insulators systems introduced a\nnew dynamic in the applications of the framework of the high e nergy physics in low energy\nsystems in special condensed matter systems. The fact that t hese systems can be described\nby Dirac equations give us new possibilities for theoretica l and experimental applications.\nIn this direction there are some effects in condensed matter sy stems still without a full\ndescription as the magnetizable systems that we described i n this work. In this sense the\nconstructionofthetheoreticalframeworksthatcan,insom elimit, beobtainedinlowenergy\nsystems is the crucial importance to understand the invaria nces and interactions in certain\nlimits. One of the important effects that we can study is relate d with the spin systems.\nSo, in this work we deal with the new framework to study the spi n systems considering\nthe Dirac equation in non-relativistic limit with torsion i nteraction[1]. Spin systems are\ngenerally connected with magnetic systems. It is well known that spin angular momentum\nisanintrinsicpropertyofquantumsystems. Whenamagnetic fieldisapplied, eachmaterial\npresents some level of magnetization, and Quantum Mechanic s says that magnetization is\nrelated to the expectation value of the spin angular momentu m operator. In the case\nof ferromagnetic materials, they can have a large magnetiza tion even under the action\nof a small magnetic field and the magnetization process is alw ays followed by hysteresis,\nand the magnetization is uniform and lined up with the magnet ic field, usually these\nmaterials exhibit a strong ordering process that results in a parallel line up the spins[2]. In\nmaterials graphene type we also can generate magnetic momen t. In this form it is possible\nto study the transport phenomena [3]. Overlapping between e lectronic wave functions are\ninteractions well understood, again thanks to Quantum Mech anics, however thereare other\nkinds of interactions occurring such as magnetocrystallin e anisotropy, connected with the\ntemperature dependence[4] and demagnetization fields [5], acting in low range. In such\nsystems if we only consider the precession we will not reach t he right limit. Certainly, the\nprecession equation has to include a damping term providing the magnetization alignment\nwith the magnetic field after a finite time [6]. In order to simu late these phenomena,\nseveral physical models have been presented. However, the L andau-Lifshitz model is still\nthe one widely used in the description of the dynamics of ferr omagnetic media. In their\npioneering work [7] in 1935, Landau and Lifshitz proposed a n ew theory based on the\nfollowing dynamical equation:\n∂tM=/vectorHeff×/vectorM+α\nM2s/vectorM×(/vectorM×/vectorHeff), (1)\nwhere/vectorHeffdenote an effective magnetic field, with the gyromagnetic rati o absorbed, inter-\nacting with the magnetization M=|/vectorM|. The first term is the precession of the magnetiza-\ntion vector around the direction of the effective magnetic fiel d and the second one describes\na damping of the dynamics. With this theory we are able to comp ute the thickness of walls\nbetween magnetic domains, and also understand the domain fo rmation in ferromagnetic\n2materials. This theory, which now goes under the name of micr omagnetics, has been in-\nstrumental in the understanding and development of magneti c memories. Landau and\nLifshitz considered the Gibbs energy G of a magnetic materia l to be composed of three\nterms: exchange, anisotropy and Zeeman energies (due to the external magnetic field), and\npostulated that the observed magnetization per unit volume M field would correspond to\na local minimum of the Gibbs energy. Later researchers added other terms to G such as\nmagnetoelastic energy and demagnetization energy. They al so derived the Landau Lifshitz\n(LL) equation using only physical arguments and not using th e calculus of variations. In\nsubsequent work, Gilbert [8] realized a more convincing for m for the damping term, based\non a variational approach, and the new combined form was then called Landau Lifshitz\nGilbert (LLG) equation, today it is a fundamental dynamic sy stem in applied magnetism.\nNowadays, the scientific and technological advances provid e a wide spectrum of ma-\nnipulations to the spin degrees of freedom. The complete for mulation for magnetization\ndynamics also include the excitation of magnons and their in teraction with other degrees\nof freedom, that remains as a challenge for modern theory of m agnetism [9]. These amaz-\ning and reliable kinds of procedures are propelling spintro nics as a consolidated sub-area\nof Condensed Matter Physics [10]. Since the experimental ad vances are increasingly pro-\nviding high-precision data, many theoretical works are bei ng presented [11, 14, 12, 13]\nand including strange materials, as the topological insula tors, connected with the mag-\nnetocondutivity [15] and graphene like structures [16] for a deeper understanding of the\nphenomenon including the spin polarization super currents for spintronics [17], holographic\nunderstanding of spin transport phenomena [18] and non-rel ativistic background [19].\nThe torsion field appears as one of the most natural extension s of General Relativity\nalong with the metric tensor, which couples to the energy-mo mentum distribution, inspects\nthe details of the spin density tensor. Actually, in General Relativity, fermions naturally\ncouple to torsion by means of their spin.\nIn this work we consider that the torsion interacts with the m atter in two types one of\nthese is present in covariant derivative that contains the s pin connection related with the\nChristoffell symbol given by the metric of the curve space time and the contorsion given\nby the torsion that have two antisymmetric index. The other c ontribution is given by the\nnon minimal spin torsion coupling that is important to the co nsistence of the theory. It is\npossible to study the non relativistic approach to the torsi on in connection with the spin\nparticles [20], in this work we only consider the torsion con tribution considering the plane\nspace-time.\nOur work is organized as follows, in Section 2, we analyse the torsion coupling in\nrelativistic limit, in Section 3 the modified version of Paul i equation is presented. Our\napproach starts with a field theoretical action where a Dirac fermion is non-minimally\ncoupled in the presence of a torsion term, a low relativistic approximation is considered\nand the equivalent Pauli equation is then obtained. We deriv e a very similar expression to\nthe Landau-Lifshitz-Gilbert (LLG) equation from our Pauli equation with torsion. Under\nspecific conditions for the magnetic moment, we show that the LLG equation can be\n3established with damping and dislocations terms.\n2 TheRelativistic andNon- Relativistic Discussions for Sp in\nCoupling with Torsion\nIn this Section, let us understand the way to describe the spi n interaction by taking into\naccount the torsion coupling. In our framework there are two terms in Dirac action, one\nof these is connected with spin current effect from the spin con nection, the other is a non-\nminimal spin torsion coupling whose effects are the subject of this work. Dirac’s equation\nis relativistic and we should justify why we use it in our mode l. Dispersion relations in\nCondensed Matter Physics (CMP) linear in the velocity appea r in a wide class of models\nand one adopts the framework of Dirac’s equation to approach them. However, the speed\nof light, c, is suitably replaced by the Fermi velocity, vF. Here, this is not what we are\ndoing. We actually start off from the Dirac’s equation and we t ake, to match with effects\nof CMP, the non-relativistic regime, for the electron moves with velocities v≤c\n300. So,\ncontrary to an analogue model where we describe the phenomen a by a sort of relativity\nwith c replaced by vF, we here consider that the non-relativistic electrons of ou r system is\na remnant of a more fundamental relativistic world. The non- relativistic limit is also more\ncomplete because it brings effects that do not directly appear in Galilean Physics. This is\nwhy we have taken the viewpoint of associating our physics to the Dirac’s equation.\n2.1 The Dirac Model for Torsion and the Spin Current Interpre tation\nIn this sub-section, we consider the microscopic discussio n that gives the explicit form of\nthe spin current in function of the gauge potential and torsi on coupling. The scenario we\nare setting up is justified by the following chain of argument s: (i) We are interested in\nspin effects. We assume that there is a space-time structures ( torsion) whose coupling with\nthe matter spin becomes relevant. But, we are actually inter ested in the possible non-\nrelativistic effects stemming from this coupling, which is mi nimal and taken into account\nin the covariant derivative though the spin connection. (ii ) The other point we consider\nis that, amongst the three irreducible torsion components, its pseudo-vector piece is the\nonly one that couples to the charged leptons. Then, with this results in mind, we realize\nthat the electron spin density may non-minimally couple, in a Pauli- like interaction, to\nthe field-strength of the torsion pseudo-vector degree of fr eedom.\nSo, our scenario is based on the relevant role space-time tor sion, here modeled by a\npseudo-vector, may place in the non-relativistic electron s of spin systems in CMP. The spin\ncurrent that we talk about is the spin magnetic moment and in g eneral is not conserved\nalone. Thequantity that is conserved is the total magnetic m oment that is the composition\nbetween both /vectorJ=/vectorJS+/vectorJL. In form that ∂µJµ= 0 where the spin current can be defined\nin related to the three component spin current as Jµ\nS=ǫµab\nρJρ\naband/vectorJLis the spin orbit\n4coupling . In analogy with the charge current, defined by the d erivative of the action in\nrelation to the gauge field Aµwe used the definition where the spin current is the derivativ e\nof the action in relation with the spin connection. We consid er here the spin current as\nthe derivative of the action in relation to the spin connecti onωab\nµthen the spin current is\ngiven by\nJµ\nab=δS\nδωabµ, (2)\nwhereωab\nµis\nωab\nµ=ea\nν∇µeνb+ea\nλΓλ\nµνeνb+ea\nλKλ\nµνeνb, (3)\nandKλ\nµνis the contorsion given by\nKλ\nµν=−1\n2(Tλ\nµ ν+Tλ\nν µ−Tλ\nµν), (4)\nwhereTλ\nµνis the torsion. In this work we consider two terms for torsion , on of these is the\ntotally anti symmetric tensor, that respecting the duality relation given by Tµνλ=ǫµνλρSρ\nwhereSρis the pseudo-vector part of torsion.1The other term that we consider is the 2-\nform tensor TµνwhereTµν=∂µSν−∂νSµthis term is analog to the field strength of the\nelectromagnetic gauge potential Aµthat in our case is changed to pseudo-vector Sµ. The\ninvariant fermionic action that contained these contribut ions for torsion is given by\nS=/integraldisplay\nd4xi¯ψ(γµDµ+λTµνΣµν+m)ψ, (5)\nwhere the covariant derivative is Dµ=Dµ−iηωab\nµΣabthat contain the covariant gauge\nderivativeDµ=∂µ−ieAµand the spin connection covariant derivative.\nWe consider the flat space-time where the only contribution f or the spin connection is\nthe contortion. In this form we have a spin current given by\nJµ\nab=1\n2¯ψγµΣabψ. (6)\nWe consider the ansatz where ωab\nµcontain the total antisymmetric part of the contorsion\ngiven by\nωκλ\nµ=µǫκλρ\nµSρ. (7)\nWe used the splitting γµΣκλ=ǫµ\nκλργργ5+δµ\nκγλ−δµ\nλγκThat give us the current in the\nform\nJµ\nαλ=δS\nδΓabµ=1\n2ǫµ\nαλρ¯ψγργ5ψ=Ji\njk+J0\nij (8)\n1Considering space times with torsion Tα\nβγ, the afine connection is not symmetric, Tα\nβγ= Γα\nβγ−Γα\nγβ, and\nwe can split it into three irreducible components, where one of them is the pseudo-trace Sκ=1\n6ǫαβγκTαβγ.\n5where the currents are\nJi\njk=1\n2ǫi\njk¯ψγ0γ5ψ; (9)\nJ0\nij=1\n2ǫijk¯ψγkγ5ψ, (10)\nthen the current part of the action coming from the covariant derivative is given by\nScurr=−η/integraldisplay\nd4xγµγ5Sµ=η/integraldisplay\nd4x/vectorS·/vectorJ. (11)\nThe action (5) considers the temporal component of the torsi on pseudo-vector S0= 0,\nwe have can be written as\nS=/integraldisplay\nd4xi¯ψ/parenleftBig\nγµ∂µ+igγµAµ+iηγµγ5Sµ+λTµνΣµν+m/parenrightBig\nψ, (12)\nOur sort of gravity background does not exhibit metric fluctu ations. The space-time is\ntaken to be flat, and we propose a scenario such that the type of gravitational background\nis parametrized by the torsion pseudo-trace Sµ, whose origin may be traced back to one\ngeometrical defect.\n2.2 Dirac equation in presence of torsion\nNow, we discuss the Dirac equation given by eq. (12). From the action above, taking the\nvariation with respect to ( δS/δ¯ψ), a modified Dirac’s equation reads as below:\n[iγµ∂µ−ηγµγ5Sµ−eγµAµ+λΣµν∂µSν+m]ψ= 0. (13)\nFor a vanishing λ−parameter, the equation (13) has been carefully studied in [ 21, 22, 23,\n24, 26].\nThe generation, manipulation, and detection of a spin curre nt, as well as the flow of\nelectron spins, are the main challenges in the field of spintr onics, which involves the study\nof active control and manipulation of the spin degree of free dom in solid-state systems,\n[27, 10, 28]. A spin current interacts with magnetization by exchanging the spin-angular\nmomentum, enabling the direct manipulation of magnetizati on without using magnetic\nfields [29, 30]. The interaction between spin currents and ma gnetization provides also\na method for spin current generation from magnetization pre cession, which is the spin\npumping [31, 32]. We showed in last Sections that there are tw o type of deformations\ninduced by the torsion. Both of these can be generating a spin current.\nAfter a suitable separation of components ( µ= 0,1,2,3), the equation of motion can\nbe written as,\ni∂tψ=iαi∂iψ+ηγ5S0ψ−ηαiγ5Siψ+eA0ψ+\n−eαiAiψ−iλ\n4ǫijkβγ5αk∂iSjψ+βmψ. (14)\n6Definingagauge-invariant momentum, πj=i∂j−eAj, andusingthatΣ ij=−i\n4ǫijkγ5αk\nwithγ0=β, the effective Hamiltonian takes the form,\nH=αkπk+ηγ5S0−ηαkγ5Sk+eA0+\n−iλ\n4ǫijkβγ5αk∂iSj+βm. (15)\nwhere we have the matricial definitions:\nαi=/parenleftbigg0σi\nσi0/parenrightbigg\n,γ5=/parenleftbigg0 1\n1 0/parenrightbigg\n,β=/parenleftbigg1 0\n0−1/parenrightbigg\n, (16)\nIn the Heisenberg picture, the position, /vector x, and momentum, /vector π, operators obey two\ndifferent kinds of relations; we consider the torsion as a func tion of position only, S=S(/vector x),\nso that\n˙/vector x=/vector α\n˙/vector π=e(/vector α×/vectorB)+e/vectorE+ηγ5∂\n∂x(/vector α·/vectorS)ˆx. (17)\nOne reproduces the usual relation for ˙/vector x, while the equation for ˙/vector πpresents a new term\napparently giving some tiny correction to the Lorentz force .\nHowever, if we consider the torsion in a broader context, now as a momentum- and\nposition-dependent background field, S=S(/vector x,/vectork), we have to deal with the following\npicture,\n˙/vector x=/vector α−ηγ5∂\n∂k(/vector α·/vectorS)ˆk\n˙/vector π=e(/vector α×/vectorB)+e/vectorE+ηγ5∂\n∂x(/vector α·/vectorS)ˆx+\n+eηγ5∂A\n∂x∂\n∂k(/vector α·/vectorS)ˆk. (18)\nThe two sets of dynamical equations above are clearly showin g us the small corrections\ninducedbythetorsionterm. Nevertheless, wearestill here intherelativistic domain, andit\nis necessary change this framework for a better understandi ng of the SHE phenomenology.\nFor this reason, in next Section we are going to approach the s ystem by going over into its\nnon-relativistic regimen\n2.3 Non-Relativistic Approach with torsion\nIn this sub-section, we consider the Dirac equation in its no n-relativistic limit. One im-\nportant requirement for the Dirac equation is that it reprod uces what we know from non-\nrelativistic quantum mechanics. We can show that, in the non -relativistic limit, two com-\nponents of the Dirac spinor are large and two are quite small. To make contact with the\n7non-relativistic description , we go back to the equations w ritten in terms of ϕandχof\nthe four component spinor ψ=eimt√\n2m/parenleftbiggϕ\nχ/parenrightbigg\n, just prior to the introduction of the γma-\ntrices. we obtain two equation on of these for ϕand the other for χ. We can solved in\nχand substiuted in the Dirac equation given by (12) and take th e non-relativistic regi-\nmen (|/vector p|<< m). So, in this physical landscape, from now and hereafter, ou r goal is to\nconsider a low-relativistic approximation based on an exte nded Pauli equation version by\nincluding torsion as presented before. Employing the Hamil tonian (15), we could carry\nout our calculations in the framework of the Fouldy-Wouthuy sen transformations; however\nfor the sake of our approximation at lowest order in v/c, we take that SHE is adequately\nwell described by the low-relativistic Pauli equation. We a re considering that the electron\nvelocities are in the range of Fermi’s velocity. In this case , we arrive at the version given\nbelow for the Pauli’s equation:\ni∂ϕ\n∂t=/bracketleftBig(/vector p−e/vectorA)2\n2m−e\n2m(/vector σeff·/vectorB)+eA0−(σ·/vectorSeff)+\n−λ\n8m(/vector∇×/vectorS)·(/vector σ×/vector p)+iλ\n8m(/vector∇×/vectorS)·/vector p+\n−eλ\n8m(/vector∇×/vectorS)·(/vector σ×/vectorA)+ieλ\n8m(/vector∇×/vectorS)·/vectorA/bracketrightBig\nϕ. (19)\nThe equation above displays the usual Pauli terms, but corre cted by new terms due to\nthe torsion coupling. The second and the fourth contributio ns in the RHS of eq.(19) can\nbe thought of as effective terms for /vector σand/vectorS, respectively given by\nσeff=/vector σ+η\n2m/vectorS+iη\n2m(/vector σ×/vectorS) (20)\n/vectorSeff=η/vectorS+iλ\n4(/vector∇×/vectorS). (21)\nThe fifth contribution in the RHS is proportional to the Rashb a SO coupling term; this\nterm yields an important effect on the behavior of spin.\n3 From the Modified Pauli Equation to Unfold in LLG\nIn this Section, we consider the magnetization equation der ivation given by Dirac non-\nrelativistic limit take into account the presence of torsio n. Let us start by considering the\nmodified Pauli equation eq.(19) and find the magnetization eq uation. By using the Landau\ngauge/vectorA=H/vector xand taking that /vector x·/vector σ= 0 (the spins are aligned orthogonally to the plane\nof motion), give us the Hamiltonian of the full system in the n on-relativistic limit as:\n8H=(/vector p−e/vectorA)2\n2m−e\n2m(/vector σeff·/vectorB)+eA0−(σ·/vectorSeff)−λ\n8m(/vector∇×/vectorS)·(/vector σ×/vector p)+iλ\n8m(/vector∇×/vectorS)·/vector p,\n(22)\nwith/vectorB(t) =µ0/vectorH(t), whereµ0is the gyromagnetic ratio. The Pauli equation associated\nwith (22) reads as follows below\ni∂ϕ\n∂t=/bracketleftBig(/vector p−eA)2\n2m−e\n2m(/vector σeff·/vectorH)+eA0−(/vector σ·/vectorSeff)+\n−λ\n8m(/vector∇×/vectorS)·(/vector σ×/vector p))+iλ\n8m(/vector∇×/vectorS)·p/bracketrightBig\nϕ. (23)\nLet us consider the magnetization vector equation related w in the spin magnetic mo-\nment/vector µ=e\n2m/vector σ. In our approach we consider the magnetization is defined by /vectorM=\n(/vector µϕ)†ϕ−ϕ†(/vector µϕ) whereϕgiven by Pauli equation (23) and ϕ†ϕ= 1 and/vectorˆSϕ=/vectorSϕ,\nwith the notation/vectorˆSis a torsion operator and /vectorSis the torsion autovalue. We have, by\nthe manipulation of Pauli equation the magnetization equat ion associated with a fermionic\nstate when we applied a external magnetic field /vectorHconsidering the Pauli product algebra\nas1\n2(σiσj−σjσi) =iǫijkσkand1\n2(σiσj+σjσi) =δij. The magnetization equation that\narrive that is\n∂/vectorM\n∂t=/vectorM×/vectorH+η/vectorM×/vectorS+β(/vectorM×/vectorL) +\n+ηe\n2m2(/vectorS×/vectorH) +eλ\n2m(/vector∇×/vectorS), (24)\nwith the magnetic moment given by /vectorL=/vector r×/vector pand/vectorSas the torsion pseudo-vector. We can\nobserved that there are two terms that arrived by the covaria nt derivative Dµdefined by\nthe coupling constant ηand the other is the parameter that arrived by non-minimal sp in\ntorsion coupling with the coupling constant λ. Where the effect of the new terms given\nwhenη/ne}ationslash= 0 andλ/ne}ationslash= 0.\nWe consider the scalar product of the magnetization /vectorM, the magnetic field /vectorHand the\ntorsion pseudo-vector /vectorSwith the equation (28) and we obtain2\n∂t/bracketleftBig1\n2(/vectorM·/vectorM)/bracketrightBig\n=ηe\n2m2/bracketleftBig\n/vectorM·(/vectorS×/vectorH)+λm\nη/vectorM·(/vector∇×/vectorS)/bracketrightBig\n; (25)\n2We used the vectorial relating given by A·(B×C) =B·(C×A) =C·(A×B) the other is\nA×(B×C) = (A·C)B−(A·B)Cand∇·(A×B) =B·(∇×A)−A·(∇×B).\n9∂t/bracketleftBig1\n2(/vectorH·/vectorM)/bracketrightBig\n=/bracketleftBig\nη/vectorM·(/vectorS×/vectorH)+β/vectorM·(/vectorL×/vectorH) +\n+e\n2mλ/vectorH·(/vector∇×/vectorS)/bracketrightBig\n; (26)\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n=/vectorM·(/vectorS×/vectorH)+β/vectorM·(/vectorL×/vectorH). (27)\n∂t/bracketleftBig1\n2(/vectorL·/vectorM)/bracketrightBig\n=/vectorM·(/vectorL×/vectorH)+η/vectorM·(/vectorL×/vectorS) +\n+ηe\n2m2/vectorS·(/vectorL×/vectorH)+eλ\n2m/vectorL·(/vector∇×/vectorS),. (28)\nWith the equations dysplayed in (25)-(27), it is possible to inspect the general behavior of\nthe magnitude of the magnetization, /vectorM, that precesses around the magnetic field, /vectorH. In\nour framework, the magnetization also precesses around the torsion vector /vectorM·/vectorS. Without\ntorsion, we have∂/vectorM\n∂t=/vectorM×/vectorH, so that/vectorM·/vectorM= constant and /vectorM·/vectorH= constant as in the\nusual case of the electron under the action of a time-depende nt external magnetic field,\nwith the Zeeman term given by the Hamiltonian HM=/vectorM·/vectorH.\n3.1 Planar torsion analysis with damping\nHere, we intend to analyze some possibilities of solutions t o the magnetization that respect\nthe conditions given by (25)-( 27). The magnitude of the magn etization is not constant in\ngeneral, as we can see in equation (25), but, if this quantity is constant, there comes out a\nconstraint given by\n/vectorM·(/vectorS×/vectorH) =−λm\nη/vectorM·(/vector∇×/vectorS). (29)\nIf we considerd(/vectorL·/vectorM)\ndt= 0, we have\n/vectorS·(/vectorL×/vectorH) =−mλ\nη/vectorL·(/vector∇×/vectorS). (30)\nThis expression describes us the case where /vectorM·/vectorH/ne}ationslash= 0 and/vectorS·/vectorM/ne}ationslash= 0; then, there is a the\ndamping angle in both directions given by the precession aro und the magnetic field /vectorHand\naround the torsion pseudo-vector /vectorS:\n∂t/bracketleftBig1\n2(/vectorH·/vectorM)/bracketrightBig\n=λm(e\n2m2/vectorH−/vectorM)·(/vector∇×/vectorS); (31)\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n=−λm\nη/vectorM·(/vector∇×/vectorS). (32)\n10Let us consider the first proposal in a very particular and ver y simple case for a planar\ntorsion field, /vectorS=1\n2χ(xˆy−yˆx); this choice allows us to realize the curl of torsion as an\neffective magnetic field, /vector∇×/vectorS=/vectorBeff=χˆz.\nIf we pick up the configuration of Fig. 1, we find the relation of the angle between the\nmagnetic field /vectorHand the magnetization /vectorM.\nFigure 1: Magnetization vector rotating around the magneti c field/vectorHwith damping given\nby the dynamics of the angle ζ. The system {M,H}rotates around the vector /vectorSin the xy-\nplane also with damping given by the angle φ. We consider φ=ωφtwithωφ=ωθ=2λmχ\nηS;\nwith this configuration ζ=ωζt=λχm\nηHM/parenleftBig\nH+ηM/parenrightBig\nt.\nWe started off by discussing the case where the torsion is plan ar with the magnitude\nof the magnetization being constant, /vectorM·/vectorM= 0, and\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n=−λχm\nη/vectorM·/vector z,. (33)\nthis gives us the magnetic momentum precession around the to rsion. For consistency, we\nshow that this result is compatible with the equation\n∂t/bracketleftBig1\n2(/vectorH·/vectorM)/bracketrightBig\n=λmχ(e\n2m2/vectorH−/vectorM)·/vector z. (34)\nIn the case of the Fig. 1, the magnetization precesses around the magnetic field and around\nthe planar torsion vector both with damping.\n113.2 Helix-Damping Sharped Effect in a Planar Torsion Configur ation\nNow, let us consider the most general case, where the magnitu de of the magnetization is\nnot constant, but with ( /vectorL·/vectorM) = 0. The configuration is considered in Fig. 2. With the\nFigure 2: In this picture we show the effect of the torsion in mag netization dynamics. The\ngreen vector is the magnetization vector and the blue vector is the external magnetic field.\nexpressions (25)-(27), we can readily write the magnitude o f the magnetization\n∂t/bracketleftBig1\n2(/vectorM·/vectorM)/bracketrightBig\n=eη\n2m2/bracketleftBig\n∂t/bracketleftBig1\n2(/vectorS·/vectorM)/bracketrightBig\n+λm\nη/vectorM·(/vector∇×/vectorS)/bracketrightBig\n. (35)\nBy using of the equation (35) and considering ∂t/bracketleftBig\n1\n2(/vectorS·/vectorM)/bracketrightBig\n/ne}ationslash= 0, we can see, the example\nof the Fig. 3, that ∂t/bracketleftBig\n1\n2(/vectorM·/vectorM)/bracketrightBig\n/ne}ationslash= 0. This possibility gives us that the magnitude of\nmagnetization is not constant, as in the usual LLG. This effect is the effect of torsion that\ngives us that the rotational lines do not return around thems elves.\nEquation (35 ) does not involve the explicit dependence of th e magnetic field. We\nchoose to work out the equation\n∂t/bracketleftBig1\n2(/vectorHeff·/vectorM)/bracketrightBig\n=e\n2mλχ/vectorH·/vector z, (36)\nwhere/vectorHeff=/vectorH−η/vectorSgives us the explicit form of the magnetic field interaction w ith the\n12Figure 3: In this draw we show the effect of the torsion in magnet ization dynamics. The\ngreen vector is the magnetization vector and the blue vector is the external magnetic field.\nIn this representation we used |/vectorM|=M(t),|/vectorS|= constant and |/vectorH|= constant with\nωθ=2mλχ\nηSthenM(t) =λeχ\nωθmsinωθt.\nmagnetization.\nWe notice that this quantity is different from zero, then the an gle between the magnetic\nfield and the magnetization is not constant; this yields us th e damping precessing effect of\nthe magnetization vector around the magnetic field. The comp osition between these two\neffects, dislocation and damping, is what we refer to as the hel ix-sharped with damping,\neffect where the damping effect can be see in Fig. 4. We can show tha t there are two\nmagnetization effects: the damping given by the longitudinal magnetization function m(t)l\nand dislocations given by the longitudinal magnetization f unctionm(t)tas we can see in\nFig. 2. The trajectory of the magnetization is the conical in creasing spiral, where the\nmodulus of magnetization increases with the time. In the tor sion plan the behavior is\ngiven by Fig3.\n13Figure 4: Damping behavior in torsion plane. Show the behavi or of theφ=ωφtdynamic.\n4 Concluding Remarks\nIn this work, we have considered that the magnetization equa tion is a non-relativistic\nremnant of the non-relativistic limit of the Dirac equation with torsion couplings. We\nhave considered two types of couplings: one of these related with the spin current in Dirac\nequation, defined by the spin connection. When we derived the action in relation with\nthe spin connection we obtain the spin current, this descrip tion is analog to the charged\ncurrent when we have the derivation of the action in relation with the gauge field.\nWerefertotheothertermasthenon-minimaltorsiontermand Itgives ustherotational\nof the torsion. We have analyzed this term in the general cont ext and observed that it is\npossible to recover the Landau Lifshitz in the case were the t orsion is zero. Then, we can\npoint out that the non-relativistic limit of the Dirac equat ion reproduces the usual case\nwhere the magnetization vector precesses around the magnet ic field. When we introduce\nthe torsion terms we analyze, in the general regime the magni tude of magnetization /vectorM·/vectorM,\nthe precession of the magnetization around the magnetic fiel d/vectorH·/vectorM, and the precession\nof the magnetization around the torsion pseudo-vector /vectorS·/vectorMis not constant.\nWhen the magnitude of the magnetization is constant, in the c ase where the torsion is\nplanar, there are two possible magnetization precessions o ne around the magnetic field and\nother around the planar torsion pseudo-vector. In both dyna mics, there occurs damping.\nAn interesting example has been analyzed in Fig. 1, where we s how that it is possible\nto realized an apparatus in some experimental device. In thi s sense, our framework can\nreproduce the LLG equation. The most general approach shoul d consider that the mag-\nnitude of the magnetization is not constant. In this case, as we can see from Fig. 2, the\nloop drawn by the magnetization damping but the is not remain in the same plane. This\n14effect is typically a torsion effect, were the lines are not close d. This effect seems to be\nlike a dislocation in the material that presents topologica l defects like solitons and vortices.\nBoth dislocation and damping give us what we refer to as the he lix-damping sharp effect,\nwich is a new feature of the models with torsion[33].\nWe can observethat this resultis thenewfeatureintroduced bytheplanartorsion, ifwe\nconsider the comparation with dampingand dislocations ter ms presented in LLG equation.\nThis may help in the task of setting up new apparatuses and may be experimental purposes\nto explore such characteristics in this phenomenon. We have found that /vectorM·/vectorM= constant,\nits consequence is the dislocation effect. The damping effect is the usual one, where the\nangle dynamic can crease and decrease with the time. In this w ork we does not study the\npolarization of the spins that is subject of next work when we will consider these systems in\nterms or the spinup and spin down dynamic. In the literature, this effect is named pumped\nspin current [34], and we shall study the possibility of this current when the system is in\na helix-sharped configuration[35].\nReferences\n[1] A. Dyrdal, J. Barnas, Phys. Rev. B 92, 165404 (2015)\n[2] Z. Wang, C. Tang, R. Sachs, Y. Barlas, and J. Shi, Phys. Rev . Lett.114, 016603\n(2015).\n[3] K. M. McCreary, A. G. Swartz, W. Han, J. Fabian, R.K. Kawak ami Phys. Rev. Lett.\n109, 186604 (2012)\n[4] I. A. Zhuravlev, V. P. Antropov andK. D. Belashchenko, Ph ys.Rev. Lett. 115, 217201\n(2015).\n[5] V. Flovik, F. Maci, J. M. Hernndez, R. Brucas, M. Hanson an d E. Wahlstrm, Phys.\nRev. B92, 104406 (2015).\n[6] I. Turek, J. Kudrnovsky and V. Drchal, Physical Review B 92, 214407 (2015).\n[7] L.D. Landau, E.M. 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E 91, 033111 (2015)\n[35] TaikiYoda, TakehitoYokoyama, ShuichiMurakami, Scie ntificReports5, 12024(2015).\n17"    },    {        "title": "1602.06201v2.A_systematic_study_of_magnetodynamic_properties_at_finite_temperatures_in_doped_permalloy_from_first_principles_calculations.pdf",        "content": "arXiv:1602.06201v2  [cond-mat.mtrl-sci]  23 Jun 2016A systematic study of magnetodynamic properties at finite te mperatures in doped\npermalloy from first principles calculations\nFan Pan,1,2,∗Jonathan Chico,3Johan Hellsvik,1Anna Delin,1,2,3Anders Bergman,3and Lars Bergqvist1,2\n1Department of Materials and Nano Physics, School of Informa tion and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, SE-16440 K ista, Sweden\n2Swedish e-Science Research Center (SeRC), KTH Royal Instit ute of Technology, SE-10044 Stockholm, Sweden\n3Department of Physics and Astronomy, Materials Theory Divi sion,\nUppsala University, Box 516, SE-75120 Uppsala, Sweden\n(Dated: June 24, 2018)\nBy means of first principles calculations, we have systemati cally investigated how the magnetody-\nnamicproperties Gilbert damping, magnetization andexcha ngestiffness areaffected whenpermalloy\n(Py) (Fe 0.19Ni0.81) is doped with 4d or 5d transition metal impurities. We find th at the trends in\nthe Gilbert damping can be understood from relatively few ba sic parameters such as the density of\nstates at the Fermi level, the spin-orbit coupling and the im purity concentration. The temperature\ndependence of the Gilbert damping is found to be very weak whi ch we relate to the lack of intraband\ntransitions in alloys. Doping with 4 delements has no major impact on the studied Gilbert damping,\napart from diluting the host. However, the 5 delements have a profound effect on the damping and\nallows it to be tuned over a large interval while maintaining the magnetization and exchange stiff-\nness. As regards spin stiffness, doping with early transitio n metals results in considerable softening,\nwhereas late transition metals have a minor impact. Our resu lt agree well with earlier calculations\nwhere available. In comparison to experiments, the compute d Gilbert damping appears slightly\nunderestimated while the spin stiffness show good general ag reement.\nI. INTRODUCTION\nSpintronics and magnonic applications have attracted\na large degree of attention due to the potential of cre-\nating devices with reduced energy consumption and im-\nprovedperformancecomparedtotraditionalsemiconduc-\ntor devices1–3. An important ingredient for understand-\ning and improving the performance of these devices is\na good knowledge of the magnetic properties. In this\nstudy, we focus on the saturation magnetization Ms, the\nexchange stiffness Aand the Gilbert damping α4. The\nlatter is related to the energy dissipation rate of which a\nmagnetic system returns to its equilibrium state from an\nexcited state, e.g. after the system has been subjected to\nanexternalstimuliisuchasanelectricalcurrentwhichal-\ntersits magneticstate. The threeparameters, Ms,Aand\nαdescribe the magnetodynamical properties of the sys-\ntem of interest. Ultimately one would like to have com-\nplete independent control and tunability of these proper-\nties. In this study, the magnetodynamical properties of\nPermalloy (Py) doped with transition metal impurities\nare systematically investigated within the same compu-\ntational framework.\nThe capability of tuning the damping for a material\nwith such a technological importance as Py is important\nfor the development of possible new devices in spintron-\nics and magnonics. The understanding of how transi-\ntion metals or rare earth dopants can affect the prop-\nerties of Py has been the focus of a number of recent\nexperimental studies5–9. Typically in these studies, the\nferromagnetic resonance10(FMR) technique is employed\nandαandMsare extracted from the linewidth of the\nuniform precession mode while Ais extracted from the\nfirst perpendicularstandingspin-wavemode11,12. On thetheory side, calculations of Gilbert damping from first-\nprinciples density functional theory methods have only\nrecently become possible due to the complexity of such\ncalculations. Two main approaches have emerged, the\nbreathing Fermi surface model13,14and the torque corre-\nlation models15,16. Common to both approaches is that\nspin-orbit coupling along with the density of states at\nthe Fermi level are the main driving forces behind the\ndamping. The breathing Fermi surface model only takes\nonly into account intraband transitions while torque cor-\nrelation model also includes interband transitions. The\ntorque correlation model in its original form contains a\nfree parameter, namely the scattering relaxation time.\nBrataaset. al17later lifted this restriction by employing\nscattering theory and linear response theory. The result-\ning formalism provides a firm foundation of calculating\nαquantitative from first principles methods and allows\nfurther investigations of the source of damping. Gilbert\ndamping in pure Py as well as doping with selected el-\nements have been calculated in the past9,18–20, however\nno systematic study of the magnetodynamic properties\nwithin the same computational framework has been con-\nducted which the present paper aims to address.\nThe paper is outlined as follows: In Section II we\npresent the formalism and details of the calculations, in\nSection III we present the results of our study and in\nSection IV we summarize our findings and provide an\noutlook.2\nII. THEORY\nA. Crystal structure of permalloy and treatment of\ndisorder in the first-principles calculations\nPure Permalloy (Py), an alloy consisting of iron (Fe)\nandnickel(Ni) withcompositionFe 0.19Ni0.81, crystallizes\nin the face centered cubic (fcc) crystal structure, where\nFe and Ni atoms are randomly distributed. Additional\ndoping with 4 dand 5dimpurities (M) substitutes Fe (or\nNi) sothat it becomes a threecomponent alloywith com-\nposition Py 1−xMx, wherexis the concentration of the\ndopant.\nAll first principles calculations in this study were\nperformed using the spin polarized relativistic (SPR)\nKorringa-Kohn-Rostoker (KKR)21Green’s function\n(GF) approach as implemented in the SPR-KKR\nsoftware22. The generalized gradient approximation\n(GGA)23wasusedintheparametrizationoftheexchange\ncorrelation potential and both the core and valence elec-\ntrons were solved using the fully relativistic Dirac equa-\ntion. The broken symmetry associated with the chemical\nsubstitution in the system was treated using the coherent\npotential approximation (CPA)24,25.\nB. Calculation of magnetodynamical properties of\nalloys: Gilbert damping within linear response\ntheory and spin stiffness\nOne of the merits with the KKR-CPA method is that\nit has a natural way of incorporating calculations of re-\nsponse properties using linear response formalism17,19,20.\nThe formalism for calculating Gilbert damping in the\npresent first principles method has been derived in Refs.\n[17] and [20], here we only give a brief outline of the\nmost important points. The damping can be related as\nthe dissipation rate of the magnetic energy which in turn\ncan be associated to the Landau-Lifshitz-Gilbert (LLG)\nequation4, leading to the expression\n˙E=Heff·dM\ndτ=1\nγ2˙ˆm[˜G(m)˙ˆm], (1)\nwhereˆm=M/Msdenotesthe normalizedmagnetization\nvector,Msthe saturation magnetization, γthe gyromag-\nnetic ratio and ˜G(m) the Gilbert relaxation rate tensor.\nPerturbing a magnetic moment from its equilibrium\nstate by a small deviation, ˆm(τ) =ˆm0+u(τ), gives an\nalternative expression of the dissipation rate by employ-\ning linear response theory\n˙Edis=π/planckover2pi1/summationdisplay\nij/summationdisplay\nµν˙uµ˙uν∝angbracketleftψi|∂ˆH\n∂uµ|ψj∝angbracketright∝angbracketleftψj|∂ˆH\n∂uν|ψi∝angbracketright×\nδ(EF−Ei)δ(EF−Ej),(2)where the δ-functions restrict the summation over\neigenstates to the Fermi level which can be rewrit-\nten in terms of Green’s function as Im G+(EF) =\n−π/summationtext\ni|ψi∝angbracketright∝angbracketleftψi|δ(EF−Ei). By comparing Eqs. (1) and\n(2), the Gilbert damping parameter αis obtained, which\nis dimensionless and is related to the Gilbert relaxation\ntensorα=˜G/(γMs). This can be expressed as a trans-\nport Kubo-Greenwood-like equation26,27in terms of the\nretarded single-particle Green’s functions\nαµν=−/planckover2pi1γ\nπMsTrace/angbracketleftBig∂ˆH\n∂uµImG+(EF)∂ˆH\n∂uνImG+(EF)/angbracketrightBig\nc,\n(3)\nwhere∝angbracketleft...∝angbracketrightcdenotes a configurational average. For the\ncubic systems treated in this study, the tensorial form of\nthe damping can with no loss of generality be replaced\nwith a scalar damping parameter. Thermal effects from\natomicdisplacementsandspinfluctuationswereincluded\nusing the alloy-analogy model28within CPA.\nThe spin-wave stiffness Dis defined as the curvature\nof the spin wave dispersion spectrum at small wave vec-\ntors (ω(q)≈Dq2).Din turn is directly related to the\nexchange interactions in the Heisenberg model which are\nobtained using the LKAG formalism29,30such that\nD=2\n3/summationdisplay\nijJijR2\nij√mimj, (4)\nwhereJijis the interatomic exchangeparameterbetween\nthei-thandj-thmagneticmoment, Rijthe distancecon-\nnecting the atomic sites with index iandjandmi(mj)\nthe magneticmoment atsite i(j). It is worthnotingthat\nEq. (4) only holds for cubic systems as treated here, for\nlower symmetries the relation needs modifications. The\nexchange couplings in metallic systems are typically long\nranged and could have oscillations of ferromagnetic and\nantiferromagnetic character, such as present in RKKY\ntype interactions. Due to the oscillations in exchange in-\nteractions, care is needed to reach numerical convergence\nof the series in Eq. (4) and it is achieved following the\nmethodology as outlined in Refs. 31 and 32.\nC. Calculation of finite temperature magnetic\nproperties\nOnce the exchange interactions within the Heisen-\nberg model have been calculated, we obtained finite\ntemperature properties from Metropolis33Monte Carlo\nsimulations as implemented in the UppASD software\npackage34,35. In particular, the temperature dependent\nmagnetization was obtained, and enters the expression\nfor micromagnetic exchange stiffness A, defined as36–39\nA(T) =DM(T)\n2gµB, (5)3\nwhereµBis the Bohr magneton, gis the Land´ e g-factor\nandM(T) the magnetization at temperature T.\nD. Details of the calculations\nForeachconcentrationofthedifferentimpuritiesinPy,\nthe lattice parameter was optimized by varying the vol-\nume and finding the energy minimum. The k-point mesh\nfor the self consistent calculations and exchange interac-\ntions was set to 223giving around 800 k-points in the ir-\nreducible wedge of the Brillouin zone (IBZ). The Gilbert\ndamping calculation requires a very fine mesh to resolve\nallthe Fermisurfacefeaturesandthereforeasignificantly\ndenser k-point mesh of 2283(∼1.0×106k-points in\nIBZ) was employed in these calculations to ensure nu-\nmericalconvergence. Moreover,vertexcorrections40were\nincluded in the damping calculationssince it has been re-\nvealedtobeimportantinpreviousstudies20forobtaining\nquantitative results.\nIII. RESULTS\nA. Equilibrium volumes and induced magnetic\nmoments\n6.656.706.756.806.856.906.957.00\nNbMoTcRuRhPdAgTaWReOsIrPtAuLattice constant (Bohr)   \n10%M\n15%M\nPy\nFIG. 1. Calculated equilibrium volumes of Py-M, where M\nstands for a 4 d(left) or 5 d(right) transition metal. Values\nfor 10% and 15% doping concentrations are shown. Reference\nvalue of pure Py is diplayed with a dashed line.\nFigure 1 shows the calculated equilibrium volume of\ndoped Pyfor twodifferent concentrations(10%and 15%)\nofimpuritiesfrom the 4 dand5dseriesofthe PeriodicTa-\nble. Firstofall, itis notedthat thevolumeincreaseswith\nthe concentration, and the volume within a series (4 dor\n5d) has a parabolicshape with minimum in the middle of\nthe series. This is expected since bonding states are con-secutivelyfilledand maximizedinthe middle ofthe series\nandthusthebondingstrengthreachesamaximum. Mov-\ning further through the series, anti-bonding states start\nto fill, giving rise to weaker bonding and larger equilib-\nriumvolumes. Thisisconsistentwiththeatomicvolumes\nwithin the two series41.\n0.500.600.700.800.901.001.10Total moment ( µΒ)\nM 5%\n10%\n15%\n20%\nPy\n-0.4-0.20.00.20.40.6\nNbMoTcRuRhPdAgTaWReOsIrPtAuLocal moment ( µΒ)M\nFIG.2. (Upper)Totalmagneticmoment(spinandorbital)for\ndifferent impurities and concentrations. Reference value f or\npure Py marked with a dashed line. (Lower) Local impurity\nmagnetic moment for Py 0.95M0.05\nThe local moments of the host atoms are only weakly\ndependent on the type of impurity atom present. More-\nover, the magnetic moments are dominated by the spin\nmomentµSwhile the orbital moments µLare much\nsmaller. As an example, in pure Py without additional\ndoping, the spin (orbital) moments of Fe is calculated to\n≈2.64 (0.05) µBand for Ni ≈0.64 (0.05) µB, respec-\ntively. This adds up to an average spin (orbital) moment\nof≈1.04(0.05)µBby taking into account the concentra-\ntion of Fe and Ni in Py. The total moment is analyzed\nin more detail in Fig. 2 (upper panel). As mentioned\nabove, one would like to achieve tunable and indepen-\ndent control of the saturation magnetization. Reducing\nthe magnetization reduces the radiative extrinsic damp-\ning but could at the same time affect the other properties\nin an unwanted manner. In many situations, one strive\nfor keeping the value of the total moment (saturation\nmagnetization) at least similar to pure Py, even for the\ndoped systems. It is immediately clear from Fig. 2 that\ndoping elements late in the series are the most preferable\nin that respect, for instance Rh and Pd in the 4 dseries\nand Ir, Pt and Au in the 5 dseries.\nIn Fig. 2 (lower panel) we show the local impurity\nmagnetic moments for 5% impurities in Py. In the be-\nginning of the 4 d(5d) series, the impurity atoms have an\nantiferromagnetic coupling, reflected in the negative mo-\nments compared to the host (Fe and Ni) atoms while lat-4\nter in the series couples ferromagnetically (positive mo-\nments). The antiferromagnetic coupling may not be pre-\nferred since it will tend to soften the magnetic properties\nand maybe even cause more complicated non-collinear\nmagnetic configurations to occur.\nB. Band structure\nSince Py and doped-Py are random alloys, they lack\ntranslational symmetry and calculations using normal\nband structure methods are more challenging due to the\nneed for large supercells. However, employing CPA re-\nstores the translational symmetry and more importantly,\nthe band structure of disordered systems can be ana-\nlyzedthroughtheBlochspectralfunction (BSF) A(E,k),\nwhich can be seen as a wave vector k-dependent density\nof states (DOS) function. For ordered systems the BSF\nis aδ-like function at energy E( k) while for disordered\nsystems each peak has an associated broadening with a\nlinewidth proportional to the amount of disorder scatter-\ning. In the upper panel of Fig. 3 the calculated BSF for\npure Py is displayed. Despite being a disordered system,\nthe electron bands are rather sharp below the Fermi level\nwhile in the vicinity ofthe Fermi level the bands becomes\nmuch more diffuse indicating that most of the disorder\nscattering takes place around these energies.\nIf Py is doped with 20% Pt impurities, the positions\nof the electron bands do not change much as shown by\nthe BSF in the lower panel of Fig. 3. The most strik-\ning change is the large increase of the disorder scatter-\ning compared to than Py causing diffuse electron bands\nthroughout the Brillouin zone and energies. However,\nexactly at the Fermi level the differences between the\ndoped and undoped system is not very pronounced and\nthesestatesarethe mostimportantforthe determination\nof the Gilbert damping, as seen from Eq. 3.\nC. Gilbert damping: effect of doping\nThe calculated Gilbert damping of the doped Py sys-\ntems fordifferent concentrationsofimpurities isshownin\nFig. 4 (upper panel). The 4 dimpurities only marginally\ninfluence the damping while the 5 dimpurities dramati-\ncally change the damping. The first observation is that\nweobtainverygoodagreementasinthe previousstudy19\nfor the 5dseries with 10% impurities, howevernot so sur-\nprising since we use same methodology. Secondly, the\nmost dramatic effect on damping upon doping is for the\ncase of Py doped with 20% Os impurities in which the\ndamping increases with approximately 800% compared\nto pure Py, as previously reported in Ref. [20]. How-\never, in the present study we have systematically var-\nied the impurity elements and concentrations and tried\nto identify trends over a large interval. Compared to\nexperiments5, the calculated values of the Gilbert damp-\ning are consistently underestimated. However it is worth\nFIG. 3. The Bloch spectral function A(E,k) of Py (upper\npanel) and Py doped with 20% Pt impurities (lower panel).\nThe Fermi level is indicated with a horizontal black line at\nzero energy.\nremembering that calculations only shows the intrinsic\npart of the damping while experiments may still have\nsome additional portion of extrinsic damping left such\nas Eddy current damping and radiation damping, since\nit is difficult to fully separate the different contributions.\nMoreover,incalculationsacompleterandomdistribution\nof atoms is assumed while there may be sample inhomo-\ngeneities such as clustering in the real samples.\nFrom most theoretical models, the two main material\nproperties that determine the damping are the density\nof states (DOS) at the Fermi level and the strength of\nthe spin-orbit coupling. In the following, we first inves-\ntigate separately how these properties affect the damp-\ning and later the combination of the two. In the lower\npanel of Fig. 4, the total DOS and the impurity-DOS are\ndisplayed for 10% impurity concentration of 4 dand 5d\nseries transition metals. In the both 4 dand 5dseries the\nimpurity-DOS exhibits a maximum in the middle of the\nseries. However, the value of the DOS are similar for the\n4dand 5dseries and therefore cannot solely explain the\nlarge difference in damping found between the two series.5\nFor the 4dseries, the calculated damping is not directly\nproportional to the DOS while there is a significant cor-\nrelation of the DOS and damping in the 5 dseries.\n0.000.010.020.030.04Gilbert damping αM 5%\n10%\n15%\n20%\nPy\nExp.5%\n0.200.400.600.801.001.201.40\nNbMoTcRuRhPdAgTaWReOsIrPtAun(EF) (sts./eV)\nM\nPy-M\nFIG. 4. (Upper) Calculated Gilbert damping parameter for\nPy+M in different concentrations of 4 dand 5dtransition\nmetal M at low temperatures ( T= 10K). Experimental re-\nsults from Ref. [5] measured at room temperature are dis-\nplayed by solid squares and dashed line indicate reference\nvalue for pure Py. (Lower) Total (blue) and impurity (black)\ndensity of states at the Fermi level EFfor 10% impurities in\nPy.\nIn order to analyze the separate influence of spin-orbit\ncoupling on the damping, we show in upper panel of\nFig. 5 the spin-orbit parameter ξ∝1\nrdV(r)\ndr, where V(r)\nis the radial potential, of the impurity d-states. The\ncalculations include all relativistic effects by solving the\nDirac equation but here we have specifically extracted\nthe main contribution from the spin-orbit coupling. As\nexpected, the spin-orbit parameter increases with atomic\nnumberZ, and is therefore considerably larger in the\n5dseries compared to the 4 dseries. This is the most\nlikely explanation why the damping is found to be larger\nin the 5dseries than the 4 dseries. However, within a\nsingle element in either the 4d or 5d series, the damp-\ning is quadratically dependent on the relatve strength of\nthe spin-orbit strength20. The calculated values of the\nspin-orbit parameter are in good agreement with previ-\nous calculations42,43and reaches large values of 0.6-0.9\neV for the late 5 delements Ir,Pt and Au while all val-ues are below 0.3 eV for the 4 dseries. If the damping\nacross elements would only be proportional to the spin-\norbit coupling, then the damping would monotonously\nincrease with atomic number and since this is not what\nhappens, we conclude that there is a delicate balance be-\ntween spin-orbit coupling and DOS that determines the\ndamping which is further highlighted through a qualita-\ntive analysis of the involved scattering processes.\n0.000.300.600.90ξ (eV)Spin-orbit coupling\n0.000.010.02\nNbMoTcRuRhPdAgTaWReOsIrPtAuα (Norm.)TC model\nCalculation\nFIG. 5. Upper: the spin-orbit parameter of d-electrons of\nthe impurity atoms. Lower: qualitative comparison between\ncalculations and torque correlation (TC) model for damping\nwith 10% impurity concentration.\nIn the torque correlation model, the dominant con-\ntribution to damping is through the scattering44,45and\ntakes the following form\nα=1\nγMs(γ\n2)2n(EF)ξ2(g−2)2/τ, (6)\nwhereτis the relaxation time between scattering events,\nandgthe Lande g-factor, for small orbital contributions,\ncan be related as46g= 2(1 +µL\nµS). We assume that τ\nis the same for all impurities, which is clearly an ap-\nproximation but calculating τis beyond the scope of the\npresent study. By normalizing the damping from Eq. (6)\nsuch that the value for Os (10% concentration) coincides\nwith the first principles calculations, we obtain a quali-\ntatively comparison between the model and calculations,\nas illustrated in lower panel of Fig. 5. It confirms the6\ntrend in which 5 dseries lead to a larger damping than\nthe4dseriesandcapturesqualitativelythemainfeatures.\nHowever, the peak value of the damping within the 5 d\nseries in the TC model occurs for Ir while calculations\ngive Os as in experiment. Another model developed for\nlow dimensional magnetic systems such as adatoms and\nclusters suggests that the damping is proportional to the\nproduct of majority and minority density of states at the\nFermi level47. It produces a parabolic trend but with\nmaximum at incorrect position and fails to capture the\nincreased damping of the 5 delements.\nTo further analyse the role of impurity atoms on the\ndamping we also performed calculations where instead of\nimpurities we added vacancies in the system, i.e. void\natoms. The results are shown in Fig. 6 where damping\nas a function of concentrationofAg (4 d), Os (5d) and va-\ncancies are compared to each other along with Os results\nfrom experimental5and previous calculations. Surpris-\ningly, vacancies have more or less the same effect as Ag\nwith the damping practically constant when increasing\nconcentration. Since Ag has a zero moment, small spin-\norbit coupling and small density of states at the Fermi\nlevel, the net effect of Ag from a damping (or scatter-\ning) point of view is mainly diluting the host similar to\nadding vacancies. In contrast, in the Os case, being a\n5dmetal, there is a strong dependence on the concentra-\ntion that was previously analyzed in terms of density of\nstates and Os having a strong spin-orbit coupling. Our\nresults from Os is slightly lower than the previous re-\nported values19,20, despite using same software. How-\never, the most likely reason for the small discrepancy is\nthe use of different exchange-correlationpotentials in the\ntwo cases.\n0.000.010.020.030.040.050.060.070.08\n 0  5  10  15  20Gilbert damping α\n(%) of MPyOs\nPyAg\nPyVac\n10%Os ref(theo.1)\n15%Os ref(theo.2)\nPyOs ref(expt.)\nFIG. 6. Calculated Gilbert damping as a function of Os,\nAg and vacancy (Vac) concentration in Py. Open red circle:\ncalculation from Ref. [19], solid red circle: calculation f rom\nRef. [20] and red solid square: experimental data from Ref. [ 5]D. Gilbert damping: effect of temperature\nIntheprevioussectionwestudiedhowthedampingde-\npends on the electronic structure and spin-orbit coupling\nat low temperatures. However, with increasing tempera-\nture additional scattering mechanisms contribute to the\ndamping, most importantly phonon and magnon scatter-\ning. The phonon scattering is indirectly taken into ac-\ncount by including a number of independent atomic dis-\nplacementsbringingtheatomsoutfromtheirequilibrium\npositions and magnon scattering is indirectly included by\nreducing the magnetic moment for a few configurations\nand then average over all atomic and magnetic configu-\nrations within CPA. It should be noted that the present\nmethodology using the alloy-analogymodel28has limita-\ntions for pure systems at very low temperatures where\nthe damping diverges, but we are far from that situation\nin this study since all systems have intrinsic chemical\ndisorder. However, the limitations for pure systems can\nbe lifted using a more advanced treatment using explicit\ncalculation of the dynamical susceptibility48.\n0.000.010.02α(T)\n1.001.25\n 0 50 100 150 200 250 300 350 400α(T)pho+mag/α(T)pho\nTemperature (K) Py+20% Mo\nPy+20% Rh\nPy+20% W\nPy+20% Pt\nFIG. 7. Gilbert damping parameter including temperature\neffects from both atomic displacements and spin fluctuations\n(upper panel). The effect of spin fluctuations on the Gilbert\ndamping (lower panel), see text.\nThe temperature dependence of damping for a few se-\nlected systems is displayed in Fig. 7 where both atomic\ndisplacements and spin fluctuations are taken into ac-\ncount. From the 4 d(5d) series, we choose to show results7\nfor Mo and Rh (W and Pt), where Mo (W) has a small\nantiferromagnetic moment and Rh (Pt) a sizeable ferro-\nmagnetic moment, from Fig. 2. All systems display an\noverall weak temperature dependence on damping which\nonly marginally increases with temperature, as shown\nin upper panel of Fig. 7. However, in order to sepa-\nrate the temperature contributions from atomic displace-\nments and spin fluctuations, we show the ratio between\nthe total damping and damping where only atomic dis-\nplacements are taken into account in the lower panel of\nFig. 7. The two systems with sizable moments (Rh and\nPt), clearly have a dominant contribution from spin fluc-\ntuations when the moments are reduced upon increased\nscattering due to temperature. In contrast, the two sys-\ntems with (small) antiferromagnetic moments (Mo and\nW), the effect of the spin fluctuations on the damping is\nnegligible and atomic displacements are solely responsi-\nble. The weak temperature dependence found in these\ndoped Py systems is somewhat surprising since in pure\nmetals like Fe and Ni, a strong temperature dependence\nhas been both measured and calculated20, however data\nfor other random alloy systems is scarce.\nThe temperature dependence of damping from the\nband structure is often attributed to interband and\nintraband transitions which arises from the torque-\ncorrelation model. Intraband transitions has conduc-\ntivity like dependence on temperature while interband\nshows resistivty-like dependence. The weak overall de-\npendence found in the systems in Fig. 7 suggests lack of\nintraband transitions but a more detailed analysis of the\nband structure and thermal disorder are left for a future\nstudy.\nE. Spin-wave stiffness and exchange stiffness\nIn the previous sections, we investigated saturation\nmagnetization and damping and we are therefore left\nwith the exchange stiffness. The calculated spin-wave\nstiffnessDatT= 0 K, from Eq. 4, is displayed in the up-\nperpanelofFig.8. Dcanbedirectlymeasuredfromneu-\ntron scattering experiments but as far as we are aware,\nno such data exist. For the late elements in the 4 dand\n5dseries, the spin wave stiffness is maximized and have\nvalues rather similar to pure Py, however with a reduc-\ntion of approximately 20%. In micromagnetic modelling,\nit is common to use the exchange stiffness Ainstead of\nD.Ais proportional to D, from Eq. 5, and the sole\ntemperature dependence of Atherefore comes from the\nmagnetization. In the lower panel of Fig. 8, we show the\ncalculated room temperature ( T= 300 K) values of A,\ntogether with values for pure Py and available experi-\nmental data. In the beginning of the 4 d(5d) series, the\nexchangestiffnessbecomes smalluponincreasingconcen-\ntration of impurities and the systems are magnetically\nvery soft. It follows from the fact that magnetization\nis small because the systems are close to their ordering\ntemperature. Contrary, for the late elements in the 4 d100200300400500Spin−wave stiffness\nD (meV Å2)5% M\n10% M15% M\nPy\n  5 10 15\nNbMoTcRuRhPdAgTaWReOsIrPtAuExchange stiffness\nA (pJ/M)\nPy(expt.) 15%(expt.)\nFIG. 8. Spin-wave stiffness Dof Py-M in the ground state\n(top) and exchange stiffness constant Aat room temperature\nT= 300K (bottom) as a function of doping concentration.\nThe strict dashed lines show the reference value of pure Py\nfrom calculation and experiments. The scattered dots indi-\ncate the experimental data for Py+15%M (Ag/Pt/Au) from\nRef.9\n(5d)series,themagnetizationhasalargefinitevalueeven\nat room temperature and thereforethe exchangestiffness\nalso has a large value, howeverreduced by approximately\n15% compared to pure Py.\nIV. SUMMARY AND CONCLUSIONS\nAsystematic study ofthe intrinsicmagnetic properties\nof transition metal doped Py has been presented. It is\nfound that the Gilbert damping is strongly dependent on\nthe spin-orbit coupling of the impurity atoms and more\nweakly dependent on the density of states that deter-\nmines disorder scattering. The strong influence of the\nspin-orbit coupling makes the 5 delements much more\neffective to change the Gilbert damping and more sen-\nsible to the concentration. As a result, the damping\ncan be increased by an order of magnitude compared to\nundoped Py. Overall, the damping features are quali-\ntatively rather well explained by the torque correlation\nmodel, yet it misses some quantitative predictive power\nthatonlyfirstprinciplesresultscanprovide. Moreover,it8\nisfoundthatthedampingoverallhasaweaktemperature\ndependence, howeverit is slightly enhanced with temper-\nature due to increased scattering caused by atomic dis-\nplacements and spin fluctuations. Elements in the begin-\nning of the 4 dor 5dseries are found to strongly influence\nthe magnetization and exchange stiffness due to antifer-\nromagnetic coupling between impurity and host atoms.\nIncontrast,elementsinthe endofthe4 dor5dserieskeep\nthe magnetization and exchange stiffness rather similar\nto undoped Py. 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Woltersdorf, Magnetic Nanos-\ntructures: Spin dynamics and spin transport (Springer,\n2013).46A. J. P. Meyer and G. Asch, Journal of Applied Physics\n32(1961).\n47S. Lounis, M. dos Santos Dias, and B. Schweflinghaus,\nPhys. Rev. B 91, 104420 (2015).\n48A. T. Costa and R. B. Muniz,\nPhys. Rev. B 92, 014419 (2015).\n49M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J. Silva,\nH. T. Nembach, O. Eriksson, O. Karis, and J. M. Shaw,\narXiv:1512:3610 1512, 3610 (2015)."    },    {        "title": "1501.03973v1.Direct_measurement_of_the_magnetic_anisotropy_field_in_Mn__Ga_and_Mn__Co__Ga_Heusler_films.pdf",        "content": "arXiv:1501.03973v1  [cond-mat.mtrl-sci]  16 Jan 2015Direct measurement of the magnetic anisotropy\nfield in Mn–Ga and Mn–Co–Ga Heusler films\nCiar´ an Fowley1, Siham Ouardi2, Takahide Kubota3,\nYildirim Oguz1, Andreas Neudert1, Kilian Lenz1,\nVolker Sluka1, J¨ urgen Lindner1, Joseph M. Law4,\nShigemi Mizukami3, Gerhard H. Fecher2, Claudia Felser2\nand Alina M. Deac1\n1Institute of Ion Beam Physics and Materials Research. Helmh oltz-Zentrum\nDresden-Rossendorf, Bautzner Landstrasse 400, 01328 Dres den, Germany.\n2Max-Planck-Institut f¨ ur Chemische Physik fester Stoffe, N ¨ othnitzer Str. 40,\n01187 Dresden.\n3WPI-Advanced Institute for Materials Research (WPI-AIMR) , Tohoku\nUniversity, Sendai 980-8577, Japan\n4High Magnetic Field Laboratory, Helmholtz-Zentrum Dresde n-Rossendorf,\nBautzner Landstrasse 400, 01328 Dresden, Germany.\nE-mail:c.fowley@hzdr.de\nAbstract.\nThe static and dynamic magnetic properties of tetragonally distorted Mn–\nGa based alloys were investigated. Static properties are de termined in magnetic\nfields up to 6.5 T using SQUID magnetometry. For the pure Mn 1.6Ga film,\nthe saturation magnetisation is 0.36 MA/m and the coercivit y is 0.29 T.\nPartial substitution of Mn by Co results in Mn 2.6Co0.3Ga1.1. The saturation\nmagnetisation of those films drops to 0.2 MA/m and the coerciv ity is increased\nto 1 T.\nTime-resolved magneto-optical Kerr effect (TR-MOKE) is use d to probe the\nhigh-frequency dynamics of Mn–Ga. The ferromagnetic reson ance frequency\nextrapolated to zero-field is found to be 125 GHz with a Gilber t damping, α,\nof 0.019. The anisotropy field is determined from both SQUID a nd TR-MOKE\nto be 4.5 T, corresponding to an effective anisotropy density of 0.81 MJ/m3.\nGiven the large anisotropy field of the Mn 2.6Co0.3Ga1.1film, pulsed magnetic\nfields up to 60 T are used to determine the field strength requir ed to saturate the\nfilminthe plane. Forthis, the extraordinary Halleffectwas e mployed as aprobe of\nthe local magnetisation. By integrating the reconstructed in–plane magnetisation\ncurve, the effective anisotropy energy density for Mn 2.6Co0.3Ga1.1is determined\nto be 1.23 MJ/m3.\nSubmitted to: J. Phys. D: Appl. Phys.Direct measurement of the anisotropy field in Mn–Co–Ga 2\n1. Introduction\nThere has been a recent resurge in research on Heusler alloys, in pa rticular Mn-\nbased ferrimagnetic compounds, for both magnetic storage and s pin-transfer-torque\napplications due to the ability to widely tune the magnetic properties w ith varying\ncomposition [1–6]. The addition of Co to Mn-Ga allows the subtle tuning o f magnetic\nproperties, such as uniaxial anisotropy and saturation magnetisa tion [4]. These films\ncan possess very high uniaxial anisotropy [7,8]. They are promising f or future rare-\nearth free permanent magnets [9]; highly stable magnetic recordin g elements scalable\ndown to 10 nm bit size [3]; spin-polarised electrodes for tunnel magne toresistance\ndevices [10,11]; as well as active elements in next generation spin-tr ansfer torque\ndevices such as THz-band spin transfer oscillators, due to their hig h ferromagnetic\nresonance frequencies and low Gilbert damping [7].\nInthesefilms, anisotropyfieldsofseveraltensofTesla’shavebee nreported[3,4,6].\nCombinedwiththeirlowmagnetisation,typicallybelow0.5MA/m, traditio nalSQUID\nmagnetometry and conventional ferromagnetic resonance (FMR ) techniques are not\nso well suited for magnetic characterisation. For the majority of r ecent reports the\nanisotropy field is extrapolated from the intersection of the low field linear slope of\nthe hard-axis magnetisation curve to the easy-axis saturation ma gnetisation. This\nintersection occurs at a field which is, in general, beyond the machine limit. For\nSQUIDmagnetometrythisrequirescarefulsubtractionofthe dia magneticbackground\nsignal [12]. It should also be pointed out that most studies to date ha ve also been\nperformed on thick samples, while more technologically relevant thinn er films have\nshown reduced anisotropies [12]. Finally, measurement responses t o standard SQUID-\nand FMR-based studies are also directly proportional to the total magnetic moment\npresent. An alternativeapproachis to exploit electricalor optical detection techniques\nto characterise such materials. The direct measurement of the an isotropy field is\nespeciallyrequiredifthereareadditional,forexamplein-plane,aniso tropycomponents\nto be considered in the material under investigation.\nTime-resolved magneto-optical Kerr effect (TR-MOKE) has alread y been\nsuccessfully used to characterise the high frequency dynamics of Mn–Ga thin films,\nshowing precession frequencies between 200 GHz and 300 GHz [7]. T he frequency\nof oscillation as a function of applied field can be fit with the Kittel form ula.\nSuch measurements therefore yield not only the ferromagentic re sonance (FMR)\nfrequency and the Gilbert damping parameter, α, but also the effective anisotropy\nfield and energy density ( µ0HkandKeff, respectively). Provided the saturation\nmagnetisation is known, the intrinsic uniaxial anisotropy Kucan be determined from\nKeff=Ku−1\n2µ0M2\nS, where1\n2µ0M2\nSrepresents the demagnetising energy of an\nextended thin-film.\nThe extraordinary Hall effect (EHE) is a very useful characterisa tion tool for\nthese perpendicularly magnetised materials [13]. When a current is a pplied along a\ncertain direction in a film, the transverse resistivity ( ρxy) is directly proportional to\ntheout–of–planemagnetisationcomponent( Mz)viatheextraordinaryHallcoefficient,\nREHE[14]. For perpendicular anisotropymaterials, if we apply an external field along\ntheeasyaxisofthefilmandswitchthemagnetisation,andtherefor eρxy, wewillobtain\nan electrical equivalent of the magnetic hysteresis loop. Similarly, wh en the field is\napplied along the hard axis, the saturation of the magnetisation can be seen as a\ngradual change and saturation of ρxy. From the hard axis data, the anisotropy field,\nµ0Hk, can be determined. EHE allows for characterisation of these high- anisotropyDirect measurement of the anisotropy field in Mn–Co–Ga 3\nfilms beyond what is normally accessible from standard magnetometr y for several\nreasons: firstly, it is a transport technique and not a magnetomet ry technique, which\nmeans it can be used to measure the magnetic response of volumes o f material and/or\npatterned structures which would be otherwise undetectable; se condly, an inherent\nadvantage is that ρxyexhibits an inverse thickness dependence meaning that the EHE\nsignal is larger for thinner films; thirdly, REHEin ferromagnetic materials is large\nmeaning that even though MSmay be low, ρxycan be high. These advantages make\nEHE an ideal probe of the magnetisation at lowerthickness and/or c onfined geometry.\nIn the search for materials with higher perpendicular anisotropy an d\nlower saturation magnetisation for spin-transfer-torque applica tions, alternative\ncharacterisation techniques, such as those outlined above, will be come increasingly\nmore important. To exemplify this, we investigate L10Mn1.6Ga (Mn–Ga) and\nMn2.6Co0.3Ga1.1(Mn–Co–Ga) using TR-MOKE and EHE in high magnetic fields.\nBoth materials posses high uniaxial anisotropy. In particular, the c hosen composition\nof Mn–Co–Ga has been shown to exhibit very low saturation magnetis ation while\nretaining high anisotropy [4]. They therefore represent ideal samp les for determining\nthe usefulness of the techniques as the properties are favourab le for applications while\nat the same time may prove difficult to determine with more traditional methods,\nespecially at reduced thicknesses [13]. We show that, in particular, E HE at high\nmagnetic fields is an ideal method for determining the anisotropy field of such Heusler\nsystems.\n2. Experimental details\n \n(a) Mn-Ga (b) Mn-Co-Ga MgO (001) substrate Cr buffer (40 nm ) Mn 1.6 Ga (30 nm) \nMgO (001) substrate Mn 2.6 Co 0.3 Ga 1.1 (30 nm) z\nx || H M\nFigure 1. Sketch of the Mn 1.6Ga (a) and Mn 2.6Co0.3Ga1.1(b) film stacks. The\ntopmost layers are Mg, MgO, and AlO xto protect the sample from oxidation (see\ntext for thicknesses).\nTetragonal Mn–Ga and Mn–Co–Ga thin films were grown by ultra-high -vacuum\nsputtering on single crystal MgO(001) substrates. Specific deta ils on sample\nfabrication and growth conditions can be found in Reference [4]. The stacking\nstructures of the multi-layered films are:\nMgO(100) substrate / Cr(40) / Mn 1.6Ga(30) / Mg(0.4) / MgO(2.0) / AlO x(1.3) and\nMgO(100) substrate / Mn 2.6Co0.3Ga1.1(30) / Mg(0.4) / MgO(2.0) / AlO x(1.3)\nas sketched in Figures 1(a)) and (b), respectively. The thickness , in nm, is appendedDirect measurement of the anisotropy field in Mn–Co–Ga 4\nin brackets after each material. The topmost three layers were ad ded to simulate a\ntunnelling barrier and to protect the sample from oxidation.\nLow-field magnetization data were obtained by magnetic field depend ent\nmagnetisation measurements at 300 K, up to an applied external fie ld of 6.5 T using\nconventional SQUID magnetometry. TR-MOKE was used to charac terise the high\nfrequency dynamic properties of the Mn 1.6Ga film. An 800 nm pump beam with a\npower of 31.2 mW at the sample, leads to ultra-fast demagnetisation of the sample.\nThe laser pulses were 100 fs in length with a repetition rate of 5.2 MHz. A fixed-\ndelay probe beam of 400 nm, with a power of 105 µW at the sample is then used\nto probe the excited dynamics via lock-in detection. The diameter of the spot sizes\nof the pump and probe beams were 17 µm and 5 µm, respectively. The pump and\nprobe fluences, calculated from the incident power, beam diameter and repetition\nfrequency 1.32 mJ/cm2and 0.05 mJ/cm2, respectively. Time resolution is obtained\nby a variable delay line on the pump beam which allows for the measureme nt of\nchanges in magnetisation both before and after the demagnetisat ion process.\nEHE was measured initially using a DC current of 1 mA at room temperat ure\nin a magnetotransport set-up capable of applying fields up to 1.6 T. S ubsequent EHE\nmeasurements were performed in a cryostat at 77 K using pulsed ma gnetic fields\nat the High Magnetic Field Laboratory located in the Helmholtz-Zentr um Dresden-\nRossendorf. The pulsed magnet which was used had a rise time of 7 ms , a fall time of\n24 ms and a maximum field of approximately 60 T. An sinusoidal AC curre nt of 1 mA\nwas applied to the sample with a frequency of 47.62 kHz. The transve rse voltage was\nmeasured during the magnetic field pulse and the signal was locked-in post experiment\nvia a digital lock-in program.\nEHE scans along the easy and hard axes, in combination with the evalu atedMS\nfrom SQUID, allows for the direct determination of the anisotropy fi eld,µ0Hk, as well\nas the effective anisotropy energy, Keff, by integration of normalised magnetisation\nloops.\n3. Magnetic properties\nThe initial magnetic characterisation of the thin films using SQUID is sh own in\nFigure 2. The pure Mn–Ga film – shown in figure 2 a) – exhibits sharp hys teretic\nswitching when the field is applied in the out–of–plane direction (closed red circles)\nwith a coercive field, µ0Hc, of 290 mT. The data show clearly that the easy axis\nis along the film normal. A small in-plane component appears in the hard axis\nmeasurementsconsistentwith anintrinsiccantedmomenton the2 bMn sub-lattice[6].\nThe saturation magnetisation, MS, is 0.36 MA/m. In–plane (open black squares)\nmagnetisation measurements give an anisotropy field, µ0HK, of 4.5 T, which was also\nverifiedbyvibratingsamplemagnetometry(VSM) up to14T (notsho wnhere). These\nvalues correspond to an anisotropy energy density Keffof 0.81 MJ/m3.\nThe addition of Co into Mn–Ga, shown in figure 2 b), leads to a reductio n of the\nMSto 0.2 MA/m and an accompanied increase of µ0Hcto 928 mT. The anisotropy\nfield is also increased to a value beyond 5 T. For the in–plane curve, th e same\ndiamagnetic background as in the out–of–plane measurement was s ubtracted from\nthe raw data, however as can be seen from the data, an accurate determination of\nthe anisotropy field is not possible. VSM magnetometry up to 14 T was not able\nto confirm the anisotropy field due to a low magnetisation signal (not shown). We\nnote that, as opposed to the Mn–Ga film we have a soft magnetic com ponent in bothDirect measurement of the anisotropy field in Mn–Co–Ga 5\n/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s32\n/s32/s97/s41\n/s77/s110\n/s49/s46/s54/s51/s71/s97/s77/s97/s103/s110/s101/s116/s105/s115/s97/s116/s105/s111/s110/s32/s32/s32 /s77 /s40/s72 /s41/s32/s91/s77/s65/s47/s109/s93\n/s65/s112/s112/s108/s105/s101/s100/s32/s105/s110/s100/s117/s99/s116/s105/s111/s110/s32/s102/s105/s101/s108/s100/s32/s32/s32\n/s48/s72 /s32/s91/s84/s93/s45/s52 /s45/s50 /s48 /s50 /s52\n/s32/s32\n/s32/s32/s32/s32/s105/s110/s45/s112/s108/s97/s110/s101\n/s32/s32/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s98/s41\n/s77/s110\n/s50/s46/s54/s67/s111\n/s48/s46/s51/s71/s97\n/s49/s46/s49\nFigure 2. Static magnetic properties at 300 K for a) Mn–Ga and b) Mn–\nCo–Ga films. Both films exhibit strong perpendicular magneti c anisotropy, with\napproximately square hysteresis loops and Mr/MS≈1. The addition of Co\nresults in a drop of the magnetisation accompanied by an incr ease in both the\ncoercive field and the anisotropy field (increased beyond 5 T) .\nthe in-plane and out-of-plane magnetisation curves indicating that we do not have\nthe same canted moment as in the pure film. Rather, the data seem t o indicate a\nsegregated phase lacking any anisotropy axis. This could be initially at tributed to Co\nclusters in the film or a lack of full epitaxial growth due to the lack of s eed layer.\nTheM(H) hysteresiscurves of both materials are nearly rectangular. The energy\nproduct is thus given by E=BrxHc, whereBrandHcare the remanence and\nthe coercive field, respectively. The out–of–plane energy produc ts are 104 kJ/m3for\nMn1.6Ga and 165 kJ/m3for Mn 2.6Co0.3Ga1.1. The addition of Co leads to a reduction\nof the maximum energy product, BHmax, from about 40 to 10 kJ/m3. The values of\nthe energy product and BHmaxfor Mn 2.6Co0.3Ga1.1are of the same order as bulk\nMn3Ga (cf. Reference [15]).\nFigure 3 shows the EHE curves obtained from 5 mm ×5 mm square films\nmeasured in the van der Pauw geometry. For both films, the coerciv ity is identical\nto that obtained from SQUID measurements. The magnitude of ρxyfor Mn–Ga is\nmuch lower than that obtained for the Mn–Co–Ga film. Although the u nderlying\nCr layer is responsible for current shunting, it is anyway expected t hat pure Mn–Ga\nfilms show a lower Hall signal with improving crystal quality [13]. It can be seen that\nthe EHE curve for the Co–Mn–Ga films does not trace out the additio nal change in\nmagnetisation close to zero field as seen in figure 2 (b). As has been c alculated for\nCo-doped Mn–Ga films, Co substitutes Mn randomly at both the 2 band 4dpositions,\nfills the minority band at EFand leads to the localisation of electrons in the minority\nband [16]. Assuming that the soft magnetic component seen in SQUID is related\nsolely to Co substitution, it is likely that, due to the increased electro n localisation,\nthe contribution of Co to the magnetotransport is diminished, ther efore the same soft\nphase is not reproduced in the EHE measurement.Direct measurement of the anisotropy field in Mn–Co–Ga 6\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s32/s77/s110/s45/s67/s111/s45/s71/s97/s72/s97/s108/s108/s32/s114/s101/s115/s105/s115/s116/s105/s118/s105/s114/s121/s32/s32/s32\n/s120/s121/s40/s72 /s41/s32/s91 /s99/s109/s93\n/s65/s112/s112/s108/s105/s101/s100/s32/s105/s110/s100/s117/s99/s116/s105/s111/s110/s32/s102/s105/s101/s108/s100/s32/s32/s32\n/s48/s72 /s32/s91/s84/s93/s45/s48/s46/s48/s51/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s32\n/s32/s77/s110/s45/s71/s97\nFigure 3. ExtraordinaryHalleffect data measured atroomtemperature forMn–\nGa and Mn–Co–Ga films. The coercive fields match exactly with t hose obtained\nfrom SQUID in figure 2. Note that the magnitudes of ρxyfor the two films differ\nby about two orders of magnitude.\n4. Time-resolved MOKE\nTR-MOKE was used to evaluate the effective anisotropy of the Mn–G a film. A\nmagneticfieldof0.65to1.3Twasappliedat60◦tothe filmnormalinordertocantthe\nmagnetisation away from the easy axis. This provides a projection o f the precession\nalong the film normal which is then measured in the polar MOKE geometr y.\nFigure 4 a) shows several TR-MOKE spectra at different applied mag netic fields\nas well as the fitted curves from which the frequency and damping a re extracted. It\ncan be seen in this figure that the magnetisation, after the initial de magnetisation\npulse att= 0, oscillates around the effective field direction leading to a charact eristic\noscillation of the optical signal. The ferromagnetic resonance freq uency (fres) and the\nGilbert damping parameter ( α) are extracted by fitting these curves with the Kittel\nformulafortheuniformmode. fresandαareplottedinfigure4b) andc), respectively.\nThe precession frequency was found to vary between 136 GHz and 142 GHz in the\naforementioned field range. The slope of the fitted line in figure 4 b) is 11.4 GHz/T.\nThe Gilbert dampingparameterisfound tobe roughlyindependent of the appliedfield\nand has an average value of 0 .019. By fitting fresversus applied field the effective\nanisotropy of the film can also be evaluated. The extracted value of value of 4.5 T\nobtainedfromfittingisinexcellentagreementwiththevalueobtained fromtheSQUID\nmeasurement. This, combined with a single peak in the FFT of the spec tra, indicate\nthat the uniform precession mode is the dominant contribution to th e TR-MOKE\nsignal. The values of fresandαobtained for this film also compare well to those in\nthe literature [7]. Furthermore, they demonstrate the potential use of high anisotropy\nalloys as microwave oscillators beyond 100 GHz.\n5. High-field EHE\nAs previously mentioned, the anisotropy field of the Mn–Co–Ga film is b eyond the\naccessible range of both the SQUID and the VSM (maximum field of 6.5 T and\n14 T, respectively). For the Mn–Co–Ga sample, it was also not possib le to obtainDirect measurement of the anisotropy field in Mn–Co–Ga 7\n/s49/s51/s53/s49/s52/s48/s49/s52/s53\n/s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/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/s32/s32\n/s49/s46/s50/s53/s32/s84\n/s49/s46/s48/s55/s32/s84\n/s48/s46/s56/s56/s32/s84/s100/s77 /s32/s40 /s72 /s41\n/s84/s105/s109/s101/s32/s32/s32 /s116/s32/s91/s112/s115/s93/s97/s41\n/s98/s41\n/s32\n/s32/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s32/s32 /s102/s40/s72 /s41/s32/s91/s71/s72/s122/s93\n/s99/s41\n/s32/s32/s68/s97/s109/s112/s105/s110/s103/s32 /s40/s72 /s41\n/s65/s112/s112/s108/s105/s101/s100/s32/s105/s110/s100/s117/s99/s116/s105/s111/s110/s32/s102/s105/s101/s108/s100/s32/s32/s32\n/s48/s72 /s32/s91/s84/s93\nFigure 4. a) TR-MOKE spectra (bold lines) together with fitting curves (thin\nlines) for the Mn–Ga film at various fields applied 60◦to the film normal. b)\nFrequency and c) damping as a function of applied field streng th. Lines in b) and\nc) result from linear fits.\nany ferromagnetic resonance data from the TR-MOKE which is most likely due to\nthe inability to obtain a large enough canting angle for polar MOKE dete ction. We\ntherefore determine the anisotropy field using EHE as a detection m ethod.\nFigure 5 a) shows the EHE curves for both in–plane and out–of–plan e applied\nfields up to 35 T. The EHE response with the field applied out–of–plane (open red\ntriangles) is, again, identical to that obtained from SQUID measure ments with sharp\nswitchingofthemagnetisationvisibleatapproximately1T.Whenthem agneticfieldis\napplied in the plane of the sample (closed black triangles) the magnetis ation gradually\ncantstotheplaneofthefilm. SincetheEHEsignalisonlysensitivetot heout–of–plane\ncomponent of magnetisation, Mz, the in–plane component of the magnetisation, Mx,\nmust be reconstructed. This is achieved using the transform, sin( cos−1(Mz)) because\nthe projection of the magnetisation along zis simply cos( φ), where φis the angle of\nthe magnetisation to the film normal (see Figure 5 b) inset and Figure 1b)).\nFigure 5 b) shows the both the Mzcomponent (open red circles), the\nreconstructed in–plane Mxcomponent (open black squares), and the canting angle, φ\n(blue dashed line). Here, both MxandMzhave been normalised to the maximum. It\ncan be seen from the figure that the Mxcomponent is almost totally saturated at 12 T\nand is completely saturated at 18 T. The anisotropy energy is obtain ed by evaluatingDirect measurement of the anisotropy field in Mn–Co–Ga 8\n/s45/s51/s48 /s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48 /s51/s48/s45/s50/s46/s48/s45/s49/s46/s48/s48/s46/s48/s49/s46/s48/s50/s46/s48\n/s32/s73/s110/s45/s112/s108/s97/s110/s101\n/s32/s32\n/s32/s32/s79/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s72/s97/s108/s108/s32/s114/s101/s115/s105/s115/s116/s105/s118/s105/s116/s121/s32 /s32/s32\n/s120/s121/s40/s72 /s41/s32/s91 /s99/s109/s93\n/s48 /s49/s53 /s51/s48 /s52/s53 /s54/s48/s48/s46/s48/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48\n/s32/s32\n/s32/s32 /s77\n/s120\n/s32/s32/s77\n/s122/s77\n/s105/s32/s47/s32 /s77\n/s83\n/s65/s112/s112/s108/s105/s101/s100/s32/s105/s110/s100/s117/s99/s116/s105/s111/s110/s32/s102/s105/s101/s108/s100/s32/s32/s32\n/s48/s72 /s32/s91/s84/s93/s72 /s32/s124/s124/s32 /s120/s32\n/s77/s122\n/s45/s57/s48/s176/s45/s54/s48/s176/s45/s51/s48/s176/s48/s176/s51/s48/s176/s54/s48/s176/s57/s48/s176/s98/s41\n/s32/s32/s32/s65/s110/s103/s108/s101/s32/s32/s32/s97/s41\nFigure 5. a) EHE response of the Mn–Co–Ga film, measured with in–plane a nd\nout–of–plane fields up to 35 T. b) reconstructed in–plane mag netisation curve\n(Mx/MS) and angle of magnetisation ( φ) as a function of applied field up to\napproximately 60 T.\nthe integral/integraltextMS\n0µ0Hx.dMxbetween zero and saturation, i.e the area enclosed by\nthe reconstructed Mxcurve and the y-axis, and multiplying this by MS. Beyond\nsaturation the Mzcomponent of magnetisation begins to increase. This is attributed\nto a small offset between the applied field direction and the film plane. A lthough the\nchange is rather large in Mzit is corresponds to a misalignment of less than 5◦.\nFrom the integration of the area between the reconstructed in-p lane and out-of-\nplane loops a value of Keffof 1.23 MJ/m3is obtained. Although the TR-MOKE\nresults were inconclusive with the Mn–Co–Ga sample, the given value o f anisotropy\nfield and saturation magnetization yield precession frequencies of o rder 350 GHz.\n6. Conclusion\nWe have investigated static and dynamic magnetic properties of Mn 1.6Ga and\nMn2.6Co0.3Ga1.1films. As expected, these materials possess a low saturation\nmagnetisation and high uniaxial anisotropy, usually beyond the acce ssible field range\navailable in typical SQUID magnetometers. Consequently, standar d magnetometry\nwas combined with both time resolved magneto-optical Kerr effect a nd extraordinary\nHall effect in high magneticfields to ascertainexact values for the ma gnetic anisotropyDirect measurement of the anisotropy field in Mn–Co–Ga 9\nof both types of thin films.\nFor the pure Mn 1.6Ga alloy, we find a magnetic anisotropy energy of 0.81 MJ/m3,\nwith a saturation magnetisation of 0.36 MA/m. The partial substitut ion of Mn by Co\nincreases the effective anisotropy energy to 1.23 MJ/m3and decreases the saturation\nmagnetisation of 0.2 MA/m.\nTime-resolved magneto-optical Kerr effect and extraordinary Ha ll effect were\nutilised to directly probe the anisotropy fields of both materials. The measured values\nare 4.5 T and 18 T for Mn 1.6Ga and Mn 2.6Co0.3Ga1.1, respectively.\nIn summary, we have shown that both time-resolved magneto-opt ical Kerr-\neffect and the extraordinary Hall effect in high magnetic fields are ex tremely useful\ntechniques in determining the anisotropy energy for materials which cannot be\nsaturated in standard magnetometers such as SQUID. Such indire ct techniques can be\nreadily applied to technologically relevant high anistropy materials for spin-transfer-\ntorque applications.\nAcknowledgments\nC.F. would like to acknowledge Dr. Marc Uhlarz for fruitful discussion s. Financial\nsupport by DfG-JST is gratefully acknowledged (projects P 1.3-A a nd P 2.1-A of the\nResearchUnitFOR1464 ASPIMATT ). Partofthisworkwasfunded byEuroMagNET\nunder EU contract no. 228043.Direct measurement of the anisotropy field in Mn–Co–Ga 10\n[1] B. Balke, G. H. Fecher, J. Winterlik, C. Felser, M. C. M. Al ves, F. Bernardi, and J. Morais.\nPhys. Rev. B , 77:054406, 2008.\n[2] V. Alijani, J. Winterlik, G. H. Fecher, and C. Felser. Appl. Phys. Lett. , 99:222510, 2011.\n[3] H. Kurt, K. Rode, M. Venkatesan, P. Stamenov, and J. M. D. C oey.Phys. Rev. B , 83(2):020405,\n2011.\n[4] S. Ouardi, T. Kubota, G. H. Fecher, R. Stinshoff, S. Mizuka mi, T. Miyazaki, E. Ikenaga, and\nC. Felser. Appl. Phys. Lett. , 101(24):242406, 2012.\n[5] T. Kubota, S. Ouardi, S. Mizukami, G. H. Fecher, C. Felser , Y. Ando, and T. Miyazaki. J.\nAppl. Phys. , 113:17C723, 2013.\n[6] K. Rode, N. Baadji, D. Betto, Y.-C. Lau, H. Kurt, M. Venkat esan, P. Stamenov, S. Sanvito,\nand J. M. D. Coey. Phys. Rev. B , 87:184429, 2013.\n[7] 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(11):117201,\n2011.\n[8] H. Kurt, K. Rode, P. Stamenov, M. Venkatesan, Yong-Chang Lau, E. Fonda, and J. M. D. Coey.\nPhys. Rev. Lett. , 112:027201, 2014.\n[9] J. M. D. Coey. J. Phys.: Condens. Matter , 26:064211, 2014.\n[10] T. Kubota, Y. Miura, D. Watanabe, S. Mizukami, F. Wu, H. N aganuma, X. Zhang, M. Oogane,\nM. Shirai, Y. Ando, and T. Miyazaki. Appl. Phys. Express , 4:043002, 2011.\n[11] Q. L. Ma, T. Kubota, S. Mizukami, X. M. Zhang, H. Naganuma , M. Oogane, Y. Ando, and\nT. Miyazaki. Appl. Phys. Lett. , 101(3):032402, 2012.\n[12] A. K¨ ohler, I. Knez, D. Ebke, C. Felser, and S. S. P. Parki n.Appl. Phys. Lett. , 103(16):162406,\n2013.\n[13] M. Glas, D. Ebke, I. M. Imort, P. Thomas, and G. Reiss. J. Magn. Magn. Mater , 333(0):134,\n2013.\n[14] N.Nagaosa, J.Sinova, S.Onoda, A.H.MacDonald, and N.P .Ong.Rev. Mod. Phys. , 82(2):1539,\n2010.\n[15] J. Winterlik, B. , Balke, G. H. Fecher, and C. Felser. Appl. Phys. Lett. , 90:152504, 2007.\n[16] A. Chadov, J. Kiss, and C. Felser. Adv. Func. Mat. , 23:832, 2013."    },    {        "title": "1808.04385v2.Gilbert_damping_phenomenology_for_two_sublattice_magnets.pdf",        "content": "Gilbert damping phenomenology for two-sublattice magnets\nAkashdeep Kamra,1,\u0003Roberto E. Troncoso,1Wolfgang Belzig,2and Arne Brataas1,y\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Department of Physics, University of Konstanz, D-78457 Konstanz, Germany\nAbstract\nWe present a systematic phenomenological description of Gilbert damping in two-sublattice mag-\nnets. Our theory covers the full range of materials from ferro- via ferri- to antiferromagnets. Fol-\nlowing a Rayleigh dissipation functional approach within a Lagrangian classical \feld formulation,\nthe theory captures intra- as well as cross-sublattice terms in the Gilbert damping, parameterized\nby a 2\u00022 matrix. When spin-pumping into an adjacent conductor causes dissipation, we obtain\nthe corresponding Gilbert damping matrix in terms of the interfacial spin-mixing conductances.\nOur model reproduces the experimentally observed enhancement of the ferromagnetic resonance\nlinewidth in a ferrimagnet close to its compensation temperature without requiring an increased\nGilbert parameter. It also predicts new contributions to damping in an antiferromagnet and sug-\ngests the resonance linewidths as a direct probe of the sublattice asymmetry, which may stem from\nboundary or bulk.\n1arXiv:1808.04385v2  [cond-mat.mtrl-sci]  6 Nov 2018I. INTRODUCTION\nThe fundamental connection1between magnetic moment and spin angular momentum\nunderlies the important role for magnets in nearly all spin-based concepts. An applied mag-\nnetic \feld provides the means to manipulate the state of a ferromagnet (FM), and thus the\nassociated spin. Conversely, a spin-polarized current absorbed by the FM a\u000bects its mag-\nnetization2{5. Exploiting a related phenomenon, switching the state of an antiferromagnet\n(AFM) has also been achieved6. Emboldened by this newly gained control, there has been\nan upsurge of interest in AFMs7{10, which o\u000ber several advantages over FMs. These include\nthe absence of stray \felds and a larger anisotropy-induced gap in the magnon spectrum. The\ntwo-sublattice nature of the AFMs further lends itself to phenomena distinct from FMs11.\nConcurrently, ferrimagnets (FiMs) have been manifesting their niche in a wide range of\nphenomena such as ultrafast switching12{14and low-dissipation spin transport15{22. A class of\nFiMs exhibits the so-called compensation temperature23{28, at which the net magnetization\nvanishes, similar to the case of AFMs. Despite a vanishing magnetization in the compensated\nstate, most properties remain distinct from that of AFMs29. Thus, these materials can be\ntuned to mimic FMs and AFMs via the temperature. In conjunction with the possibility of a\nseparate angular-momentum compensation, when the magnetization does not vanish but the\ntotal spin does, FiMs provide a remarkably rich platform for physics and applications. An\nincreased complexity in the theoretical description29,30hence accompanies these structurally\ncomplicated materials, and may be held responsible for comparatively fewer theoretical\nstudies. Nevertheless, a two-sublattice model with distinct parameters for each sublattice\nqualitatively captures all the phenomena mentioned above.\nDissipation strongly in\ruences the response of a magnet to a stimulus and is thus cen-\ntral to the study of magnetic phenomena such as switching, domain wall motion and spin\ntransport. Nevertheless, magnetic damping has conventionally been investigated via the\nferromagnetic resonance (FMR) linewidth. It is accounted for phenomenologically in the\nLandau-Lifshitz description of the magnetization dynamics via the so-called Gilbert damp-\ning term31, which produces a good agreement with experiments for a wide range of systems.\nThe Gilbert damping represents the viscous contribution and may be `derived' within a\nLagrangian formulation of classical \feld theory by including the Rayleigh dissipation func-\ntional31. While the magnetic damping for FMs has been studied in great detail29,31{35,\n2from phenomenological descriptions to microscopic models, a systematic development of an\nanalogous description for ferri- and antiferromagnets has been lacking in literature. Further-\nmore, recent theoretical results on spin pumping in two-sublattice magnets36and damping\nin AFMs37suggest an important role for the previously disregarded29cross-sublattice terms\nin Gilbert damping, and thus set the stage for the present study. Yuan and co-workers have\nrecently presented a step in this direction focussing on spin torques in AFMs38.\nHere, we formulate the magnetization dynamics equations in a general two-sublattice\nmagnet following the classical Lagrangian approach that has previously been employed for\nFMs31. The Gilbert damping is included phenomenologically via a Rayleigh dissipation\nfunctional appropriately generalized to the two-sublattice system, which motivates intra-\nas well as cross-sublattice terms. The Gilbert damping parameter thus becomes a 2 \u00022\nmatrix, in contrast with its scalar form for a single-sublattice FM. Solving the system of\nequations for spatially homogeneous modes in a collinear ground state, we obtain the decay\nrates of the two eigenmodes \fnding direct pathways towards probing the dissipation mech-\nanism and asymmetries in the system. Consistent with recent experiments28,39, we \fnd an\nenhancement in the decay rates39close to the magnetization compensation in a FiM with\nan unaltered damping matrix28. The general description is found to be consistent with the\nspin pumping mediated damping in the magnet34{36, and allows for relating the Gilbert\ndamping matrix with the interfacial spin-mixing conductances. Focusing on AFMs, we ex-\npress the magnetization dynamics in terms of the Neel variable thus clarifying the origin\nof the di\u000berent damping terms in the corresponding dynamical equations38,40. Apart from\nthe usually considered terms, we \fnd additional contributions for the case when sublattice-\nsymmetry is broken in the AFM36,41{45. Thus, FMR linewidth measurements o\u000ber a direct,\nparameter-free means of probing the sublattice asymmetry in AFMs, complementary to the\nspin pumping shot noise36.\nThis paper is organized as follows. We derive the Landau-Lifshitz-Gilbert (LLG) equa-\ntions for the two-sublattice model in Sec. II. The ensuing equations are solved for the\nresonance frequencies and decay rates of the uniform modes in a collinear magnet in Sec.\nIII. Section IV presents the application of the phenomenology to describe a compensated\nferrimagnet and spin pumping mediated Gilbert damping. The case of AFMs is discussed in\nSec. V. We comment on the validity and possible generalizations of the theory in Sec. VI.\nThe paper is concluded with a summary in Sec. VII. The discussion of a generalized Rayleigh\n3dissipation functional and properties of the damping matrix is deferred to the appendix.\nII. MAGNETIZATION DYNAMICS AND GILBERT DAMPING\nWe consider a two-sublattice magnet described by classical magnetization \felds MMMA\u0011\nMMMA(rrr;t) andMMMB\u0011MMMB(rrr;t) corresponding to the sublattices AandB. The system is\ncharacterized by a magnetic free energy F[MMMA;MMMB] with the magnetization \felds assumed\nto be of constant magnitudes MA0andMB0. Here, the notation F[ ] is employed to emphasize\nthat the free energy is a functional over the magnetization \felds, i.e. an integration of the\nfree energy density over space.\nThe undamped magnetization dynamics is described by equating the time derivative of\nthe spin angular momentum associated with the magnetization to the torque experienced\nby it. The resulting Landau-Lifshitz equations for the two \felds may be written as:\nd\ndt\u0012MMMA;B\n\u0000j\rA;Bj\u0013\n=\u0000_MMMA;B\nj\rA;Bj=MMMA;B\u0002\u00160HHHA;B; (1)\nwhere\rA;B(<0) are the gyromagnetic ratios for the two sublattices, and HHHA;Bare the\ne\u000bective magnetic \felds experienced by the respective magnetizations. This expression of\nangular momentum \row may be derived systematically within the Lagrangian classical \feld\ntheory31. The same formalism also allows to account for a restricted form of damping via\nthe so-called dissipation functional R[_MMMA;_MMMB] in the generalized equations of motion:\nd\ndt\u000eL[\u0001]\n\u000e_MMMA;B\u0000\u000eL[\u0001]\n\u000eMMMA;B=\u0000\u000eR[_MMMA;_MMMB]\n\u000e_MMMA;B; (2)\nwhereL[\u0001]\u0011 L [MMMA;MMMB;_MMMA;_MMMB] is the Lagrangian of the magnetic system. Here,\n\u000eL[\u0001]=\u000eMMMArepresents the functional derivative of the Lagrangian with respect to the var-\nious components of MMMA, and so on. The left hand side of Eq. (2) above represents the\nconservative dynamics of the magnet and reproduces Eq. (1) with31\n\u00160HHHA;B=\u0000\u000eF[MMMA;MMMB]\n\u000eMMMA;B; (3)\nwhile the right hand side accounts for the damping.\nThe Gilbert damping is captured by a viscous Rayleigh dissipation functional parame-\nterized by a symmetric matrix \u0011ijwithfi;jg=fA;Bg:\nR[_MMMA;_MMMB] =Z\nVd3r\u0010\u0011AA\n2_MMMA\u0001_MMMA+\u0011BB\n2_MMMB\u0001_MMMB+\u0011AB_MMMA\u0001_MMMB\u0011\n; (4)\n4whereVis the volume of the magnet. The above form of the functional assumes the damping\nto be spatially homogeneous, isotropic, and independent of the equilibrium con\fguration.\nA more general form with a lower symmetry is discussed in appendix A. Including the\ndissipation functional via Eq. (2) leads to the following replacements in the equations of\nmotion (1):\n\u00160HHHA!\u00160HHHA\u0000\u0011AA_MMMA\u0000\u0011AB_MMMB; (5)\n\u00160HHHB!\u00160HHHB\u0000\u0011BB_MMMB\u0000\u0011AB_MMMA: (6)\nHence, the LLG equations for the two-sublattice magnet become:\n_MMMA=\u0000j\rAj(MMMA\u0002\u00160HHHA) +j\rAj\u0011AA\u0010\nMMMA\u0002_MMMA\u0011\n+j\rAj\u0011AB\u0010\nMMMA\u0002_MMMB\u0011\n; (7)\n_MMMB=\u0000j\rBj(MMMB\u0002\u00160HHHB) +j\rBj\u0011AB\u0010\nMMMB\u0002_MMMA\u0011\n+j\rBj\u0011BB\u0010\nMMMB\u0002_MMMB\u0011\n:(8)\nThese can further be expressed in terms of the unit vectors ^mmmA;B=MMMA;B=MA0;B0:\n_^mmmA=\u0000j\rAj(^mmmA\u0002\u00160HHHA) +\u000bAA\u0010\nmmmA\u0002_^mmmA\u0011\n+\u000bAB\u0010\n^mmmA\u0002_^mmmB\u0011\n; (9)\n_^mmmB=\u0000j\rBj(^mmmB\u0002\u00160HHHB) +\u000bBA\u0010\n^mmmB\u0002_^mmmA\u0011\n+\u000bBB\u0010\n^mmmB\u0002_^mmmB\u0011\n; (10)\nthereby introducing the Gilbert damping matrix ~ \u000bfor a two-sublattice system:\n~\u000b=0\n@\u000bAA\u000bAB\n\u000bBA\u000bBB1\nA=0\n@j\rAj\u0011AAMA0j\rAj\u0011ABMB0\nj\rBj\u0011ABMA0j\rBj\u0011BBMB01\nA; (11)\n\u000bAB\n\u000bBA=j\rAjMB0\nj\rBjMA0: (12)\nAs elaborated in appendix B, the positivity of the dissipation functional implies that the\neigenvalues and the determinant of ~ \u000bmust be non-negative, which is equivalent to the\nfollowing conditions:\n\u0011AA;\u0011BB\u00150; \u0011 AA\u0011BB\u0015\u00112\nAB=)\u000bAA;\u000bBB\u00150; \u000b AA\u000bBB\u0015\u000bAB\u000bBA: (13)\nThus, Eqs. (9) and (10) constitute the main result of this section, and introduce the damping\nmatrix [Eq. (11)] along with the constraints imposed on it [Eq. (12) and (13)] by the\nunderlying formalism.\n5III. UNIFORM MODES IN COLLINEAR GROUND STATE\nIn this section, we employ the phenomenology introduced above to evaluate the resonance\nfrequencies and the decay rates of the spatially homogeneous modes that can be probed in a\ntypical FMR experiment. We thus work in the macrospin approximation, i.e. magnetizations\nare assumed to be spatially invariant. Considering an antiferromagnetic coupling J(>0)\nbetween the two sublattices and parameterizing uniaxial easy-axis anisotropies via KA;B(>\n0), the free energy assumes the form:\nF[MMMA;MMMB] =Z\nVd3r\u0002\n\u0000\u00160H0(MAz+MBz)\u0000KAM2\nAz\u0000KBM2\nBz+JMMMA\u0001MMMB\u0003\n;(14)\nwhereH0^zzzis the applied magnetic \feld. The magnet is assumed to be in a collinear ground\nstate:MMMA=MA0^zzzandMMMB=\u0000MB0^zzzwithMA0>M B0. Employing Eq. (3) to evaluate the\ne\u000bective \felds, the magnetization dynamics is expressed via the LLG equations (9) and (10).\nConsidering MMMA=MAx^xxx+MAy^yyy+MA0^zzz,MMMB=MBx^xxx+MBy^yyy\u0000MB0^zzzwithjMAx;Ayj\u001c\nMA0,jMBx;Byj \u001cMB0, we linearize the resulting dynamical equations. Converting to\nFourier space via MAx=MAxexp (i!t) etc. and switching to circular basis via MA\u0006(B\u0006)=\nMAx(Bx)\u0006iMAy(By), we obtain two sets of coupled equations expressed succinctly as:\n0\n@\u0006!\u0000\nA\u0000i!\u000b AA\u0000\u0010\nj\rAjJMA0+i!\u000b ABMA0\nMB0\u0011\n\u0010\nj\rBjJMB0+i!\u000b BAMB0\nMA0\u0011\n\u0006!+ \n B+i!\u000b BB1\nA0\n@MA\u0006\nMB\u00061\nA=0\n@0\n01\nA; (15)\nwhere we de\fne \n A\u0011j\rAj(JMB0+ 2KAMA0+\u00160H0) and \n B\u0011j\rBj(JMA0+ 2KBMB0\u0000\n\u00160H0). Substituting !=!r\u0006+i!i\u0006into the ensuing secular equation, we obtain the\nresonance frequencies !r\u0006to the zeroth order and the corresponding decay rates !i\u0006to the\n\frst order in the damping matrix elements:\n!r\u0006=\u0006(\nA\u0000\nB) +p\n(\nA+ \n B)2\u00004J2j\rAjj\rBjMA0MB0\n2; (16)\n!i\u0006\n!r\u0006=\u0006!r\u0006(\u000bAA\u0000\u000bBB) +\u000bAA\nB+\u000bBB\nA\u00002Jj\rBjMA0\u000bAB\n!r++!r\u0000: (17)\nIn the expression above, Eq. (16) and Eq. (17), we have chosen the positive solutions of\nthe secular equations for the resonance frequencies. The negative solutions are equal in\nmagnitude to the positive ones and physically represent the same two modes. The positive-\npolarized mode in our notation corresponds to the typical ferromagnetic resonance mode,\nwhile the negative-polarized solution is sometimes termed `antiferromagnetic resonance'25.\n6020406080100120\n0 0.2 0.4 0.6 0.8 10.040.050.060.070.080.090.1FIG. 1. Resonance frequencies and normalized decay rates vs. the applied \feld for a quasi-\nferromagnet ( MA0= 5MB0).j\rAj=j\rBj= 1;1:5;0:5 correspond to solid, dashed and dash-dotted\nlines respectively. The curves in blue and red respectively depict the + and \u0000modes. The damping\nparameters employed are \u000bAA= 0:06,\u000bBB= 0:04 and\u000bAB= 0.\nIn order to avoid confusion with the ferromagnetic or antiferromagnetic nature of the un-\nderlying material, we call the two resonances as positive- and negative-polarized. The decay\nrates can further be expressed in the following form:\n!i\u0006\n!r\u0006=\u0016\u000b(\nA+ \n B)\u00002Jj\rBjMA0\u000bAB\n!r++!r\u0000\u0006\u0001\u0016\u000b; (18)\nwith \u0016\u000b\u0011(\u000bAA+\u000bBB)=2 and \u0001\u0016\u000b\u0011(\u000bAA\u0000\u000bBB)=2. Eq. (18) constitutes the main result\nof this section and demonstrates that (i) asymmetric damping in the two sublattices is\nmanifested directly in the normalized decay rates of the two modes (Figs. 1 and 2), and\n(ii) o\u000b-diagonal components of the damping matrix may reduce the decay rates (Fig. 2).\nFurthermore, it is consistent with and reproduces the mode-dependence of the decay rates\nobserved in the numerical studies of some metallic AFMs37.\nTo gain further insight into the results presented in Eqs. (16) and (18), we plot the\n7resonance frequencies and the normalized decay rates vs. the applied magnetic \feld for a\ntypical quasi-ferromagnet, such as yttrium iron garnet, in Fig. 1. The parameters employed\nin the plot arej\rBj= 1:8\u00021011,MB0= 105,KA=KB= 10\u00007, andJ= 10\u00005in SI units,\nand have been chosen to represent the typical order of magnitude without pertaining to a\nspeci\fc material. The plus-polarized mode is lower in energy and is raised with an increasing\napplied magnetic \feld. The reverse is true for the minus-polarized mode whose relatively\nlarge frequency makes it inaccessible to typical ferromagnetic resonance experiments. As\nanticipated from Eq. (18), the normalized decay rates for the two modes di\u000ber when \u000bAA6=\n\u000bBB. Furthermore, the normalized decay rates are independent of the applied \feld for\nsymmetric gyromagnetic ratios for the two sublattices. Alternately, a measurement of the\nnormalized decay rate for the plus-polarized mode is able to probe the sublattice asymmetry\nin the gyromagnetic ratios. Thus it provides essential information about the sublattices\nwithout requiring the measurement of the large frequency minus-polarized mode.\nIV. SPECIFIC APPLICATIONS\nWe now examine two cases of interest: (i) the mode decay rate in a ferrimagnet close to\nits compensation temperature, and (ii) the Gilbert damping matrix due to spin pumping\ninto an adjacent conductor.\nA. Compensated ferrimagnets\nFMR experiments carried out on gadolinium iron garnet23,39\fnd an enhancement in the\nlinewidth, and hence the mode decay rate, as the temperature approaches the compensation\ncondition, i.e. when the two e\u000bective46sublattices have equal saturation magnetizations.\nThese experiments have conventionally been interpreted in terms of an e\u000bective single-\nsublattice model thereby ascribing the enhancement in the decay rate to an increase in the\nscalar Gilbert damping constant allowed within the single-sublattice model24. In contrast,\nexperiments probing the Gilbert parameter in a di\u000berent FiM via domain wall velocity\n\fnd it to be essentially unchanged around compensation28. Here, we analyze FMR in a\ncompensated FiM using the two-sublattice phenomenology developed above and thus address\nthis apparent inconsistency.\n8020406080100120\n1 2 3 4 500.050.10.150.20.250.3FIG. 2. Resonance frequencies and normalized decay rates vs. relative saturation magnetizations\nof the sublattices. The curves which are not labeled as + or \u0000represent the common normalized\ndecay rates for both modes. The parameters employed are the same as for Fig. 1 with \rA=\rB.\nThe compensation behavior of a FiM may be captured within our model by allowing\nMA0to vary while keeping MB0\fxed. The mode frequencies and normalized decay rates\nare examined with respect to the saturation magnetization variation in Fig. 2. We \fnd an\nenhancement in the normalized decay rate, consistent with the FMR experiments23,39, for a\n\fxed Gilbert damping matrix. The single-sublattice interpretation ascribes this change to a\nmodi\fcation of the e\u000bective Gilbert damping parameter24, which is equal to the normalized\ndecay rate within that model. In contrast, the latter is given by Eq. (18) within the\ntwo-sublattice model and evolves with the magnetization without requiring a modi\fcation\nin the Gilbert damping matrix. Speci\fcally, the enhancement in decay rate observed at\nthe compensation point is analogous to the so-called exchange enhancement of damping in\nAFMs47. Close to compensation, the FiM mimics an AFM to some extent.\nWe note that while the spherical samples employed in Ref. 23 are captured well by our\nsimple free energy expression [Eq. (14)], the interfacial and shape anisotropies of the thin\n9\flm sample employed in Ref. 39 may result in additional contributions to decay rates. The\nsimilarity of the observed linewidth trends for the two kinds of samples suggests that these\nadditional anisotropy e\u000bects may not underlie the observed damping enhancement. Quan-\ntitatively accounting for these thin \flm e\u000bects requires a numerical analysis, as discussed\nin Sec VI below, and is beyond the scope of the present work. Furthermore, domain forma-\ntion may result in additional damping contributions not captured within our single-domain\nmodel.\nB. Spin pumping mediated Gilbert damping\nSpin pumping34from a FM into an adjacent conductor has been studied in great detail35\nand has emerged as a key method for injecting pure spin currents into conductors48. The\nangular momentum thus lost into the conductor results in a contribution to the magnetic\ndamping on top of the intrinsic dissipation in the bulk of the magnet. A variant of spin\npumping has also been found to be the dominant cause of dissipation in metallic magnets37.\nThus, we evaluate the Gilbert damping matrix arising due to spin pumping from a two-\nsublattice magnet36into an adjacent conductor acting as an ideal spin sink.\nWithin the macrospin approximation, the total spin contained by the magnet is given by:\nSSS=\u0000MA0V^mmmA\nj\rAj\u0000MB0V^mmmB\nj\rBj: (19)\nThe spin pumping current emitted by the two-sublattice magnet has the following general\nform36:\nIIIs=~\neX\ni;j=fA;BgGij\u0010\n^mmmi\u0002_^mmmj\u0011\n; (20)\nwithGAB=GBA, where the spin-mixing conductances Gijmay be evaluated within di\u000berent\nmicroscopic models36,49{51. Equating the spin pumping current to \u0000_SSSand employing Eqs.\n(9) and (10), the spin pumping contribution to the Gilbert damping matrix becomes:\n\u000b0\nij=~Gijj\rij\neMi0V; (21)\nwhich in turn implies\n\u00110\nij=~Gij\neMi0Mj0V; (22)\n10for the corresponding dissipation functional. The resulting Gilbert damping matrix is found\nto be consistent with its general form and constraints formulated in Sec. II. Thus, employing\nthe phenomenology developed above, we are able to directly relate the magnetic damping in\na two-sublattice magnet to the spin-mixing conductance of its interface with a conductor.\nV. ANTIFERROMAGNETS\nDue to their special place with high symmetry in the two-sublattice model as well as the\nrecent upsurge of interest7{10,52{54, we devote the present section to a focused discussion on\nAFMs in the context of the general results obtained above. It is often convenient to describe\nthe AFM in terms of a di\u000berent set of variables:\nmmm=^mmmA+^mmmB\n2; nnn=^mmmA\u0000^mmmB\n2: (23)\nIn contrast with ^mmmAand ^mmmB,mmmandnnnare not unit vectors in general. The dynamical\nequations for mmmandnnnmay be formulated by developing the entire \feld theory, starting with\nthe free energy functional, in terms of mmmandnnn. Such a formulation, including damping,\nhas been accomplished by Hals and coworkers40. Here, we circumvent such a repetition and\ndirectly express the corresponding dynamical equations by employing Eqs. (9) and (10) into\nEq. (23):\n_mmm=\u0000(mmm\u0002\rm\u00160HHHm)\u0000(nnn\u0002\rn\u00160HHHn) +X\np;q=fm;ng\u000bm\npq(ppp\u0002_qqq); (24)\n_nnn=\u0000(mmm\u0002\rn\u00160HHHn)\u0000(nnn\u0002\rm\u00160HHHm) +X\np;q=fm;ng\u000bn\npq(ppp\u0002_qqq); (25)\nwith\n\rm\u00160HHHm\u0011j\rAj\u00160HHHA+j\rBj\u00160HHHB\n2; (26)\n\rn\u00160HHHn\u0011j\rAj\u00160HHHA\u0000j\rBj\u00160HHHB\n2; (27)\n\u000bm\nmm=\u000bn\nnm=\u000bAA+\u000bBB+\u000bAB+\u000bBA\n2; (28)\n\u000bm\nmn=\u000bn\nnn=\u000bAA\u0000\u000bBB\u0000\u000bAB+\u000bBA\n2; (29)\n\u000bm\nnn=\u000bn\nmn=\u000bAA+\u000bBB\u0000\u000bAB\u0000\u000bBA\n2; (30)\n\u000bm\nnm=\u000bn\nmm=\u000bAA\u0000\u000bBB+\u000bAB\u0000\u000bBA\n2: (31)\n11A general physical signi\fcance, analogous to \rA;B, may not be associated with \rm;nwhich\nmerely serve the purpose of notation here. The equations obtained above manifest new\ndamping terms in addition to the ones that are typically considered in the description\nof AFMs. Accounting for the sublattice symmetry of the antiferromagnetic bulk while\nallowing for the damping to be asymmetric, we may assume \rA=\rBandMA0=MB0, with\n\u0016\u000b\u0011(\u000bAA+\u000bBB)=2, \u0001\u0016\u000b\u0011(\u000bAA\u0000\u000bBB)=2, and\u000bAB=\u000bBA\u0011\u000bod. Thus, the damping\nparameters simplify to\n\u000bm\nmm=\u000bn\nnm=\u0016\u000b+\u000bod; (32)\n\u000bm\nmn=\u000bn\nnn=\u0001\u0016\u000b; (33)\n\u000bm\nnn=\u000bn\nmn=\u0016\u000b\u0000\u000bod; (34)\n\u000bm\nnm=\u000bn\nmm=\u0001\u0016\u000b; (35)\nthereby eliminating the \\new\" terms in the damping when \u000bAA=\u000bBB. However, the sublat-\ntice symmetry may not be applicable to AFMs, such as FeMn, with non-identical sublattices.\nFurthermore, the sublattice symmetry of the AFM may be broken at the interface41{43via,\nfor example, spin mixing conductances36,45,55resulting in \u000bAA6=\u000bBB.\nThe resonance frequencies and normalized decay rates [Eqs. (16) and (18)] take a simpler\nform for AFMs. Substituting KA=KB\u0011K,\rA=\rB\u0011\r, andMA0=MB0\u0011M0:\n!r\u0006=\u0006j\rj\u00160H0+ 2j\rjM0p\n(J+K)K; (36)\n!i\u0006\n!r\u0006=J(\u0016\u000b\u0000\u000bod) + 2K\u0016\u000b\n2p\n(J+K)K\u0006\u0001\u0016\u000b\u0019(\u0016\u000b\u0000\u000bod)\n2r\nJ\nK+ \u0016\u000br\nK\nJ\u0006\u0001\u0016\u000b; (37)\nwhere we have employed J\u001dKin the \fnal simpli\fcation. The term /p\nK=J has typically\nbeen disregarded on the grounds K\u001cJ. However, recent numerical studies of damping in\nseveral AFMs37\fnd \u0016\u000b\u001d\u0016\u000b\u0000\u000bod>0 thus suggesting that this term should be comparable\nto the one proportional top\nJ=K and hence may not be disregarded. The expression above\nalso suggests measurement of the normalized decay rates as a means of detecting the sublat-\ntice asymmetry in damping. For AFMs symmetrical in the bulk, such an asymmetry may\narise due to the corresponding asymmetry in the interfacial spin-mixing conductance36,45,55.\nThus, decay rate measurements o\u000ber a method to detect and quantify such interfacial e\u000bects\ncomplementary to the spin pumping shot noise measurements suggested earlier36.\n12VI. DISCUSSION\nWe have presented a phenomenological description of Gilbert damping in two-sublattice\nmagnets and demonstrated how it can be exploited to describe and characterize the system\ne\u000bectively. We now comment on the limitations and possible generalizations of the formal-\nism presented herein. To begin with, the two-sublattice model is the simplest description of\nferri- and antiferromagnets. It has been successful in capturing a wide range of phenomenon.\nHowever, recent measurements of magnetization dynamics in nickel oxide could only be ex-\nplained using an eight-sublattice model56. The temperature dependence of the spin Seebeck\ne\u000bect in yttrium iron garnet also required accounting for more than two magnon bands57.\nA generalization of our formalism to a N-sublattice model is straightforward and can be\nachieved via a Rayleigh dissipation functional with N2terms, counting \u0011ijand\u0011jias sepa-\nrate terms. The ensuing Gilbert damping matrix will be N \u0002N while obeying the positive\ndeterminant constraint analogous to Eq. (13).\nIn our description of the collinear magnet [Eq. (14)], we have disregarded contributions\nto the free energy which break the uniaxial symmetry of the system about the z-axis. Such\nterms arise due to spin-nonconserving interactions58, such as dipolar \felds and magnetocrys-\ntalline anisotropies, and lead to a mixing between the plus- and minus-polarized modes30.\nIncluding these contributions converts the two uncoupled 2 \u00022 matrix equations [(15)] into\na single 4\u00024 matrix equation rendering the solution analytically intractable. A detailed\nanalysis of these contributions30shows that their e\u000bect is most prominent when the two\nmodes are quasi-degenerate, and may be disregarded in a \frst approximation.\nIn evaluating the resonance frequencies and the decay rates [Eqs. (16) and (18)], we\nhave assumed the elements of the damping matrix to be small. A precise statement of the\nassumption employed is !i\u001c!r, which simply translates to \u000b\u001c1 for a single-sublattice\nferromagnet. In contrast, the constraint imposed on the damping matrix within the two-\nsublattice model by the assumption of small normalized decay rate is more stringent [Eq.\n(18)]. For example, this assumption for an AFM with \u000bAB= \u0001\u0016\u000b= 0 requires \u0016 \u000b\u001c\np\nK=J\u001c1. This stringent constraint may not be satis\fed in most AFMs37, thereby\nbringing the simple Lorentzian shape description of the FMR into question. It can also be\nseen from Fig. 2 that the assumption of a small normalized decay rate is not very good for\nthe chosen parameters.\n13VII. SUMMARY\nWe have developed a systematic phenomenological description of the Gilbert damping\nin a two-sublattice magnet via inclusion of a Rayleigh dissipation functional within the La-\ngrangian formulation of the magnetization dynamics. Employing general expressions based\non symmetry, we \fnd cross-sublattice Gilbert damping terms in the LLG equations in con-\nsistence with other recent \fndings36{38. Exploiting the phenomenology, we explain the en-\nhancement of damping23,39in a compensated ferrimagnet without requiring an increase in\nthe damping parameters28. We also demonstrate approaches to probe the various forms\nof sublattice asymmetries. Our work provides a uni\fed description of ferro- via ferri- to\nantiferromagnets and allows for understanding a broad range of materials and experiments\nthat have emerged into focus in the recent years.\nACKNOWLEDGMENTS\nA. K. thanks Hannes Maier-Flaig and Kathrin Ganzhorn for valuable discussions. We\nacknowledge \fnancial support from the Alexander von Humboldt Foundation, the Research\nCouncil of Norway through its Centers of Excellence funding scheme, project 262633, \\QuS-\npin\", and the DFG through SFB 767 and SPP 1538.\nAppendix A: Generalized Rayleigh dissipation functional\nAs compared to the considerations in Sec. II, a more general approach to parameterizing\nthe dissipation functional is given by:\nR[_MMMA;_MMMB] =1\n2Z\nVZ\nVd3r0d3rX\np;q=fA;BgX\ni;j=fx;y;zg_Mpi(rrr)\u0011ij\npq(rrr;rrr0)_Mqj(rrr0): (A1)\nThis form allows to capture the damping in an environment with a reduced symmetry.\nHowever, the larger number of parameters also makes it di\u000ecult to extract them reliably\nvia typical experiments. The above general form reduces to the case considered in Sec. II\nwhen\u0011ij\npq(rrr;rrr0) =\u0011pq\u000eij\u000e(rrr\u0000rrr0) and\u0011pq=\u0011qp. Furthermore, the coe\u000ecients \u0011ij\npqmay depend\nuponMMMA(rrr) andMMMB(rrr) as has been found in recent numerical studies of Gilbert damping\nin AFMs37.\n14Appendix B: Damping matrix\nThe Rayleigh dissipation functional considered in the main text is given by:\nR[_MMMA;_MMMB] =Z\nVd3r\u0010\u0011AA\n2_MMMA\u0001_MMMA+\u0011BB\n2_MMMB\u0001_MMMB+\u0011AB_MMMA\u0001_MMMB\u0011\n; (B1)\nwhich may be brought into the following concise form with the notation~_MMM\u0011[_MMMA_MMMB]|:\nR[_MMMA;_MMMB] =1\n2Z\nVd3r~_MMM|~\u0011~_MMM; (B2)\nwhere ~\u0011is the appropriate matrix given by:\n~\u0011=0\n@\u0011AA\u0011AB\n\u0011AB\u0011BB1\nA: (B3)\nConsidering an orthogonal transformation~_MMM=~Q~_M, the dissipation functional can be\nbrought to a diagonal form\nR[_MMMA;_MMMB] =1\n2Z\nVd3r~_M|~Q|~\u0011~Q~_M; (B4)\nwhere ~Q|~\u0011~Qis assumed to be diagonal. 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Treboniu Laurean, 400271, Cluj -Napoca, Romania  \n  \n \nAbstract  \nPerpendicular magnetic anisotropy  (PMA)  in ultrathin magnetic structures  is a key ingredient for the \ndevelopment of electrically controlled spintronic devices. Due to their relatively large spin -polarization , \nhigh Curie temperature  and low Gilbert damping t he Co -based full Heusler alloys are of special \nimportance  from  a scientific and application s point of view. Here, we study the mechanisms responsible \nfor the PMA in Pt/Co -based full Heusler alloy /MgO  thin films structures . We show that the  ultrathin  \nHeusler films exhibit  strong PMA even in the absence of magnetic annealing. By means of ferromagne tic \nresonance experiments, we demonstrate that the effective magnetization  shows a two -regime behavior  \ndepending on the thickness of the Heusler layers . Using  Auger spectroscopy measurements,  we evidence \ninterdiffusion at the underlayer/Heusler interface  and the formation of an interfacial CoFe -rich layer which \ncauses  the two -regime behavior. In the case of the ultrathin films, th e interfacial CoFe -rich layer promotes \nthe strong PMA through the electronic hybridization of the metal alloy  and oxygen  orbitals across the \nferromagnet /MgO interface . In addition,  the interfacial CoFe -rich layer  it is also generating  an increase  of \nthe Gilbert damping  for the ultrathin films  beyond the spin -pumping effect . Our results  illustrate  that the \nstrong PMA is not an intrinsic property of the Heusler/MgO interface but  it is actively influenced by the \ninterdiffusion, which can be tuned by a proper choice of the underlayer material, as we show for the case \nof the Pt, Ta and Cr underlayers.  \n \n \n                                                 \na) mihai.gabor@phys.utcluj.ro  2 \n  \n \nIntroduction  \n \nUltrathin films structures showing perpendicular magnetic anisotropy (PMA) are under intensive \nresearch for the development of electrically controlled spintronic devices. Particularly, current induced \nspin–orbit torques (SOTs) in heavy -metal/ferromagnet (FM) heterostructures showing PMA are used to \ntrigger  the magnetization switching1,2. Besides , the antisymmetric  interfacial Dzyaloshinskii -Moriya3,4 \ninteraction ( iDMI)  in similar  PMA  architectures , if strong enough, can lead to the formation of special \nchiral structures like skyrmions5 which are  drivable  by electrical currents6. In the case of  the spin transfer \ntorque magnet ic random -access  memories (STT -MRAMs), considered a s a poten tial replacement for the \nsemiconductor -based  ones, the use of strong PMA materials is required for increased thermal stability7, \nwhile high spin polarization and low Gilbert damping is needed  to obtain large magnetoresistive ratios \nand efficient curre nt induced STT switching8,9.  \nCobalt-based full Heusler alloys are a special class of ferromagnetic materials that attract an increased \nscientific interest , since the ir theoretical  prediction of  half-metallicity10. These compounds are described \nby the formula Co2YZ, where Y is a transition metal,  or a mixture of two trans ition metals , and Z is a main \ngroup , or a mixture of two main group  sp element s. Large magnetoresistive ratios are experimentally \ndemonstrated in certain Co2YZ based in -plane11-18 and out -of-plane19,20 magnetized magnetic tunnel \njunctions  (MTJs)  and r elative ly low Gilbert dampin g parameters were  determined for some compounds21-\n28. Furthermore , PMA was evidenced for  Co2FeAl/ MgO19,29-33, Co 2FeAl 0.5Si0.5/MgO34-37, \nCo2FeSi/MgO38,39 or Co2FexMn 1-xSi/MgO40,41 structures with different  non-magnetic  underla yers. In \nsome of the cases, an annealing stage was necessary to induce PMA, while  for the other the perpendicular \nmagnetiz ation  was achieved even in the as -deposited state. The origin of the strong  PMA in th is type of \nstructures is still under debate . It could be related to both the oxidation at the Heusler/MgO interface42,43 \nand to the spin -orbit interaction effects at the heavy -metal underlayer/Heusler interface44,45. Moreover, it \nwas recently pointed out that in t he case of Co 2FeAl/MgO the diffusion of Al towards the MgO layer \nduring annealing  plays an important role for the  stabiliz ation of  the PMA46,47. The precise knowledge and \ncontrol of the mechanisms responsible for PMA is essential in order to be able to develop viable  spintronic \napplication s. Therefore, i n this paper, we  study  the underlying physics governing the  PMA for Co2FeAl, \nCo2FeAl 0.5Si0.5, Co2FeSi and Co2Fe0.5Mn 0.5Si Heusler alloy thin films  sandwiched between Pt and  MgO \nlayers. We show that b elow  a certain  critical thickness all the Heusler films  show strong PMA even in the 3 \n absence of magnetic annealing . Additionally, using ferromagnetic resonance experiments , we demonstrate \nthat, depending on the thickness  of the Heusler layers,  the effective perpendicular magnetic anisotropy \nshows a  two-regime  behavior.  After excluding other possible mechanism s, we evidence  using Auger \nspectroscopy measu rements, that the diffusion of the lighter elements towards the Pt underlayer and the \nformation of an interfacial CoFe -rich layer causes the  two-regime behavior . In the case of the ultrathin \nfilms , this interfacial CoFe -rich layer  promotes the  strong  PMA through the hybridization of the [Co,Fe] \n3𝑑𝑧2 and O 2𝑝𝑧 orbital s at the interface  and is also responsible  for the increased Gilbert  damping . Our \nstudy reveals that the strong PMA is not intrinsic to the Heusler/MgO interface . It is strongly  influenced \nby the interdiffusion  and can be adjusted  by a proper choice of the underlayer  material , as we show for \nthe case of the Pt, Ta and Cr  underlayers.  \n \nExperimental   \n \nAll the samples studied here were grown at room temperature on thermally oxidized silicon substrates \nin a magnetron sputtering system having a base pressure lower than 2×10-8 Torr.  The main samples have \nthe following structure: Si/SiO 2//Ta (3 nm )/Pt (4 nm)/ FM (0.8-10 nm)/MgO (1 nm)/ Ta (3 nm), where FM \nstands for  Co2FeAl (CFA), Co 2FeAl 0.5Si0.5 (CFAS), Co 2FeSi (CFS), Co 2Fe0.5Mn 0.5Si (CFMS) or CoFeB \n(CFB) , depending on the sample. Additional samples were grown,  and their  structure will be discussed \nlater in the text. The metallic layers were deposited  by dc sputtering  under an argon pressure of 1 mTorr, \nwhile the MgO layer was grown by rf sputtering under an argon pressure of 10 mTorr. The Heusler alloy s \nthin films were sputtered  from stoichiometric targets. The 3 nm thick Ta buffer layer was grown directly \non the substrate to minimize  the roughness and to facilitate the (111) texturing of the  upper Pt  layer. The \n1 nm thick MgO layer was deposited to induce perpendicular m agnetic anisotropy on the Heusler thin \nfilm7. An additional  3 nm th ick Ta capping layer was sputtered  to protect the samples from oxidation due \nto air exposure. The structure of the samples was characterized by x -ray diffraction (XRD) using a four -\ncircle diffractometer. The static magnetic properties have been investigated using a Vibrating Sample \nMagnetometer (VSM) , while  the dynamic magnetic properties by using a TE(011) cavity  Ferromagnetic \nResonance (FMR)  setup working in X -band  (9.79 GHz ). Auger spectra have been recorded  in derivative \nmode, using  a cylindrical mirror analyzer spectrometer working at an electron beam energy of 3keV.  \nDepth profile analysis have been performed by successive  recording of the Auger spectra and Ar ion \nsputter -etching of the surface of the samples by  using a relatively low  ion energy of 600 eV.    \n 4 \n  \n \nResults and discussions  \n \nFigure 1 (a) shows  2θ/ω x-ray diffraction patterns recorded for four representative  Pt (4 nm)/  Co2YZ \n(10 nm)/  MgO (1 nm)  samples . Irrespective of the Heusler composition, the patterns show the (111) and \n(222) peaks  belonging to the Pt  layer , the (022) peak  arising from the Heusler films  and the  (001) peak of \nthe Si substrate . This indicates that the Pt layer has a (111) out -of-plane texture, while the Heusler films \nare (011) out -of-plane textured.  Laue oscillations are observable around the (111) Pt reflection which \nconfirms the good crystalline quality for the Pt films48. Moreover, ϕ -scan measurements (not shown here) \nindicate that both the Pt underlayer and the Heusler fil ms have no in -plane texturing but  show an  in-plane \nisotropic distribution  of the crystallites . No peak belon ging to the Ta capping layer was observed, \nindicating  that the film is in an amorphous or nanocrystalline state.  \nThe static magnetic properties of our films were characterized by VSM measurements . Figure 2 show s \nhysteresis loops measured with the magnetic field applied perpendicular  to the plane of the samples, for \nrepresentative Heusler films thickness es. In order to remove th e substrate diamagnetic contribution,  we \nfitted the large field data with a linear function and extract ed the linear slope  from the raw data.  Regardless  \nof the ir composition , all the Heusler films show a similar behavior. Above a critical spin-reorientation \ntransition thickness  the samples show in-plane magnetic anisotropy . This is indicated by the shape of the \nhysteresis loop s in Fig. 2  (a)–(d), which is typical for a hard axis of magnetization, showing a continuous \nrotation of the magnetization up to saturation.  Below th is critical thickness, the samples show PMA , which \nis attested by  the square shaped hysteresis loops in Fig. 2 (e) –(h). We also determined the saturation \nmagnetization  (𝑀𝑆) and the effective thickness es of the ferr omagnetic layers  using hysteresis loop \nmeasurements  and the procedure described in 31.  The effective thicknesses of the ferromagnetic layers \nare used throughout  the paper and the 𝑀𝑆 is found  to be  790 ± 70 emu/cm3, 660 ± 50 emu/cm3, 935 ± 75 \nemu/cm3 and 895 ± 75 emu/cm3 for CFS, CFMS, CFAS and CFA samples, respectively.   \nIn order to get more insights on the magnetic anisotropy properties of our films, we have performed \nFMR measurements  with the magnetic field applied at diffe rent θH angles  (defined in the inset of Fig. 4) \nwith respect to the normal direction of the layers . Figure 3 shows typical FMR spectra for various fie ld \nangle s recorded  for a 2.4 nm thick Pt/CFAS sample . We define the resonance field HR as the intersection \nof the spectr um with the base line, and the linewidth HPP as the distance between the positive and negative \npeaks of the spectrum. Figure 4 shows the θH dependence of the HR and of the linewidth HPP for the 2.4 5 \n nm thick Pt/CFAS sample. In order to extract the relevant FMR parameters, we analyzed the θH \ndependence of the FMR spectrum using  a model in which the total energy per unit volume is given by  \n𝐸=−𝑀𝑆𝐻cos(𝜃𝐻���𝜃𝑀)+2𝜋𝑀𝑆2cos2𝜃𝑀−𝐾⊥cos2𝜃𝑀,                             (1) \nwhere the first term is the Zeeman energy, the second term is the demagnetizing energy, and the last term \nis the magnetic anisotropy energy.  The 𝑀𝑆 is the saturation magnetization, 𝜃𝐻 and 𝜃𝑀 are the field and \nmagnetization angles defined in the inset of Fig. 4, and the 𝐾⊥ is the effective perpendicular magnetic \nanisotropy constant. From eq. 1 and the Landau -Lifshitz -Gilbert equation, one can der ive the resonance \ncondition as49 \n(𝜔\n𝛾)2\n=𝐻1×𝐻2,                                                                (2)  \nwhere 𝜔 is the angular frequency of the microwave , 𝛾 is the gyromagnetic ratio , given by 𝛾=𝑔𝜇𝐵ℏ where \n𝑔 is the  Landé  g-factor, 𝜇𝐵 is the Bohr magneton and ℏ is the reduced Plan ck constant , and with 𝐻1 and \n𝐻2 given by   \n𝐻1=𝐻𝑅cos(𝜃𝐻−𝜃𝑀)−4𝜋𝑀effcos2𝜃𝑀,                                             (3) \n𝐻2=𝐻𝑅cos(𝜃𝐻−𝜃𝑀)−4𝜋𝑀effcos2𝜃𝑀,                                           (4) \nwhere 4𝜋𝑀eff is the effective magnetization defined as  4𝜋𝑀eff=4𝜋𝑀𝑆−2𝐾⊥𝑀𝑆⁄ and 𝐻𝑅 is the \nresonance field.  For each value of  𝜃𝐻, the 𝜃𝑀 at resonance is calculated from the energy minimum \ncondition 𝜕𝐸 𝜕𝜃𝑀=0 ⁄ . Hence , the 𝐻𝑅 dependence on 𝜃𝐻 can be fitted by Eq. (2)-(4) using 4𝜋𝑀eff and \n𝑔 as adjustable parameters . A typical fit curve is s hown in Fig. 4(a).  \n Figure 5 shows the 𝑔 factor depen dence on the thickness of the Heusler layers for samples with \ndifferent Heusler layer composition s. Depending on the thickness , two regimes are discernable . For \nrelatively  large thickness es, above 2.5-3 nm, the  𝑔 factor shows rather  constant value s between 2.07 and \n2.11, depending on the type of the Heusler layer . For lower thickness es, 𝑔 shows a monotonous decrease, \nregardless  of the Heusler  layer composition. This is an interface effect and it is usually attributed to the \nfact that at the interfaces , due to the symmetry breaking, the orbital motion is n o longer entirely  quenched  \nand will contribute to the gyromagnetic ratio50,51. Another possibility, which cannot be exclude d in our \ncase, is the reduction of the 𝑔 factor due to intermixing between the ferromagnetic  Heusler layer and non -\nmagnetic materials at the interfaces50. \n Figure 6 shows the effective magnetization 4𝜋𝑀eff dependence on the inverse thickness of the    \nferromagnetic layer for samples with different composition s. It is to be mentioned  that the 4𝜋𝑀eff was \ndetermined from FMR experiments only for samples with in-plane magnetic anisotropy (positive 4𝜋𝑀eff). \nIn the case of ultrathin samples showing perpendicular magnetic anisotropy  (negative 4𝜋𝑀eff), due to the 6 \n strong linewidth enhancement , it was not possible to obtain reliable resonance curves.  Therefore, in this \ncase the 4𝜋𝑀eff was estimated from VSM measurements.  Generally, it is considered  that the effective \nperpendicular magnetic anisotropy co nstant  𝐾⊥ can be written as the sum of a volume  (𝐾𝑉), which includes \nmagneto -crystalline and strain related anisotropies, and a surface (𝐾𝑆) contribution : 𝐾⊥=𝐾𝑉+𝐾𝑆𝑡⁄, \nwhere 𝑡 is the thickness of the ferromagnetic layer . Thus, the effective magnetization can be written as  \n \n4𝜋𝑀eff=(4𝜋𝑀𝑆−2𝐾𝑉\n𝑀𝑆)−2𝐾𝑆\n𝑀𝑆1\n𝑡.                                                             (5) \n \nThe above relation implies a linear dependence of  the effective magnetization on the inverse thickness of \nthe ferromagnetic layer.  However, as shown in Fig.6  the Heusler samples do not show a single linear \ndependence for the  entire  thickness range, but two regimes above and below a  certain critical thickness.  \nUsing the 𝑀𝑆 values determined from VSM measurements and b y fitting the experimental data in the large  \nthickness regime to eq. (5) , we extract a surface  anisotropy constant 𝐾𝑆 for the CFA  and CFAS of 0.24  ± \n0.03 erg/ cm2 and 0.22 ± 0.02 erg/cm2 and a volume contribution 𝐾𝑉 of (1.27 ± 0. 69)×106 erg/cm3 and \n(1.51 ± 0.7)×106 erg/cm3, respectively. In the case of the CFMS and CFS, the 𝐾𝑆 was negligible small \nwithin the error bars  and the 𝐾𝑉 was found to be (0.44 ± 0.38)×106 erg/cm3 and ( 0.51 ± 0. 4)×106 erg/cm3, \nrespectively. Using the as extracted values of th e anisotropy constants , we can calculate , for example,  in \nthe case of the CFAS samples a spin-reorientation tra nsition thickness of around 0.55 nm. This is clearly \nnot in agreement with the experimental data , as seen from Fig. 2 and 6 , already  a 1 nm thick CFAS film \nshows strong PMA  and it is spontaneous perpendicular ly magnetized . This is a consequence of the fact \nthat the  1 nm thick CFAS film  falls within  the second anisotropy regime below the critical thickness . The \noccurrence  of this second anisotropy regime with larger effective perpendicular magnetic anisotropy can \nhave several explanations.  For such thin films one must always consider the possible influences of the \nsurface roughness. If the roughness is relatively large , an in -plane demagnetization field will develop at \nthe edges of the terraces  which will reduce the shape anisotropy and fav or perpendicular magneti zation . \nThis is equivalent  to the  emergence  of an additional dipolar surface anisotropy contribution52. The \nroughness is  a parameter which is not easily quantifiable  experimentally  in such thin  multilayer structures. \nHowever, it is reasonable to expect to be comparable  for similar  heterostructures in which the Heusler \nalloy film is replaced with a CFB layer . Atomic force microscopy topography images (not shown here) \nrecorded for heterostructure w ith CFB and CFAS layers are featureless and show a similar RMS \nroughness.  As such, if the low thickness  anisotropy  regime is due to the roughness it must be observable \nalso in the case of CFB samples. However, this is not the case , as shown in Fig. 6, the CFB samples show 7 \n a single linear behavior for the whole range of  thickness . Fitting the data to eq. (5), allowed us to extract  \nfor CFB samples  a surface anisotropy contribution 𝐾𝑆 of 0.79 ± 0.04 erg/cm2 and a  negligible small 𝐾𝑉 \nvolume contribution , in line with previous reports7,53. These findings suggest  that the roughness is not \nresponsible for the two regimes behavior observed in the case of the Heusler samples.  \nAnother possible  physical mechanism  which can explain the presence of the two regimes is  the strain \nvariation  due to coherent –incoherent  growth transition54,55. Within this model, below the critical thickness , \nthe ferromagnetic layer grows uniformly strained in order to account for the lattice  misfit with the adjacent \nlayer s. Above the critical thickness, the strains are partially relaxed through the formation  of misfit \ndislocations.  The changes in the magnetoelastic anisotropy contributions corresponding to this structural \ntransition  can be responsible for t he presence of the two regimes54,55. This scenario is likely in the case of \nthe Heusler samples , since both the bottom Pt layer  the upper Heusler film  grow  out-of-plane textured.  In \norder to test this hypothesis, we have deposited two additional sets of samples. The first set consisted  of \nSi/SiO 2//Ta ( 6 nm)/ CFAS  (tCFAS)/MgO (1 nm)/Ta (3 nm)  samples . The motivation to grow this type of \nsamples was to obtain Heusler films with no out -of-plane texturing. Indeed, x -ray diffra ction measurement  \n[Fig. 1(b)], performed on a Ta/CFAS sample with a Heusler layer thickness of 10 nm , did not indicate the \npresence of any diffraction peak s, except  for the one belonging to the Si substrate. This suggest that both \nthe Ta and the CFAS films are either nanocrystalline or amorphous . Thus , in this type of structure we do \nnot expect the presence of the coherent –incoherent growth transition. The second set of samples consisted \nof epitaxial  MgO  (001)//Cr ( 4 nm)/CFAS  (tCFAS)/MgO (1 nm)/Ta (3 n m) structures . The x -ray diffraction \nmeasurement [Fig. 1 (b)], performed on a Cr/CFAS sample with a Heusler layer thickness of 10 nm, \nindicate s the exclusive presence of the (001) type reflections from the MgO substrate and the Cr and CFAS \nlayers . This confirms the epitaxial growth of the stacks, except for the Ta capping layer, which is \namorphous. Having in view the epitaxial growth we might expect for these samples a possible coherent –\nincoherent growth transition, eventually at higher CFAS thick nesses having in view the relative low \nmismatch between the CFAS lattice and the 45° in-plane rotated Cr lattice  (0.7%). The effective \nmagnetization dependence on the inverse thickness of the ferromagnetic layer for amorphous Ta/CFAS \nand epitaxial Cr/CFAS samples alongside with the Pt/CFAS samples is shown in Fig. 7 . Is to be mentioned \nthat in the for the Ta/CFAS samples the PMA  was obtained for thicknesses  below 1.6 nm , while for the \nCr/CFAS the PMA was not achieved even for thicknesses down to 1 nm. Interestingly, in the case of the \nepitaxial Cr/CFAS samples , for which one might expect possible coherent –incoherent growth transition, \na single linear behavior for the whole thickness range is observ ed. In the case amorphous  Ta/CFAS \nsamples, for which the coherent –incoherent growth transition is not expected, a two-regimes behavior  can 8 \n be distinguished . Although we cannot rule a possible coherent -incoherent growth transition  at larger \nthicknesses, t he results indicate this mechanism is not responsible for the two regimes behavior that we \nobserve  at relatively low thicknesses and other mechanism s must be at play.  \n By fitting the high thickness regime data from Fig.7  to eq. (5)  we extracted for Ta/CFAS  samples a \nsurface anisotropy contribution 𝐾𝑆 of 0.27 ± 0.08 erg/cm2. The volume contribution,  𝐾𝑉, was determine d \nto the be negligible small, as expected for untextured films. Remarkably, the 𝐾𝑆 for the Ta/CFAS samples \nis similar within the error bar s to one obtained for  the Pt/CFAS samples.  Moreover, even in the low \nthickness regime the 𝐾𝑆 might be assumed  similar for the two sets of samples. However, we must  consider  \nthe large uncertainty  having in view the  sparse  data points available for fitting  in the low thickness regime. \nEven so, the clear difference between the two sets of samples is that the Ta/CFAS one shows a  larger \ncritical thickness  (around 2.4 nm ) that separates  the two anisotropy regimes , as compared t o the Pt/CFAS  \none (around 1.5 nm) . This suggests that the possible mechanism responsible for the two -regime behavior \nmight be related to the atomic diffusion at the  Pt/CFAS and  Ta/CFAS interface s. It is well known that Ta \nis prone to diffusion of light elements56. Therefore , a larger critical thickness for the Ta/CFAS samples \nwill imply a larger  atomic  diffusion at the Ta/CFAS interface  compared to the Pt/CFAS one .  \nTo test th e hypothesis  of the interdiffusion , we performed Auger electron spectroscopy (AES) analyses \non th e three set of samples : Pt/CFAS, Cr/CFAS and Ta/CFAS. AES is a surface sensitive technique which \ncan give information about the chemical composition of the surface  with a depth detection limit of 1 -2 \nnm. We started from 10 nm thick CFAS layer samples  and first Ar ion etched the CFAS films down to 4 \nnm thickness and recorded the AES spectra.  Subsequently, t he Ar ion etching and AES spectra recording \nwas repeated  in steps of 1 nm until reaching the underlaye r/CFAS interface.  The etching rate of CFAS \nwas previously calibrated using  ex-situ x-ray reflectometry measurements.  Figure 8 (a) shows two spectra \nrecorded for the Pt/CFAS sample, one after etching the  CFAS layer down to 4 nm (Pt/CFAS 4 nm) and \nthe other  one after etching the CFAS layer down to 1 nm of thickness (Pt/CFAS 1 nm). In the case of the \nPt/CFAS 4 nm spectr um the peaks of Co and Fe are visible alongside with the peaks fro m Al and Si. The \ninset of Fig. 8(a) depicts a n enlargement of the Pt/CF AS 4 nm spectrum around the peaks of Al and Si. \nThe amplitude of the Co and Fe peaks is m uch larger than the amplitude of the of Al and Si ones. This is \ndue to the higher  concentration and higher  Auger relative sensitivity of the  Co and Fe compared to the Al \nand Si. In the case of the Pt/CFAS 1 nm the spectrum  shows the peaks from Co and Fe , with a lower \namplitude,  and the peaks from the Pt underlayer. The presence of the Co and Fe peaks together with the \nPt peaks is not surprising . It is owed to the possible interdiffusion layer at the interface  and to the finite \ndepth resolution  of the  AES which probes both the CFAS layer and the Pt underlayer. The Al and Si peaks 9 \n are not observable , which can be associated  to the relatively low amplitude of the Al and Si falling below \nthe detection limit of the measurement. To test this possibility,  we acquired  Auger  spectra in a narrow \nenergy window around the Al peak , using a longer acquisition time and averaging 10 spectra for e ach \nrecoded  spectrum.  We selected the Al peak and not the Si one because of its larger ampli tude. These \nspectra  recorded for the Pt/CFAS, Ta/CFAS and  Cr/CFAS samples after etching the CFAS layer down to \n4, 3, 2 and 1 nm are shown in Fig.8  (b)-(d). In the case of the Pt/CFAS sample the Al peak is observable \nfor CFAS thickness es down to 1 nm , while in the case of Ta /CFAS  for thicknesses down to 2 nm. \nInterestingly, in the case of the Cr/CFAS sample the Al peak is visible even for a CFAS thickness of 1  \nnm, although with lower amplitude. These findings suggest th at at the underlayer/CFAS interface there is \na diffusion of the light er elements (Al and most likely also Si)  towards the underlayer, with different \ndegree, depe nding  on the nature of the underla yer. As shown schematically in Fig. 8, due to th is lighter \nelements diffusion a CoFe -rich layer form s at the underlayer/CFAS interface. The extent  of the CoFe rich \nlayer depends on the nature of the underlayer . It has the largest thickness for the Ta underlayer (between \n2 and 3 nm) , it is decreasing for the Pt underlayer (between 1 and 2 nm) and it is most likely non -existing \nor extremely thin (below 1 nm) in the case of the Cr underlayer.  \nThe presence of th e CoFe r ich layer agrees  with our findings concerning the occurrence  of the high \nand the low effective PMA  regime s depend ing on the thickness of the Heusler layer. In the case of  the \nPt/CFAS /MgO  samples , the low effective PMA  regime  occurs  for a CFAS layer thickness above  1.6 nm. \nIn this case, the bottom interface consists of Pt/CoFe -rich layer, while the top one of CFAS/MgO . In \nprinciple, both interfaces could contribute to PMA  through Co-O hybridization in the case of the Co-\nterminated CFA S/MgO interface43 or through the d –d hybridization between the  spin-split Co 3d bands \nand the Pt layer 5d bands with large  spin-orbit  coupling44,45. However, their contribution to PMA is small \nand, as we previously mentioned,  would not stabilize perpendicular magnetization except for extremely \nthin CFAS layer s. In the case  of the  high effective  PMA regime (below 1.6 nm) , the bottom interface is \nsimilar consisting of Pt/CoFe -rich layer  and will contribute negligibl y to PMA . However, the top interfa ce \nis now constituted of CoFe -rich layer/MgO  and will induce strong PMA through the hybridization of the \n[Co,Fe] 3𝑑𝑧2 and O 2𝑝𝑧 orbitals42. The premise that the strong PMA is induced by the CoFe -rich \nlayer/MgO interface  is also consistent with our observat ions regarding the dependence of the magnetic \nanisotropy on the nature of the underlayer. As seen in Fig. 7, i n the case of the Cr/CFAS samples, where \nno CoFe -rich layer was evidenced, there is only one anisotropy regime with a relatively low effective \nPMA . In the case of the Ta/CFAS sa mples, the high effective PMA regime is present starting from a larger \nCFAS thickness , as compared to de case of Pt/CFAS samples , which is in agreement  with the thicker 10 \n CoFe -rich layer observed for the Ta/CFAS relative to the Pt/CFAS ones.  It is to be mentioned that in the \ncase of Ru/CFA/MgO and Cr/CFA/Mg O annealed samples  Al diffusion towards the MgO but not towards \nthe underlayer was previously observed46,47. The lack of Al diffusion towards the Cr underlayer is in \nagreement with our findings. In the case of the aforementioned studies, the thermal  annealing of the \nsamples was necessary  to facilitate the Al diffusion  and to achieve strong PMA. In our case, for the Pt and \nTa underlayer , we attain  strong PMA in the low thickness regime without the need of thermal annealing. \nThis indicates that for the Pt and Ta underlayer s the [Al,Si] diffusion takes place during the growth of the \nCFAS film, which results in the formation of the interfacial CoFe -rich layer  directly during deposition . A \nfurther deposition of MgO on this CoFe -rich layer will generate the strong PMA  through the hybridizati on \nof the [Co,Fe] 3𝑑𝑧2 and O 2𝑝𝑧 orbitals42. Having in view the si milar behavior  of the magnetic anisotropy \nfor the CFA, CFAS, CFMS and CFS Heusler alloys thin films that we study here , it is reasonable to \nassume that in all the cases there is  a diffusion of the lighter elements  (Al, Si) towards the Pt underlayer \nand the formation of the CoFe -rich interfacial  layer , which , when MgO is deposited on top,  will give rise \nto the strong PMA in the low thickness regime.  \nWe now discuss the thickness  dependence of the Gilbert damping parameter extracted from  the θH \ndependence of the linewidth HPP. It is known  that generally the linewidth is given by a sum o f extrinsic  \nand intrinsic contr ibution  as49,57-59: \n𝐻PP=𝐻PPint+𝐻PPext,                                                                (5)  \n𝐻PPint=𝛼(𝐻1+𝐻2)|d𝐻𝑅\nd(𝜔𝛾⁄)|,                                                       (6) \n𝐻PPext=|d𝐻𝑅\nd(4𝜋𝑀eff)|Δ(4𝜋𝑀eff)+|d𝐻𝑅\nd𝜃𝐻|Δ𝜃𝐻+Δ𝐻TMS,                            (7) \nwhere, 𝛼 is the intrinsic Gilbert dampi ng parameter and the three  terms in equation (7) are the linewidth \nenhancement due to the anisotropy distribution , due to deviation from planarity of the films  and due to the \ntwo-magnon scattering. In the case of our films, the θH dependence of the linewidth HPP is well fitted \nusing only the intrinsic contribution  and the extrinsic enhancement due to the anisotropy distribution. For \nthis, |d𝐻𝑅d(𝜔𝛾⁄) ⁄ |  and |d𝐻𝑅d(4𝜋𝑀eff) ⁄ |  are numerically calculated using Eqs. (1)-(4) and the HPP vs. \nθH experimental dependence is fi tted to Eq. (5) using 𝛼 and Δ(4𝜋𝑀eff) as adjustable parameters49. An \nexample of a  fit curve is depicted  in Fig. 4(b)  for the case of the 2.4 nm thick Pt/CFAS sample . Figure 9 \nshows the 𝛼 dependence on the inverse ferromagnetic layer  (1/t) thickness for the Pt/CFA, Pt/CFS, \nPt/CFMS , Pt/CF AS and Pt/CFB samples.  We will first discuss the case of CFB, where  a linear dependence \nis observed . The linear increase of the Gilbert damping  parameter with 1/t is expected  and it is due to the \nangular momentum loss due to the spin pumping effect in the Pt layer. In this type of structures it  was 11 \n shown60 that the total damping is given by 𝛼=𝛼0+𝛼SP𝑡⁄, where  𝛼0 is the Gilbert damping of the \nferromagnetic film and 𝛼SP is due to the spin pu mping effect.  By linear fitting the data in Fig. 9  we obtain \na Gilbert damping parameter for the CFB of 0.0028 ± 0.0003 , in agreement with other reports61,62. In the \ncase of the CFAS films, the linear dependence  is observed only for the large thickness region  and by fitting \nthis data we obtain a Gilbert da mping parameter of 0.00 53 ± 0.00 12, consistent wit h previously reported \nvalue s for relative ly thick er films25. The low thickness data deviates from the  linear dependence . This \nbehavior is similar for all the other studied Heusler films, with the low thickness deviation being even \nmore pronounced.  The strong increase of the damping can be related to  the [Al,Si] diffusion and the  \nformation of the interfacial CoFe -rich layer. Since the [Al,Si]  diffusion  is more important for thinner films , \nit will have a stronger impact on the chemical composition relative to the thicker ones. The relatively  small \ndamping of the Co based full Heusl er alloys is a consequence of the ir specific electronic structure21. \nConsequently , deviations from the  correct stoichiometry , which is expected to have a n important  effect \non the electronic structure , will lead to a strong increase of the damping, as shown, for example, by ab-\ninitio calculation in the case of Al deficient CFA films47. Therefore, the increase of the damping beyond \nthe spin pumping effect for the thinner  Heusler films is explained by  the interfacial CoFe -rich layer \nformat ion.  \n \nConclusions  \n \nWe have studied the mechanism s responsible for PMA in the case of  Co2FeAl, Co 2FeAl 0.5Si0.5, Co 2FeSi \nand Co 2Fe0.5Mn 0.5Si Heusler alloy thin films sandwiched between Pt and MgO layers.  We showed that \nthe ultrathin Heusler films exhibit  strong PMA irrespective of their composition. The effective \nmagnetization displays  a two -regime behavior depending on the thickness of the Heusler layers.  The two -\nregime behavior is generated  by the formation of a n CoFe -rich layer at the underlayer/Heusler  interface \ndue to the interdiffusion. The strong PMA observed in the case of the ultrathin films can be explained by \nthe electronic hybridization of the CoFe -rich metal lic layer  and oxygen orbitals across the \nferromagne t/MgO interface . The formation of the  interfacial  CoFe -rich layer  causes the increase of the \nGilbert damping  coefficient  beyond the spin pumping for the ultrathin Heusler films. Our results illustrate \nthat the strong PMA is not an intrinsic property of the  Heusler/MgO interface,  but it is actively influenced \nby the interdiffusion, which can be tuned by a proper choice of the underlayer material.  \n \n  12 \n FIG. 1. (a) 2θ/ω x-ray diffraction patterns recorded for four representative  Pt/Co 2YZ/MgO samples having \na thickness of the Heusler layer of 10 nm.  The patterns show the (111) and (222) peaks belonging to the \nPt layer , the (022) peak from the Heusler films and the (001) peak of the Si substrate.  (b) 2θ/ω x-ray \ndiffraction patterns for the Ta/CFAS (10 nm)/MgO and Cr/CFAS (10 nm)/MgO samples indicating the \namorphous or epitaxial  growth of the CFAS layer , respectively.  \n13 \n FIG. 2. Hysteresis  loops measured with the magnetic field applied perpendicular to the plane of the \nsamples . Depending on the thickness of the Heusler layers, the samples show  in-plane magnetic anisotropy \n(a)-(d) or perpendicular magnetic anisotropy (e) -(h).  \n  \n14 \n FIG. 3. Typical  FMR spectra measured at 9.79 GHz  for different θH field angles for a 2.4 nm thick \nPt/CFAS sample.  \n15 \n FIG. 4. (a) Resonance field HR and (b) linewidth HPP dependence on the θH field angle for a 2.4 nm thick \nPt/CFAS sample . The inset s hows a schematic of the measurement geometry. The points  stand for  \nexperimental data while the lines represent  the result of the theoretical fits, as described in text.  \n \n  \n16 \n FIG. 5. \ng factor dependence on the thickness of the Heusler layers for samples with different Heusler layer \ncomposition.  \n \n  \n  \n17 \n FIG. 6. The effective magnetization \n4effM dependence on the inverse thickness of the  ferromagnetic \nlayer for samples with different composition s. The points are experimental data while the lines are linear \nfits. In the case of the Heusler samples two linear fits correspond to the two anisotropy regimes.   \n \n18 \n FIG. 7. The effective magnetization  \n4effM dependence on the inverse thickness of the ferromagnetic \nlayer for amorphous Ta/CFAS and epitaxial Cr/CFAS samples.  The data for Pt/CFAS is also shown for \ncomparison. The points are experimental data while the lines are linear fits.  \n \n19 \n FIG. 8. (a) AES spectra recoded for the Pt/CFAS sample  after etching the CFAS layer down to 4 and 1 \nnm, respectively . The inset shows a zoom around de Al and Si peaks. AES spectra recorded around the \nAl peak  after etching the CFAS layer down to 4, 3, 2 and 1 nm for the (b) Pt/CFAS, (c) Ta/CFAS and \n(d) Cr/CFAS samples.  Schematic representation of the [Al,Si] diffusion to wards the underlayer and the \ninterfacial CoFe -rich layer formation . \n \n \n20 \n FIG. 9. Gilbert damping parameter ( 𝛼) dependence on the inverse ferromagnetic layer (1/t) thickness for \nthe Pt/CFA, Pt/CFAS , Pt/CFMS, Pt/CFS and Pt/CFB samples.  The points are experim ental data while \nthe lines are linear fits for  Pt/CFB and Pt/CFAS samples. 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Tiusan,  \nJournal of Physics D: Applied Physics 51 (4), 0 45002 (2018).  \n \n "    },    {        "title": "2308.07236v3.Temperature_Evolution_of_Magnon_Propagation_Length_in_Tm__3_Fe__5_O___12___Thin_Films__Roles_of_Magnetic_Anisotropy_and_Gilbert_Damping.pdf",        "content": "  \n1 \n Temperature Evolution of Magnon Propagation Length in \nTm 3Fe5O12 Thin F ilms: Roles  of Magnetic Anisotropy and Gilbert \nDamping  \n \nAmit Chanda1, Christian Holzmann2, Noah S chulz1, Aladin Ullrich2, Derick DeTellem1, \nManfred Albrecht2*, Miela J. Gross3, Caroline A. Ross3*, Dario  A. Arena1, Manh -Huong \nPhan1, and Hari haran  Srikanth1* \n1Department of Physics, University of South Florida, Tampa, Florida 33620, USA  \n2Institute of Physics, University of Augsburg, 86159 Augsburg, Germany  \n3Department of Materials Science and Engineering, Massachusetts Institute of Technology, \nCambridge, Massachusetts 02139, USA  \n \n*Corresponding authors: manfred.albrecht@physik.uni -augsburg.de ; caross@mit.edu ; \nsharihar@usf.edu  \n \nKeywords:  Longitudinal spin Seebeck effect, Inverse spin Hall effect, Magnon propagation length, \nGilbert damping, Magnetic anisotropy , Rare -earth iron garnet  \n \nABSTRACT  \nThe magnon propagation length , 〈𝜉〉 of a ferro -/ferrimagnet (FM) is one of the key factors that  \ncontrols the generation and propagation of thermally -driven magnonic spin current in FM/heavy \nmetal ( HM) bilayer based spincaloritronic devices . For the development of a complete physical \npicture of thermally -driven magnon transport in FM/HM bilayers over a wide temperature range,    \n2 \n it is of utmost importance to understand  the respective roles of temperature -dependent Gilbert \ndamping (𝛼) and effective magnetic anisotropy (𝐾𝑒𝑓𝑓) in controlling  the temperature evolution of \n〈����〉. Here, we report  a comprehensive investigation  of the temperature -dependent longitudinal spin \nSeebeck effect (LSSE), radio frequency transverse susceptibility, and broadband ferromagnetic \nresonance measurements  on Tm 3Fe5O12 (TmIG )/Pt bilayers grown on different substrates. We \nobserve a remarkable drop  in the LSSE voltage below 200 K independent  of TmIG film thickness \nand substrate choice . This is attribute d to the noticeable increases in effective magnetic anisotropy \nfield, 𝐻𝐾𝑒𝑓𝑓 (∝𝐾𝑒𝑓𝑓) and 𝛼 that occur within the same temperature range. From the TmIG  \nthickness dependence of the LSSE voltage, we determined the temperature dependence of 〈𝜉〉 and \nhighlighted its correlation with the temperature -dependent  𝐻𝐾𝑒𝑓𝑓 and 𝛼 in TmIG/Pt  bilayers , which \nwill be beneficial for the development of rare -earth iron garnet -based efficient spincaloritronic \nnanodevices . \n \n \n \n \n \n \n \n \n   \n3 \n 1. INTRODUCTION  \nIn recent years, interface -engineered bilayer thin films have gained intense attention of the \nmaterials science community because of their multifunctionality and emergent  physical properties \nranging from  ferroelectricity1 and magnetism2 to spin -electronics3. Bilayers  comprised of  \ninsulating rare -earth iron garnet (REIG ) and heavy metal (HM) form  the most appealing platform  \nto generate, transmit , and detect pure spin currents  in the field of spin -based -electronics4–6. The \ninterplay of damping and magnon propagation length ( 〈𝜉〉) of the REIG layer and spin-orbit \ncoupling (SOC)  of the HM layer leads to a wide range of emergent  spintronic phenomena in t his \nfascinating class of heterostructures , including  the spin Hall effect7, spin -orbit torque8,9, spin-\npumping  effect  (SPE)10, and the longitudinal spin Seebeck effect  (LSSE)11–13.  The discovery of \nthe SSE14 instigated  a new generation of spintronic nano devices facilitating  electrical energy \nharvesting from renewable thermal energy wherein a magnonic  spin current is thermally generated \nand electrically detected by applying a temperature gradient across a magnetic insulator (MI) /HM \nbilayer15. Unlike magnetostatic spin waves with millimeter -range propagation lengths, 〈𝜉〉 for \nthermally generated magnons is significantly smaller, a few hundreds of nanometers16. In the \nframework of an atomistic spin model based on linear spin -wave theory, it was theoretically \nshown17,18 that thermally generated magnons have a broad frequency (𝑓) distribution with \n𝑓𝑚𝑖𝑛𝑖𝑚𝑢𝑚=2𝐾𝑒𝑓𝑓[ℎ(1+𝛼2)] ⁄  and 𝑓𝑚𝑎𝑥𝑖𝑚𝑢𝑚 =4𝐾𝑒𝑓𝑓[ℎ(1+𝛼2)] ⁄ , where ℎ is the Planck \nconstant, 𝐾𝑒𝑓𝑓 is the effective magnetic anisotropy constant and 𝛼 is the Gilbert damping \nparameter. While the high -f magnons experience stronger damping, low -f magnons possess a very \nlow group velocity, and hence, the majority of the thermally generated magnons become damped \non shorter length -scales17,18. Therefore, only the subthermal magnons, i.e., the low-f magnons   \n4 \n dominate the long -range thermo -spin transport19–21. Within this hypothesis, it was predicted that \n〈𝜉〉 is inversely proportional to both 𝛼 and √𝐾𝑒𝑓𝑓.17,18 \n \nY3Fe5O12 (YIG) has been a widely explored  MI for generating and transmitting pure spin \ncurrents due to its ultra -low damping  (𝛼 ≈ 10-4-10-5) and large  〈𝜉〉 (~100-200 nm) 11,13,17. This has \nled to a drastic  increase in research over the last few decade s, aimed at enhancing the spin current \ninjection efficiency across the MI/HM  interface  by reducing the conductivity mismatch between \nthe MI and HM layers  by introducing  atomically thin semiconducting interlayers22–28 and \nenhancing the  interfacial  spin-mixing conductance .29–31 Hariharan’s group has explored the roles \nof bulk and surface magnetic anisotrop ies in LSSE in different REIG -based MI/HM bilayers12,13, \nwhereas a recent study highlights  the influence of damping  on SPE and LSSE  in a compensated \nferrimagnetic  insulator .32 It has also been demonstrated that the LSSE in YIG/Pt bilayers varies \ninversely with intrinsic Gilbert damping of the YIG films, however, the LSSE coefficient does not \nshow any significant correlation with the enhanced damping due to SPE in YIG/Pt bilayers.33 All \nthese studies highlight the important role s of both magnetic anisotropy and Gilbert damping in \nthermally generated magnon propagation in MI/HM bilayers .  \n \nBy investigating  the YIG thickness dependence of the local LSSE measurements in YIG/Pt, \nGuo et al.34 determined the temperature ( T) dependence of 〈𝜉〉 and found a scaling behavior of \n〈𝜉〉 ∝ 𝑇−1. On the contrary , by employing non -local measurement geometries, Cornelissen et al. \ndemonstrated that the magnon diffusion length  for thermally driven magnonic spin currents (𝜆𝑡ℎ𝑚) \nof YIG decreases with decreasing temperature over a broad temperature range .35 Gomez -Perez et \nal. reported similar observation s and demonstrate d that the temperature dependen ce of  𝜆𝑡ℎ𝑚 is   \n5 \n independent of  the YIG thickness .19 The different trends of the temperature dependent \ncharacteristic critical length scales for thermally generated magnon propagation  in YIG observ ed \nby different groups indicate s distinct temperature evolutions of 𝛼 and 𝐾𝑒𝑓𝑓 in the YIG films grown \nby these groups. In other words, different thin film growth conditions and sample dependent \nchanges in the physical properties can give rise to different temperature dependences of both 𝛼 \nand 𝐾𝑒𝑓𝑓 and hence 〈𝜉〉. For the development of a complete physical picture of LSSE in these \nREIGs over a wide temperature range,  it is of utmost importance to comprehend  the respective \nroles of  both 𝛼 and 𝐾𝑒𝑓𝑓 simultaneously in determining the temperature evolution of  〈𝜉〉, which \nremains largely unexplored.   \n \nAlthough YIG is considered as a benchmark system for LSSE,11,34 there is only  a limited \nnumber of studies that explore temperature dependent LSSE in other iron garnets .12,32,36,37 For \nexample, Gd3Fe5O12 (GdIG) which is a ferrimagnetic insulator with magnetic compensation \ntemperature (𝑇𝐶𝑜𝑚𝑝 ) close to room temperature, shows a sign -inversion in the LSSE voltage12 as \nwell as in the spin -Hall anomalous Hall effect38 around its magnetic compensation. However, the \nGilbert damping in GdIG diverges over a broad temperature range around its  𝑇𝐶𝑜𝑚𝑝  which makes \nit difficult to probe the temperature evolution  of 𝛼 and its  contribution  towards 〈𝜉〉 over a wide \ntemperature range around the 𝑇𝐶𝑜𝑚𝑝 .32 Apart from YIG and GdIG , there has been a renaissance of \nresearch interest in another member of the REIG family: Tm 3Fe5O12 (TmIG) due to its  wide -\nranging extraordinary magnetic properties6 e.g., strain -tunable perpendicular magnetic anisotropy \n(PMA),39 chiral and topological spin textures,40 and interfacial Dzyaloshinskii -Moriya \ninteraction40,41 combined with low coercivity6 which make this system a promising candidate for \nnumerous efficient spintronic applications, such as spin -orbit torque induced magnetization   \n6 \n switching,8,42 current -induced domain -wall motion,43 and spin Hall –topological Hall effect s44,45. \nRecently, the LSSE has been investigated in TmIG/Pt bilayers with PMA at room temperature, \nand shown to exhibit high interfacial spin transparency and spin -to-charge conversion efficiency \nat the TmIG/Pt interface46. TmIG has a higher Gilbert damping parameter (≈10−2)6 compared to \nYIG, and unlike GdIG, TmIG does not exhibit any magnetic compensation in the temperature \nrange between 1.5 and 300 K47,48, which allows us to probe the relative contribution of 𝛼 towards  \nthe temperature evolution of  〈𝜉〉 and hence the LSSE  over a broad temperature range close to  the \nroom temperature . However, the temperature evolution of LSSE and hence 〈𝜉〉 as well as their \nrelation ship with 𝛼 and 𝐾𝑒𝑓𝑓 in TmIG/Pt bilayers are yet to be explored , which would be of critical \nimportance for REIG -based efficient magnonic device applications . Here , we have performed a \ncomprehensive investigation of the temperature -dependent LSSE, radio frequency (RF) transverse \nsusceptibility (TS), and broadband ferromagnetic resonance (FMR) of TmIG /Pt bilayers grown on \ndifferent substrates . From the TmIG thickness dependence of the LSSE voltage, we determined \nthe temperature dependence of 〈𝜉〉 and highlighted  its correlation with the temperature -dependent \neffective magnetic anisotropy field, 𝐻𝐾𝑒𝑓𝑓 (∝𝐾𝑒𝑓𝑓) and 𝛼 in TmIG/Pt  bilayers.  \n \n2. RESULTS AND DISCUSSION  \n2. 1. Structural Characterization  \nSingle -crystalline TmIG films with different thicknesses were grown on (111) -oriented \nGd3Sc2Ga3O12 (GSGG) and Gd 3Ga5O12 (GGG) substrates by pulsed laser deposition ( see \nMethods ). The high crystalline quality of the TmIG films was confirmed by X-ray diffraction \n(XRD). Figure 1 (a) shows the 𝜃−2𝜃 X-ray diffractograms of the GSGG/TmIG( 𝑡) films with \ndifferent TmIG film thickness  𝑡 (t = 236, 150, 89, 73, 46 and 28 nm) .    \n7 \n  \n \nFigure 1 . Structural and Morphological characterization.  (a) 𝜃−2𝜃 X-ray diffractogram  of \nthe GSGG/TmIG( 𝑡) films with different film thickness  𝑡 (t = 236, 150, 89, 73, 46 and 28 nm).  The \nreciprocal space maps recorded in the vicinity of the (642) reflection for (b) GSGG/TmIG( 30 nm) \nand (c) GSGG/TmIG(205 nm) films . For the thinner film (30  nm), the TmIG  film peak matches \nthe IP lattice constant of the GSGG substrate, whereas  for the thicker film (205nm), the TmIG film \nis largely relaxed.  \n \n The substrate choice and the TmIG film thickness influence  the strain state of the film . \nFigs. 1 (b) and ( c) show the reciprocal space maps in the vicinity of the (642) reflection for the \nGSGG/TmIG ( 30 nm) and GSGG/TmIG (205 nm) films, respectively . For the thinner film (30  \nnm), the TmIG film 𝑞𝑥 matches the in -plane (IP) lattice spacing  of the GSGG substrate  indicating \ncoherent growth , and the out -of-plane (OOP) lattice spacing is smaller than that of the substrate \n(higher 𝑞𝑍), consistent with the smaller unit cell volume for TmIG compared to GSGG. However,  \nthe thicker film (205  nm) is relaxed in plane with smaller IP and OOP lattice spacing than that of \nthe substrate, and its peak position is close to that of bulk TmIG. The 𝜃−2𝜃 scans show  a decrease \nin the OOP spacing (increase in 2𝜃) for thinner films . These trends are consistent with the TmIG \n  \n8 \n initially growing with an IP lattice match  to the substrate and hence a tensile IP strain (and a \nmagnetoelastic anisotropy favoring PMA), but the strain relaxes as the film thickness increases. \nThe thickest films, which are strain -relaxed, have a slightly higher OOP lattice spacing compared \nto bul k according to Fig. 1 (a) which suggests the presence of oxygen vacancies or Tm:Fe ratio \nexceeding 0.6, which can occur in thin films and raise the unit cell volume.  All the films show a \nsmooth surface morphology with a low root -mean -square roughness below 0.5nm, as visible in \natomic force microscopy (AFM) images for the GSGG/TmIG( 46nm), GGG/TmIG(44nm) and \nsGGG/TmIG(75nm) films shown in  the Supplementary Fig ure 1. \n \n A cross -section of an about 220 nm thick TmIG film on GSGG substrate, covered with \na 5 nm Pt layer, was analyzed by scanning transmission electron microscopy (STEM). Fig. 2 (a) \nshows a low magnification STEM image of the whole layer stack. An annular detector with a small \ncollector angle (24 -48 mrad) was used to highlight strain (Bragg) contrast over mass (Z) contrast \n49. The TmIG film shows columnar features attributed to strain contrast. An atomically resolved  \nSTEM image at the TmIG film -Pt interface ( Fig. 2 (b)) reveals a single crystalline TmIG film under \nthe polycrystalline Pt layer, with the bright spots indicating columns of Tm and Fe. The STEM \nimage of an area within the TmIG film close to the Pt interface shows the presence of a planar \ndefect in which s elected lattice planes are highlighted by colored lines in Fig. 2 (c). Such planar \ndefects could be associated with  partial dislocations or atomic level disorder, which are common \nin REIGs.50,51 \n \n   \n9 \n  \n \nFigure 2. Cross -sectional scanning transmission electron microscopy (STEM) analysis of the \nGSGG/TmIG(220 nm)/Pt(5nm) film.   (a) TEM image of the layer stack recorded by an annular \ndetector with a small collector angle (24 -48 mrad), highlighting strain (Bragg) contrast over mass \n(Z) contrast, (b) shows an atomic -resolution STEM image of the TmIG -Pt interface with [110] \nzone axis , while (c) shows an area within the TmIG film. The colored lines highlight a planar \ndefect . (d) electron energy loss spectroscopy ( EELS) scan at the Fe L3 and L2 edges. The measured \nenergy loss spectra are displayed as data points, exempl ified for positions close to  the garnet -\nsubstrate and garnet -Pt interfaces, with the fitted functions presented as colored lines. (e) The \nthickness dependent Fe L3 peak position and FWHM is extracted.  \n \n \n \n \n  \n10 \n 2. 2. Correlation between  Thermo -Spin Transport and Magnetism   \nFig. 3(a) shows the schematic illustration of our LSSE measurement  configuration . Simultaneous \napplication of a  vertical (+ z-axis) T-gradient ( 𝛁𝑻⃗⃗⃗⃗⃗ ) and an in-plane (x-axis) DC magnetic field \n(𝝁𝟎𝑯⃗⃗⃗⃗⃗⃗⃗⃗ ) across the TmIG fil m causes diffusion of thermally -excited magnons and develops a spatial \ngradient of magnon accumulation along the direction of 𝛁𝑻⃗⃗⃗⃗⃗ .52 The accumulated magnons close to \nthe TmIG/Pt  interface transfer spin angular momenta to the electrons of the adjacent Pt layer52. \nThe injected spin current density  is, 𝑱𝑺⃗⃗⃗ ∝−𝑆𝐿𝑆𝑆𝐸𝛁𝑻⃗⃗⃗⃗⃗ , where 𝑆𝐿𝑆𝑆𝐸 is the LSSE coefficient52,53. The \nspin current injected into the Pt layer along the z-direction is converted into a charge current , 𝑱𝑪⃗⃗⃗  =\n (2𝑒\nℏ)𝜃𝑆𝐻𝑃𝑡(𝑱𝑺⃗⃗⃗ × 𝝈𝑺⃗⃗⃗⃗⃗ ) along the y-direction via the inverse spin Hall effect (ISHE), where e, ℏ, 𝜃𝑆𝐻𝑃𝑡, \nand  𝝈𝑺⃗⃗⃗⃗⃗  are the electron ic charge, the reduced Planck’s constant , the spin Hall angle of Pt, and the \nspin-polarization vector, respectively . The corresponding LSSE voltage  is52,54,55  \n 𝑉𝐿𝑆𝑆𝐸= 𝑅𝑦𝐿𝑦𝐷𝑃𝑡(2𝑒\nℏ)𝜃𝑆𝐻𝑃𝑡| 𝐽𝑆|tanh(𝑡𝑃𝑡\n2𝐷𝑃𝑡),    (1) \nwhere, 𝑅𝑦,𝐿𝑦,𝐷𝑃𝑡,and 𝑡𝑃𝑡 represent the electrical resistance between the contact -leads, t he \ndistance between the contact -leads, the spin  diffusion length of Pt, and the Pt layer  thickness , \nrespectively .  \n \nFig. 3(b) shows the magnetic field  (H) dependen t ISHE  voltage,  𝑉𝐼𝑆𝐻𝐸(𝐻) for \nGSGG/TmIG(236  nm)/Pt (5 nm) for different values of the temperature  difference between the hot \n(𝑇ℎ𝑜𝑡) and cold ( 𝑇𝑐𝑜𝑙𝑑) blocks, ∆𝑇=(𝑇ℎ𝑜𝑡−𝑇𝑐𝑜𝑙𝑑), at a fixed average sample temperature 𝑇=\n 𝑇ℎ𝑜𝑡+𝑇𝑐𝑜𝑙𝑑\n2 = 295 K. For all Δ𝑇, 𝑉𝐼𝑆𝐻𝐸(𝐻) exhibit s a nearly square -shaped hysteresis  loop. The inset \nof Fig. 3(b) plots  the ∆𝑇-dependence of the background -corrected LSSE voltage,  𝑉𝐿𝑆𝑆𝐸(Δ𝑇)=  \n11 \n  [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑠𝑎𝑡,Δ𝑇)−𝑉𝐼𝑆𝐻𝐸(−𝜇0𝐻𝑠𝑎𝑡,Δ𝑇)\n2], where 𝜇0𝐻𝑠𝑎𝑡 is the  saturation field . Clearly,  𝑉𝐿𝑆𝑆𝐸 increases  \nlinearly with ∆𝑇 as expected from Eqn. 1 .12  \n \nFigure 3. Magnetism and longitudinal spin Seebeck effect (LSSE) in \nGSGG/TmIG(236nm)/Pt(5nm) film.  (a) Schematic illustration of the experimental configuration \nfor LSSE measurements.  A temperature gradient ( 𝛁𝑻⃗⃗⃗⃗⃗ ) is applied along the + z axis and an in -plane \n(IP) dc magnetic field ( 𝝁𝟎𝑯⃗⃗⃗⃗⃗⃗⃗⃗ ) is applied along the + x axis. The inverse spin Hall effect (ISHE) \ninduced voltage ( 𝑉𝐼𝑆𝐻𝐸) is measured along the y-axis. (b) 𝑉𝐼𝑆𝐻𝐸(𝐻) loops for different values of \nthe temperature difference ∆𝑇 at a fixed average sample temperature 𝑇 = 295 K. The inset shows \na linear ∆𝑇-dependence of the background -corrected LSSE voltage. (c) 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops \nmeasured at selected temperatures in the range 120 K ≤ T ≤ 295 K for Δ𝑇 = +10 K. (d) The IP \nM(H) hysteresis loops at selected temperatures.  \n \n  \n12 \n Fig. 3 (c) shows the 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops for GSGG/TmIG(236 nm)/Pt(5 nm) \nmeasured at selected temperatures for Δ𝑇= +10 K. Clearly, |𝑉𝐼𝑆𝐻𝐸(𝜇0𝐻𝑠𝑎𝑡)| significantly \ndecreases, and the hysteresis loop broadens at low temperatures, especially below 200 K. To \ncorrelate thermo -spin transport with the bulk magnetic properties, in Fig. 3 (d), we show the \nmagnetic field dependence of magnetization, 𝑀(𝐻) at selected temperatures for GSGG/TmIG(236 \nnm)/Pt(5 nm) measured while scanning an in -plane (IP) magnetic field. It is evident that with \nlowering the temperature, the saturation magnetization (𝑀𝑆) decreases and t he coercivity ( 𝐻𝐶) \nincreases with a corresponding increase in the magnetic anisotropy, especially below 200 K. This \nobservation is also in agreement with the T-dependent magnetic force microscopy (MFM) results \nshown in Supplementary Figure 2 , which clearly reveals that the root mean square (RMS) value \nof the phase shift, Δ𝜙𝑅𝑀𝑆 decreases significantly between 300 and 150 K indicating changes in the \nmagnetic domain structure at low -T. \n \nThe decrease in 𝑀𝑆 at low -T is well-known in TmIG48,56 and is a result of the increasing \nmoment of the Tm3+ ion at low -T, which competes with the net moment of the Fe3+ ions ( i.e., the \ndodecahedral Tm3+ moment opposes the net moment of the tetrahedral and octahedral Fe3+ \nmoments).  Based on the molecular -field-coefficient theory developed by Dionne57, we have \nperformed molecular -field simulations58,59 to determine 𝑀𝑆(T) for TmIG ( see Supplementary  \nFigure 3(n)) which is consistent with our experimental observation of the decrease in 𝑀𝑆 at low -\nT. It is apparent from Figs. 3 (c) and (d) that the temperature evolution of 𝑉𝐼𝑆𝐻𝐸 signal follows that \nof 𝑀𝑆. To further explore the correlation between 𝑉𝐼𝑆𝐻𝐸 and 𝑀𝑆, magnetometry and LSSE \nmeasurements were repeated on the GSGG/TmIG( t)/Pt(5 nm) sample series  with different TmIG \nfilm thicknesses (28 nm≤𝑡≤236 nm). Films with  46 nm≤𝑡≤236 nm possess IP easy -axes   \n13 \n while the 28 nm film has an OOP easy -axis of magnetization , which  was confirmed via IP -\nmagnetometry and OOP p-MOKE measurements (see Supplementary  Figure 3(e)). The total \nmagnetic anisotropy of a (111) -oriented TmIG fi lm, neglecting growth and interfacial anisotropies,  \nhas contributions from shape anisotropy ( 𝐾𝑠ℎ𝑎𝑝𝑒), cubic magnetocrystalline anisotropy ( 𝐾𝑚𝑐), and \nmagnetoelastic anisotropy ( 𝐾𝑚𝑒)47,49,60 i.e., 𝐾𝑒𝑓𝑓=𝐾𝑠ℎ𝑎𝑝𝑒+𝐾𝑚𝑐 + 𝐾𝑚𝑒=−1\n2 𝜇0𝑀𝑆2−𝐾1\n12−\n9\n4𝜆111𝑐44(𝜋\n2−𝛽), where K1 is the magnetocrystalline anisotropy coefficient, 𝜆111 is the \nmagnetostriction along the [111] direction,  𝑐44 is the shear modulus and 𝛽 is the cor ner angle of \nthe rhombohedrally -distorted unit cell. For a negative magnetostriction ( 𝜆111  = −5.2×10−6 for \nbulk TmIG47), the tensile IP  strain, which results from the difference in lattice parameters \n(𝑎𝐺𝑆𝐺𝐺=12.57 Å and 𝑎𝑇𝑚𝐼𝐺=12.32 Å) promotes PMA ( 𝐾𝑒𝑓𝑓>0).49,60,61  PMA is expected for \nfully -strained films (28  nm), but strain -relaxation in thicker films reduces the magnetoelastic \ncontribution , and the easy -axis reorients to IP  direction60. \n \nFigs. 4 (a) and (b) depict the  𝑉𝐼𝑆𝐻𝐸(𝐻) loop on the left y-scale and corresponding 𝑀(𝐻) \nloop on the right y-scale at 295 K for the thicknesses: 𝑡=236 and 28 nm,respectively . The \n𝑀(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis -loops for all other  thicknesses are shown in the Supplementary \nFigures 3 and 4. Clearly, the 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis -loops for all the thicknesses mimic the \ncorresponding 𝑀(𝐻) loops. Note that, unlike YIG -slab, there is no surface magnetic anisotropy \ninduced anomalous low field feature in the 𝑉𝐼𝑆𝐻𝐸(𝐻) loop for any of our TmIG thin films. This is \npossibly because the thickness of the TmIG films is smaller than their average magnetic domain \nsize62. This is why the YIG thin films also do not show any low field anomalous feature in the \n𝑉𝐼𝑆𝐻𝐸(𝐻) loops52. Additionally, the 𝑉𝐼𝑆𝐻𝐸(𝐻) loop for our TmIG film with PMA ( the 28 nm film)  \nat 295 K is quite similar to that of a TmIG thin film with PMA at room temperature reported in the   \n14 \n literature46. In Figs. 4 (c) and (d), we demonstrate the T-dependence of the background -corrected \nLSSE voltage,  𝑉𝐿𝑆𝑆𝐸(𝑇)=𝑉𝐼𝑆𝐻𝐸(𝑇,+𝜇0𝐻𝑠𝑎𝑡)−𝑉𝐼𝑆𝐻𝐸(𝑇,−𝜇0𝐻𝑠𝑎𝑡)\n2 for Δ𝑇= +10 K  on the left y-scale and \ncorresponding 𝑀𝑆(𝑇) on the right y-scale for GSGG/TmIG( 236 nm )/Pt(5 nm) and \nGSGG/TmIG( 28 nm )/Pt(5 nm) , respectively. Interestingly, 𝑉𝐿𝑆𝑆𝐸(𝑇) and 𝑀𝑆(𝑇) for both films \ndrop remarkably below the T-window of 180 -200 K. We observed a similar trend in 𝑉𝐿𝑆𝑆𝐸(𝑇) and \n𝑀𝑆(𝑇) for all GSGG/TmIG( t)/Pt(5 nm) films with other thicknesses (see Supplementary Figures \n3 and 4 ). These results indicate that this behavior is intrinsic to TmIG.  \n \nFigure 4. Longitudinal spin Seebeck effect in GSGG/TmIG( t)/Pt(5  nm) films.  The 𝑉𝐼𝑆𝐻𝐸(𝐻) \nhysteresis loops on the left y-scale and the IP 𝑀(𝐻) loops on the right y-scale at  T = 295 K for \nGSGG/TmIG( t)/Pt films for t = (a) 236 nm, and (b) 28 nm . The temperature dependence of the \nbackground -corrected LSSE voltage,  𝑉𝐿𝑆𝑆𝐸(𝑇) on the left y-scale and temperature dependence of \nsaturation magnetization, 𝑀𝑆(𝑇) on the right y-scale for the GSGG/TmIG( t)/Pt(5 nm) films  for t = \n(c) 236 nm and (d) 28 nm, for Δ𝑇 = +10 K . \n  \n15 \n Note that the yellow and grey background colors in all the  graphs throughout the \nmanuscript are used to highlight significant changes in physical parameters between  high (yellow)  \nand low (grey) temperature regions. Additionallly, we have used the sky blue background color in \nsome of the specific graphs (especially temperature dependence of 𝑉𝐿𝑆𝑆𝐸(𝑇) and 𝑀𝑆(𝑇)) to \nindicate considerable changes in the corresponding physical parameters occurring around the \nnarrow temperature window: 180 K ≤𝑇 ≤200 K. However, we have used a gradual transition \nfrom yellow to grey background in rest of the graphs where the changes in the physical parameters \nare less significant in the temperature window: 180 K ≤𝑇 ≤200 K. \n \nNext, we discuss  the additional voltage contribution s due to the magnetic proximity  effect \n(MPE) -induced anomalous Nernst effect (ANE) as well as MPE –induced LSSE in the Pt layer. \nThe MPE  leads to  a magnetic moment in a few atomic layers of Pt close to the TmIG/Pt \ninterface.63,64 In the presence of a vertical temperature gradient, a tra nsverse voltage is generated \nin the proximitized Pt layer  due to ANE which adds to the LSSE voltage.  Furthermore, due to the \ntemperature gradient, spin currents are generated inside the magnetized Pt layer, which induces an \nadditional IP charge current at the proximitized Pt/nonmagnetic Pt interface via the ISHE and \ntherefore contributes to the LSSE signa l.65 In an earlier study, Bougiatioti et al.,63 showed that the \nMPE -induced ANE in the proximitized Pt layer is only significant for a conducting FM/Pt bilayer \nbut negligible for semiconducting FM/Pt bilayers and becomes zero for insulating FM/Pt bilayers. \nSince TmIG is insulating, the contribution of the MPE -induced ANE in the proximitized Pt layer \ntowards the total LSSE signal can be neglected throughout the measured temperature range66. \nFurthermore , since  the LSSE voltage decreases  with decreasing thickness of the magnetic layer ,16 \nand the thickness of the proximitized Pt layer is very small, the MPE -induced LSSE contribution   \n16 \n due to the proximitized Pt layer  can also be neglected66. Therefore, the total voltage measured \nacross the TmIG/Pt bilayers is considered to be solely contributed by the intrinsic LSSE of the \nTmIG films.  \n \n2. 3. Analysis of the Thickness Dependent Longitudinal Spin Seebeck Effect  \nTo ascertain the origin of the decrease in 𝑉𝐿𝑆𝑆𝐸 below 180 -200 K in our TmIG  films, it is \nessential to determine the temperature evolution of 〈𝜉〉 which signifies the critical length -scale for \nthe thermally -generated magnons of a magnetic thin film16,18,34. For an effective determination of \nthe temperature dependence of 〈𝜉〉, the contributions of the the thermal resistances of the substrate \nand the grease layers as well as the interfacial thermal resistances need to be considered.67 To \nquantify the temperature evolution of 〈𝜉〉 for our TmIG/Pt bilayer films, we have employed a \nmodel proposed by Jimenez -Cavero et al.,68 according to which t he total temperature difference \n(Δ𝑇) across the GSGG/TmIG/Pt heterostructure can be expressed as a linear combination of \ntemperature drops in the Pt layer, at the TmIG/Pt interface, in the TmIG  layer, at the GSGG/TmIG  \ninterface and across the GSGG substrate  as well as in the N -grease layers (thickness ≈ 1 m) on \nboth sides of the GSGG/TmIG/Pt heterostructures as ,68 ∆𝑇= ∆𝑇𝑃𝑡+∆𝑇Pt\nTmIG+∆𝑇TmIG+\n ∆𝑇TmIG\nGSGG+∆𝑇GSGG+2.∆𝑇N−Grease  (see Fig. 5 (a)). Assuming negligible drops in ∆𝑇 in the Pt layer \nand at the GSGG/TmIG  and Pt/TmIG  interface ,68,69 the total temperature difference can be \napproximately written as, ∆𝑇= ∆𝑇Pt\nTmIG+∆𝑇TmIG+∆𝑇GSGG+2.∆𝑇𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 . Considering these \ncontributions, the temperature drops in the TmIG layer and at the TmIG/Pt interface can be written \nas,68,69 ∆𝑇TmIG = 𝛥𝑇\n [1+𝜅𝑇𝑚𝐼𝐺\n𝑡𝑇𝑚𝐼𝐺(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒\n𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 + 𝑡𝐺𝑆𝐺𝐺\n𝜅𝐺𝑆𝐺𝐺)] and ∆𝑇Pt\nTmIG= [(𝜅𝐺𝑆𝐺𝐺𝜅𝑇𝑚𝐼𝐺)𝑅𝑖𝑛𝑡\n(𝜅𝑇𝑚𝐼𝐺𝑡𝐺𝑆𝐺𝐺+𝜅𝐺𝑆𝐺𝐺𝑡𝑇𝑚𝐼𝐺)]𝛥𝑇, \nrespectively. The bulk (𝑉𝐿𝑆𝑆𝐸𝑏)and interfacial (𝑉𝐿𝑆𝑆𝐸𝑖) contributions to the LSSE voltage can then   \n17 \n be expressed as, 𝑉𝐿𝑆𝑆𝐸𝑏=𝑆𝐿𝑆𝑆𝐸𝑏.∆𝑇TmIG.𝐿𝑦=\n[(𝐴\n𝑡𝑇𝑚𝐼𝐺){cosh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)−1\nsinh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)}]{𝛥𝑇\n[1+𝜅𝑇𝑚𝐼𝐺\n𝑡𝑇𝑚𝐼𝐺(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒\n𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 + 𝑡𝐺𝑆𝐺𝐺\n𝜅𝐺𝑆𝐺𝐺)]}𝐿𝑦 and 𝑉𝐿𝑆𝑆𝐸𝑖=𝑆𝐿𝑆𝑆𝐸𝑖.∆𝑇Pt\nTmIG.𝐿𝑦=\n𝑆𝐿𝑆𝑆𝐸𝑖.[(𝜅𝐺𝑆𝐺𝐺𝜅𝑇𝑚𝐼𝐺)𝑅𝑖𝑛𝑡𝛥𝑇\n(𝜅𝑇𝑚𝐼𝐺𝑡𝐺𝑆𝐺𝐺+𝜅𝐺𝑆𝐺𝐺𝑡𝑇𝑚𝐼𝐺)]𝐿𝑦, respectively. Here,  𝑆𝐿𝑆𝑆𝐸𝑏=[(𝐴\n𝑡𝑇𝑚𝐼𝐺){cosh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)−1\nsinh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)}] and 𝑆𝑖𝑛𝑡 \ndenote  the bulk and interfacial LSSE coefficient s for TmIG and TmIG/Pt interface , respectively,  \n𝑡𝑇𝑚𝐼𝐺(𝑡𝐺𝑆𝐺𝐺) is the thickness of TmIG film (GSGG substrate), 𝜅𝑇𝑚𝐼𝐺  and 𝜅𝐺𝑆𝐺𝐺 are the thermal \nconductivity  of TmIG and GSGG respectively, 𝑡N−Grease  and 𝜅N−Grease  are the thickness and \nthermal conductivity of the N -grease layers, 𝑅𝑖𝑛𝑡 is the interfacial thermal -resistance at the \nTmIG/Pt interface  and 𝐴 is a constant .68. The approximate values of 𝜅N−Grease , 𝜅𝑇𝑚𝐼𝐺  and 𝜅𝐺𝑆𝐺𝐺 \nat different temperatures are obtained from the literature70–75. Note that , we have ignored the \ninterfacial thermal resistances between the N -grease and the hot/cold plates as well as between the \nsample and N -grease layers .76 Threfore , the total LSSE voltage across GSGG/TmIG/Pt can be \nexpressed as,68 \n𝑉𝐿𝑆𝑆𝐸(𝑡𝑇𝑚𝐼𝐺)= 𝑉𝐿𝑆𝑆𝐸𝑖(𝑡𝑇𝑚𝐼𝐺)+𝑉𝐿𝑆𝑆𝐸𝑏(𝑡𝑇𝑚𝐼𝐺)= [𝑆𝑖𝑛𝑡{(𝜅𝐺𝑆𝐺𝐺𝜅𝑇𝑚𝐼𝐺)𝑅𝑖𝑛𝑡\n(𝜅𝑇𝑚𝐼𝐺𝑡𝐺𝑆𝐺𝐺+𝜅𝐺𝑆𝐺𝐺𝑡𝑇𝑚𝐼𝐺)}𝐿𝑦𝛥𝑇+\n[(𝐴\n𝑡𝑇𝑚𝐼𝐺){cosh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)−1\nsinh(𝑡𝑇𝑚𝐼𝐺\n〈𝜉〉)}]{𝛥𝑇\n[1+𝜅𝑇𝑚𝐼𝐺\n𝑡𝑇𝑚𝐼𝐺(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒\n𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 + 𝑡𝐺𝑆𝐺𝐺\n𝜅𝐺𝑆𝐺𝐺)]}𝐿𝑦]     (2) \n \nIn Fig. 5 (b), we demonstrate the 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops at T = 295 K for the \nGSGG/TmIG/Pt films with different 𝑡𝑇𝑚𝐼𝐺  in the range 28 nm≤𝑡≤236 nm. Clearly, \n|𝑉𝐼𝑆𝐻𝐸(𝜇0𝐻𝑠𝑎𝑡)| decreases significantly with decreasing 𝑡𝑇𝑚𝐼𝐺 . Therefore, we fitted the thickness \ndependent LSSE voltage at different temperatures with Eqn. 2  to evaluate the temperature \ndependence of 〈𝜉〉 for our GSGG/TmIG/Pt films. It has recently been shown77 that the 𝑀𝑆 also   \n18 \n needs be considered to evaluate 〈𝜉〉 from the LSSE voltage by normalizing the LSSE voltage by \n𝑀𝑆. In Fig. 5 (c), we show the thickness -dependence of the background -corrected modified LSSE \nvoltage,𝑉𝐿𝑆𝑆𝐸(𝑡𝑇𝑚𝐼𝐺)\n∆𝑇.𝑀𝑆, at selected temperatures fitted to Eqn. 2 . From the fits, we obtained 〈𝜉〉 = 62 \n± 5 nm  for the TmIG film at 295 K, which is smaller than that of YIG thin films  grown by PLD  \n(90–140 nm)16, but higher than that for GdIG  thin films (45±8 nm)12.  \n \nFigure 5. Thickness D epende nt LSSE and Magnon Propagation L ength in \nGSGG/TmIG( t)/Pt(5  nm) films.  (a) Schematic illustration of heat flow across the \nGSGG/TmIG( t)/Pt(5 nm) films.  (b) The 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops for GSGG/TmIG( t)/Pt films \nwith different thicknesses at T = 295 K for Δ𝑇 = +10 K . (c) The thickness dependence of the \nnormalized background corrected LSSE voltage,  𝑉𝐿𝑆𝑆𝐸(𝑡)Δ𝑇.𝑀𝑆⁄  at three selected temperatures \nT = 295, 200 , and 140 K fitted with Eqn. (2) . (d) The temperature dependence of the magnon \npropagation length, 〈𝜉〉 obtained from the fits.  \n  \n19 \n Fig. 5 (d) demonstrates the T-dependence of 〈𝜉〉 obtained from the fit of 𝑉𝐿𝑆𝑆𝐸(𝑡𝑇𝑚𝐼𝐺)\n∆𝑇.𝑀𝑆 for \nGSGG/TmIG( t)/Pt(5nm)  films . Interestingly,  〈𝜉〉 decreases gradually with decreasing temperature \nand shows a comparatively faster decrease at low temperatures, especially below 200 K. Our \nobservation is strikingly different than that reported by Guo et al.34 for YIG/Pt bilayers. From the \nYIG thickness dependence of the local LSSE measurements in YIG/Pt, they determined the \ntemperature dependence of 〈𝜉〉 and found a scaling behavior of 〈𝜉〉 ∝ 𝑇−1.34 However, by \nemploying nonlocal measurement geometries, Cornelissen et al., demonstrated that 〈𝜉〉 (and hence, \nthe magnon diffusion length) for YIG/Pt decreases with decreasing temperature over a broad \ntemperature range35, similar to what we observed in our TmIG/Pt bilayers. Gomez -Perez et al.,19 \nalso observed similar behavior of the magnon diffusion length in YIG/Pt. However, none of these \nstudies indicated significant change in 〈𝜉〉 at low temperatures . Therefore, the observed \ntemperature evolution of 〈𝜉〉 presented in this study is intrinsic to TmIG. To rule out possible \neffects of strain o n 𝑉𝐿𝑆𝑆𝐸(𝑇), we performed LSSE measurements on TmIG films grown on \ndifferent substrates  (see Supplementary  Figures 5 and 6). It is evident that 𝑀𝑆(𝑇) and 𝑉𝐿𝑆𝑆𝐸(𝑇) \nfor the Gd3Ga5O12(GGG) /TmIG(44  nm)/Pt (5 nm) and \n(Gd 2.6Ca0.4)(Ga 4.1Mg 0.25Zr0.65)O12(sGGG )/TmIG(40  nm)/Pt (5 nm) films (see Supplementary \nFigure 7) exhibit the same trend as GSGG/TmIG(46  nm)/Pt(5  nm). More specifically , both  \n𝑉𝐿𝑆𝑆𝐸(𝑇) and 𝑀𝑆(𝑇) drop be low 180 -200 K for all the TmIG films independent  of substrate choice.  \n \nTo interpret  the decrease in 〈𝜉〉 at lo w temperatures , we recall that 〈𝜉〉 of a magnetic \nmaterial with lattice constant 𝑎0 (considering simple cubic structure) is related to the Gilbert \ndamping parameter ( 𝛼), the effective magnetic anisotropy constant ( 𝐾𝑒𝑓𝑓), and the strength of the   \n20 \n Heisenberg exchange interaction between nearest neighbors ( 𝐽𝑒𝑥) through the relation17,18 〈𝜉〉=\n 𝑎0\n2𝛼.√𝐽𝑒𝑥\n2𝐾𝑒𝑓𝑓. As discussed before,  𝐾𝑒𝑓𝑓=𝐾𝑚𝑒−1\n2 𝜇0𝑀𝑆2−𝐾1\n12. Therefore, we can express 〈𝜉〉 as, \n〈𝜉〉= 𝑎0\n2𝛼.√𝐽𝑒𝑥\n2(𝐾𝑚𝑒−𝐾1\n12−1\n2 𝜇0𝑀𝑆2)                                 (3) \nEqn. 3 indicates that  (i) 〈𝜉〉∝ (1\n𝛼), and (ii) a decrease in 𝑀𝑆 also suppresses 〈𝜉〉. Since the \neffective anisotropy field, 𝐻𝐾𝑒𝑓𝑓 ∝𝐾𝑒𝑓𝑓, Eqn. 3 can be alternatively written as 〈𝜉〉=𝑎0\n2𝛼.√𝐽𝑒𝑥\n2𝐾𝑒𝑓𝑓 ∝\n1\n𝛼.(𝐻𝐾𝑒𝑓𝑓)1/2  , which indicates  that 〈𝜉〉 is inver sely proportional to the square -root of 𝐻𝐾𝑒𝑓𝑓. This \nimplie s that  the temperature evolution of  〈𝜉〉 is intrinsically dependent on both the physical \nquantities:  𝛼 and 𝐻𝐾𝑒𝑓𝑓. To determine the roles of 𝛼 and 𝐻𝐾𝑒𝑓𝑓in the temperature evolution of 〈𝜉〉, \nwe have performed radio frequency (RF) transverse susceptibility (TS) and broadband \nferromagnetic resonance (FMR) measurements, respectively on the TmIG films, which have been \ndiscussed in the following sections.  \n \n2. 4. Radio Frequency Transverse Susceptibility and Magnetic Anisotropy  \nRF TS measurements were performed  to determine the  temperature evolution of 𝐻𝐾𝑒𝑓𝑓 in \nthe TmIG films. The  magnetic field dependence ( 𝐻𝐷𝐶) of TS, 𝜒𝑇(𝐻𝐷𝐶), is known to exhibit \npeaks/cusps at the effective anisotropy fields, ±𝐻𝐾𝑒𝑓𝑓.78,79 The schematic illustration of our TS \nmeasurement configuration is shown in Fig. 6 (a). T he RF magnetic field, HRF is parallel to the film \nsurface and 𝐻𝐷𝐶 points perpendicular  to it. All the TS data in this paper are presented as the \nrelative change in  𝜒𝑇(𝐻𝐷𝐶), which we define as ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶)=𝜒𝑇(𝐻𝐷𝐶)−𝜒𝑇(𝐻𝐷𝐶=𝐻𝐷𝐶𝑠𝑎𝑡)\n𝜒𝑇(𝐻𝐷𝐶𝑠𝑎𝑡), where \n𝜒𝑇(𝐻𝐷𝐶= 𝐻𝐷𝐶𝑠𝑎𝑡) is the value of 𝜒𝑇(𝐻𝐷𝐶) at the saturation field (  𝐻𝐷𝐶𝑠𝑎𝑡). Bipolar field -scans of   \n21 \n ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶) for the GSGG/TmIG( 236 nm )/Pt film at 295 and 100 K are shown in Fig. 6 (b), which \nclearly indicates an increase in 𝐻𝐾𝑒𝑓𝑓 at low -T.  \n \nFigure 6. RF Transverse Susceptibility and Magnetic Anisotropy in GSGG/TmIG( t)/Pt(5  \nnm) films. (a)  The schematic illustration of our RF transverse susceptibility measurement. (b) \nComparison of the bipolar field scans ( +𝐻𝐷𝐶𝑚𝑎𝑥→−𝐻𝐷𝐶𝑚𝑎𝑥→+𝐻𝐷𝐶𝑚𝑎𝑥) of transverse susceptibility at \nT = 295 and 100 K for the GSGG/TmIG( 236 nm)/Pt film measured with  configuration 𝐻𝐷𝐶⊥\nfilm surface  (IP easy axis) . (c) Fitting of our ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶) data for the GSGG/TmIG( 236nm )/Pt film \nat 295 K with the Eqn. 4 . (d) Temperature dependence of the effective anisotropy field ( 𝐻𝐾𝑒𝑓𝑓) for \nthe GSGG/TmIG(236  nm)/Pt(5  nm) film  obtained from the transverse susceptibility (TS) \nmeasurements on the left y-scale and corresponding 𝑉𝐿𝑆𝑆𝐸(𝑇) for the same film on the right y-scale.  \n \n  \n22 \n For an accurate determination of 𝐻𝐾𝑒𝑓𝑓from the field dependent TS curves, we fitted the \nline shapes for the TS curves with the following expression,79,80  \n∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶)= ∆𝜒𝑆𝑦𝑚(∆𝐻\n2)2\n(𝐻𝐷𝐶−𝐻𝐾𝑒𝑓𝑓)2\n+(∆𝐻\n2)2+∆𝜒𝐴𝑠𝑦𝑚∆𝐻\n2(𝐻𝐷𝐶−𝐻𝐾𝑒𝑓𝑓)\n(𝐻𝐷𝐶 −𝐻𝐾𝑒𝑓𝑓)2\n+(∆𝐻\n2)2+∆𝜒0    (4) \nwhere, ∆𝐻 is the linewidth of the ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶) spectrum,  ∆𝜒𝑆𝑦𝑚 and ∆𝜒𝐴𝑠𝑦𝑚  are the coefficients of \nsymmetric and antisymmetric Lorentzian functions and ∆𝜒0 is the constant offset parameter. Fig. \n6(c) shows the fitting of our ∆𝜒𝑇\n𝜒𝑇(𝐻𝐷𝐶) data for the GSGG/TmIG( 236 nm )/Pt film at 295 K with \nthe Eqn. 4 . As shown on the left y-scale of Fig. 6 (d), 𝐻𝐾𝑒𝑓𝑓(𝑇) increases throughout the measured \ntemperature range but the increase in 𝐻𝐾𝑒𝑓𝑓 is comparatively faster below the temperature range: \n180-200 K, which coincides with the remarkable drop in 𝑉𝐿𝑆𝑆𝐸. Similar behavior was also \nobserved for other film thicknesses (see Supplementary Figure 8 ). \n \nA significant increase  in magnetocrystalline anisotropy at low -T has been  reported in \nvarious REIGs, which was interpreted  in the framework of the single -ion anisotropy model \nconsidering the collective influence of the crystal and exchange fields of the REIG  on the energy \nlevels of the individual magnetic ions81. Typically,  𝐾1 increases by ≈ 80 -100% between 300 and \n150 K in most of the REIGs .81 Furthermore,  𝜆111 for TmIG increases from −5.2 ×10−6 at 300 K \nto −17.4 ×10−6 at 150 K which gives rise to enhanced contribution of  𝐾𝑚𝑒 towards 𝐾𝑒𝑓𝑓 in \nTmIG films at low temperatures.82,83 Shumate Jr. et al.,84 observed a rapid increase in 𝐻𝐾𝑒𝑓𝑓 and \ncoercive field at low temperatures in mixed REIGs. An increase in 𝐻𝐾𝑒𝑓𝑓and a corresponding \ndecrease in 𝑉𝐿𝑆𝑆𝐸 below 175 K was also observed in YIG/Pt13, which was attributed  to the single -\nion anisotropy of Fe2+ ions85. To gain knowledge on the oxidation state of Fe in our TmIG films,   \n23 \n electron energy loss spectroscopy (EELS) was conducted during the cross -sectional TEM study \ndescribed earlier. Fig. 2 (d) shows two EELS spectra, recorded at the Fe L3 and L2 edges, and at \npositions close to the film -substrate and the film -Pt interface. The spectra are fitted following86,87, \nshown by colored lines, using a Gauß ian profile and a combination of a power -law background \nand a double -step function (arctangent) with a fixed step -ratio. Fig. 2 (e) shows the extracted \nthickness -dependent Fe L3 peak position alongside the corresponding FWHM. While an exact \nquantification of the Fe oxidation state distribution using the EELS Fe L3 peak position or L3/L2 \nwhite -line ratio is challenging, the presence of different oxidation states can be  indicated \nqualitatively by a shift in the peak position  because  Fe2+ ions contribute at slightly lower energies \ncompared to Fe3+ ions86–89. However, in our measured spectra, a constant peak position at about \n710.1 eV and a constant FWHM of about 2.3 eV across the whole film thickness is observed. Our \nobservation strongly hints at the presence of only one Fe oxidation state, namely the Fe3+ ion and \nhence, we can rule out the contribution of single ion anisotropy of Fe2+ ions towards the increased \nmagnetic anisotropy. This is also in agreement with recent studies on Tb-rich TbIG thin films58,90 \nwhich reveal very low Fe2+ ion concentrations . Therefore, the increase in 𝐻𝐾𝑒𝑓𝑓 below 200  K in the \nTmIG films may arise from single -ion anisotrop ies of the Tm3+ and Fe3+ ions81 as well as  from the \nenhanced contributions of 𝐾1 and 𝐾𝑚𝑒 towards 𝐾𝑒𝑓𝑓 at low temperatures81–83.  \n \n2. 5. Magnetization Dynamics and Broadband Ferromagnetic Resonance  \nNext, we examine the temperature evolution of 𝛼 and its influence on 〈𝜉〉 through \nbroadband IP FMR measurements. Fig. 7 (a) shows the field -derivative of the microwave (MW) \npower absorption spectra (𝑑𝑃\n𝑑𝐻) as a function of the IP DC magnetic field for a fixed frequency f  = \n12 GHz at selected temperatures for the GSGG/ TmIG( 236nm ) film . As temperature decreases, the   \n24 \n (𝑑𝑃\n𝑑𝐻) lineshape noticeably broadens and the resonance field 𝐻𝑟𝑒𝑠 shifts to higher field values. The \nlinewidth of the (𝑑𝑃\n𝑑𝐻) lineshape becomes so broad at low temperatures that we were unable to \ndetect the FMR signal below 160 K. We observed the same behavior for the GSGG/TmIG( 236 \nnm)/Pt(5 nm)  film, as shown in the Supplementary Figure 9 . Fig. 7 (b) shows the (𝑑𝑃\n𝑑𝐻) lineshapes \nfor the GSGG/TmIG( 236 nm ) film for different frequencies in the range 6 GHz ≤𝑓 ≤20 GHz \nat 295K fitted with a linear combination of symmetric and antisymmetric Lorentzian function \nderivatives as,91 \n   𝑑𝑃\n𝑑𝐻= 𝑃𝑆𝑦𝑚∆𝐻\n2(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)\n[(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻\n2)2\n]2+𝑃𝐴𝑠𝑦𝑚(∆𝐻\n2)2\n−(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2\n[(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻\n2)2\n]2+𝑃0               (5) \nwhere, 𝐻𝑟𝑒𝑠 is the resonance field, ∆𝐻 is the linewidth of the 𝑑𝑃\n𝑑𝐻 lineshapes, 𝑃𝑆𝑦𝑚 and \n𝑃𝐴𝑠𝑦𝑚  are the coefficients of the symmetric and antisymmetric Lorentzian derivatives, respectively, \nand 𝑃0 is a constant offset parameter.  The fitted curves are shown by solid lines in Fig. 7 (b). Using \nthe values of 𝐻𝑟𝑒𝑠 obtained from the fitting of the 𝑑𝑃\n𝑑𝐻 lineshapes,  we fitted the f-𝐻𝑟𝑒𝑠 curves at \ndifferent temperatures using the Kittel equation for magnetic thin films with IP magnetic field,92 \nwhich is expressed as 𝑓= 𝛾𝜇0\n2𝜋√𝐻𝑟𝑒𝑠(𝐻𝑟𝑒𝑠+𝑀𝑒𝑓𝑓), where  𝑀𝑒𝑓𝑓 is the effective magnetization, \n𝛾\n2𝜋= 𝑔𝑒𝑓𝑓 𝜇𝐵\nℏ is the gyromagnetic ratio, 𝜇𝐵 is the Bohr magneton, 𝑔𝑒𝑓𝑓 is the effective Landé g-\nfactor, and ℏ is the reduced Planck’s constant. Fig. 7 (d) demonstrates the fitting of the f-𝐻𝑟𝑒𝑠 \ncurves at T = 295, 200, and 160 K. We found that 𝑔𝑒𝑓𝑓 = 1.642 ± 0.002 at T = 295 K for our \nGSGG/TmIG( 236 nm ) film, which is significantly lower than that of the free electron value ( 𝑔𝑒𝑓𝑓 \n= 2.002), but close to the bulk TmIG value ( 𝑔𝑒𝑓𝑓 = 1.63)93 as well as that for TmIG thin films \n(𝑔𝑒𝑓𝑓 ≈ 1.57)6,94. Furthermore, as shown in Fig. 7( e), 𝑔𝑒𝑓𝑓 for our GSGG/TmIG( 236 nm ) film    \n25 \n decreases gradually with decreasing temperature . We observed similar behavior of 𝑔𝑒𝑓𝑓 for the \nGSGG/TmIG( 236 nm )/Pt(5 nm) film, and well as for other TmIG film thicknesses (see \nSupplementary Figures 9 and 10 ). \n \nFigure 7. Broadband Ferromagnetic Resonance.  (a) The field derivative of microwave (MW) \npower absorption spectra ( 𝑑𝑃\n𝑑𝐻 line shapes) for the GSGG/TmIG( 236 nm ) film at a fixed \nfrequency ( f = 12 GHz) in the range 160 K ≤ T ≤ 295 K . (b) 𝑑𝑃\n𝑑𝐻 line shapes  at different frequencies \nbetween f = 6 - 20 GHz fitted with the linear combination of symmetric and anti -symmetric \nLorentzian function derivatives for the GSGG/TmIG( 236 nm ) film  at T = 295 K . (c) Frequency \ndependence of linewidth,  ∆𝐻 at different temperatures for the GSGG/TmIG( 236 nm ) film with \nlinear fit. (d) The f-𝐻𝑟𝑒𝑠 curves at T = 295, 200, and 160 K along with Kittel fits. (e) Temperature \ndependence of the Gilbert damping parameter,  𝛼𝑇𝑚𝐼𝐺 , the inhomogeneous broadening, ∆𝐻0 and \nthe effective Landé g-factor for the GSGG/TmIG( 236 nm) film.  \n \n  \n26 \n Finally, to quantify the temperature dependence of the Gilbert damping parameter  (𝛼𝑇𝑚𝐼𝐺 ), \nwe fitted the ∆𝐻-f curves at different temperatures using the expression ,95 ∆𝐻=∆𝐻0+4𝜋𝛼\n𝛾𝜇0𝑓, \nwhere  ∆𝐻0 is the  frequency -independent contribution to the linewidth, known as the \ninhomogeneous  broadening  linewidth . From the fits ( see Fig. 7(c)), we obtained   ����𝑇𝑚𝐼𝐺 = 0.0103 \n± 0.002 at 295  K for  our GSGG/TmIG( 236 nm) film which is close to the previously reported \nvalues of 𝛼 (≈ 0.0132 -0.0146) for TmIG films6,96. Most importantly,  𝛼𝑇𝑚𝐼𝐺  increases gradually \nwith decreasing temperature but shows a comparatively faster  increase at low temperatures, \nespecially below ≈ 200 K (Fig. 7(e)). A similar increase in 𝛼 at low-T has also been observed in \nGSGG/TmIG( 236 nm)/Pt(5 nm), GSGG /TmIG( 46 nm)/Pt(5 nm) and GGG/TmIG( 44 nm)/Pt(5 \nnm) films  (see Supplementary  Figures 9 and 10), indicating that this behav ior is independent of \nTmIG film thickness and substrate choice.  In compensated ferrimagnetic insulators, e.g., GdIG, 𝛼 \nincreases drastically  close to the magnetic compensation temperature32. However, most of the  \nearlier reports indicate that TmIG films do not show magnetic compensation in the temperature \nrange between 1.5 and 300 K.47,48 Since our TmIG  films also do not show magnetic compensation \nin the measured temperature range, the increased value of  𝛼𝑇𝑚𝐼𝐺  at low temperatures in our TmIG \nfilms has a different origin . Sizeable increase s in 𝛼 and ∆𝐻 at low  temperatures  were also  observed \nin YIG and different  REIGs92,97 –99 including TmIG46, which was primarily  attributed to Fe2+ and/or \nRE3+ impurity relaxation mechanisms. However, our EELS study  confirms the absence of Fe2+ \nions, and therefore, we can rule out the possibility of Fe2+ impurity relaxation in our TmIG films.  \nTherefore, t he increased damping at low temperatures in our TmIG films may be associated with \nenhanced magnon scattering by defects ,100–104 and slowly relaxing Tm3+ ions92,105. It is known that \nthe contribution of slowly relaxing RE impurity ions towards damping is proportional to the orbital \nmoment  (L) of the RE3+ ions105–107, suggesting  that this mechanism applies to Tm3+ (L = 5).    \n27 \n 2. 6. Correlating Magnon Propagation Length with Magnetic Anisotropy and Gilbert \nDamping   \nIn the previous sections, we have demonstrated that both 𝐻𝐾𝑒𝑓𝑓and  𝛼𝑇𝑚𝐼𝐺  for our TmIG \nfilms show  clear increases at low temperatures, especially below 200 K.  It is known that the \nmagnon energy -gap (ℏ𝜔𝑀) is related to 𝐾𝑒𝑓𝑓 through the expression: ℏ𝜔𝑀∝2𝐾𝑒𝑓𝑓17,18. \nTherefore, an increase in  𝐻𝐾𝑒𝑓𝑓 (and hence, 𝐾𝑒𝑓𝑓) below 200 K enhances ℏ𝜔𝑀 giving rise to only \nhigh-frequency  magnon propagation with shorter 〈𝜉〉. Since only the subthermal magnons, i.e., the \nlow frequency  magnons are primarily responsible for the long -range thermo -spin transport and \ncontributes towards LSSE19–21, the 𝑉𝐿𝑆𝑆𝐸 signal also decreases below 200 K in our TmIG films12,13. \nThis also explains the noticeable decrease in 〈𝜉〉 below 200 K, as the maximum value of the \nfrequency -dependent propagation length is 〈𝜉〉𝑚𝑎𝑥∝1\n√ℏ𝜔𝑀𝑚𝑖𝑛 , where ℏ𝜔𝑀𝑚𝑖𝑛 is the  minimum value \nof ℏ𝜔𝑀, and ℏ𝜔𝑀𝑚𝑖𝑛 ∝2𝐾𝑒𝑓𝑓.17 Therefore, according to the expression17,18 〈𝜉〉=𝑎0\n2𝛼.√𝐽𝑒𝑥\n2𝐾𝑒𝑓𝑓 ∝\n1\n𝛼.(𝐻𝐾𝑒𝑓𝑓)1/2 , the observed decrease in 〈𝜉〉 and hence, the 𝑉𝐿𝑆𝑆𝐸 signal at low temperatures, \nespecially below 200 K in our TmIG films has contributions from the temperature evolutions of \nboth 𝐻𝐾𝑒𝑓𝑓and  𝛼𝑇𝑚𝐼𝐺 . The roles of magnetic anisotropy and damping in LSSE in different REIG -\nbased MI/HM bilayers have been explored by different groups12,13,32,33. All these studies indicated \nthat the LSSE signal strength varies inversely with both magnetic anisotropy and damping. In this \nmanuscript, we have not only highlighted the roles of the temperature evolutions of both magnetic \nanisotropy and damping in cont rolling the temperature dependent LSSE effect in TmIG/Pt bilayers, \nbut also attempted to establish possible correlations between 〈𝜉〉, 𝐻𝐾𝑒𝑓𝑓and 𝛼. Since 〈𝜉〉 is intrinsic \nto a magnetic film and hence independent of the thickness of the magnetic film19, it is convenient   \n28 \n to directly correlate 〈𝜉〉 with the physical parameters 𝐻𝐾𝑒𝑓𝑓and 𝛼 of individual magnetic films with \ndifferent thicknesses. We display the temperature dependence of 〈𝜉〉 on the left -y scales, and the \ntemperature evolutions of 𝐻𝐾𝑒𝑓𝑓and  𝛼𝑇𝑚𝐼𝐺  are shown on the right y-scales of Figs. 8 (a) and (b), \nrespectively. It is evident that the prominent drop in 〈𝜉〉 below 200 K in the TmIG/Pt bilayers is \nassociated with the noticeable increases in 𝐻𝐾𝑒𝑓𝑓 and 𝛼 that occur within the same temperature \nrange.  \n \nFigure 8. Temperature evolution of magnon propagation length and its correlation with \nmagnetic anisotropy and Gilbert damping:  (a) and (b)  Temperature dependence of 〈𝜉〉 on the \nleft-y scales, and the temperature evolutions of 𝐻𝐾𝑒𝑓𝑓and  𝛼𝑇𝑚𝐼𝐺  are shown on the right y-scales , \nrespectively. 〈𝜉〉 as a function of  (c) √𝐻𝐾𝑒𝑓𝑓 and (d)  𝛼𝑇𝑚𝐼𝐺  for the GSGG/TmIG(236  nm) film \nobtained from the temperature evolutions of 〈𝜉〉, 𝐻𝐾𝑒𝑓𝑓 and  𝛼𝑇𝑚𝐼𝐺 . \n  \n29 \n For a clearer understanding of the direct correlation between 〈𝜉〉 and 𝐻𝐾𝑒𝑓𝑓 in our TmIG/Pt \nbilayer films , we have plotted 〈𝜉〉 as a function of √𝐻𝐾𝑒𝑓𝑓 for the GSGG/TmIG(236  nm)/Pt film in \nFig. 8 (c) obtained from the temperature evolutions of 〈𝜉〉 and 𝐻𝐾𝑒𝑓𝑓. 〈𝜉〉 varies inversely with \n√𝐻𝐾𝑒𝑓𝑓 in the measured temperature range,  which is consistent with the expression 〈𝜉〉 ∝\n1\n𝛼.(𝐻𝐾𝑒𝑓𝑓)1/2. Similarly, we have plotted 〈𝜉〉 as a function of  𝛼𝑇𝑚𝐼𝐺  for the GSGG/TmIG(236  nm)/Pt \nfilm in Fig. 8 (d) obtained from the temperature evolutions of 〈𝜉〉 and  𝛼𝑇𝑚𝐼𝐺 . An inverse \ncorrelation between 〈𝜉〉 and  𝛼𝑇𝑚𝐼𝐺  in the measured temperature range  is also apparent from this \nplot, and hence , in agreement  with the aforementioned theoretical expression. To establish a more \naccurate correlation between the parameters 〈𝜉〉, 𝐻𝐾𝑒𝑓𝑓and 𝛼, one needs to  fix 𝐻𝐾𝑒𝑓𝑓(𝛼), and then \nevaluate 〈𝜉〉 for different values of 𝛼 (𝐻𝐾𝑒𝑓𝑓). It is however challenging  to change 𝐻𝐾𝑒𝑓𝑓 of a \nmagnetic material without varying 𝛼 significantly. Nevertheless, we have observed concurrent \nremarkable drops in the LSSE voltage as well as 〈𝜉〉 below 200 K in our TmIG/Pt bilayers \nregardless of TmIG film thickness and substrate choice and correlated the temperature evolution \nof 〈𝜉〉 with the noticeable increases in 𝐻𝐾𝑒𝑓𝑓 and 𝛼 that occur within the same temperature range. \nIt is important  to note that FMR  probes  only the zone  center  magnons  in the GHz  range  whereas  \nthe subthermal  magnons  which  primarily  contribute  towards  the LSSE  signal  belong  to the THz \nregime34. Therefore,  the behavior  of LSSE  cannot  be determined  by FMR  excited  by GHz -range  \nmicrowaves.  As shown  by Chang  et al.,33 the LSSE  voltage  varies  inversely  with 𝛼. Furthermore,  \nthe correlation  between  〈𝜉〉 and 𝛼 was predicted  theoretically18 but never  shown  experimentally.  \nAs indicated  in this report,  an experimental  demonstration  of the correlation  between  〈𝜉〉, 𝐻𝐾𝑒𝑓𝑓 \nand 𝛼 would  be beneficial  to fabricate  efficient  spincaloritronic  devices  with higher  〈𝜉〉 by tuning    \n30 \n these  fundamental  parameters.  However,  for deeper  understanding  of LSSE  and 〈𝜉〉, their magnon  \nfrequency  dependences  need  to be highlighted . \n \n2. 7. Magnon Frequency Dependence s of the LSSE Voltage and Magnon Propagation Length  \n As discussed  before,  the low energy  subthermal  magnons  with longer  〈𝜉〉 are primarily  \nresponsible  for LSSE.  These  low frequency  magnons  are partially  frozen  out by the application  of \nexternal  magnetic  field because  of increased  magnon  energy  gap due to the Zeeman  effect.20,108 \nTherefore,  〈𝜉〉 and hence  the LSSE  signal  is strongly  suppressed  by the application  of high \nmagnetic  field.20,108 However,  the field induced  suppression  is dependent  on the thickness  of the \nmagnetic  film.34 If the film thickness  is lower  than 〈𝜉〉, the low frequency  subthermal  magnons  \ncannot  recognize  the local  temperature  gradient  and do not participate  in LSSE.  In that case,  only \nhigh frequency  magnons  with shorter  〈𝜉〉 and much  higher  energy  than the Zeeman  energy  \ncontribute  towards  the LSSE  signal  and hence  the field induced  suppression  of LSSE  becomes  \nnegligible.20,108 However,  if the film thickness  is higher  than 〈𝜉〉, most  of the low frequency  \nmagnons  contribute  towards  LSSE  and hence,  the field induced  suppression  becomes  more  \nsignificant.20,108 As we have  observed  in our TmIG  films  that the trend  of temperature  dependent  \nLSSE  signal  is nearly  independent  of the substrate  choice,  the temperature  dependent  〈𝜉〉 is not \nsupposed  to change  significantly  with the substrate  choice .    \n31 \n  \nFigure 9. Magnetic field induced suppression of the LSSE voltage : 𝑉𝐼𝑆𝐻𝐸(𝐻) loops for the \nsGGG/TmIG(40nm)/Pt nm film at T = (a) 295, (b) 200 and (c) 140 K measured up to high \nmagnetic field of 𝜇0𝐻=9 T. 𝑉𝐼𝑆𝐻𝐸(𝐻) loops for the sGGG/TmIG(75nm)/Pt nm film at T = (d) \n295, (e) 200 and (f) 140 K measured up to high magnetic field of 𝜇0𝐻=9 T. 𝑉𝐿𝑆𝑆𝐸(𝑇,𝜇0𝐻=9T) \nand 𝑉𝐿𝑆𝑆𝐸𝑚𝑎𝑥(𝑇)  for (g) 40 nm and (h) 75 nm films. (i) Temperature dependence of 𝛿𝑉𝐿𝑆𝑆𝐸(%) for \n40 nm and 75 nm films.  \n \n Therefore,  (i) to verify  the influence  of thickness  on the field induced  suppression  of the \nLSSE  signal  in TmIG  films  and (ii) to confirm  whether  〈𝜉〉 of the GSGG/TmIG/Pt films obtained \nby analyzing the low field LSSE signal matches closely with that of the sGGG/TmIG/Pt films , we \nperformed  the high field LSSE  measurements  on the sGGG/TmIG/Pt  films  with thicknesses  of 40 \nand 75 nm. Figs. 9 (a)-(c) demonstrate the  𝑉𝐼𝑆𝐻𝐸(𝐻) loops for the 40 nm film at T = 295, 200 and \n140 K measured up to high magnetic field of 𝜇0𝐻=9 T. It can be seen that the LSSE  signal  for \nthe 40 nm film does not show  prominent  suppression  at 9 T at 295 K . However,  as temperature  \ndecreases  below  200 K, the suppression  of the LSSE  signal  becomes  noticeable.  On the other  hand,  \n  \n32 \n as seen in Figs. 9 (d)-(f), the LSSE  signal  for the 75 nm film shows  significant  suppression  even  at \n295 K and the suppression  of LSSE  signal  enhances  with decreasing  temperature.  The more  intense  \nsuppression  of the LSSE  signal  in the 75 nm film compared  to the 40 nm film at all temperatures  \nbetween  295 and 140 K is also evident  from  𝑉𝐿𝑆𝑆𝐸(𝑇) for these two films shown in Figs. 9 (g) and \n(h). We have also estimated the percentage change in 𝑉𝐿𝑆𝑆𝐸 by the application of 9 T magnetic \nfield, which we define as, 𝛿𝑉𝐿𝑆𝑆𝐸(%)=[𝑉𝐿𝑆𝑆𝐸(9 T)−𝑉𝐿𝑆𝑆𝐸𝑚𝑎𝑥\n𝑉𝐿𝑆𝑆𝐸𝑚𝑎𝑥]×100% , where, 𝑉𝐿𝑆𝑆𝐸(9 T) is the \nabsolute value of 𝑉𝐿𝑆𝑆𝐸 at 9 T magnetic field and 𝑉𝐿𝑆𝑆𝐸𝑚𝑎𝑥 is the value of 𝑉𝐿𝑆𝑆𝐸 at the maximum point \nof the 𝑉𝐼𝑆𝐻𝐸(𝐻) loop. As shown in  Figs. 9 (i), |𝛿𝑉𝐿𝑆𝑆𝐸| for the 75 nm film is nearly 14% at 295 K \nbut increases to ≈ 32% at 140 K. On the other hand, |𝛿𝑉𝐿𝑆𝑆𝐸| for the 40 nm film is negligible at \n295 K but increases to ≈ 7% at 140 K. These  results  indicate  that 〈𝜉〉 for the sGGG/TmIG/Pt  films  \nat 295 K is between  40 and 75 nm, which  is close  to the value  of 〈𝜉〉 obtained  for the \nGSGG/TmIG/Pt  films.  Since  〈𝜉〉 decreases  at low temperatures  and becomes  smaller  than 40 nm \nbelow  150 K, the sGGG/TmIG(40nm)/Pt  film shows  significant  field induced  suppression  of 𝑉𝐿𝑆𝑆𝐸 \nat low temperatures.  Similarly,  since  〈𝜉〉 at low temperatures  is much  smaller  than 75 nm, the field \ninduced  suppression  of 𝑉𝐿𝑆𝑆𝐸 is also large at low temperatures for the 75 nm film. Note  that in case \nof YIG/Pt  films,  the magnetic  field induced  suppression  of the LSSE  signal  diminishes  with \ndecreasing  temperature,34 whereas,  an opposite  trend  has been  observed  in case of TmIG . Such  \nbehavior  can be explained  by different  trends  of the temperature  dependent  〈𝜉〉  in YIG and TmIG.  \nAs explained  by Guo et al.,34 the temperature  induced  enhancement  of 〈𝜉〉 neutralizes  the field \ninduced  suppression  of 〈𝜉〉, and because  of these  two competing  factors,  the field induced  \nsuppression  of the LSSE  voltage  is less prominent  at low temperatures  in YIG/Pt  films.  On the \ncontrary,  the combined  effects  of the temperature  induced  reduction  in 〈𝜉〉 observed  in our   \n33 \n TmIG/Pt  films  and field induced  suppression  of 〈𝜉〉 give rise to stronger  field induced  suppression  \nof the LSSE  voltage  at lower  temperatures.  \n \nFigure 10. Magnon frequency dispersion for TmIG : (a) Magnon frequency dispersion for TmIG \nfor 𝜇0𝐻=0 T and 9 T magnetic fields at T = 295 K. (b)  Comparison  of the magnon  frequency  \ndispersion  for YIG and TmIG  at room  temperature  for 𝜇0𝐻=0 T. Magnon frequency dispersion \nof TmIG at different temperatures for  (c) 𝜇0𝐻=0 T and (d) 9 T magnetic fields . \n \nNext, to have a qualitative understanding of the magnon frequency dependences of the \nLSSE signal and 〈𝜉〉, we have estimated the magnon frequency dispersion for TmIG in the absence \nand in presence of high magnetic field of 9T. According  to the classical  Heisenberg  ferromagnet  \nmodel  for spin waves,  the parabolic  magnon  frequency  dispersion  at low energies  can be expressed  \nas,17,20,109 ℏ𝜔𝑘= 𝑔𝑒𝑓𝑓𝜇𝐵𝐻+𝐷𝑆𝑊.𝑘2𝑎02+𝐸𝑎𝑛𝑖(𝐾𝑒𝑓𝑓), where  the first term represents  the \nZeeman  energy  gap due to the application  of external  magnetic  field,  the second  term is associated  \n  \n34 \n with spin wave  stiffness  (𝐷𝑆𝑊 is the spin wave  stiffness  constant),  and the third  term represents  \nthe contribution  of effective  magnetic  anisotropy  energy,  𝐸𝑎𝑛𝑖(𝐾𝑒𝑓𝑓). Here,  the value  of 𝐷𝑆𝑊 is \ntaken  as that of YIG,  i.e., 𝐷𝑆𝑊𝑎02=4.2 ×10−29 erg.cm2 at room  temperature.108 Using  the \nexpression  𝐾𝑒𝑓𝑓=𝐾𝑠ℎ𝑎𝑝𝑒+𝐾𝑚𝑐 + 𝐾𝑚𝑒, we determined the temperature dependence of \n𝐸𝑎𝑛𝑖(𝐾𝑒𝑓𝑓) for TmIG.  Here, the temperature variation of  𝐾𝑚𝑒 was obtained from the temperature \ndependence of 𝜆111 reported in the literature83 (see Supplementary Figures 8(e) ). Since the shear \nmodulus, 𝑐44 in REIGs is weakly dependent on the rare -earth species, the value of 𝑐44 is taken as \nthat of YIG (76.4 GPa at room temperature).47 The temperature dependence of 𝐾𝑠ℎ𝑎𝑝𝑒 was \nestimated from the temperature variation of 𝑀𝑆 (see Supplementary Figures 8(f) ). We assumed \nconstant value of 𝐾𝑚𝑐=0.058 kJ/m3 throughout the measured temperature range.6 As shown in \nSupplementary Figures 8(f),  𝐾𝑒𝑓𝑓 is positive for 𝑇≤300 K (𝐾𝑒𝑓𝑓=20 kJ/m3) and its absolute \nvalue increases considerably with decreasing temperature. Fig. 10 (a) shows the magnon frequency \ndispersion for TmIG for 𝜇0𝐻=0 T and 9 T magnetic fields at T = 295 K. Clearly,  the high \nmagnetic  field opens  a magnon  energy  gap in the low frequency  regime  (much  smaller  than \nthermal  energy  at room  temperature)  indicating  the suppression  of 〈𝜉〉 and hence  significant  \nreduction  of the LSSE  signal  at high magnetic  fields.  As shown  in Fig. 10 (b), we have  compared  \nthe magnon  frequency  dispersion  for YIG and TmIG  at room  temperature  for 𝜇0𝐻=0 T. It is \nevident  that the opening  of magnon  energy  gap in TmIG  is higher  than in YIG even  in the absence  \nof external  magnetic  field,  which  is mainly  caused  by the effective  magnetic  anisotropy.  Note  that \n𝐾𝑒𝑓𝑓 of TmIG is higher than that of YIG.110 The higher  value  of magnon  energy  gap in TmIG  \ncompared  to YIG thus indicates  the higher  possibility  of freezing  out of the low energy  subthermal  \nmagnons  in TmIG . This,  along  with higher  value  of 𝛼 in TmIG  contributes  to the lower  value  of \n〈𝜉〉 and hence  the LSSE  voltage  in TmIG  compared  to YIG16. Furthermore,  the value  of 𝑔𝑒𝑓𝑓 at   \n35 \n room  temperature  is lower  in TmIG  (≈1.63)93 than in YIG (≈ 2.046)111. Therefore,  for a given  \napplied  magnetic  field strength,  the magnitude  of the magnon  energy  gap due to Zeeman  effect  \nwill be different  in TmIG  than in YIG.  Moreover,  𝛼 in TmIG6,96 is nearly  two orders  of magnitude  \nhigher  than in YIG112. In other  words,  different  values  of 𝐾𝑒𝑓𝑓, 𝑔𝑒𝑓𝑓 and 𝛼 as well as their different \ntemperature dependences give rise to different temperature profiles of 〈𝜉〉 in TmIG and YIG. In \nFigs. 10 (c) and (d), we show the magnon frequency dispersion of TmIG at different temperatures \nfor 𝜇0𝐻=0 T and 9 T magnetic fields, respectively. Clearly, the magnon energy gap increases \nwith decreasing temperature due to enhanced magnetic anisotropy at low temperatures. \nApplication of 9 T magnetic field increases the magnon energy gap further due to Zeeman ef fect. \nThese results help explain  the observed decrease in 〈𝜉〉 and enhanced magnetic field induced \nsuppression of the LSSE signal at low temperatures in TmIG.20,108 \n \nWe believe that our findings will attract the attention of the spintronic community for \nfurther exploration of long-range thermo -spin transport in different REIG based magnetic thin \nfilms and heterostructures for tunable spincaloritronic efficiency by manipulating 𝐻𝐾𝑒𝑓𝑓 and 𝛼. For \nexample, 𝐻𝐾𝑒𝑓𝑓 of the REIG thin films grown on piezoelectric substrates can be modulated by \napplying a gate voltage,113 which can eventually influence 〈𝜉〉 and hence the spincaloritronic \nefficiency.  Therefore, our study also  provides a step towards the development of efficient \nspincaloritronic devices based on voltage controlled LSSE.  \n \n3. CONCLUSION  \nIn summary, we have performed a comprehensive investigation of the temperature \ndependent LSSE, RF transverse susceptibility, and broadband FMR measurements  on TmIG /Pt   \n36 \n bilayers grown on different substrates. The decrease  in the LSSE volta ge below 200 K independent  \nof TmIG film thickness and substrate choice is attribute d to the increases in 𝐻𝐾𝑒𝑓𝑓 and 𝛼 that occur \nwithin the same temperature range. From the TmIG thickness dependence of the LSSE voltage, \nwe determined the temperature dependence of 〈𝜉〉 and highlighted its correlation with the \ntemperature dependent 𝐻𝐾𝑒𝑓𝑓 and 𝛼 in TmIG/Pt  bilayers, which will be beneficial for the \ndevelopment of REIG -based spincaloritronic nanodevices . Furthermore, the enhanced suppression \nof the LSSE voltage by the application of high magnetic field at low temperatures together with \nthe temperature evolution of  magnon frequency dispersion in TmIG estimated from the \ntemperature dependent 𝐾𝑒𝑓𝑓 and 𝛼 support our observation of the decrement of 〈𝜉〉 at low \ntemperatures in the TmIG/Pt bilayers.  \n \n \n \n \n \n \n \n \n \n \n \n \n   \n37 \n 4. METHODS  \nThin film  growth  and structural/morphological characterization : Single -crystalline TmIG thin \nfilms were deposited by pulsed laser deposition (PLD), using two different PLD setups. The thin \nfilms were grown epitaxially on different (111) -oriented substrates, including GGG ( Gd3Ga5O12), \nGSGG ( Gd3Sc2Ga3O12), and sGGG ( (Gd 2.6Ca0.4)(Ga 4.1Mg 0.25Zr0.65)O12). Substrates with (111) \norientation are chosen so that the magnetoelastic anisotropy of the TmIG films favors PMA.  Using \nthe first PLD setup, films with varying thickness between 28 nm and 236 nm were grown on GGG \nand GSGG substrates. A KrF excimer laser with a wavelength of 248 nm , a fluence of 3 -4 J/cm²,  \nand a repetition rate of 2 Hz is used. Before the first deposition, t he TmIG target was preablated \ninside the PLD chamber with more than 104 pulses.  All substrates were annealed for 8 h at 1250°C \nin oxygen atmosphere prior to the film deposition to provide a high substrate surface quality . \nGrowth conditions were selected to  achieve stoichiometric, single -crystalline thin films with a \nsmooth surface of about 0.2 -0.3 nm in root -mean square roughness (RMS).  For all films, the \nsubstrate was heated to 595°C  during the film deposition , monitored by a thermocouple inside the \nsubstrate holder. The TmIG thin films were grown  at a rate of 0.01 − 0.02 nm/s, in the presence of \nan oxygen background atmosphere of 0.05 mbar. After the deposition, the samples were  cooled to \nroom temperature at approximately 5 K/min , maintaining the oxygen atmosphere.  A layer of 5 nm \nPt was deposited at room temperature ex-situ on the garnet films by DC magnetron sputtering \nusing a shadow mask. The TmIG films were annealed at 400°C for 1 h inside the sputter chamb er \nprior to the Pt deposition to avoid surface contamination114.To complement these samples, TmIG \nfilms with thicknesses 75 and 40 nm were grown on sGGG substrates using a second  PLD setup . \nThe laser wavelength was 248 nm at 10 Hz, the fluence 1.3 J/cm2, and the substrate  temperature   \n38 \n was ~750 ˚C with an oxygen pressure of 0.2 mbar. Samples were cooled at 20 K/min in 0.2 mbar \noxygen.  \n \nThe film surface morphology was investigated by atomic force microscopy (AFM), while \nthe structural properties of the thin films were identified by x -ray diffraction (XRD) using \nmonochromatic Cu Kα radiation. The film thickness was evaluated from the Laue oscillations (for \nthe thinne r films) and by spectroscopic ellipsometry. Further, a cross -sectional high resolution \nscanning transmission electron microscopy (HR -STEM) was conducted, using a JEOL NEOARM \nF200 operated at an electron energy of 200 keV. Electron energy loss spectra (EELS) were \nobtained using a GATAN Continuum S EE LS spectrometer. The cross -sectional sample was \nprepared by mechanical dimpling and ion polishing.  Interdiffusion between the TmIG film and \nsubstrate is expected to be limited to a depth of order 1 -3 nm41 and its effects are neglected for the \nfilm thicknesses used in this study.  See Supplementary Figure 1 (e) for energy dispersive X -ray \nspectroscopy (EDX) using transmission  electron microscopy (TEM)  performed on the \nGSGG/TmIG(20 5nm) film.   \n \nTemperature dependent MFM measurements : Temperature dependent MFM measurements were \nperformed on a Hitachi 5300E system. All measurements were done  under high vacuum (P ≤ 10-6 \nTorr). MFM measurements utilized HQ: NSC18/Co -Cr/Al BS tips, which were magnetized out -\nof-plane with respect to the tip surface via a permanent magnet. Films were first magnetized to \ntheir saturation  magnetization by being placed in a 1T static magnetic field, in -plane with the film \nsurface. After that AC demagnetization of the film was implemented before init iating the MFM \nscans. After scans were performed, a parabolic background was subtracted, which arises from the   \n39 \n film not being completely flat on the sample stage. Then, line artifacts were subtracted before \nfinally applying a small Gaussian averaging/sharpening filter over the whole image. Phase \nstandard deviation was determined by fitting a Gaussian to the image p hase distribution and \nextracting the standard deviation from the fit parameters.  \n \nMagnetometry : The magnetic properties of the samples were measured using a superconducting \nquantum interference device - vibrating sample magnetometer (SQUID -VSM) at temperatures \nbetween 10 K and 350 K. A linear background stemming from the paramagnetic substrate was \nthereby subtracted. Due to a trapped remanent field inside the superconducting coils, the measured \nmagnetic field was corrected using a paramagnetic reference sample. Additionally, a polar \nmagneto -optical Kerr effect (MOKE) setup was used to record out -of-plane hysteresis loops at \nroom temperature.  The molecular field coefficient ( MFC ) model was a Python -coded version of \nDionne’s model115 using molecular field coefficients57. \n \nLongitudinal spin Seebeck effect measurements : The longitudinal spin Seebeck effect (LSSE) \nwas measured over a broad temperature window of  120 K ≤ T ≤ 295 K using a custom -built setup \nassembled on a universal PPMS sample puck. During the LSSE measurements, the films were \nsandwiched between two copper blocks, as shown in Fig. 3(a).  The s ame sample geometry was \nused for all films and the distance between the contact leads on the Pt surface were fixed at Ly = 3 \nmm for all films.  A single layer of thin Kapton tape was thermally affixed to the naked surfaces of \nthe top (cold) and bottom (hot) copper blocks. To ensure a good thermal link between the film \nsurface and the Kapton tape (thermally conducting and electrically insulating) attached to the top \nand bottom blocks, cryogenic Apiezon N -grease was used. Additionally, the Kapton tape   \n40 \n electrically insulated the cold (hot) blocks from the top (bottom) surface of the films . The \ntemperatures of both these blocks were controlled individually by two separate temperature \ncontrollers (Scientific Instruments Model no. 9700) to achieve an ultra -stable temperature \ndifference ( ∆𝑇) with [∆𝑇]𝐸𝑟𝑟𝑜𝑟  < ± 2 mK. The top block (cold) was thermally anchored to the  base \nof the  PPMS puck  using two molybdenum screws whereas a 4 -mm-thick Teflon block was \nsandwiched between the puck  base and the hot block (bottom) to maintain a temperature difference \nof ~ 10 K between the hot block and the PPMS base. A resistive chip -heater (PT -100 RTD sensor) \nand a calibrated Si -diode thermometer (DT-621-HR silicon diode sensor) were attached to each of \nthese blocks to efficiently control and sense the temperature. The heaters  and thermometers \nattached to the copper blocks were connected to the temperature controllers in such a manner that \na temperature gradient develops along the + z-direction that generates a temperature difference, ∆𝑇, \nbetween the top (cold) and bottom (hot) copper blocks. For a given temperature gradient, the in -\nplane voltage generated along the y-direction across the Pt layer due to the ISHE ( 𝑉𝐼𝑆𝐻𝐸) was \nrecorded by a Keithley 2182a nanovoltmeter while sweeping an external in -plane DC magnetic \nfield from positive to negative values along the x-direction. The Ohmic contacts for the voltage \nmeasurements were made by electrically anchoring  a pair of  ultra-thin gold wires (25 µm diameter) \nto the Pt layer  by high quality conducting silver paint (SPI Supplies ). \n \nTransverse susceptibility measurements : The temperature evolution of effective magnetic \nanisotropy in the GSGG/TmIG/Pt film was measured by employing a radio frequency (RF) \ntransverse susceptibility (TS) technique using a home -built self -resonant tunnel diode oscillator \n(TDO) circuit with a resonance frequency of 12 MHz and sensitivity of ±10 Hz. A physical \nproperty measurement system (PPMS) was employed as a platform to scan the external DC   \n41 \n magnetic field ( HDC) and temperature. Before the TS measurements, the film was mounted inside \nan inductor coil (L), which is a component of an LC tank circuit. The entire tank circuit was placed \noutside the PPMS except the coil , L, which was positioned at the base of the PPMS sample \nchamber using a multi -purpose PPMS probe insert ed in such a manner that the axial RF magnetic \nfield ( HRF) of amplitude ~ 10 Oe produced inside the coil was always parallel to the film surface, \nbut perpendicular to HDC. For the T mIG with IP easy axis, 𝐻𝐷𝐶⊥film surface , whereas for the \nfilms with OOP easy axis, 𝐻𝐷𝐶∥film surface. When the sample is subject to both HRF and HDC, \nthe dynamic susceptibility of the sample changes which in turn changes the inductance of the coil \nand, hence, the resonance frequency of the LC tank circuit. The relative change in the resonance \nfrequency is proportional to the relative change in the transverse susceptibility of the sample. \nTherefore, TS as a function of HDC was acquired by monitoring the shift in the resonance frequency \nof the TDO -oscillator circuit by employing an Agilent frequency counter . \n \nBroadband f erromagnetic resonance measurements : Broadband ferromagnetic resonance \n(FMR) measurements ( 𝑓 = 6-20 GHz) were performed using a broadband FMR spectrometer \n(NanOscTM Phase -FMR  Spectrometer , Quantum Design Inc., USA) integrated to a Dynacool \nPPMS. The TmIG film was firmly affixed  on the surface of a commercial 200-μm-wide coplanar \nwaveguide (CPW)  (also provided by  NanOscTM Phase -FMR Spectrometer, Quantum Design Inc., \nUSA ) using Kapton tape . The TmIG films were placed faced down on the CPW so that the CPW \ncan efficiently transmit the MW signal from the RF source over a broad f-range. The role of the \nKapton tape is to electrically insulate the films from the CPW. An in-plane  RF mag netic field, 𝐻𝑅𝐹 \nis generated in close vicinity to the CPW. In presence of an appropriate external in-plane  DC \nmagnetic field, 𝐻𝐷𝐶 provided by the superconducting magnet of the PPMS applied along the   \n42 \n direction of the MW current flowing through the CPW(𝐻𝐷𝐶���𝐻𝑅𝐹) and frequency, 𝐻𝑅𝐹 \nresonantly excites the TmIG film. The spectrometer employs lock -in detection and records the \nfield derivative of the power absorbed ( 𝑑𝑃/𝑑𝐻) by the film when it is excited by a  microwave \n(MW)  electromagnetic field generated by injecting a  MW current to the CPW . \n \nACKNOWLEDGEMENTS  \nFinancial support by the US Department of Energy, Office of Basic Energy Sciences, Division of \nMaterials Science and Engineering under Award No. DE -FG02 -07ER46438 at USF and by the \nGerman Research Foundation (DFG) within project No. 318592081AL618/37 -1 at U Augsburg \nare gratefully acknowledged. CR acknowledges support of NSF award DMR 1808190 and \n1954606.  \n \nCONFLICT OF INTEREST  \nThe authors have no conflicts to disclose.  \n \nDATA AVAILABILITY  \nThe data that support the findings of this study are available from the corresponding author upon \nreasonable request.  \n \n    \n43 \n REFERENCES  \n(1) Shirsath, S. E.; Cazorla, C.; Lu, T.; Zhang, L.; Tay, Y. Y.; Lou, X.; Liu, Y.; Li, S.; Wang, \nD. 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Appl. \nPhys.  1970 , 41 (12), 4874 –4881.  \n "    },    {        "title": "1709.10365v1.Non_local_Gilbert_damping_tensor_within_the_torque_torque_correlation_model.pdf",        "content": "Non-local Gilbert damping tensor within the torque-torque correlation model\nDanny Thonig,1,\u0003Yaroslav Kvashnin,1Olle Eriksson,1, 2and Manuel Pereiro1\n1Department of Physics and Astronomy, Material Theory, Uppsala University, SE-75120 Uppsala, Sweden\n2School of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden\n(Dated: July 19, 2018)\nAn essential property of magnetic devices is the relaxation rate in magnetic switching which\ndepends strongly on the damping in the magnetisation dynamics. It was recently measured that\ndamping depends on the magnetic texture and, consequently, is a non-local quantity. The damping\nenters the Landau-Lifshitz-Gilbert equation as the phenomenological Gilbert damping parameter\n\u000b, that does not, in a straight forward formulation, account for non-locality. E\u000borts were spent\nrecently to obtain Gilbert damping from \frst principles for magnons of wave vector q. However,\nto the best of our knowledge, there is no report about real space non-local Gilbert damping \u000bij.\nHere, a torque-torque correlation model based on a tight binding approach is applied to the bulk\nelemental itinerant magnets and it predicts signi\fcant o\u000b-site Gilbert damping contributions, that\ncould be also negative. Supported by atomistic magnetisation dynamics simulations we reveal the\nimportance of the non-local Gilbert damping in atomistic magnetisation dynamics. This study gives\na deeper understanding of the dynamics of the magnetic moments and dissipation processes in real\nmagnetic materials. Ways of manipulating non-local damping are explored, either by temperature,\nmaterials doping or strain.\nPACS numbers: 75.10.Hk,75.40.Mg,75.78.-n\nE\u000ecient spintronics applications call for magnetic ma-\nterials with low energy dissipation when moving magnetic\ntextures, e.g. in race track memories1, skyrmion logics2,3,\nspin logics4, spin-torque nano-oscillator for neural net-\nwork applications5or, more recently, soliton devices6. In\nparticular, the dynamics of such magnetic textures |\nmagnetic domain walls, magnetic Skyrmions, or magnetic\nsolitons | is well described in terms of precession and\ndamping of the magnetic moment mias it is formulated\nin the atomistic Landau-Lifshitz-Gilbert (LLG) equation\nfor sitei\n@mi\n@t=mi\u0002\u0012\n\u0000\rBeff\ni+\u000b\nms@mi\n@t\u0013\n; (1)\nwhere\randmsare the gyromagnetic ratio and the\nmagnetic moment length, respectively. The precession\n\feldBeff\niis of quantum mechanical origin and is ob-\ntained either from e\u000bective spin-Hamilton models7or\nfrom \frst-principles8. In turn, energy dissipation is\ndominated by the ad-hoc motivated viscous damping in\nthe equation of motion scaled by the Gilbert damping\ntensor\u000b. Commonly, the Gilbert damping is used as\na scalar parameter in magnetization dynamics simula-\ntions based on the LLG equation. Strong e\u000borts were\nspend in the last decade to put the Gilbert damping\nto a \frst-principles ground derived for collinear mag-\nnetization con\fgurations. Di\u000berent methods were pro-\nposed: e.g. the breathing Fermi surface9{11, the torque-\ntorque correlation12, spin-pumping13or a linear response\nmodel14,15. Within a certain accuracy, the theoretical\nmodels allow to interpret16and reproduce experimental\ntrends17{20.\nDepending on the model, deep insight into the fun-\ndamental electronic-structure mechanism of the Gilbertdamping\u000bis provided: Damping is a Fermi-surface ef-\nfect and depending on e.g. scattering rate, damping\noccurs due to spin-\rip but also spin-conservative tran-\nsition within a degenerated (intraband, but also inter-\nband transitions) and between non-degenerated (inter-\nband transitions) electron bands. As a consequence of\nthese considerations, the Gilbert damping is proportional\nto the density of states, but it also scales with spin-orbit\ncoupling21,22. The scattering rate \u0000 for the spin-\rip tran-\nsitions is allocated to thermal, but also correlation ef-\nfects, making the Gilbert damping strongly temperature\ndependent which must be a consideration when applying\na three-temperature model for the thermal baths, say\nphonon14, electron, and spin temperature23. In particu-\nlar, damping is often related to the dynamics of a collec-\ntive precession mode (macrospin approach) driven from\nan external perturbation \feld, as it is used in ferromag-\nnetic resonance experiments (FMR)24. It is also estab-\nlished that the Gilbert damping depends on the orien-\ntation of the macrospin25and is, in addition, frequency\ndependent26.\nMore recently, the role of non-collective modes to the\nGilbert damping has been debated. F ahnle et al.27\nsuggested to consider damping in a tensorial and non-\nisotropic form via \u000bithat di\u000bers for di\u000berent sites i\nand depends on the whole magnetic con\fguration of the\nsystem. As a result, the experimentally and theoret-\nically assumed local Gilbert equation is replaced by a\nnon-local equation via non-local Gilbert damping \u000bijac-\ncounting for the most general form of Rayleigh's dissi-\npation function28. The proof of principles was given for\nmagnetic domain walls29,30, linking explicitly the Gilbert\ndamping to the gradients in the magnetic spin texture\nrm. Such spatial non-locality, in particular, for discrete\natomistic models, allows further to motivate energy dis-arXiv:1709.10365v1  [cond-mat.mtrl-sci]  29 Sep 20172\nij\nαij\nq\nFIG. 1: Schematic illustration of non-local energy dissipation\n\u000bijbetween site iandj(red balls) represented by a power\ncord in a system with spin wave (gray arrows) propagation q.\nsipation between two magnetic moments at sites iand\nj, and is represented by \u000bij, as schematically illustrated\nin Fig. 1. An analytical expression for \u000bijwas already\nproposed by various authors14,31,32, however, not much\nwork has been done on a material speci\fc, \frst-principle\ndescription of the atomistic non-local Gilbert damping\n\u000bij. An exception is the work by Gilmore et al.32who\nstudied\u000b(q) in the reciprocal space as a function of the\nmagnon wave vector qand concluded that the non-local\ndamping is negligible. Yan et al.29and Hals et al.33, on\nthe other hand, applied scattering theory according to\nBrataas et al.34to simulate non-collinearity in Gilbert\ndamping, only in reciprocal space or continuous meso-\nscopic scale. Here we come up with a technical descrip-\ntion of non-locality of the damping parameter \u000bij, in\nreal space, and provide numerical examples for elemental,\nitinerant magnets, which might be of high importance in\nthe context of ultrafast demagnetization35.\nThe paper is organized as follows: In Section I, we\nintroduce our \frst-principles model formalism based on\nthe torque-torque correlation model to study non-local\ndamping. This is applied to bulk itinerant magnets bcc\nFe, fcc Co, and fcc Ni in both reciprocal and real space\nand it is analysed in details in Section II. Here, we will\nalso apply atomistic magnetisation dynamics to outline\nthe importance in the evolution of magnetic systems. Fi-\nnally, in the last section, we conclude the paper by giving\nan outlook of our work.\nI. METHODS\nWe consider the torque-torque correlation model in-\ntroduced by Kambersk\u0013 y10and further elaborated on by\nGilmore et al.12. Here, \fnite magnetic moment rotations\ncouple to the Bloch eigenenergies \"n;kand eigenstates\njnki, characterised by the band index nat wave vec-tork, due to spin-orbit coupling. This generates a non-\nequilibrium population state (a particle-hole pair), where\nthe excited states relax towards the equilibrium distribu-\ntion (Fermi-Dirac statistics) within the time \u001cn;k=1=\u0000,\nwhich we assume is independent of nandk. In the adi-\nabatic limit, this perturbation is described by the Kubo-\nGreenwood perturbation theory and reads12,36in a non-\nlocal formulation\n\u000b\u0016\u0017(q) =g\u0019\nmsZ\n\nX\nnmT\u0016\nnk;mk+q\u0000\nT\u0017\nnk;mk+q\u0001\u0003Wnk;mk+qdk:\n(2)\nHere the integral runs over the whole Brillouin zone\nvolume \n. A frozen magnon of wave vector qis consid-\nered that is ascribed to the non-locality of \u000b. The scat-\ntering events depend on the spectral overlap Wnk;mk+q=R\n\u0011(\")Ank(\";\u0000)Amk+q(\";\u0000) d\"between two bands \"n;k\nand\"m;k+q, where the spectral width of the electronic\nbandsAnkis approximated by a Lorentzian of width \u0000.\nNote that \u0000 is a parameter in our model and can be spin-\ndependent as proposed in Ref. [37]. In other studies, this\nparameter is allocated to the self-energy of the system\nand is obtained by introducing disorder, e.g., in an al-\nloy or alloy analogy model using the coherent potential\napproximation14(CPA) or via the inclusion of electron\ncorrelation38. Thus, a principle study of the non-local\ndamping versus \u0000 can be also seen as e.g. a temperature\ndependent study of the non-local damping. \u0011=@f=@\"is\nthe derivative of the Fermi-Dirac distribution fwith re-\nspect to the energy. T\u0016\nnk;mk+q=hnkj^T\u0016jmk+qi, where\n\u0016=x;y;z , are the matrix elements of the torque oper-\nator ^T= [\u001b;Hso] obtained from variation of the mag-\nnetic moment around certain rotation axis e.\u001band\nHsoare the Pauli matrices and the spin-orbit hamilto-\nnian, respectively. In the collinear ferromagnetic limit,\ne=ezand variations occur in xandy, only, which al-\nlows to consider just one component of the torque, i.e.\n^T\u0000=^Tx\u0000i^Ty. Using Lehmann representation39, we\nrewrite the Bloch eigenstates by Green's function G, and\nde\fne the spectral function ^A= i\u0000\nGR\u0000GA\u0001\nwith the\nretarded (R) and advanced (A) Green's function,\n\u000b\u0016\u0017(q) =g\nm\u0019Z Z\n\n\u0011(\")^T\u0016^Ak\u0010\n^T\u0017\u0011y^Ak+qdkd\":(3)\nThe Fourier transformation of the Green's function G\n\fnally is used to obtain the non-local Gilbert damping\ntensor23between site iat positionriand sitejat position\nrj,\n\u000b\u0016\u0017\nij=g\nm\u0019Z\n\u0011(\")^T\u0016\ni^Aij\u0010\n^T\u0017\nj\u0011y^Ajid\": (4)\nNote that ^Aij= i\u0000\nGR\nij\u0000GA\nji\u0001\n. This result is consis-\ntent with the formulation given in Ref. [31] and Ref. [14].\nHence, the de\fnition of non-local damping in real space3\nand reciprocal space translate into each other by a\nFourier transformation,\n\u000bij=Z\n\u000b(q) e\u0000i(rj\u0000ri)\u0001qdq: (5)\nNote the obvious advantage of using Eq. (4), since it\nallows for a direct calculation of \u000bij, as opposed to tak-\ning the inverse Fourier transform of Eq. (5). For \frst-\nprinciples studies, the Green's function is obtained from\na tight binding (TB) model based on the Slater-Koster\nparameterization40. The Hamiltonian consists of on-site\npotentials, hopping terms, Zeeman energy, and spin-orbit\ncoupling (See Appendix A). The TB parameters, includ-\ning the spin-orbit coupling strength, are obtained by \ft-\nting the TB band structures to ab initio band structures\nas reported elsewhere23.\nBeyond our model study, we simulate material spe-\nci\fc non-local damping with the help of the full-potential\nlinear mu\u000en-tin orbitals (FP-LMTO) code \\RSPt\"41,42.\nFurther numerical details are provided in Appendix A.\nWith the aim to emphasize the importance of non-\nlocal Gilbert damping in the evolution of atomistic\nmagnetic moments, we performed atomistic magnetiza-\ntion dynamics by numerical solving the Landau-Lifshitz\nGilbert (LLG) equation, explicitly incorporating non-\nlocal damping23,34,43\n@mi\n@t=mi\u00020\n@\u0000\rBeff\ni+X\nj\u000bij\nmj\ns@mj\n@t1\nA:(6)\nHere, the e\u000bective \feld Beff\ni =\u0000@^H=@miis allo-\ncated to the spin Hamiltonian entails Heisenberg-like ex-\nchange coupling\u0000P\nijJijmi\u0001mjand uniaxial magneto-\ncrystalline anisotropyP\niKi(mi\u0001ei)2with the easy axis\nalongei.JijandKiare the Heisenberg exchange cou-\npling and the magneto-crystalline anisotropy constant,\nrespectively, and were obtained from \frst principles44,45.\nFurther details are provided in Appendix A.\nII. RESULTS AND DISCUSSION\nThis section is divided in three parts. In the \frst part,\nwe discuss non-local damping in reciprocal space q. The\nsecond part deals with the real space de\fnition of the\nGilbert damping \u000bij. Atomistic magnetization dynam-\nics including non-local Gilbert damping is studied in the\nthird part.\nA. Non-local damping in reciprocal space\nThe formalism derived by Kambersk\u0013 y10and Gilmore12\nin Eq. (2) represents the non-local contributions to the\nenergy dissipation in the LLG equation by the magnonwave vector q. In particular, Gilmore et al.32con-\ncluded that for transition metals at room temperature\nthe single-mode damping rate is essentially independent\nof the magnon wave vector for qbetween 0 and 1% of\nthe Brillouin zone edge. However, for very small scat-\ntering rates \u0000, Gilmore and Stiles12observed for bcc Fe,\nhcp Co and fcc Ni a strong decay of \u000bwithq, caused by\nthe weighting function Wnm(k;k+q) without any sig-\nni\fcant changes of the torque matrix elements. Within\nour model systems, we observed the same trend for bcc\nFe, fcc Co and fcc Ni. To understand the decay of the\nGilbert damping with magnon-wave vector qin more de-\ntail, we study selected paths of both the magnon qand\nelectron momentum kin the Brillouin zone at the Fermi\nenergy\"Ffor bcc Fe (q;k2\u0000!Handq;k2H!N),\nfcc Co and fcc Ni ( q;k2\u0000!Xandq;k2X!L) (see\nFig. 2, where the integrand of Eq. (2) is plotted). For\nexample, in Fe, a usually two-fold degenerated dband\n(approximately in the middle of \u0000H, marked by ( i)) gives\na signi\fcant contribution to the intraband damping for\nsmall scattering rates. There are two other contributions\nto the damping (marked by ( ii)), that are caused purely\nby interband transitions. With increasing, but small q\nthe intensities of the peaks decrease and interband tran-\nsitions become more likely. With larger q, however, more\nand more interband transitions appear which leads to an\nincrease of the peak intensity, signi\fcantly in the peaks\nmarked with ( ii). This increase could be the same or-\nder of magnitude as the pure intraband transition peak.\nSimilar trends also occur in Co as well as Ni and are\nalso observed for Fe along the path HN. Larger spectral\nwidth \u0000 increases the interband spin-\rip transitions even\nfurther (data not shown). Note that the torque-torque\ncorrelation model might fail for large values of q, since\nthe magnetic moments change so rapidly in space that\nthe adiababtic limit is violated46and electrons are not\nstationary equilibrated. The electrons do not align ac-\ncording the magnetic moment and the non-equilibrium\nelectron distribution in Eq. (2) will not fully relax. In\nparticular, the magnetic force theorem used to derive\nEq. (3) may not be valid.\nThe integration of the contributions in electron mo-\nmentum space kover the whole Brillouin zone is pre-\nsented in Fig. 3, where both `Loretzian' method given\nby Eq. (2) and Green's function method represented\nby Eq. (3) are applied. Both methods give the same\ntrend, however, di\u000ber slightly in the intraband region,\nwhich was already observed previously by the authors\nof Ref. [23]. In the `Lorentzian' approach, Eq. (2), the\nelectronic structure itself is una\u000bected by the scattering\nrate \u0000, only the width of the Lorentian used to approx-\nimateAnkis a\u000bected. In the Green function approach,\nhowever, \u0000 enters as the imaginary part of the energy\nat which the Green functions is evaluated and, conse-\nquently, broadens and shifts maxima in the spectral func-\ntion. This o\u000bset from the real energy axis provides a more\naccurate description with respect to the ab initio results\nthan the Lorentzian approach.4\nΓHq(a−1\n0)\nΓ H\nk(a−1\n0)\nFe\nΓX\nΓ X\nk(a−1\n0)\n Co\nΓX\nΓ X\nk(a−1\n0)\n Ni\n(i) (ii) (ii)\nFIG. 2: Electronic state resolved non-local Gilbert damping obtained from the integrand of Eq. (3) along selected paths in the\nBrillouin zone for bcc Fe, fcc Co and fcc Ni. The scattering rate used is \u0000 = 0 :01 eV. The abscissa (both top and bottom in\neach panels) shows the momentum path of the electron k, where the ordinate (left and right in each panel) shows the magnon\npropagation vector q. The two `triangle' in each panel should be viewed separately where the magnon momentum changes\naccordingly (along the same path) to the electron momentum.\nWithin the limits of our simpli\fed electronic structure\ntight binding method, we obtained qualitatively similar\ntrends as observed by Gilmore et al.32: a dramatic de-\ncrease in the damping at low scattering rates \u0000 (intra-\nband region). This trend is common for all here ob-\nserved itinerant magnets typically in a narrow region\n0<jqj<0:02a\u00001\n0, but also for di\u000berent magnon propa-\ngation directions. For larger jqj>0:02a\u00001\n0the damping\ncould again increase (not shown here). The decay of \u000b\nis only observable below a certain threshold scattering\nrate \u0000, typically where intra- and interband contribu-\ntion equally contributing to the Gilbert damping. As\nalready found by Gilmore et al.32and Thonig et al.23,\nthis point is materials speci\fc. In the interband regime,\nhowever, damping is independent of the magnon propa-\ngator, caused by already allowed transition between the\nelectron bands due to band broadening. Marginal vari-\nations in the decay with respect to the direction of q\n(Inset of Fig. 3) are revealed, which was not reported be-\nfore. Such behaviour is caused by the break of the space\ngroup symmetry due to spin-orbit coupling and a selected\nglobal spin-quantization axis along z-direction, but also\ndue to the non-cubic symmetry of Gkfork6= 0. As a re-\nsult, e.g., in Ni the non-local damping decays faster along\n\u0000Kthan in \u0000X. This will be discussed more in detail in\nthe next section.\nWe also investigated the scaling of the non-local\nGilbert damping with respect to the spin-orbit coupling\nstrength\u0018dof the d-states (see Appendix B). We observe\nan e\u000bect that previously has not been discussed, namely\nthat the non-local damping has a di\u000berent exponential\nscaling with respect to the spin-orbit coupling constant\nfor di\u000berentjqj. In the case where qis close to the Bril-\nlouin zone center (in particular q= 0),\u000b/\u00183\ndwhereas\nfor wave vectors jqj>0:02a\u00001\n0,\u000b/\u00182\nd. For largeq,\ntypically interband transitions dominate the scatteringmechanism, as we show above and which is known to\nscale proportional to \u00182. Here in particular, the \u00182will\nbe caused only by the torque operator in Eq. (2). On the\nother hand, this indicates that spin-mixing transitions\nbecome less important because there is not contribution\nin\u0018from the spectral function entering to the damping\n\u000b(q).\nThe validity of the Kambserk\u0013 y model becomes ar-\nguable for\u00183scaling, as it was already proved by Costa\net al.47and Edwards48, since it causes the unphysical\nand strong diverging intraband contribution at very low\ntemperature (small \u0000). Note that there is no experi-\nmental evidence of such a trend, most likely due to that\nsample impurities also in\ruence \u0000. Furthermore, various\nother methods postulate that the Gilbert damping for\nq= 0 scales like \u00182 9,15,22. Hence, the current applied\ntheory, Eq. (3), seems to be valid only in the long-wave\nlimit, where we found \u00182-scaling. On the other hand,\nEdwards48proved that the long-wave length limit ( \u00182-\nscaling) hold also in the short-range limit if one account\nonly for transition that conserve the spin (`pure' spin\nstates), as we show for Co in Fig. 11 of Appendix C. The\ntrends\u000bversusjqjas described above changes drastically\nfor the `corrected' Kambersk\u0013 y formula: the interband re-\ngion is not a\u000bected by these corrections. In the intraband\nregion, however, the divergent behaviour of \u000bdisappears\nand the Gilbert damping monotonically increases with\nlarger magnon wave vector and over the whole Brillouin\nzone. This trend is in good agreement with Ref. [29].\nFor the case, where q= 0, we even reproduced the re-\nsults reported in Ref. [21]; in the limit of small scattering\nrates the damping is constant, which was also reported\nbefore in experiment49,50. Furthermore, the anisotropy\nof\u000b(q) with respect to the direction of q(as discussed\nfor the insets of Fig. 3) increases by accounting only for\npure-spin states (not shown here). Both agreement with5\n510−22Fe\n0.000\n0.025\n0.050\n0.075\n0.100\nq: Γ→H\n2510−2α(q)Co\nq: Γ→X\n510−225\n10−310−210−110+0\nΓ (eV)Ni\nq: Γ→X\nFIG. 3: (Color online) Non-local Gilbert damping as a func-\ntion of the spectral width \u0000 for di\u000berent reciprocal wave vector\nq(indicated by di\u000berent colors and in units a\u00001\n0). Note that q\nprovided here are in direct coordinates and only the direction\ndi\u000bers between the di\u000berent elementals, itinerant magnets.\nThe non-local damping is shown for bcc Fe (top panel) along\n\u0000!H, for fcc Co (middle panel) along \u0000 !X, and for fcc Ni\n(bottom panel) along \u0000 !X. It is obtained from `Lorentzian'\n(Eq. (2), circles) and Green's function (Eq. (3), triangles)\nmethod. The directional dependence of \u000bfor \u0000 = 0:01 eV is\nshown in the inset.\nexperiment and previous theory motivate to consider \u00182-\nscaling for all \u0000.\nB. Non-local damping in real space\nAtomistic spin-dynamics, as stated in Section I (see\nEq. (6)), that includes non-local damping requires\nGilbert damping in real-space, e.g. in the form \u000bij. This\npoint is addressed in this section. Such non-local con-\ntributions are not excluded in the Rayleigh dissipation\nfunctional, applied by Gilbert to derive the dissipation\ncontribution in the equation of motion51(see Fig. 4).\nDissipation is dominated by the on-site contribution\n-101 Fe\nαii= 3.552·10−3\n˜αii= 3.559·10−3\n-101αij·10−4Co\nαii= 3.593·10−3\n˜αii= 3.662·10−3\n-10\n1 2 3 4 5 6\nrij/a0Ni\nαii= 2.164·10−2\n˜αii= 2.319·10−2FIG. 4: (Color online) Real-space Gilbert damping \u000bijas\na function of the distance rijbetween two sites iandjfor\nbcc Fe, fcc Co, and fcc Ni. Both the `corrected' Kambersk\u0013 y\n(red circles) and the Kambersk\u0013 y (blue squares) approach is\nconsidered. The distance is normalised to the lattice constant\na0. The on-site damping \u000biiis shown in the \fgure label. The\ngrey dotted line indicates the zero line. The spectral width is\n\u0000 = 0:005 eV.\n\u000biiin the itinerant magnets investigated here. For both\nFe (\u000bii= 3:55\u000110\u00003) and Co ( \u000bii= 3:59\u000110\u00003) the\non-site damping contribution is similar, whereas for Ni\n\u000biiis one order of magnitude higher. O\u000b-site contri-\nbutionsi6=jare one-order of magnitude smaller than\nthe on-site part and can be even negative. Such neg-\native damping is discernible also in Ref. [52], however,\nit was not further addressed by the authors. Due to\nthe presence of the spin-orbit coupling and a preferred\nglobal spin-quantization axis (in z-direction), the cubic\nsymmetry of the considered itinerant magnets is broken\nand, thus, the Gilbert damping is anisotropic with re-\nspect to the sites j(see also Fig. 5 left panel). For ex-\nample, in Co, four of the in-plane nearest neighbours\n(NN) are\u000bNN\u0019\u00004:3\u000110\u00005, while the other eight are\n\u000bNN\u0019\u00002:5\u000110\u00005. However, in Ni the trend is opposite:\nthe out-of-plane damping ( \u000bNN\u0019\u00001:6\u000110\u00003) is smaller\nthan the in-plane damping ( \u000bNN\u0019 \u00001:2\u000110\u00003). In-\nvolving more neighbours, the magnitude of the non-local6\ndamping is found to decay as 1=r2and, consequently, it\nis di\u000berent than the Heisenberg exchange parameter that\nasymptotically decays in RKKY-fashion as Jij/1=r353.\nFor the Heisenberg exchange, the two Green's functions\nas well as the energy integration in the Lichtenstein-\nKatsnelson-Antropov-Gubanov formula54scales liker\u00001\nij,\nG\u001b\nij/ei(k\u001b\u0001rij+\b\u001b)\njrijj(7)\nwhereas for simplicity we consider here a single-band\nmodel but the results can be generalized also to the multi-\nband case and where \b\u001bdenotes a phase factor for spin\n\u001b=\";#. For the non-local damping the energy integra-\ntion is omitted due to the properties of \u0011in Eq. (4) and,\nthus,\n\u000bij/sin\u0002\nk\"\u0001rij+ \b\"\u0003\nsin\u0002\nk#\u0001rij+ \b#\u0003\njrijj2:(8)\nThis spatial dependency of \u000bijsuperimposed with\nRuderman-Kittel-Kasuya-Yosida (RKKY) oscillations\nwas also found in Ref. [52] for a model system.\nFor Ni, dissipation is very much short range, whereas in\nFe and Co `damping peaks' also occur at larger distances\n(e.g. for Fe at rij= 5:1a0and for Co at rij= 3:4a0).\nThe `long-rangeness' depends strongly on the parameter\n\u0000 (not shown here). As it was already observed for the\nHeisenberg exchange interaction Jij44, stronger thermal\ne\u000bects represented by \u0000 will reduce the correlation length\nbetween two magnetic moments at site iandj. The same\ntrend is observed for damping: larger \u0000 causes smaller\ndissipation correlation length and, thus, a faster decay\nof non-local damping in space rij. Di\u000berent from the\nHeisenberg exchange, the absolute value of the non-local\ndamping typically decreases with \u0000 as it is demonstrated\nin Fig. 5.\nNote that the change of the magnetic moment length\nis not considered in the results discussed so far. The\nanisotropy with respect to the sites iandjof the non-\nlocal Gilbert damping continues in the whole range of the\nscattering rate \u0000 and is controlled by it. For instance, the\nsecond nearest neighbours damping in Co and Ni become\ndegenerated at \u0000 = 0 :5 eV, where the anisotropy between\n\frst-nearest neighbour sites increase. Our results show\nalso that the sign of \u000bijis a\u000bected by \u0000 (as shown in\nFig. 5 left panel). Controlling the broadening of Bloch\nspectral functions \u0000 is in principal possible to evaluate\nfrom theory, but more importantly it is accessible from\nexperimental probes such as angular resolved photoelec-\ntron spectroscopy and two-photon electron spectroscopy.\nThe importance of non-locality in the Gilbert damping\ndepend strongly on the material (as shown in Fig. 5 right\npanel). It is important to note that the total | de\fned as\n\u000btot=P\nj\u000bijfor arbitrary i|, but also the local ( i=j)\nand the non-local ( i6=j) part of the Gilbert damping do\nnot violate the thermodynamic principles by gaining an-\ngular momentum (negative total damping). For Fe, the\n-101\n1. NN.\n2. NN.Fe\n34567αii\nαtot=/summationtext\njαijαq=0.1a−1\n0αq=0\n-10αij·10−4Co\n123456\nαij·10−3\n-15-10-50\n10−210−1\nΓ (eV)Ni\n5101520\n10−210−1\nΓ (eV)FIG. 5: (Color online) First (circles) and second nearest\nneighbour (triangles) Gilbert damping (left panel) as well as\non-site (circles) and total Gilbert (right panel) as a function of\nthe spectral width \u0000 for the itinerant magnets Fe, Co, and Ni.\nIn particular for Co, the results obtained from tight binding\nare compared with \frst-principles density functional theory\nresults (gray open circles). Solid lines (right panel) shows the\nGilbert damping obtained for the magnon wave vectors q= 0\n(blue line) and q= 0:1a\u00001\n0(red line). Dotted lines are added\nto guide the eye. Note that since cubic symmetry is broken\n(see text), there are two sets of nearest neighbor parameters\nand two sets of next nearest neighbor parameters (left panel)\nfor any choice of \u0000.\nlocal and total damping are of the same order for all\n\u0000, where in Co and Ni the local and non-local damp-\ning are equally important. The trends coming from our\ntight binding electron structure were also reproduced by\nour all-electron \frst-principles simulation, for both de-\npendency on the spectral broadening \u0000 (Fig. 5 gray open\ncircles) but also site resolved non-local damping in the\nintraband region (see Appendix A), in particular for fcc\nCo.\nWe compare also the non-local damping obtain from\nthe real and reciprocal space. For this, we used Eq. (3)\nby simulating Nq= 15\u000215\u000215 points in the \frst magnon\nBrillouin zone qand Fourier-transformed it (Fig. 6). For7\n-1.0-0.50.00.51.0αij·10−4\n5 10 15 20 25 30\nrij/a0FFT(α(q));αii= 0.003481\nFFT(G(k));αii= 0.003855\nFIG. 6: (Color online) Comparing non-local Gilbert damping\nobtained by Eq. (5) (red symbols) and Eq. (4) (blue symbols)\nin fcc Co for \u0000 = 0 :005 eV. The dotted line indicates zero\nvalue.\nboth approaches, we obtain good agreement, corroborat-\ning our methodology and possible applications in both\nspaces. The non-local damping for the \frst three nearest\nneighbour shells turn out to converge rapidly with Nq,\nwhile it does not converge so quickly for larger distances\nrij. The critical region around the \u0000-point in the Bril-\nlouin zone is suppressed in the integration over q. On\nthe other hand, the relation \u000btot=P\nj\u000bij=\u000b(q= 0)\nfor arbitrary ishould be valid, which is however violated\nin the intraband region as shown in Fig. 5 (compare tri-\nangles and blue line in Fig. 5): The real space damping\nis constant for small \u0000 and follows the long-wavelength\nlimit (compare triangles and red line in Fig. 5) rather\nthan the divergent ferromagnetic mode ( q= 0). Two\nexplanations are possible: i)convergence with respect to\nthe real space summation and ii)a di\u000berent scaling in\nboth models with respect to the spin-orbit coupling. For\ni), we carefully checked the convergence with the summa-\ntion cut-o\u000b (see Appendix D) and found even a lowering\nof the total damping for larger cut-o\u000b. However, the non-\nlocal damping is very long-range and, consequently, con-\nvergence will be achieved only at a cut-o\u000b radius >>9a0.\nForii), we checked the scaling of the real space Gilbert\ndamping with the spin-orbit coupling of the d-states\n(see Appendix B). Opposite to the `non-corrected' Kam-\nbersk\u0013 y formula in reciprocal space, which scales like\n\u00183\nd, we \fnd\u00182\ndfor the real space damping. This indi-\ncates that the spin-\rip scattering hosted in the real-space\nGreen's function is suppressed. To corroborate this state-\nment further, we applied the corrections proposed by\nEdwards48to our real space formula Eq. (4), which by\ndefault assumes \u00182(Fig. 4, red dots). Both methods, cor-\nrected and non-corrected Eq. (4), agree quite well. The\nsmall discrepancies are due to increased hybridisations\nand band inversion between p and d- states due to spin-\norbit coupling in the `non-corrected' case.\nFinally, we address other ways than temperature (here\nrepresented by \u0000), to manipulate the non-local damping.\nIt is well established in literature already for Heisenberg\nexchange and the magneto crystalline anisotropy that\n-0.40.00.40.81.2αij·10−4\n1 2 3 4 5 6 7\nrij/a0αii= 3.49·10−3αii= 3.43·10−3FIG. 7: (Color online) Non-local Gilbert damping as a func-\ntion of the normalized distancerij=a0for a tetragonal dis-\ntorted bcc Fe crystal structure. Here,c=a= 1:025 (red circles)\nandc=a= 1:05 (blue circles) is considered. \u0000 is put to 0 :01 eV.\nThe zero value is indicated by dotted lines.\ncompressive or tensial strain can be used to tune the mag-\nnetic phase stability and to design multiferroic materials.\nIn an analogous way, also non-local damping depends on\ndistortions in the crystal (see Fig. 7).\nHere, we applied non-volume conserved tetragonal\nstrain along the caxis. The local damping \u000biiis marginal\nbiased. Relative to the values of the undistorted case,\na stronger e\u000bect is observed for the non-local part, in\nparticular for the \frst few neighbours. Since we do a\nnon-volume conserved distortion, the in-plane second NN\ncomponent of the non-local damping is constant. The\ndamping is in general decreasing with increasing distor-\ntion, however, a change in the sign of the damping can\nalso occur (e.g. for the third NN). The rate of change\nin damping is not linear. In particular, the nearest-\nneighbour rate is about \u000e\u000b\u00190:4\u000110\u00005for 2:5% dis-\ntortion, and 2 :9\u000110\u00005for 5% from the undistorted case.\nFor the second nearest neighbour, the rate is even big-\nger (3:0\u000110\u00005for 2:5%, 6:9\u000110\u00005for 5%). For neigh-\nbours larger than rij= 3a0, the change is less signi\fcant\n(\u00000:6\u000110\u00005for 2:5%,\u00000:7\u000110\u00005for 5%). The strongly\nstrain dependent damping motivates even higher-order\ncoupled damping contributions obtained from Taylor ex-\npanding the damping contribution around the equilib-\nrium position \u000b0\nij:\u000bij=\u000b0\nij+@\u000bij=@uk\u0001uk+:::. Note that\nthis is in analogy to the magnetic exchange interaction55\n(exchange striction) and a natural name for it would\nbe `dissipation striction'. This opens new ways to dis-\nsipatively couple spin and lattice reservoir in combined\ndynamics55, to the best of our knowledge not considered\nin todays ab-initio modelling of atomistic magnetisation\ndynamics.\nC. Atomistic magnetisation dynamics\nThe question about the importance of non-local damp-\ning in atomistic magnetization dynamics (ASD) remains.8\n0.40.50.60.70.80.91.0M\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nt(ps)0.5\n0.1\n0.05\n0.01αtot\nαij\n0.5 1.0 1.5 2.0 2.5 3.0\nt(ps)Fe\nCo\nFIG. 8: (Color online) Evolution of the average magnetic mo-\nmentMduring remagnetization in bcc Fe (left panel) and\nfcc Co (right panel) for di\u000berent damping strength according\nto the spectral width \u0000 (di\u000berent colors) and both, full non-\nlocal\u000bij(solid line) and total, purely local \u000btot(dashed line)\nGilbert damping.\nFor this purpose, we performed zero-temperature ASD\nfor bcc Fe and fcc Co bulk and analysed changes in the\naverage magnetization during relaxation from a totally\nrandom magnetic con\fguration, for which the total mo-\nment was zero (Fig. 8)\nRelated to the spectral width, the velocity for remag-\nnetisation changes and is higher, the bigger the e\u000bective\nGilbert damping is. For comparison, we performed also\nASD simulations based on Eq. (2) with a scalar, purely\nlocal damping \u000btot(dotted lines). For Fe, it turned out\nthat accounting for the non-local damping causes a slight\ndecrease in the remagnetization time, however, is overall\nnot important for relaxation processes. This is under-\nstandable by comparing the particular damping values\nin Fig. 5, right panel, in which the non-local part ap-\npear negligible. On the other hand, for Co the e\u000bect\non the relaxation process is much more signi\fcant, since\nthe non-local Gilbert damping reduces the local contribu-\ntion drastically (see Fig. 5, right panel). This `negative'\nnon-local part ( i6=j) in\u000bijdecelerates the relaxation\nprocess and the relaxation time is drastically increased\nby a factor of 10. Note that a `positive' non-local part\nwill accelerate the relaxation, which is of high interest for\nultrafast switching processes.\nIII. CONCLUDING REMARKS\nIn conclusion, we have evaluated the non-locality of\nthe Gilbert damping parameter in both reciprocal and\nreal space for elemental, itinerant magnets bcc Fe, fcc\nCo and fcc Ni. In particular in the reciprocal space,\nour results are in good agreement with values given in\nthe literature32. The here studied real space damping\nwas considered on an atomistic level and it motivates\nto account for the full, non-local Gilbert damping in\nmagnetization dynamic, e.g. at surfaces56or for nano-\nstructures57. We revealed that non-local damping canbe negative, has a spatial anisotropy, quadratically scales\nwith spin-orbit coupling, and decays in space as r\u00002\nij.\nDetailed comparison between real and reciprocal states\nidenti\fed the importance of the corrections proposed by\nEdwards48and, consequently, overcome the limits of the\nKambersk\u0013 y formula showing an unphysical and experi-\nmental not proved divergent behaviour at low tempera-\nture. We further promote ways of manipulating non-local\nGilbert damping, either by temperature, materials dop-\ning or strain, and motivating `dissipation striction' terms,\nthat opens a fundamental new root in the coupling be-\ntween spin and lattice reservoirs.\nOur studies are the starting point for even further in-\nvestigations: Although we mimic temperature by the\nspectral broadening \u0000, a precise mapping of \u0000 to spin\nand phonon temperature is still missing, according to\nRefs. [14,23]. Even at zero temperature, we revealed a\nsigni\fcant e\u000bect of the non-local Gilbert damping to the\nmagnetization dynamics, but the in\ruence of non-local\ndamping to \fnite temperature analysis or even to low-\ndimensional structures has to be demonstrated.\nIV. ACKNOWLEDGEMENTS\nThe authors thank Lars Bergqvist, Lars Nordstr om,\nJustin Shaw, and Jonas Fransson for fruitful discus-\nsions. O.E. acknowledges the support from Swedish Re-\nsearch Council (VR), eSSENCE, and the KAW Founda-\ntion (Grants No. 2012.0031 and No. 2013.0020).\nAppendix A: Numerical details\nWe performkintegration with up to 1 :25\u0001106mesh\npoints (500\u0002500\u0002500) in the \frst Brillouin zone for bulk.\nThe energy integration is evaluated at the Fermi level\nonly. For our principles studies, we performed a Slater-\nKoster parameterised40tight binding (TB) calculations58\nof the torque-torque correlation model as well as for the\nGreen's function model. Here, the TB parameters have\nbeen obtained by \ftting the electronic structures to those\nof a \frst-principles fully relativistic multiple scattering\nKorringa-Kohn-Rostoker (KKR) method using a genetic\nalgorithm. The details of the \ftting and the tight binding\nparameters are listed elsewhere23,59. This puts our model\non a \frm, \frst-principles ground.\nThe tight binding Hamiltonian60H=H0+Hmag+\nHsoccontains on-site energies and hopping elements H0,\nthe spin-orbit coupling Hsoc=\u0010S\u0001Land the Zeeman\ntermHmag=1=2B\u0001\u001b. The Green's function is obtained\nbyG= (\"+ i\u0000\u0000H)\u00001, allows in principle to consider\ndisorder in terms of spin and phonon as well as alloys23.\nThe bulk Greenian Gijin real space between site iandj\nis obtained by Fourier transformation. Despite the fact\nthat the tight binding approach is limited in accuracy, it\nproduces good agreement with \frst principle band struc-\nture calculations for energies smaller than \"F+ 5 eV.9\n-1.5-1.0-0.50.00.51.01.5\n5 10 15 20 25 30\nrij(Bohr radii)Γ≈0.01eVTB\nTBe\nDFT\nαDFT\nii= 3.9846·10−3\nαTB\nii= 3.6018·10−3-1.5-1.0-0.50.00.51.01.5\nΓ≈0.005eV\nαDFT\nii= 3.965·10−3\nαTB\nii= 3.5469·10−3αij·10−4\nFIG. 9: (Colour online) Comparison of non-local damping ob-\ntained from the Tight Binding method (TB) (red \flled sym-\nbols), Tight Binding with Edwards correction (TBe) (blue\n\flled symbols) and the linear mu\u000en tin orbital method (DFT)\n(open symbols) for fcc Co. Two di\u000berent spectral broadenings\nare chosen.\nEquation (4) was also evaluated within the DFT and\nlinear mu\u000en-tin orbital method (LMTO) based code\nRSPt. The calculations were done for a k-point mesh\nof 1283k-points. We used three types of basis func-\ntions, characterised by di\u000berent kinetic energies with\n\u00142= 0:1;\u00000:8;\u00001:7 Ry to describe 4 s, 4pand 3dstates.\nThe damping constants were calculated between the 3 d\norbitals, obtained using using mu\u000en-tin head projection\nscheme61. Both the \frst principles and tight binding im-\nplementation of the non-local Gilbert damping agree well\n(see Fig. 9).\nNote that due to numerical reasons, the values of\n\u0000 used for the comparisons are slightly di\u000berent in\nboth electronic structure methods. Furthermore, in the\nLMTO method the orbitals are projected to d-orbitals\nonly, which lead to small discrepancies in the damping.\nThe atomistic magnetization dynamics is also per-\nformed within the Cahmd simulation package58. To\nreproduce bulk properties, periodic boundary condi-\ntions and a su\u000eciently large cluster (10 \u000210\u000210)\nare employed. The numerical time step is \u0001 t=\n0:1 fs. The exchange coupling constants Jijare\nobtained from the Liechtenstein-Kastnelson-Antropov-\nGubanovski (LKAG) formula implemented in the \frst-\nprinciples fully relativistic multiple scattering Korringa-\nKohn-Rostoker (KKR) method39. On the other hand,\nthe magneto-crystalline anisotropy is used as a \fxed pa-\nrameter with K= 50\u0016eV.\n012345678α·10−3\n0.0 0.02 0.04 0.06 0.08 0.1\nξd(eV)2.02.22.42.62.83.03.2γ\n0.0 0.1 0.2 0.3 0.4\nq(a−1\n0)-12-10-8-6-4-20αnn·10−5\n01234567\nαos·10−3 1.945\n1.797\n1.848\n1.950\n1.848\n1.797\n1.950FIG. 10: (Color online) Gilbert damping \u000bas a function of\nthe spin-orbit coupling for the d-states in fcc Co. Lower panel\nshows the Gilbert damping in reciprocal space for di\u000berent\nq=jqjvalues (di\u000berent gray colours) along the \u0000 !Xpath.\nThe upper panel exhibits the on-site \u000bos(red dotes and lines)\nand nearest-neighbour \u000bnn(gray dots and lines) damping.\nThe solid line is the exponential \ft of the data point. The\ninset shows the \ftted exponents \rwith respect wave vector\nq. The colour of the dots is adjusted to the particular branch\nin the main \fgure. The spectral width is \u0000 = 0 :005 eV.\nAppendix B: Spin-orbit coupling scaling in real and\nreciprocal space\nKambersk\u0013 y's formula is valid only for quadratic spin-\norbit coupling scaling21,47, which implies only scattering\nbetween states that preserve the spin. This mechanism\nwas explicitly accounted by Edwards48by neglecting the\nspin-orbit coupling contribution in the `host' Green's\nfunction. It is predicted for the coherent mode ( q= 0)21\nthat this overcomes the unphysical and not experimen-\ntally veri\fed divergent Gilbert damping for low tem-\nperature. Thus, the methodology requires to prove the\nfunctional dependency of the (non-local) Gilbert damp-\ning with respect to the spin-orbit coupling constant \u0018\n(Fig. 10). Since damping is a Fermi-surface e\u000bects, it\nis su\u000ecient to consider only the spin-orbit coupling of\nthe d-states. The real space Gilbert damping \u000bij/\u0018\r\nscales for both on-site and nearest-neighbour sites with\n\r\u00192. For the reciprocal space, however, the scaling is\nmore complex and \rdepends on the magnon wave vec-\ntorq(inset in Fig. 10). In the long-wavelength limit,\nthe Kambersk\u0013 y formula is valid, where for the ferromag-\nnetic magnon mode with \r\u00193 the Kambersk\u0013 y formula\nis inde\fnite according to Edwards48.10\n10−32510−2α(q)\n10−310−210−110+0\nΓ (eV)0.000\n0.025\n0.050\n0.075\n0.100\nq: Γ→XCo\nFIG. 11: (Colour online) Comparison of reciprocal non-local\ndamping with (squares) or without (circles) corrections pro-\nposed by Costa et al.47and Edwards48for Co and di\u000berent\nspectral broadening \u0000. Di\u000berent colours represent di\u000berent\nmagnon propagation vectors q.\nAppendix C: Intraband corrections\nFrom the same reason as discussed in Section B, the\nrole of the correction proposed by Edwards48for magnon\npropagations di\u000berent than zero is unclear and need to\nbe studied. Hence, we included the correction of Ed-\nward also to Eq. (3) (Fig. 11). The exclusion of the spin-\norbit coupling (SOC) in the `host' clearly makes a major\nqualitative and quantitative change: Although the in-\nterband transitions are una\u000bected, interband transitions\nare mainly suppressed, as it was already discussed by\nBarati et al.21. However, the intraband contributions are\nnot totally removed for small \u0000. For very small scat-\ntering rates, the damping is constant. Opposite to the\n`non-corrected' Kambersk\u0013 y formula, the increase of the\nmagnon wave number qgives an increase in the non-\nlocal damping which is in agreement to the observation\nmade by Yuan et al.29, but also with the analytical modelproposed in Ref. [52] for small q. This behaviour was ob-\nserved for all itinerant magnets studied here.\nAppendix D: Comparison real and reciprocal\nGilbert damping\nThe non-local damping scales like r\u00002\nijwith the dis-\ntance between the sites iandj, and is, thus, very long\nrange. In order to compare \u000btot=P\nj2Rcut\u000bijfor arbi-\ntraryiwith\u000b(q= 0), we have to specify the cut-o\u000b ra-\ndius of the summation in real space (Fig. 12). The inter-\nband transitions (\u0000 >0:05 eV) are already converged for\nsmall cut-o\u000b radii Rcut= 3a0. Intraband transitions, on\nthe other hand, converge weakly with Rcutto the recipro-\ncal space value \u000b(q= 0). Note that \u000b(q= 0) is obtained\nfrom the corrected formalism. Even with Rcut= 9a0\nwhich is proportional to \u001930000 atoms, we have not\n0.81.21.62.0αtot·10−3\n4 5 6 7 8 9\nRcut/a00.005\n0.1\nFIG. 12: Total Gilbert damping \u000btotfor fcc Co as a function\nof summation cut-o\u000b radius for two spectral width \u0000, one in\nintraband (\u0000 = 0 :005 eV, red dottes and lines) and one in the\ninterband (\u0000 = 0 :1 eV, blue dottes and lines) region. The\ndotted and solid lines indicates the reciprocal value \u000b(q= 0)\nwith and without SOC corrections, respectively.\nobtain convergence.\n\u0003Electronic address: danny.thonig@physics.uu.se\n1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008), URL http://www.sciencemag.org/cgi/doi/\n10.1126/science.1145799 .\n2J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nature\nNanotech 8, 742 (2013), URL http://www.nature.com/\ndoifinder/10.1038/nnano.2013.176 .\n3A. Fert, V. Cros, and J. 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One consequence of this is the possibility\nof the Cooper-pair spin current analogous to the magnon spin current in magnetic insulators, the\nanalogy also extending to the existence of the Gilbert damping of the collective spin-triplet dynamics.\nThe recently fabricated heterostructure of the thin \flm of the itinerant ferromagnet SrRuO 3on\nthe bulk Sr 2RuO 4, the best-known candidate material for the spin-triplet superconductor, o\u000bers a\npromising platform for generating such spin current. We will show how such heterostructure allows\nus to not only realize the long-range spin valve but also electrically drive the collective spin mode\nof the spin-triplet order parameter. Our proposal represents both a new realization of the spin\nsuper\ruidity and a transport signature of the spin-triplet superconductivity.\n^x^y φ\nJzsp(a) ^d\nφ\nJzsp^n\nJ ↑↑\nJ ↓↓(b) (c)\nFIG. 1. Schematic illustration of the analogy between the\nmagnetic insulator and the spin-triplet superconductor. (a)\nThe planar spiraling of the magnetic order parameter ^ nleads\nto spin current. (b) The same phenomena occurs for that of\nthe spin component ^dof the spin-triplet superconductor order\nparameter, (c) the dual picture of which is the counter\row of\nthe spin up-up and down-down pairs.\nIntroduction : Harnessing spin rather than charge in\nelectronic devices has been a major topic in solid state\nphysics, which not only has been utilized for various\nmemory devices but is also expected to play a key role\nin processing quantum information [1]. In order for vari-\nous spin devices to function robustly, the long-range spin\ntransport needs to be achieved. Metallic wires, however,\ntypically do not transport spins beyond the spin-di\u000busion\nlength due to the single electron spin relaxation [2].\nIn recent years, it has been shown that the exponential\ndamping can be circumvented in the spin transport via\ncollective magnetic excitations. For example, easy-plane\n(ferro- and antiferro-)magnetic insulators, as the U(1)\norder parameter can characterize them, may be consid-\nered analogous to the conventional super\ruid [3{5]. As\nFig. 1 (a) illustrates schematically, the planar spiraling\nof the magnetic order parameter in such magnetic insu-\nlators can give rise to the spin supercurrent, just as the\nphase gradient of the conventional super\ruid gives rise\nto the mass supercurrent; in this sense these magnetic\ninsulators can be regarded as spin super\ruids [6].\nInterestingly, there exists a class of super\ruids and su-\nperconductors which can support both mass and spin su-\npercurrent. Such super\ruids and superconductors wouldneed to involve both spin ordering and gauge symme-\ntry breaking. This occurs in the condensate of both\nthe spin-1 bosons [7] and the spin-triplet Cooper pairs\nof3He atoms [8, 9] or electrons [10, 11]; in the latter\ncase, the dissipationless spin current would be carried by\nthe Cooper pairs. While the vortices with spin supercur-\nrent circulation have been observed in all theses systems\n[12, 13], the bulk spin supercurrent has not been detected\nin the superconductor.\nIn this Letter, we will show how this existence of spin\nsuper\ruidity in the spin-triplet superconductor allows\nnot only the long-range spin current but also electrically\nexciting the spin wave in the bulk. For realizing these\nphenomena, we propose a two-terminal setup with volt-\nage bias between ferromagnetic metal leads in contact\nwith the spin-triplet superconductor. While the static\norder-parameter case [14] can be essentially reduced to\nthe Blonder-Tinkham-Klapwijk type formalism [15] for\nthe interfacial transport, here we need to complement\nit with the appropriate equations of motion for the col-\nlective spin dynamics in the superconductor. Recently,\na thin \flm of the itinerant ferromagnet SrRuO 3has\nbeen epitaxially deposited on the bulk Sr 2RuO 4, the best\nknown candidate material for the spin-triplet supercon-\nductor [16], yielding, due to their structural compatibil-\nity, an atomically smooth and highly conductive interface\n[17] with a strong Andreev conductance [18]. This makes\nSr2RuO 4and SrRuO 3the most suitable candidate mate-\nrials for the bulk and the leads, respectively, of our setup\n[19]. For the remainder of this paper, we will \frst show\nhow the simplest e\u000bective spin Hamiltonian for the spin-\ntriplet superconductor and the resulting spin dynamics\nare analogous to those of the antiferromagnetic insula-\ntor; then, we will discuss the magnetoresistance for the\nDC bias voltage and the coupling between the AC bias\nvoltage and the spin wave.\nGeneral considerations : We \frst point out the close\nanalogy between the spin order parameter of the antifer-arXiv:1802.01599v1  [cond-mat.supr-con]  5 Feb 20182\nromagnet and the spin-triplet superconductor. De\fned\ni(d\u0001\u001b)\u001by=\u0014\u0000dx+idydz\ndzdx+idy\u0015\n\u0011\u0014\u0001\"\"\u0001\"#\n\u0001#\"\u0001##\u0015\n;(1)\nthe d-vector of the spin-triplet pairing, which\nparametrizes the Cooper-pair spin state, be-\nhaves similarly under spin rotations to the N\u0013 eel\norder parameter of an antiferromagnet, i.e.,\n[Si(r);dj(r0)] =i\u0016h\u000fijk\u000e(r\u0000r0)dk(r) and [di;dj] = 0\nfor the condensate spin S(unlike the magnetization,\nneither the N\u0013 eel order parameter nor the d-vector\ngenerate the spin rotation in themselves) [8, 9, 11].\nGiven that the commutation relations establish S\u0002^d\nas the conjugate momentum to din both cases, it is\nnatural that the simplest e\u000bective Hamiltonian for the\nspin-triplet superconductor ^d-vector,\nH=1\n2Z\ndr[A(r^d)2+K^d2\nz+\r2\neS2=\u001f]; (2)\nwhere\reis the electron gyromagnetic ratio, Athe^d-\nvector sti\u000bness, and \u001fthe magnetic susceptibility, should\nbe equivalent to that of the antiferromagnet N\u0013 eel order\nparameter, once we identify the ^d-vector with the N\u0013 eel\norder parameter [4]. In the latter, antiferromagnetic case,\na (xy) planar texture of the orientational order param-\neter^ n!(cos\u001e;sin\u001e;0) is associated with a collective\n(z-polarized) spin current Jz/z\u0001^ n\u0002@i^ n!@i\u001e\row-\ning in theith direction. While this extends directly to\nour spin-triplet case, Eq. (1) gives the intuitive dual pic-\nture of Fig. 1 (c) for the planar spiraling of the d-vector,\ni.e.,^d= (cos\u000b;sin\u000b;0). Namely, as the phase of \u0001 \"\"\n(\u0001##) is given by \u001ec\u0007\u000b(where\u001ecis the overall phase\nof the superconductor), the spiraling of the d-vector on\nthexyplane as shown in Fig. 1 (b), or the gradient of\n\u000b, would imply the counter\row of the spin up-up and\ndown-down pairs. The resultant ( z-polarized) spin cur-\nrent is/\u0000r\u000b. Given the same commutation relation\nand the same e\u000bective Hamiltonian, it is natural that, in\nabsence of dissipation, the equations of motion for these\ntwo cases, the Leggett equations the ^d-vector [8, 9, 20]\nand the Landau-Lifshitz type equation for the N\u0013 eel order\nparameter, are identical.\nWe further argue that both cases have the same phe-\nnomenological form of dissipation as well. For the case of\nthe N\u0013 eel order parameter ^ n, such dissipation, /\u000b(@t^ n)2,\nknown generally as Gilbert damping for collective mag-\nnetic dynamics, has been understood phenomenologically\n[4, 21, 22]. That such dissipation has not been featured in\nthe 3He super\ruid literature can be attributed not to the\nintrinsic nature of the spin-triplet pairing but rather to\nthe very weak relativistic spin-orbit coupling of the 3He\natoms originating solely from the nuclear dipole-dipole\ninteraction [8]. In contrast, electrons in Sr 2RuO 4are\nsubject to the Ru atomic spin-orbit coupling [23] esti-\nmated to be of order 0.1 eV [24]. In this work, we willconsider the decay rate of \u000bn\u0016h\r2\ne=\u001ffor the condensate\nspin, the addition of which makes the Leggett equations\nof motion for spin [25] equivalent to the Landau-Lifshitz-\nGilbert type equations for antiferromagnets:\n@t^d=\u0000^d\u0002\r2\ne\n\u001fS;\n@tS=^d\u0002(Ar2^d\u0000K^dz^ z\u0000\u000bn\u0016h@t^d); (3)\nwhere\u000bis the dimensionless Gilbert damping parame-\nter andnthe Cooper-pair density. This set of equations\nshows how the e\u000bective Hamiltonian of Eq. (2) provides\nthe simplest method for considering the local ^d-vector\ndynamics, including the spin-wave excitation and the col-\nlective dissipation.\nFor the boundary conditions, at the interface between\nthe ferromagnetic lead and the spin-triplet superconduc-\ntor, we consider a two-channel interface conductance due\nto the spins aligned or anti-aligned to the lead magne-\ntization We note, in this regard, that the SrRuO 3thin\n\flm has a very high transport spin polarization, with\na 3-to-1 ratio between the majority and minority spin\nchannels [26{28], while the magnetization gets enhanced\nin the heterostructure [17]. In this Letter, for the sake of\nsimplicity, we shall only consider the case where the lead\nmagnetizations are collinear. Furthermore, the d-vector\nof the bulk spin-triplet superconductor will be taken to be\nperpendicular to the lead magnetization, i.e., the Cooper\npairs are equal-spin paired along the quantization axis\nparallel to the magnetization; it has been claimed for\nthe Sr 2RuO 4superconductor, based on the c-axis NMR\nmeasurement, that its d-vector can be rotated into the\nab-plane by applying magnetic \feld larger than 200 G\n[29], well below the upper critical \feld.\nLong-range spin valve : The simplest physics that can\narise in our two-terminal setup is the spin-valve magne-\ntoresistance due to the relative alignment of the leads.\nWe consider the case where the spin-triplet supercon-\nductor has the easy-plane anisotropy, that is, K > 0\nin Eq. (2), while the lead magnetization is perpendic-\nular to this plane; as already mentioned, the former\ncan be realized for the SrRuO 3/Sr2RuO 4heterostruc-\nture by applying a \u0015200 G \feld along the c-axis. In\nthis case, we can take ^dzto be a small parameter in ^d=\n(q\n1\u0000^d2zcos\u001ez;q\n1\u0000^d2zsin\u001ez;^dz) andjSx;yj\u001cjSzj. In\nsuch a case, [ \u001ez(r);Sz(r0)] =i\u0016h\u000e(r\u0000r0) gives us the con-\njugate pair, leading to the equations of motion\n@t\u001ez=\r2\ne\n\u001fSz; @tSz=Ar2\u001ez\u0000\u000bn\u0016h@t\u001ez;(4)\nwhere the \frst equation is a spin analogue of the Joseph-\nson relation and the second is the spin continuity equa-\ntion with the relaxation term. Note that we measure\nSzwith respect to its equilibrium value. One con\frms\nthe condensate spin imbalance relaxation time to be3\nHa\nVLIVR\n^\n^yz\nx^\nI\nVLVR\nz^\n^y\n^x\nFIG. 2. The setup for the DC voltage bias for the spin valve\n(upper) and the AC bias voltage for the spin-wave detection\n(lower), where ^ x;^ y;^ zcoincide with the crystalline a; b; c -axes,\nrespectively. For the upper \fgure, the lead magnetization is\nalong the c-axis, with the applied magnetic \feld Ha\u0015200 G\nalong the c-axis giving us the easy plane d-vector con\fgura-\ntion on the ab-plane, hence the spiraling in the ab-plane. For\nthe lower \fgure, the lead magnetization is along the a-axis; as\nthe easy-axis d-vector anisotropy favors the alignment along\nthec-axis, in the absence of an applied \fled, the AC bias volt-\nage gives us the low-frequency standing wave of the d-vector\noscillating around the c-axis in the bc-plane.\n\u001f=\u000bn \u0016h\r2\nefrom Eq. (4) through deriving @tSz+r\u0001Jsp\nz=\n\u0000\u000bn\u0016h\r2\neSz=\u001f, where Jsp\nz=\u0000Ar\u001ez. It is also impor-\ntant to note here that the magnitude of the d-vector\nanisotropy Khas no e\u000bect on the in-plane d-vector pre-\ncession, which allows us to ignore the fact that our ap-\nplied \feld gives us the Abrisokov vortices in the spin-\ntriplet superconductor and hence a non-uniform K.\nWe consider the spin-up current and the spin-down\ncurrent to be independent at the interface:\nI\u001b\nL;R=\u0006g\u001b\u001b\nL;R(VL;R\u0000\u0016h@t'\u001b=2e); (5)\nwhereg\u001b\u001b\nL;R's are the conductances for the \u001b-spin,IL;R\nthe\u001b-spin current into (out of) the left (right) lead, and\nVL;Rthe bias voltage of the left (right) lead; this is due\nto the spin-triplet superconductor having the equal spin\npairing axis collinear with the lead magnetization and\ntakingg\"#= 0. From Eq. (1), we see that the overall\n(or charge) phase of the superconductor is given by the\naverage of the spin up-up and the spin down-down con-\ndensate phase, \u001ec=P\n\u001b'\u001b=2, while\u001ezof Eq. (4) is given\nby\u001ez=P\n\u001b\u001b'\u001b=2. We are interested here in the steady-\nstate solution, i.e., @t'\u001b= const, for which we de\fne the\nconstant precession rate of !c\u0011P\n\u001b@t'\u001b=2 for the over-\nall phase\u001ecand \ns\u0011P\n\u001b\u001b@t'\u001b=2 for\u001ez. For such\nsolution, the following continuity conditions can be ap-plied to the charge and spin supercurrents, respectively:\nX\n\u001b(I\u001b\nL\u0000I\u001b\nR)=0;X\n\u001b\u001b(I\u001b\nL\u0000I\u001b\nR)=2\u000bne\nsSL (6)\n(Sis the bulk cross section area and Lthe spacing be-\ntween the two leads), the former from the charge con-\nservation and the latter from applying the steady-state\ncondition on Eq. (4), along with the spin current loss\n/\u000bLin the superconductior.\nThe current through the Sr 2RuO 4bulk can be ob-\ntained from the interface boundary conditions and the\ncontinuity conditions above, with the larger magni-\ntude for the parallel magnetization than the antipar-\nallel magnetization. We de\fne the total conductance\ngL;R\u0011P\n\u001bg\u001b\u001b\nL;R and the conductance polarization\npL;R\u0011P\n\u001b\u001bg\u001b\u001b\nL;R=gL;R, which de\fnes the relevant trans-\nport spin polarization. Applying the continuity condi-\ntions Eq. (6) on the interface boundary conditions Eq. (5)\nand setting VL=\u0000VR=V=2, we obtain\n\u0012\ngL+gRpLgL+pRgR\npLgL+pRgRgL+gR+g\u000b\u0013\u0012\n!c\n\ns\u0013\n=eV\n\u0016h\u0012gL\u0000gR\npLgL\u0000pRgR\u0013\n;\n(7)\nwhereg\u000b\u00114\u000bne2SL\n\u0016h. We can now obtain the dependence\nof the charge current on the conductance polarization:\nIc=X\n\u001bI\u001b=I0\u0014\n1\u0000gLgR(pL\u0000pR)2\n(gL+gR)(gL+gR+g\u000b)\u0000(pLgL+pRgR)2\u0015\n;\n(8)\nwhereI0\u0011gLgRV=(gL+gR). Note that Icis max-\nimized atpL=pR, when the steady-state angle \u001ezre-\nmains static. Di\u000berent spin polarizations at the two ends,\non the other hand, would trigger spin dynamics and re-\nsult in a nonzero dissipation rate of R=1\n2\u000bn\u0016h\n2\ns=\nR0(1\u0000Ic=I0)2=(pL\u0000pR)2per volume of the supercon-\nducting bulk, where R0= 8\u000bn(eV)2=\u0016h. Given that pL;R\nchange sign on the magnetization reversal, the above re-\nsults e\u000bectively give us the spin-valve magnetoresistance\nof our heterostructure, i.e., a larger conductance for the\nparallel magnetizations than for the antiparallel. Any\ne\u000bect that the spin-triplet pairing may have on the mag-\nnetization, hence the conductance polarization, can be\nignored when the Curie temperature of SrRuO 3(\u0018160K)\n[30] is two orders of magnitude higher than the supercon-\nducting critical temperature ( \u00181.5K) Sr 2RuO 4.\nWe emphasize that the above magnetoresistance re-\nsult is obtain solely for the current carried by Cooper\npairs. At a \fnite-temperature, quasiparticle contribu-\ntion would generally result in an exponentially-decaying\nmagnetoresistance, negligible for the lead spacing be-\nyond the spin-di\u000busion length. By contrast, the cur-\nrent of Eq. (8), which is carried by the Cooper pairs,\ngives us the\u00181=Lbehavior for the large spacing limit.\nTherefore, any magnetoresistance beyond the quasipar-\nticle spin-di\u000busion length should arise only below the su-\nperconducting transition at Tc, upon the emergence of4\n0.80.91.|Ic|/I0\n12340.50.60.70.80.91.\nω/ω0|Ic|/I0\nFIG. 3. Charge current versus frequency plotted for ~ g= 0:5,\n~L= 2, \u0000 =!0= 0:1 and ~A= 0:2, with the orange curve\nrepresenting pL=pR=pand the blue pL=\u0000pR=p. Note\nthatp= 0:8 for the top plot and p= 0:2 for the bottom plot.\na Cooper-pair condensate. For our Sr 2RuO 4/ SrRuO 3\nheterostructure, detection of magnetoresistance in the su-\nperconducting state for the lead spacing larger than the\nSr2RuO 4spin-di\u000busion length can be taken as a trans-\nport evidence for the spin-triplet superconductivity. The\nvalue of the spin-di\u000busion length itself can be extracted\nby measuring the exponential decay of the (normal) mag-\nnetoresistance, both above and below the transition.\nElectrically driven spin collective mode : For the case of\nthe easy-axis anisotropy of the d-vector, hence K < 0 in\nEq. (2),the spin collective excitation of the Cooper pairs\n[8, 9, 31, 32] will modify the supercurrent transport under\nthe AC bias voltage. We shall still continue to consider\nthe case where Eq. (5) would be valid, i.e., the equal spin\npairing axis of the spin-triplet superconductor collinear\nto the lead magnetizations. One way to satisfy this con-\ndition would be to have the lead magnetizations collinear\nto thea-axis, with no applied magnetic \feld; that would\nleave thea-axis as the equal spin pairing axis, with the\nd-vector moving on the the bc-plane. The equations of\nmotion, corresponding to spin injection polarized along\nthex-direction, are then modi\fed to\n@t\u001ex=\r2\ne\n\u001fSx; @tSx=Ar2\u001ex\u0000!2\n0\u001f\n\r2ecos\u001exsin\u001ex\u0000\u000b\u0016h@t\u001ex;\n(9)\nwhere\u001exis conjugate to Sxand!2\n0\u0011 jKj\r2\ne=\u001fis\nthe spin-wave energy gap. For the AC voltage bias\nV=V0exp(\u0000i!t), the steady-state solution for thespin phase \u001ex(x;t) =f(x) exp(\u0000i!t) and the charge\nphase\u001ec(x;t) =g(x) exp(\u0000i!t) behave di\u000berently, fo-\ncusing on the frequencies far below the plasma fre-\nquency. Hence the spin equations of motion Eq. (9)\ngives usf(x) =C+cosh\u0014x+C\u0000sinh\u0014x, wherev2\u00142=\n!2\u0000!2\n0\u0000i!\u0000, withv\u0011\rep\nA=\u001f (the^d-vector sti\u000b-\nnessAde\fned in Eq. (2)) being the spin-wave veloc-\nity and \u0000\u0011\u000bn\u0016h\r2\ne=\u001fthe damping rate. By contrast,\nthe charge current Jc(x;t) =\u0000\u001a@x\u001ec, where\u001ais the\n\u001ecsti\u000bness, should be uniform, which means we can set\n\u001ec(x;t) = const:\u0000x(Jc\n0=\u001a) exp(\u0000i!t), with a constant\nJc\n0. By imposing consistency between the current ob-\ntained from the boundary conditions of Eq. (5) and the\ndynamics of Eq. (9), we can solve for Jc\n0andC\u0006; Fig. 3\nshows the numerical results for Ic=Jc\n0Sfor the case of\nbothpL=pRandpL=\u0000pR.\nOur numerical results show that magnetoresistance be-\ncomes signi\fcant at !>\u0018!0, where the collective spin\nmode of the Cooper pairs is activated. For simplicity\nwe have set gL=gR=gand used the dimensionless\nparameters ~ g\u0011g\u0016hv=2eA,~L\u0011!0L=2v, and ~A=A=\u001a.\nFor! <! 0, in addition to barely noticeable magnetore-\nsistance, the charge current amplitude does not oscillate\nwith frequency; it remains close to the DC value I0, which\ncontrasts with the complete transport suppression ob-\ntained for the magnetic insulator [3]. In contrast, for\n! > ! 0, we see an oscillation with the !=! 0period of\nabout\u0019=~L, where the current amplitude maxima for the\nantiparallel lead magnetization occur at the current am-\nplitude minima for the parallel lead magnetization and\nvice versa. As in the ferromagnetic insulator [3], we ex-\npect that for ~L\u001c1 (whileLis still larger than the quasi-\nparticle spin-di\u000busion length), the magnetoresistance of\nEq. (8) is recovered for the static bias, i.e.,!!0.\nWe point out that the detection of the oscillation\nshown in Fig. 3 would determine the yet-unknown energy\nparameters for the spin-triplet pairing of Sr 2RuO 4. From\nthe e\u000bective Hamiltonian of Eq. (2), if we had known\naccurately the \feld Hcalong thec-axis that would ex-\nactly restore the d-vector isotropy, the gap frequency !0\nshould be just the electron Larmor frequency of this \feld\nfrom the spin equations of motion of Eq. (9). However,\nwe know no more than the upper bound Hc<200 G,\nhence only !0< \re\u0002200 G = 3:5 GHz, while the AC\nbias experiment, as shown in in Fig. 3, would allow us to\nde\fnitely identify the spin collective mode gap.\nConclusion and discussion : We have studied the DC\nand AC current transport between the itinerant ferro-\nmagnetic lead with collinear magnetization through the\nspin-triplet superconductor. We showed here that mag-\nnetoresistance can arise for both cases due to the Cooper-\npair spin transport. For the DC bias, the persistence\nof magnetoresistance for the lead spacing larger than\nthe quasiparticle spin-di\u000busion length can be taken as\na transport evidence for the spin-triplet pairing. For\nthe AC bias, the activation of magnetoresistance and5\nfrequency dependent oscillation above the threshold fre-\nquency will allow us to determine the spin anisotropy\nenergy scale. All together, our work shows both a new\nrealization of the spin super\ruidity and a transport sig-\nnature of the spin-triplet superconductivity. The recently\nfabricated SrRuO 3/Sr2RuO 4heterostructure provides a\npromising experimental setup.\nAcknowledgement : We would like to thank Young Jun\nChang, Bongju Kim, Han Gyeol Lee, Seung Ran Lee,\nYoshiteru Maeno, Tae Won Noh, S. Raghu, Manfred\nSigrist and So Takei for sharing their insights. The hos-\npitalities of Natal International Institute for Physics dur-\ning the workshop \\Collective Spin Transport in Electrical\nInsulators\" (S.B.C. and Y.T.) and of Kyoto University\nduring the workshop \\Oxide Superspin 2017\" (S.B.C.),\nwhere parts of this work has been completed, are grate-\nfully acknowledged. This research was supported by IBS-\nR009-Y1 (S.B.C. and K.H.L.), and United States Army\nResearch O\u000ece under Contract No. W911NF-14-1-0016\n(S.K.K. and Y.T.).\n\u0003sbchung0@uos.ac.kr\nyevol@physics.ucla.edu\n[1] S. A. 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B 94, 064508 (2016)."    },    {        "title": "1605.03996v1.Classical_limit_of_Rabi_nutations_in_spins_of_ferromagnets.pdf",        "content": "1 \n Title:  \n \nClassical limit  of Rabi nutation s in spins of  ferromagnet s \n \nAuthors:  \nAmir Capua1, Charles Rettner1, See-Hun Yang1, Stuart S. P. Parkin1,2 \n \nAffiliations:  \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 \nAbstract:  \n \nRabi oscillations  describe the interaction of a two-level system  with a rotating \nelectromagnetic field. As such, they  serve as the principle method for manipulating \nquantum bits. By using a combination of femtosecond laser pulses and microwave \nexcitations, we have observed the classical form  of Rabi nutations in a ferr omagnetic \nsystem  whose equations of motion mirror the case of a precessing quantum two -level \nsystem . Key to our experiments is the selection of a subset of spins that is in resonance \nwith the microwave excitation and whose coherence time is thereby extende d. Taking \nadvantage of Gilbert damping, the relaxation times are further increased  such that mode -\nlocking takes place . The observation of such Rabi nutations  is the first step towards \npotential applications based on phase -coherent spin manipulation in ferr omagnets.  \n \n \n  2 \n Main Text:  \nA practical gateway to the quantum world is provided by m acroscopic quantum systems \nthat are large cooperative ensembles (1). Superfluids (2, 3), superconductors (4, 5) and \nultracold dilute atomic vapors (6-8), are examples of such systems. Another example are \nmagnon gases in ordered ensembles of magnetic moments that form a macroscopic state \nwhere the quantum nature is unveiled  even at room temperature  (9). In ferromagnets, t he \nmacroscopic  quantum behavior asserts itself at low  temperatures (mK)  and/or small enough  \nlength scales  (nanometer)  (10-12). In that limit the angular momentum observables obey \nthe classical equations of motion. Hence , a great deal of insight  into the quantum word is \ngained from  studies  of classical analog s (13). \nAlthough isolated electron or nuclear s pin states are ideal candidates for quantum  \ninformation processing (14-18), the abundant spin states in ferromagnetic systems are not \ncurrently considered suitable for such applica tions . Their s pin states lack prot ection due to \nspin-spin and spin -lattice interactions. While these  may be overcome  in the quantum \nregime  (10), their coherent manipulation r emains unexplored, even in the  classical limit.  \nThe initialization, manipulation, and readout of spin ensembles in ferromagnetic \nsystems  requires operation in the non -adiabatic regime. This regime pertains whenever an \noscillatory field and the state repres enting the ensemble are not in equilibrium . The \nadiabatic interaction  has been primarily explored  using  ferromag netic resonance (FMR) \nmethod s where  steady state  spin precessions are driven  continuously  by an oscillatory \nmicrowave field . Similar studies  of the magnetic order  have  also been conducted by 3 \n studying the impulse  responses in the absence of the rotating  field using the  time-resolved \nmagneto -optical Kerr effect (TR -MOKE) (19-23). In this technique the free induction \ndecay response  of a ferromagnet  is triggered by an intense optical pulse which  disturbs the \nmagnetic order  and drives the system away  from the equilibrium  state.  Despite  the \nextensive studies  of spin dynamics  in ferromagnetic metals , little attention has been given  \nto non -adiabatic transitions. This mode of operation can be accessed whenever the driven \nspin precessions are disrupted and is achieved by either modifying the state of the \noscillatory magnetic field, or that of the magnetization. While the former method is more \ncommonly used, for instance by applying a \n -pulse  (24), the latter is adopted here.  \nUsing ultra short  optical pulse s to perturb a microwave dri ven ferromagnetic  system s \n(19), we study the non-adiabatic regime  and show that Rabi nutations  in their classical form  \ncan be  revealed  in a ferromagnet . We observe  a chirping of the precession frequency , and \nstudy the  ability  to manipulate the spin states in the presence of  significant inhomogeneous \nbroadening.  In agreement with Gilbert’s damping theory (25), the intrinsic relaxation \ntimes, which represent the loss of spin angular momentum to the environment , can be \nextended  by tuning the external magnetic field (26). In such cases, consecutive optical  \npulses act to synchronize the phase s of the precessing spins (27). Consequently, spin -mode -\nlocking is initiated having the form of  intense pulsations of the magnetization . Our \nexperiments reveal that the microwave signal induces  coherence in the ensemble by \nselecting a subset of spins that are driven resonantly with the field. Hence , the ensemble 4 \n dephasing is suppressed and  relaxation times that represent more closely th ose of  \nindividual spin s result. \nThe sample studied  was a Co36Fe44B20 film with a  thickness of 11 Å that was \nperpendicularly magnetized and grown by magnetron sputtering . The effective anisotropy \nfield, \n0 KeffH , was measured to be ~ 140 mT, with \n0  being the magnetic permeability . \nFrom TR-MOKE measurements of the free induction decay responses  as a function of \napplied field  (Fig. 1A) a Gilbert damping  constant , \n, of 0.023 and a distribution of the \neffective anisotropy field,  \n0 KeffH , of 17 .5 mT were determined  (28, 29). \n represents  \nthe losses of spin angular momentum without  the ensemble dephasing while \nKeffH  allows \nto determine the inhomogeneous broadening of the resonance  linewidth  (28).  \nThe basic concept of  our experiment is presented in Fig. 1 B. A microwave field is \nused to drive spin precessions in the film which are then perturbed by a femtosecond optical \npulse  while being phase locked with the microwave oscillator (30). The temporal recovery \nis recorded by a weak optical pulse probe as a func tion of a pump -probe delay time . The \nexternal magnetic field was applied in the sample plane causing precession s to occur about \nthe x-axis (see figure) while the out -of-plane component of the magnetization, mz, was \ndetected in a polar -MOKE configuration  (28).  \nA measurement of the non -adiabatic interaction is presented in Fig. 1C for three  \nvalues of externally applied field , H0, and a microwave frequency of 10 GHz. T he \nresonance field, Hres, corresponding to this frequency is \n0 resH  ~ 450 mT (Fig. S1) . It is 5 \n seen that a distinct envelope modulates the 10 GHz oscillations  of the precessional motion . \nFurther more,  this envelope exhibits a systematic behavior; the time of its minim um, as \nindicated by the arrows in the figure, increases  as H0 approaches Hres. \nThe responses for a complete  set of applied  fields, are illustrated in Fig. 1 D. At H0 \n< Hres, the shift in time of the minima  is seen clearly and forms a “valley”. At H0 > Hres, a \nmaxim um is initially formed instead , making the response asymmetric . Given that the \nconditions for the non-adiabatic  interaction prevail , the related time s of these signatures , \nfor example the time of the minima , T, should be describable by  the inverse of the \ngeneralized  Rabi  frequency . In the absence of the magnetocrystalline and demagnetization \nfields,  T is given by :  \n22\n0 0 02\nrf rfT\nHh\n  \n\n    (1) \nHere, \n , hrf, and \nrf  are the gyromagnetic ratio, microwave amplitude,  and microwave \nangular frequency,  respectively.  The times  obtained by  the Rabi formula are overlaid on \nthe measured responses  of Fig. 1D and are seen to agree  well with the observed signature s, \ndemonstrating Rabi nutations in a ferromagnet in the classical limit . Also the second  \nnutation is readily seen. It is instructive  to notice that the microwave amplit ude is only \n0rfh\n ~ 0.8 mT in these  measurement s. Hence , the main contribution to the minima time, \nT, stems from t he off-resonance term, \n00 rf H  .  6 \n  The field dependent phase responses also reveal intricate detail s of the dynamic s \nand are analyzed by plotting the dataset  of Fig. 1D  as a two -dimensional contour plot  (Fig. \n2A). Before the pump pulse arrives, a net phase shift of ~ 0.75  π is measured across the \nresonance (enlarged in Fig. 2B). This phase shift is smaller than the expected theoretical \nvalue of π and is related to the relatively large Gilbert  damping.  Surprisingly , at times  \n(pump -probe delays)  well after the  perturbation , a phase shift of  ~ 2.75 π is observed  as H0 \nis varied  (Fig. 2C). To understand  the origin of  this behavior we analyze the instantaneous \nfrequency profiles  (Fig. 2 D). Apart from the sharp  transient  at t = 0, we extract negative, \nzero, and positive chirp profiles corresponding to H0 < Hres, H0 = Hres, and H0 > Hres, \nrespectively . At long delays, the instantaneous frequency recovers to the driving frequency , \nindependent  of H0. This behavior  is explained  qualitatively  by recalling that Rabi \noscillations  can be  regarded as a beating of the  natural  transient response of the system  at \nthe angular  frequency  of \n00H with the steady state  response at  \nrf (31). Hence,  when\n00H\n < \nrf, a negative chirp initially takes place which  recovers to \nrf . The same \nexplanation holds  also for other  H0 values . This reasoning  also account s for the asymmetry \nseen in the responses of Fig. 1D.  As H0 varies, the pump pulse perturbs the magnetization \nat different points along  the precession trajectory  owing to the p hase shif t associated with \nthe resonance . The resultant  beating response then changes from a destructive nature to a \nconstructive interaction. Therefore, variation of H0 provides  a means  of controlling the \neffective  “area” (the time -integrated Rabi frequency)  of the microwave radiation.  A 7 \n theoretical description of the interaction using the Landau -Lifshitz -Gilbe rt equation is \nfurther discussed  in the supplementary  materials section.  \n An important aspect  of Rabi nutations  is the dependence of their frequency on the \nmagnitude of the microwave field. This dependence  is most readily  seen under resonance \nconditions  in which case  the angular Rabi frequency simplifies to  \n0r rf h . The \nmeasured results are shown in Fig.  3A. In contrast  to our expectation s, no dependence  of \nthe envelope  on the amplitude of the microwave  is revealed . This apparent discrepancy is \nresolved by c onsidering  the contributions to Eq. (1). The maximal applied  microwave field \namplitude was  \n0rfh  ~ 7.5 mT  while the inhomogeneous linewidth broadening  at 10 GHz , \nas derived from the value of  \nKeffH , is 10.5 mT  (28). Hence, the detuning term in Eq. (1) \nis still significant so that the Rabi frequency is mainly determined by the off-resonant  \ncontribution rather than by the microwave power . In order to observe the dependence on \nthe microwave power , the contribution of the inhomogeneous broadening must first  be \nsuppressed . This was achieved  by repeating the measurements on a  single crystal sample, \nin the form of a 4 nm thick  epitaxially grown Fe film. In contrast to the sputter deposited \nfilm, the envelope exhibits a clear dependence on the microwave amplitud e (Fig. 3 B). The  \nexpected  increase in the associated time scales describing the envelope  is seen  for \nincreasing a mplitude s as predicted by Rabi’s formula.  \n Next, we turn to s how that the train of optical  pulses can synchronize the phases of \nthe spin precessions and induce pulsation s of the magnetization, namely , spin mode -8 \n locking. This mode of operation can be reached if the responses generated by subsequent \npump pulses of the pulse train interfere. Therefore, this regime requires that the transient  \npart of the responses persist for a duration longer than  the laser  repetition time,  TR (Fig. \n4A).  \nAs follows from Gilbert’s theory for damping, the rate of transfer of spin angular \nmomentum to the lattice can be controlled by the magnitude of H0. This process is \nquantified using  the intrinsic relaxation time, \nint , and is given by  \n1\n00H   in the limit \nwhere only the externally applied field is present . Accordingly, interference effects are \nexpected  at low H0 values . The nature of the interference will then depend on the arrival \ntime of the optical pump pulse within th e microwave cycle. This time is represented by  \n \nwhich is the relative phase between the microwave signal and the pump pulse (Fig. 4A).   \nThe measured response s as a function of \n  are presented in Figs. 4B & 4C.  At high \nmagnetic field  (\n00H  = 450 mT ) and short \nint  (~ 1.1 ns) (28) compared to  the laser  \nrepetition time, TR, of 12.5 ns, the interaction of each pump pulse within the train of pulses \ncan be regarded as an isolated e vent (Fig. 4B). V ariation of \n  has no effect on the \nenvelope; the carrier wave simply  shifts within the same envelope.  In contrast, for low \nexternal magnetic fields (\n00H  = 90 mT) and correspondingly long \nint  (~ 5 ns), \ninterference occur s and the moment at which the optical pulse is sent becomes  critical  (Fig. \n4C). For \n  90\n , constru ctive interference results in a sharp pulsation of the \nmagnetization. Likewise, for additional \n180\n , at \n  270\n , pulsations of opposite polarity 9 \n are generated. However, when the phase is tuned to \n  0\n  and \n  180\n , destructive \ninterference takes place and no pulsations are observed. The existence of the pulsations \nindicates that  the spins have become synchron ized, i.e., mode -locking takes place  (27). \nIn addition to the intrinsic relaxation,  the decay of the transient response is governed \nalso by the dephasing of the inhomogeneously broadened ensemble. This process is \nrepresented by the ensemble dephasing time, \nIH , so that the effective decay time of the \nmagnetization of the entire ensemble , \neff , is given by : \nint 1/ 1/ 1/eff IH    (28, 29, 32).  \nInterestingly,  while a fundamentally different dependence on \n  is observed in the \ntwo regimes of Figs. 4B and 4C, the inhomogeneous broadening causes \neff  to be very \nsimilar  in both cases  and correspond s to ~ 0.51 ns and ~ 0.49 ns for \n00H  = 450 mT  and \n00H\n = 90 mT , respectively . This fact shows that the  long intrinsic  relaxation time , \nint , in \nthe case of low H0 (Fig. 4C) can be  sensed  despite the significant ensemble dephasing. By \nuse of the relations \nint int2/  and \n2/IH IH  for the intrinsic  resonance linewidth  \nand inhomogeneou s broaden ing, respectively,  \nint1.75 rad GHz     and \n2.15 radIH GHz   \n were  extracted for \n00H  = 450 mT , while \nint0.43 rad GHz     and \n3.66 radIH GHz   \n were  found for \n00H  = 90 mT . The se linewidths are illustrated in \nthe lower schematic  of Figs . 4B & 4C . In contrast to the high H0 case, at low H0 only a \nsubset of  spins which  exhibit long \nint  are interacting, namely, the microwave induces \ncoherence in the ensemble. The action of the oscillatory field is to s timulate  the subset of 10 \n spins that are driven resonantly , while suppress ing the off -resonance subsets. Hence, the \ninhomogeneity is overcome and \neff  extends towards its upper limit of \nint . \nThe action of  “filtering” by the microwave signal  is further emphasized by \nexamining the free induction decay responses. Similar long intrinsic relaxation times  that \nare responsible for the mode -locking with \n00H  = 90 mT, also dominate the corresponding \nTR-MOKE m easurement  at \n00H  = 100 mT (Fig. 1A ), for example . In contrast to the \nmicrowave driven measurement, t his respons e show s that the ensemble dephases within  TR \nand exhibits  no signs of coherent interference , as apparent from times around t = 0 (27). \nFurthe rmore,  a closer inspection of  the measurements at the high H0 of Fig. 1D also turn \nout to reveal slight signatures of extended coherence that last for the duration of TR, and \nare clearly not found in the corresponding TR -MOKE responses. These  signatures  are \ndiscussed in the supplementary materials section  and once more demonstrate the ability  of \nour technique to observe details that are obscured by the ensemble dephasing . \nLastly, in the high m agnetic field limit of Fig. 4B , variation of \n  was shown to \ncause  no effect on the envelope . This  observation  seemingly contradicts  the explanation \nbehind the  appearance of the asymmetric signature  of Fig. 1D  which was attributed to the \nphase shift associated with the resonanc e. However, a fundamental difference  exists  \nbetween the two measurements;  in the former  case,  the phase -shift stems from delaying the \nmicrowave  signal  with respect to the optical pulse s while  in the latter  it stems from the \nphase  response of  the resonance.  11 \n In summary, w e have demonstrated a technique that has revealed  the classical form  \nof Rabi nutations in a ferromagnetic system . Our experiments show that a “purified” sub-\nensemble  is generated  and whose dephasing is largely reduced . Extension s of the prese nt \nwork  include  studies of the quantum regime in mesoscopic ferromagnetic structures , as \nwell as  studies of more complex dynamics such as the spin -Hall effect or the spin transfer \ntorque in the non -adiabatic regime . 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The m easured signal is \nproportional to the angle of polarization rotation , \n, of the optical beam . Lower schemati c \nshows the signals in time. (C) Temporal responses of the pu mp-probe ferromagnetic \nresonance measurement for three values of H0 at 10 GHz  and microwave field amplitude \nof ~ 0.8 mT . Each trace is normalized to the peak value.  Arrows indicate the position of \nthe minimum.  The p ump pulse arrives at t = 0 ps.  (D) Measured temporal responses at 10 \nGHz for a complete range of applied fields. Each trace is normalized  individually  to the \npeak value. The solid red lines were plotted us ing the Rabi formula.  Second oscillation is \nindicated by the guiding black dashed line. Inset illustrates the motion of the magnetization  \nvector .  \n  16 \n Fig. 2  \n \n17 \n Fig. 2. Phase response . (A) Dependence of  phase responses  on the applied field  at 10 GHz . \nData set of Fig. 1D is presented  in a two-dimensional contour  plot to represent the phase  \ninformation. Each temporal response was normalized individually. Blue curved  guiding  \nline indicates the location of the “valley” in Fig. 1D.  (B) Phase r esponse prior to the \nperturbation . The figure presents a close -up of the b lack dashed area of panel (A) for times \nbetween -300 ps and -100 ps.  An overall phase shift of ~ 0.75  π is measured across the \nresonance.  Data is not normalized . (C) Phase r esponse at long delays corresponding to \nblack dashed area in panel (A) which starts at 2200 ps . Data is presented in normalized \nunits. The measured net phase shift across  the resonance is ~ 2.75 π. (D) Instantaneous \nfrequency  profiles at \n00H  values  of 424 mT (blue), 444 mT (red), and 468 mT (yellow) \ncorresponding the blue , red, and yellow dashed lines of panel (A), respectively . Inset \nillustrates a schematic of the beating of the steady state  response  of the system at \nrf  with \nthe na tural response at the angular frequency of \n00H . \n 18 \n Fig. 3  \n \n \n19 \n Fig. 3. Microwave power dependence of  the nutations . (A) Temporal responses  at \nvarious microwave field amplitude s for the CoFeB sample . Responses are presented for a \nfrequency of 10 GHz and \n00H  = 446 mT. (B) Temporal responses of the MBE grown Fe \nsample at 12 GHz and \n00H  = 143 mT.  The measurements i n (A) and (B) were carried out \nat the resonance conditions . Envelopes  of the  responses  are indicated by the guiding  dashed \nlines.  20 \n Fig. 4  \n \n \n \n21 \n Fig. 4. Phase dependent t emporal responses for long and short intrinsic relaxation \ntimes.  (A) Schematic arrangement of signals  in time . TR represents the laser  repetition time . \n(B) Dependence of the temporal response on \n  for short \nint . Data presented for a \nfrequency of 10 GHz and \n00H  = 450 mT.  The relative phase , \n, does not have  a \nsignificant effect on the envelope. S imilar behavior is recorded  also at other bias fields (not \nshown) . (C) Dependence of temporal response  on \n for long \nint . Data presented for a \nfrequency of 1  GHz and \n00H  = 90 mT . Data is shown for the CoFeB sample.  Lower \nschematic is panels  (B) and (C) illustrate  the inhomogeneous broadening. Blue solid line \nindicates the total effective  resonance  linewidth  which  includes contributions of the \ninhomogeneous broadening. Shaded resonance linewidths indicate the subgroups that are \nselected by the microwave. Red solid lines indicate the subgroups that are not interacting \nwith the microwave  signal . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 22 \n  \n \n \n \nSupplementary Materials for  \n \nClassical limit  of Rabi nutations in  spins of  ferromagnet s \n \nAmir Capua, Charles Rettner, See -Hun Yang, Stuart S. P. Parkin  \n \n \n \n \n 23 \n Materials and Methods  \n \nMaterials and Fabrication  \nThe CoFeB film in this study was prepared on thermally oxidized Si(100) substrate and \nconsisted of the following structure , starting from the substrate side : SiO 2 (250)/Ta (100)/CoFeB \n(11)/MgO (11)/Ta (30) (numbers are in nominal thicknesses in angstroms). The MgO layer was \ndeposited by RF sputtering . The sample was annealed at a temperature of  \n275\nC  for 30 minutes \nwhile applying a 1 T field in  the out -of-plane direction. In -plane and out -of-plane magnetization \nloops are shown in Fig. S1. The single crystalline 40 Å thick Fe film was grown on MgO(100) \nusing molecular beam epitaxy.  \nFor the pump -probe ferromagnetic resonance measurement the samples were patterned to a \nmagnetic island of  \n220 20 m  using electron -beam lithography. A shorted Au microwire serving \nas an RF -transmission line was patterned at a distance of \n1 m  away from the island by lift -off. \n \nFerromagnetic resonance pump -probe measurement  \nA Ti:Sapphire oscillato r emitting ~ 70 fs pulses  at 800 nm  having energy of ~ 5 nJ per pulse  \nwas used for the optical measurements. The beam was focused to a spot size of approximately 10 \nm\n. The probe pulses were attenuated by 20 dB relative to the pump.  The time jitter between the \noptical pump and the microwave signal was measured to be smaller than 1 ps. All measurements \nwere carried out at room temperature.  The maximum microwave power applied was 1 W and \ncorresponded to an amplitude of ~ 7.5 mT.  \nIn the  pump -probe ferromagnetic resonance measurements a double lock -in detection scheme \nwas used by modulating the microwave signal at 50 KHz and the optical probe at 1 KHz.  In order \nto exert sufficient torque by the optical pump, the external magnetic field wa s applied at an angle \nof \n4\n  away from the sample plane. The same arrangement was applied also in the TR -MOKE \nmeasurements.  \n \nNumerical simulation  \nCalculation of the non -adiabatic interaction was carried out by numerically integrating  the \nLandau -Lifshitz -Gilbert equation.  Since the calculation does not account for the inhomogeneous \nbroadening, it describes the experiment in a qualitative manner.  In the calculation, the steady \nprecessional state was first obtained before applying the pe rturbation.  Two sources for the \nperturbation were introduced that gave the best results: quenching of the magnetization and \nintroduction of a momentary  phenomenological  magnetic field. The latter was required in order to \nreproduce the phase response at pos itive times near t = 0, namely the curvature in the vertical \ncontours of Fig. S2 appearing at times that immediately follow the pump. In Fig. S2 , the wave \nfronts shift to later times as the field increases to a value of ~ 440 mT. When the field is further \nincreased , the wave fronts shift to earlier  times . Since the magnetization acquires a phase shift that \nis associated with the resonance at a field of about 450 mT, presenting an additional \nphenomenological magnetic field causes the magnetization to alter i ts motion. This additional \ntorque was applied in the form of a 3 ps pulsed magnetic field of 60 mT which lied in the film \nplane orthogonal to the axis of precession.  The recovery profile of the magnetization after 24 \n quenching consisted of two time constants of 50 ps and 500 ps while the modulation depth was \n5%.  \nThe simulation result is shown in Fig. S3. Imprints of nutations on the amplitude of the \nprecessions are readily seen. The formation of the “valley” as in Fig. 1D is also observed. This \nvalley however  appears also at field values which are larger than the resonance field in contrast to \nthe measurement. At magnetic fields near resonance and immediately after zero time, similar \ncontours to the ones shown in Fig. S2 are seen.  \n \nExtraction of decay times f rom TR -MOKE measurements  \nExtraction of the ensemble dephasing times and the intrinsic spin relaxation times from TR -\nMOKE measurements was based on the analysis presented in  Ref. (29). Accordingly, \n  and \nKeffH\n were obtained by fitting the measured effective linewidths, \neff , with the equation:  \n \n0\n0 0 0 02\n00\n0\n0 0 0 022\n002         for        \n2\n2     for         eff Keff Keff keff\nKeff\nKeff Keff\neff Keff Keff\nKeffKeffHH H H H H\nH H H\nHH HH H H HHH HH  \n       \n\n\n       \n. \n          \n \nHere \n2/eff eff , and \neff  is the overall decay time of the precessional motion as measured in \nthe TR -MOKE experiment. This analysis is valid for H0 much larger or much smaller than HKeff. \nThe first terms of the equation represent the intrinsic linewidth \nint  while the second terms \nrepresent the inhomogeneous broadening \nIH . For H0 ~ HKeff , as for the case where \n00 90 mT H\n, \nint and \nIH  were extracted using a numerical method (32).  \n \nSignatures of extended coherence at high applied field and 10 GHz  \nSignatures of the extended coherences were even found in the phase responses at 10 GHz \n(Fig. 2A) where \neff and \nint  are shorter. At negative times, before the pump pulse arrives, the phase \nin Fig. 2B shifts slightly to negative values as H0 increases to \n00H  ~ 450 mT.  This response \ndiffers from the responses measured when the optical pump is completely turned off (Fig. S 4), \nindicating the slight remnant coherence from the previous cycle. Moreover, the negative phase \nshift resembles the behavior of the phases seen immed iately after t = 0 (Fig. S 2) implying a link \nbetween the responses manifested by long lasting coherence despite the short \neff  and \nint  (27). A \nnon-coherent process, such as a thermal process would have had an equal effect for all H0 and \nwould not have affected the phase in the manner observed. Once more, the corresponding TR -\nMOKE traces show no sign of the coherent interaction after TR, demonstrating the ability to \nobserve details that were obscured by the ensemble dephasing.  \n 25 \n  Figures  \n \n \n  \nFig. S1.  \nCoFeB film characterization. (A) In-plane and out -of-plane magnetization loops.  (B) Frequency \nvs. applied field as measured in a TR -MOKE experiment. The magnetic field was applied at an \nangle of \n4\n  away from the sample plane. B lack dashed line indicates HKeff. \n \n \n26 \n  \nFig. S 2 \nClose -up of Fig. 2A for times between for times between 100 ps and 300 ps. A negative phase \nshift is seen as the field increases to a value of 440 mT  after which a positive phase shift occurs \nwhen the field is further increased.  \n  \n27 \n  \n \nFig. S 3 \nCalculation of the out -of-plane component of the magnetization, mz. The r esponse at each bias \nfield was normalized independently to reach a maximum value of unity.  \n \n  \n28 \n  \nFig. S 4 \nResonance response without the optical pump for the CoFeB sample. Measurement shows the \nout-of-plane component of the magnetization, mz, at 10 GHz. The phase increases monotonically \nwith the field in contrast to Fig. 2B of  the main text. Data  is not normalized.  \n \n \n \n"    },    {        "title": "2004.01245v2.Stable_solitons_in_a_nearly_PT_symmetric_ferromagnet_with_spin_transfer_torque.pdf",        "content": "arXiv:2004.01245v2  [nlin.PS]  6 Apr 2020Stable solitons in a nearly PT-symmetric ferromagnet with spin-transfer torque\nI V Barashenkov∗\nCentre for Theoretical and Mathematical Physics, Universi ty of Cape Town,\nRondebosch 7701, South Africa and Joint Institute for Nucle ar Research, Dubna, Russia\nAlexander Chernyavsky†\nCentre for Theoretical and Mathematical Physics, Universi ty of Cape Town,\nRondebosch 7701, South Africa and Department of Mathematic s and Statistics,\nUniversity of Victoria, Victoria, BC V8P 5C2, Canada\nWe consider the Landau-Lifshitz equation for the spin torqu e oscillator — a uniaxial ferromagnet\nin an external magnetic field with polarised spin current dri ven through it. In the absence of the\nGilbert damping, the equation turns out to be PT-symmetric. We interpret the PT-symmetry\nas a balance between gain and loss — and identify the gaining a nd losing modes. In the vicinity\nof the bifurcation point of a uniform static state of magneti sation, the PT-symmetric Landau-\nLifshitz equation with a small dissipative perturbation re duces to a nonlinear Schr¨ odinger equation\nwith a quadratic nonlinearity. The analysis of the Schr¨ odi nger dynamics demonstrates that the spin\ntorque oscillator supports stable magnetic solitons. The PTnear-symmetry is crucial for the soliton\nstability: the addition of a finite dissipative term to the La ndau-Lifshitz equation destabilises all\nsolitons that we have found.\nKeywords: spin torque oscillator; Landau-Lifshitz equati on; parity-time symmetry; nonlinear Schroedinger\nequation; solitons; stability\nI. INTRODUCTION\nConceived in the context of nonhermitian quantum me-\nchanics [1], the idea of parity-time ( PT) symmetry has\nproved to be useful in the whole range of applied disci-\nplines [2]. A PT-symmetric structure is an open system\nwhere dissipative losses are exactly compensated by sym-\nmetrically arranged energy gain. In optics and photon-\nics, systems with balanced gain and loss are expected to\npromote an efficient control of light, including all-optical\nlow-threshold switching [3, 4] and unidirectional invisi-\nbility [4–6]. There is a growing interest in the context\nof electronic circuitry [7], plasmonics [8], optomechanical\nsystems [9], acoustics [10] and metamaterials [11].\nThis study is concerned with yet another area where\nthe gain-loss balance gives rise to new structures and\nbehaviours, namely, the magnetism and spintronics. In\ncontrast to optics and nanophotonics, where the nonher-\nmitian effects constitute a well-established field of study,\nthe research into PT-symmetric magnetic systems is still\nin its early stages, with only a handful of models set up\nover the last several years.\nOne of the systems proposed in the literature com-\nprises two coupled ferromagnetic films, one with gain\nand the other one with loss [12]. (For an experimen-\ntal implementation of this structure, see [13].) A re-\nlated concept consists of a pair of parallel magnetic\nnanowires, with counter-propagating spin-polarized cur-\nrents [14]. In either case the corresponding mathematical\nmodel is formed by two coupled Landau-Lifshitz equa-\n∗Igor.Barashenkov@uct.ac.za\n†chernya@uvic.cations, with PTsymmetry being realised as a symme-\ntry between the corresponding magnetisation vectors. A\ntwo-spin Landau-Lifshitz system gauge-equivalent to the\nPT-symmetric nonlocal Schr¨ odinger equation is also a\nmember of this class of models [15].\nAn independent line of research concerned the dynam-\nics of a single spin under the action of the spin-transfer\ntorque. Projecting the magnetisation vector onto the\ncomplex plane stereographically and modelling the spin\ntorque by an imaginary magnetic field [16], Galda and\nVinokur have demonstrated the PT-symmetry of the re-\nsulting nonhermitian Hamiltonian [17]. (For the gener-\nalisation to spin chains, see [18]; the nonreciprocal spin\ntransfer is discussed in [19].) Unlike the two-component\nstructures of Refs [12–14], the PT-symmetry of the spin\ntorque oscillator of Galda and Vinokur is an intrinsic\nproperty of an individual spin. It results from the sys-\ntem’s invariance under the simultaneous time reversal\nand the imaginary magnetic field flip [17].\nThe structure we consider in this paper shares a num-\nber of similarities with the spin torque oscillator of Refs\n[17, 18]. (There is also a fair number of differences.) It\nconsists of two ferromagnetic layers separated by a con-\nducting film (Fig 1). The spin-polarised current flows\nfrom a layer with fixed magnetisation to a layer where\nthe magnetisation vector is free to rotate [16, 20].\nA one-dimensional uniaxial classical ferromagnet in\nthe external magnetic field is described by the Landau-\nLifshitz equation [16, 20] (also known as the Landau-\nLifshitz-Gilbert-S/suppress lonczewski equation in the current con-\ntext):\n˙M=−M×M′′−M×H−β(M·ˆz)M׈z\n−γM×M׈z+λM×˙M. (1)2\nFIG. 1. A schematic of the spin torque oscillator. An electri c\ncurrent flows through a nanowire with two ferromagnetic lay-\ners. In the thick layer (on the left) the magnetisation is fixe d\n(through large volume, large anisotropy or pinning by addi-\ntional underlayers). This causes a polarisation of the pass ing\nelectron spins. The polarised current exerts torque on the\nthin layer (on the right) where the magnetisation is governe d\nby the Landau-Lifshitz equation (1).\nHere the overdot stands for the time derivative and the\nprime indicates the derivative with respect to x. In equa-\ntion (1), the variables have been non-dimensionalised so\nthat the magnetisation vector M= (Mx,My,Mz) lies on\na unit sphere: M2= 1. The magnetic field is taken to\nbe constant and directed horizontally: H= (H0,0,0).\nThe anisotropy axis is z, with ˆz= (0,0,1). The positive\nand negative constant βcorresponds to the easy-axis and\neasy-plane anisotropy, respectively. (Note that the au-\nthors of [17, 18] considered the ferromagnet anisotropic\nalong thexaxis.) The fourth term in the right-hand side\nof (1) — the S/suppress lonczewski term — accounts for the spin\ntransfer by the current that passes through an external\nferromagnetic layer that has a fixed magnetisation in the\ndirection ˆz. The last term is the Gilbert damping term.\nThe damping coefficient λis positive; the field H0and the\ncurrent amplitude γcan also be chosen positive without\nloss of generality.\nIn this paper, we study the nonlinear dynamics of the\nlocalised solutions of the equation (1), both with small\nand finite-strength damping.\nA class of soliton solutions of the Landau-Lifshitz equa-\ntion (1) was obtained by Hoefer, Silva and Keller [21].\nThe solitons discovered by those authors are dissipative\nanalogs of the Ivanov-Kosevich magnon droplets [22].\n(For the experimental realisation, see [23].) Our setup\nhas a different geometry from the one of Hoefer et al.\nOne difference is that we consider the magnetic layer with\na parallel anisotropy while the magnon droplets require\na perpendicular one [21–23]. An additional distinction\nis that our vector His orthogonal to the direction of\nthe fixed magnetisation — while the magnetic field in\nRef [21] was not. Because of the different geometry, the\nLandau-Lifshitz equation of Ref [21] does not exhibit the\nPTinvariance. The dissipative magnon droplets are sus-\ntained through the competition of torque and damping,\nthe two actors represented by terms of different mathe-\nmatical form, rather than by a symmetric balance of twosimilar but oppositely-directed effects.\nAnother class of localised structures in the spin torque\noscillator is commonly referred to as the standing spin\nwave bullets . These have been theoretically predicted by\nSlavin and Tiberkevich [24] — outside the context of the\nLandau-Lifshitz equation. (For the experimental realisa-\ntion, see [25].) The spin wave bullets are found in the\nmagnetic layer with parallel anisotropy, when the mag-\nnetic field is directed parallel to the fixed layer’s mag-\nnetisation. The direction of the vector His what makes\nour geometry different from the setup considered in Ref\n[24]. Like the magnon droplets, the spin wave bullets are\nsustained by an asymmetric balance of the spin torque\nand finite-strength damping.\nThe paper is organised as follows. We start with the\ndemonstration of the gain-loss balance in the Landau-\nLifshitz equation with the vanishing Gilbert damping.\nThisPT-symmetric system and systems that are close\nto it will prove to have special properties in this paper,\nwhere we consider equations both with small and finite λ.\nIn section III we classify stability and bifurcation of four\nnonequivalent stationary states with uniform magnetisa-\ntion. Three of those states are found to be admissible\nas stable backgrounds for localised structures. In the\nvicinity of the bifurcation points, the dynamics of the lo-\ncalised structures are governed by quadratic Schr¨ odinger\nor Ginsburg-Landau equations, depending on whether\nthe Gilbert damping is weak or finite-strength (section\nIV). Despite the absence of the dissipative terms, our\nquadratic Schr¨ odinger equations are not conservative;\nhowever one of them obeys the PT-symmetry. Both\nGinsburg-Landau equations and their Schr¨ odinger coun-\nterparts — PT-symmetric or not — support two types\nof soliton solutions. We show that either of these types\nis only stable in the PT-symmetric situation (sections\nV-VI). Section VII summarises results of this study.\nII. GAIN-LOSS BALANCE IN THE ABSENCE\nOF GILBERT LOSSES\nThe equation (1) is nonconservative due to the pres-\nence of the spin torque and Gilbert’s dissipative term. In\nspin torque oscillators, solitons are expected to exist due\nto the energy supplied by torque being offset by finite-\nstrength dissipation [21]. However when λ= 0, the spin\nhamiltonian modelling our structure is PT-symmetric\n[17] and therefore some form of the gain-loss balance\nshould occur in this case as well, despite the absence of\nthe Gilbert damping. To uncover the gain-loss competi-\ntion intrinsic to the spin torque, we define two complex\nfields,u(x,t) andv(x,t), related to the magnetisation\nvectorMvia the Hopf map:\nMx=v∗u+u∗v, M y=i(u∗v−v∗u), Mz=|u|2−|v|2.\n(2)\nWhen the magnetisation is spatially uniform, ∂M/∂x=\n0, the equation (1) with λ= 0 can be reformulated as a3\nnonlinear Schr¨ odinger dimer:\niut+H0\n2v+β\n2(|u|2−|v|2)u=iγ|v|2u, (3)\nivt+H0\n2u−β\n2(|u|2−|v|2)v=−iγ|u|2v. (4)\nAccording to equations (3)-(4), the external energy is fed\ninto theu-mode and dissipated by its vcounterpart. The\nmagnetic field H0couplesutov, carrying out the energy\nexchange between the two modes.\nThe sustainability of the gain-loss balance in the sys-\ntem (3)-(4) is reflected by its invariance under the prod-\nuct of the PandTtransformations. Here the inversion\nPswaps the two modes around,\nP:u→v, v→u, (5)\nwhileTrepresents the reflection of time:\nT:t→ −t, u→u∗, v→v∗. (6)\nThese transformations admit a simple formulation in\nterms of the components of magnetisation (2):\nP:My→ −My, Mz→ −Mz (7)\nand\nT:t→ −t, M y→ −My. (8)\nThe involutions (7) and (8) remain relevant in the anal-\nysis of the equation (1) with the x-dependent magneti-\nsation. Here one can either leave the parity operation in\nthe form (7) or include the inversion of the xcoordinate\nin this transformation:\nP:x→ −x, M y→ −My, Mz→ −Mz. (9)\nWriting the vector equation (1) in the component form,\n\n\n˙Mx=MzM′′\ny−MyM′′\nz−βMyMz\n−γMxMz+λ(My˙Mz−Mz˙My),\n˙My=MxM′′\nz−MzM′′\nx−H0Mz+βMxMz\n−γMyMz+λ(Mz˙Mx−Mx˙Mz),\n˙Mz=MyM′′\nx−MxM′′\ny+H0My\n+γ(M2\nx+M2\ny) +λ(Mx˙My−My˙Mx),(10)\none readily checks that in the conservative limit ( γ=λ=\n0), the Landau-Lifshitz equation is invariant under the\nP- andT-involutions individually. The equation with\nthe spin torque term added ( γ/ne}ationslash= 0) is invariant under\nthe product ( PT) transformation only. Accordingly, the\nequation with the γ-term is a PT-symmetric extension of\nthe conservative Landau-Lifshitz equation. Finally, the\naddition of the Gilbert damping term ( λ/ne}ationslash= 0) breaks the\nPT-symmetry.III. UNIFORM STATIC STATES\nThe uniform static states are space- and time-\nindependent solutions of equation (1) satisfying M2= 1.\nThese are given by fixed points of the dynamical system\n\n\n˙Mx=−βMyMz−γMxMz+λ(My˙Mz−Mz˙My),\n˙My=(βMx−H0−γMy)Mz+λ(Mz˙Mx−Mx˙Mz),\n˙Mz=H0My+γ(M2\nx+M2\ny)+λ(Mx˙My−My˙Mx)\n(11)\non the surface of the unit sphere.\nOnce a fixed point M(0)has been determined, we let\nM=M(0)+δM, linearise the system (10) in δM, and\nconsider solutions of the form\nδM=meµt−ikx, (12)\nwherem= (mx,my,mz)Tis a real constant vector and k\na real wavenumber that may take values from −∞to∞.\nWe call the uniform static state unstable if at least one\nof the roots µof the associated characteristic equation\nhas a positive real part in some interval of k. Otherwise\nthe state is deemed stable.\nSince the equation (1) conserves the quantity M2, the\ndifference between/parenleftbig\nM(0)/parenrightbig2and the square of the vector\nM(0)+δMwill be time-independent:\n∂\n∂t/parenleftBig\n2δM·M(0)/parenrightBig\n= 0.\nSubstituting from (12) and assuming µ/ne}ationslash= 0, this gives\nm·M(0)= 0. (13)\nEquation (13) implies (a) that the time-dependent per-\nturbations of the uniform static states lie on the unit\nsphere; and (b) that the characteristic equation may not\nhave more than two nonzero roots, µ1andµ2. The third\nroot (µ3) has to be zero.\nApart from classifying stability of the uniform static\nstates, it is useful to know which of these solutions can\nserve as backgrounds to static magnetic solitons. To weed\nout a priori unsuitable cases, we set µ= 0 in the char-\nacteristic equation and consider k2as a new unknown\n(rather than a parameter that varies from 0 to ∞). If\nall roots (k2)nof the resulting equation are real positive,\nthere can be no localised solutions asymptotic to the uni-\nform static state M(0)asx→ ±∞ . On the other hand,\nif there is at least one negative or complex root, the solu-\ntionM(0)remains a candidate for solitons’ background.\nA. Equatorial fixed points on the unit sphere\nOne family of time-independent solutions of the system\n(11) describes a circle on the ( Mx,My)-plane:\nM2\nx+/parenleftbigg\nMy+H0\n2γ/parenrightbigg2\n=H2\n0\n4γ2, Mz= 0.4\nFIG. 2. The phase portrait of the dynamical system (11) with\nH0>/radicalbig\nβ2+γ2(a) andH0< γ(b). In (a), two dots on the\nequator of the unit sphere mark the fixed points of the vector\nfield: the western (blue) and eastern point (red). In (b), the\nblue dot indicates the northern and the red dot the southern\nfixed point. Apart from the fixed points, the figures show a\nfewrepresentativetrajectories; physically, thesecorre spondto\nspatially-uniform evolutions of magnetisation. (The port raits\nin (a) and (b) are for λ= 0.)\nImposing the constraint M2= 1 leaves us with just two\nmembers of the family:\nM(0)\nx=±/radicalBig\n1−γ2/H2\n0, M(0)\ny=−γ/H0, M(0)\nz= 0.\n(14)\nIn the system with γ/ne}ationslash= 0, these fixed points are born\nasH0is increased through the value H0=γ. Since the\npoints (14) lie on the equator of the unit sphere, we will\nbe referring to them simply as the equatorial fixed points,\nthe eastern ( M(0)\nx>0) and the western ( M(0)\nx<0) one.\nFig 2(a) depicts the equatorial fixed points in the phase\nportrait of the dynamical system (11).\nLinearising equation (1) about the uniform static state\ncorresponding to an equatorial fixed point, we obtain two\nnonzero stability eigenvalues\nµ1,2=−λ(2K−β)±/radicalbig\nλ2β2−4K(K−β)\n2(1 +λ2),(15a)\nK=k2+H0M(0)\nx.(15b)Making use of (15) it is not difficult to see that in the\neasy-plane or isotropic ferromagnet (i.e. in the situation\nwhereβ≤0), the eastern uniform static state ( M(0)\nx>0)\nis stable irrespective of the choice of γ,λandH0. On\nthe other hand, when the anisotropy is easy-axis ( β >0),\nthe eastern state is stable if\nH0≥/radicalbig\nβ2+γ2 (16)\nand unstable otherwise.\nTo check the suitability of the eastern uniform static\nstate as a background for solitons, we set µ= 0 in\nthe expression (15a); this transforms it into a quadratic\nequation for k2. Whenβ </radicalbig\nH2\n0−γ2, both roots\nof this equation are negative: k2=−/radicalbig\nH2\n0−γ2and\nk2=β−/radicalbig\nH2\n0−γ2. This implies that there is a pair of\nexponentials exp( −ik1,2x) decaying to zero as x→ −∞\nand another pair decaying as x→+∞. Therefore\nthe uniform static state (14) with M(0)\nx>0 can serve\nas a background to solitons for any set of parameters\nβ,γ,H 0,λin its stability domain.\nTurning to the west-point solution ( M(0)\nx<0 in equa-\ntion (14)), a simple analysis of the eigenvalues (15) indi-\ncates that there are wavenumbers ksuch that Re µ >0\nfor any quadruplets of β,γ,H 0andλ. Hence the west-\nern uniform static state is always unstable. We are not\nconsidering it any further.\nB. Latitudinal fixed points\nAnother one-parameter family of constant solutions\nof the equation (1) forms a vertical straight line in the\nMx,My,Mz-space:\nMx=H0β\nβ2+γ2, My=−H0γ\nβ2+γ2,−∞<Mz<∞.\nThe substitution of the above coordinates into M2= 1\nselects two fixed points on the unit sphere:\nM(0)\nx=H0β\nβ2+γ2, M(0)\ny=−H0γ\nβ2+γ2,\nM(0)\nz=±/radicalBigg\n1−H2\n0\nβ2+γ2. (17)\nSince these points lie above and below the equatorial\n(Mx,My)-plane, we will be calling them the latitudinal\nfixed points: the northern ( M(0)\nz>0) and the southern\n(M(0)\nz<0) point. See Fig.2(b).\nIn the anisotropic equation ( β/ne}ationslash= 0) the northern and\nsouthern points are born as H0is decreased through/radicalbig\nβ2+γ2. In this case, there is a parameter interval\nγ < H 0</radicalbig\nβ2+γ2where two pairs of fixed points,\nlatitudinal and equatorial, coexist.\nThe bifurcation diagram for the isotropic equation is\ndifferent. When β= 0, the latitudinal fixed points (17)5\nemerge as the eastern and western points (14) converge\nand split out of the equatorial plane. In this case, there\nis just one pair of uniform static states for any H0/ne}ationslash=γ:\nthe equatorial pair for H0> γ and the latitudinal pair\nforH0<γ.\nThe linearisation of equation (1) about the uniform\nstatic state corresponding to the latitudinal fixed-point\n(17) gives\nµ1,2=−λQ+γM(0)\nz±/radicalBig\n1\n4λ2β2h2−P(P−βh)\n1 +λ2,(18)\nwhere\nP=k2+β−γλM(0)\nz, Q =k2+β/parenleftbigg\n1−h\n2/parenrightbigg\nand\nh=H2\n0\nβ2+γ2. (19)\nA simple analysis demonstrates that when β≥0, the\nnorth-point solution ( M(0)\nz>0 in (17)) is stable regard-\nless of the values of λ≥0,H0andγ >0. As for the easy-\nplane anisotropy ( β < 0), the northern uniform static\nstate is stable only when the inequalities λ≤λcand\nH0≤Hcare satisfied simultaneously. Here\nλc=γ\n|β|√\n1−h\n1−h/2(20)\nand\nHc=/radicalBigg\n2γ(β2+γ2)\nβ2/parenleftBig/radicalbig\nβ2+γ2−γ/parenrightBig\n. (21)\nNote thatHcis smaller than/radicalbig\nγ2+β2; hence the re-\ngionH0≤Hclies entirely within the northern point’s\nexistence domain (defined by the inequality H0</radicalbig\nγ2+β2).\nFinally, we consider the eigenvalues pertaining to the\nsouthern uniform static state ( M(0)\nz<0 in (17)). Our\nconclusion here is that in the isotropic and easy-plane\nferromagnet ( β≤0), this solution is unstable regardless\nof the choice of other parameters. In the easy-axis situ-\nation (β >0), the southern state is stable if λ≥λcwith\nλcas in (20) — and unstable otherwise.\nTo determine the parameter region where the north-\nand south-point solutions can serve as backgrounds for\nsolitons, we set µ= 0 in (18). In each of the two cases,\nthe resulting quadratic equation for k2has two positive\nroots only if β <0 is satisfied along with the inequality\nH0> Hc, whereHcis as in (21). This is the only no-\ngo region for solitons. Outside this region, the quadratic\nequation has either two negative or two complex roots;\nthe corresponding uniform static states can serve as soli-\ntons’ asymptotes.\nThe bottom line is that either of the two latitudinal\nuniform static states is suitable as a background for soli-\ntons in its entire stability domain.C. Summary of uniform static states\nFor convenience of the reader, the stability properties\nof the constant solutions corresponding to the four fixed\npoints are summed up in Table I.\nBefore turning to the perturbations of these uniform\nstatic states, it is worth noting their symmetry proper-\nties. Each of the equatorial states is PT-symmetric in the\nsense that each of these two solutions is invariant under\nthe product of the transformations (9) and (8). In con-\ntrast, neither of the two latitudinal states is invariant; the\nPToperator maps the northern solution to southern and\nthe other way around. The different symmetry proper-\nties of the equatorial and longitudinal solutions will give\nrise to different invariances of equations for their small\nperturbations.\nFixed-point\nβ <0 β= 0 β >0\nsolution:\neastern stable stablestable if\nH0≥/radicalbig\nβ2+γ2\nwestern unstable unstable unstable\nnorthernstable if λ≤λcstable stable\nandH0≤Hc\nsouthern unstable unstablestable\nifλ≥λc\nTABLE I. Stability of four constant solutions of equation (1 ).\nIV. SLOW DYNAMICS NEAR BIFURCATION\nPOINTS\nA. Perturbation of equatorial fixed point\nConsider the eastern point of the pair of equatorial\nfixed points (14):\nM(0)=/parenleftBigg/radicalBigg\n1−γ2\nH2\n0,−γ\nH0,0/parenrightBigg\n. (22)\nWe assume that the parameters β,γ,λandH0lie in the\nstability domain of the uniform static state (22).\nThe plane orthogonal to the vector M(0)is spanned by\nthe vectors\nA= (0,0,1),B=/parenleftBigg\nγ\nH0,/radicalBigg\n1−γ2\nH2\n0,0/parenrightBigg\n.\nThe unit vector Mcan be expanded over the orthonormal\ntriplet{A,B,M(0)}:\nM=ηA+ξB+χM(0).6\nLettingM(x,t)→M(0)asx→ ±∞ , the coefficient fields\nη,ξandχhave the following asymptotic behaviour:\nη→0, ξ→0, χ→1 as|x| → ∞.\nThe complex field Ψ = ξ+iηsatisfies\ni˙Ψ =χΨ′′−Ψχ′′+λ(Ψ ˙χ−χ˙Ψ)−/radicalBig\nH2\n0−γ2Ψ\n+γ(χ−iηξ−1 +η2) +iβηχ, (23)\nwhereχ=/radicalbig\n1−|Ψ|2while the prime and overdot in-\ndicate the derivative with respect to xandt, respec-\ntively. Note that when λ= 0, the equation (23) is PT-\nsymmetric, that is, invariant under a composite transfor-\nmation consisting of three involutions: t→ −t,x→ −x,\nand Ψ→Ψ∗.\nAssume that H0is close to the bifurcation point of the\nuniform static state (22) — that is, H0is slightly greater\nthanγ. In this case, Ψ will depend on a hierarchy of\nslow times Tn=ǫntand stretched spatial coordinates\nXn=ǫn/2x, wheren= 1,3,5,...and the small parameter\nǫis defined by\nǫ2= 1−γ2\nH2\n0.\nIn the limit ǫ→0 the new coordinates become indepen-\ndent so we can write\n∂\n∂t=ǫD1+ǫ3D3+...;\n∂2\n∂x2=ǫ∂2\n1+ 2ǫ2∂1∂3+ǫ3(∂2\n3+ 2∂1∂5) +... ,\nwhereDn=∂/∂Tnand∂n=∂/∂Xn. Assume, in addi-\ntion, that the anisotropy constant βis of orderǫand let\nβ=ǫBwithB=O(1). Considering small ηandξ, we\nexpand\nΨ =ǫψ1+ǫ3ψ3+... .\nSubstituting the above expansions in (23), we equate\ncoefficients of like powers of ǫ. The order ǫ2gives a\nGinsburg-Landau type of equation with a quadratic non-\nlinearity:\n(i+λ)D1ψ−∂2\n1ψ+γ\n2ψ2=−γψ+B\n2(ψ−ψ∗).(24)\n(Hereψis just a short-hand notation for ψ1.)\nNote that in the derivation of (24) we took λto be\nO(1). If we, instead, let λ=O(ǫ), the dissipative term\nwould fall out of the equation (24) and we would end up\nwith a nonlinear Schr¨ odinger equation:\niD1ψ−∂2\n1ψ+γ\n2ψ2=−γψ+B\n2(ψ−ψ∗). (25)\nThe quadratic Schr¨ odinger equation (25) does not have\nthe U(1) phase invariance. However, the equation is PT-\nsymmetric, that is, invariant under the composite map\nt→ −t,x→ −x,ψ→ψ∗. As we will see in section V,\nthis discrete symmetry is enough to stabilise solitons.B. Perturbation of latitudinal fixed points\nChoosing the background in the form of one of the two\nlatitudinal fixed points\nM(0)=/parenleftbiggβ\nH0h,−γ\nH0h,±√\n1−h/parenrightbigg\n, (26)\nwe letM(x,t) approach the same point M(0)asx→ ±∞ .\nIn (26),his defined by the equation (19).\nAs in the previous subsection, we expand the magneti-\nsation vector over an orthonormal basis {A,B,M(0)}:\nM=ηA+ξB+χM(0), (27)\nwhere, this time,\nA=/parenleftbigg\n∓β\nH0/radicalbig\nh(1−h),±γ\nH0/radicalbig\nh(1−h),√\nh/parenrightbigg\nand\nB=/parenleftbiggγ\nH0√\nh,β\nH0√\nh,0/parenrightbigg\n.\nWe assume that H0is close to the bifurcation point\nwhere the northern and southern fixed points are born\n(that is,H0is slightly smaller than/radicalbig\nβ2+γ2) and define\na small parameter ǫ:\nh= 1−ǫ2.\nAs in the analysis of the equatorial fixed points, we let\nβ=ǫB, whereB=O(1). Assuming that the magnetisa-\ntionMis just a small perturbation of M(0), we expand\nthe small coefficients in (27) in powers of ǫ:\nη=ǫη1+ǫ3η3+..., ξ =ǫξ1+ǫ3ξ3+... .\nThe constraint η2+ξ2+χ2= 1 implies then\nχ= 1−ǫ2η2\n1+ξ2\n1\n2+... .\nSubstituting these expansions in the Landau-Lifshitz\nequation (10) and equating coefficients of like powers of\nǫ, the order ǫ2gives\nD1ξ1=∂2\n1η1−λD1η1−γ(η1ξ1±ξ1)\nand\nD1η1=λD1ξ1−∂2\n1ξ1+Bξ1∓γη1−γ\n2(η2\n1−ξ2\n1).\nThe above two equations can be combined into a single\nequation for the complex function ψ=ξ1+iη1:\n(i+λ)D1ψ−∂2\n1ψ+γ\n2ψ2=∓iγψ−B\n2(ψ+ψ∗).(28)\nThe Ginsburg-Landau equation (28) resembles the\nequation (24) governing the dynamics near the equato-\nrial uniform static state; however there is an important\ndifference. Namely, even if we let λ= 0 in (28) [that is,\neven if we assume that the damping is O(ǫ) or weaker\nin the Landau-Lifshitz-Gilbert equation (1)], the result-\ning nonlinear Schr¨ odinger equation will notbecomePT-\nsymmetric. This fact will have important repercussions\nfor the stability of solitons.7\n5\n4\n3\n2\n1\n0\n10\n5\n0-505\n2\n1.5\n1\n0.5\n0\n10\n5\n0-505\nFIG. 3. Instability of the fundamental soliton in the presen ce of damping. This evolution was obtained by the direct nume rical\nsimulation of the equation (30) with b= 0 and λ= 0.1. The initial condition was in the form of the soliton (31) pe rturbed by\na random perturbation within 5% of the soliton’s amplitude. The spatial interval of simulation was ( −58,58); in the plot it\nhas been cut down for visual clarity.\nV. SOLITON EXCITATIONS OF EQUATORIAL\nSTATE\nLetting\nu(x,t) =−1\n3ψ(X1,T1), x =√γ\n2X1, t =γ\n4T1,(29)\nthe Ginsburg-Landau equation (24) is cast in the form\n(i+λ)ut−uxx−6u2=−4u+b(u−u∗), (30)\nwhereb= 2B/γ. (We alert the reader that the scaled\nvariablesxandtdo not coincide with the original xandtof the Landau-Lifshitz equation (1). We are just re-\nemploying the old symbols in a new context here.)\nIn the present section we consider localised solutions of\nthe equation (30) approaching 0 as |x| → ∞ . Regardless\nofλ, the zero solution is stable if b≤2 and unstable\notherwise. This inequality agrees with the stability range\n(16) of the eastern uniform static state within the original\nLandau-Lifshitz equation. (Note that the term bu∗plays\nthe role of the parametric driver in (30) [26]; the above\nstability criterion states that the zero solution cannot\nsustain drivers with amplitudes greater than b= 2.)\nA. Fundamental soliton and its stability\nEquation (30) has a stationary soliton solution:\nus= sech2x. (31)\nTo distinguish it from localised modes with internal\nstructure, we refer to this solution as the fundamental\nsoliton — or simply sech mode . Letting\nu(x,t) =us(x) +ε[f(x) +ig(x)]eµt\nand linearising in small ε, we obtain an eigenvalue prob-\nlem\nµ(g−λf) =Hf, (32a)\n−µ(f−λg) = (H− 2b)g, (32b)\nwith the operator\nH=−d2/dx2+ 4−12 sech2x. (33)The vector eigenvalue problem (32) is reducible to a\nscalar eigenvalue problem of the form\n(H−b+µλ)2g+ (µ2−b2)g= 0.\nThe stability exponents µare roots of the quadratic equa-\ntion\n(E−b+µλ)2+µ2−b2= 0,\nwhereEis an eigenvalue of the operator H:Hy=Ey.\nThe two roots are\nµ(±)=λ(b−E)±/radicalbig\nλ2b2+E(2b−E)\n1 +λ2. (34)\nThe eigenvalues of the P¨ oschl-Teller operator (33) are\nE0=−5,E1= 0, and E2= 3, with the eigen-\nfunctionsy0= sech3x,y1= sech2xtanhxandy2=8\n1.5\n1\n0.5\n02 \u0000\n15\n105\n-505\n0\n0.1\n0.05\n0\n-0.05\n-0.1\u0001 \u0002\n15\n10\n5\n0-505\nFIG. 4. The evolution of the initial condition in the form of a gaussian, u(x,0) = exp( −x2), in the equation (30) with λ= 0\nandb= 0. Left panel: Re u; right panel: Im u. The emerging solution is a breather with a small imaginary p art and the real\npart close to the soliton (31). Note that the figure shows only a portion of the full simulation interval ( −58,58).\nsechx/parenleftbig\n1−5\n4sech2x/parenrightbig\n, respectively. The continuous spec-\ntrum occupies the semiaxis Econt≥4, with the edge\neigenfunction given by y3= tanhx/parenleftbig\n1−5\n3tanh2x/parenrightbig\n. For\neach eigenvalue En,n= 0,1,2, equation (34) yields two\nroots,µ(+)\nnandµ(−)\nn.\nIn the analysis of the roots (34) we need to distinguish\nbetween two situations: damped ( λ>0) and undamped\none (λ= 0). Assume, first, that λ >0 and let, in addi-\ntion,b≥0. It is not difficult to check that the root µ(+)\nn\nwill have a positive real part provided the corresponding\neigenvalueEnsatisfiesEn<2b. On the other hand, the\nset of three eigenvalues of the operator (33) does include\na negative eigenvalue ( E0) that satisfies E0<2bregard-\nless of the particular value of b≥0. Therefore the soliton\nhas an exponent µ(+)\n0with Reµ(+)\n0>0 for anyb≥0.\nIn the case where λ>0 butb <0, the root µ(+)\nnwill\nhave a positive real part provided EnsatisfiesEn<0.\nAs in the previous case, this inequality is satisfied by the\neigenvalueE0so that the soliton has an exponent with\nReµ(+)\n0>0 for anyb<0.\nWe conclude that the fundamental soliton of the equa-\ntion (30) is unstable in the presence of damping — re-\ngardless of the sign and magnitude of the anisotropy co-\nefficientb. Figure 3 illustrates the evolution of a weakly\nperturbed soliton in the Ginsburg-Landau equation with\nλ/ne}ationslash= 0.\nTurning to the situation with λ= 0 we assume, first,\nthatb>0. The equations (34) will give a pair of oppo-\nsite real roots µ(±)\nnif the corresponding eigenvalue sat-isfies 0< En<2band a pair of pure imaginary roots\notherwise. The only positive eigenvalue of the operator\n(33) isE2= 3; it satisfies the above inequality if b>3/2.\nIn the situation where λ= 0 butb <0, the pair of\nopposite exponents µ(±)\nnis real ifEnfalls in the interval\n2b < E n<0 and pure imaginary if Enlies outside this\ninterval. The only negative eigenvalue is E0=−5; it falls\nin the interval in question if b<−5/2.\nFinally, in the isotropic ferromagnet ( b= 0) the sta-\nbility exponents are all pure imaginary: µ(±)\nn=±iEn.\nCombining the intervals where all exponents are pure\nimaginary gives us the stability region of the undamped\nfundamental soliton in terms of the anisotropy to spin-\ncurrent ratio:\n−5\n2≤b≤3\n2. (35)\nB. Twisted modes in isotropic ferromagnet\nThe Ginsburg-Landau equation (30) with b= 0 admits\nan additional pair of localised solutions:\nuT= 2sech2(2x)±2isech(2x) tanh(2x). (36)\nThe modulus of uT(x) is bell-shaped while its phase\ngrows or decreases by πasxchanges from −∞ to +∞.\nThe solution looks like a pulse twisted by 180◦in the\n(Reu,Imu)-plane. In what follows, we refer to each of\nequations (36) as a twisted , or simply sech-tanh , mode.\nLinearising equation (30) about the twisted mode (36)\nand assuming that the small perturbation depends ontime aseµt, we arrive at an eigenvalue problem\nLf(X) =−µ\n4(λ+i)f(X) (37)9\nfor the Schr¨ odinger operator with the Scarff-II complex\npotential:\nL=−d2\ndX2+ 1−6sech2X∓6isechXtanhX. (38)\nIn (37)-(38), X= 2x.\nThePT-symmetric operator (38) has an all-real spec-\ntrum including three discrete eigenvalues [27]. Let yn\nbe the eigenfunction associated with an eigenvalue En:\nLyn=Enyn. The eigenvalue-eigenfunction pairs are\nthen given by\nE0=−5\n4, y0= (sech2X±isechXtanhX)3/2;\nE1= 0, y1= sechX(sechX±itanhX)2, (39)\nandE2= 3/4 with\ny2= (3±2isinhX)(sech2X±isechXtanhX)3/2.(40)\nEach of the eigenvalues Engives rise to a stability ex-\nponent\nµn= 4i−λ\n1 +λ2En\nin equation (37). When the dissipation coefficient λ>0,\nthe exponent pertaining to the negative eigenvalue E0\nhas a positive real part. Accordingly, the twisted modes\n(36) are unstable in the presence of damping. In con-\ntrast, when λ= 0, all exponents µn(n= 0,1,2) are pure\nimaginary so the twisted modes are stable.\nC. Oscillatory modes\nAn interesting question is whether there are any other\nstable localised structures — in particular, in the situa-\ntion where the equation (30) has zero damping. Figure\n4 illustrates the evolution of a gaussian initial condition\nu(x,0) = exp( −x2) that can be seen as a nonlinear per-\nturbation of the soliton (31). The gaussian evolves into\nan oscillatory localised structure (a kind of a breather)\nwhich remains close to the soliton (31) — but does not\napproach it as t→ ∞ . This observation suggests that\nequation (30) with λ= 0 has a family of stable time-\nperiodic spatially localised solutions, with the stationary\nsoliton (31) being just a particular member of the family.\nIt is fitting to note that the existence of breather fam-\nilies is common to nonlinear PT-symmetric equations\n[28]. Breathers prevail among the products of decay of\ngeneric localised initial conditions [28, 29].\n2\n1.5\n1\n0.5\n0\n5\n0\n-5-505\n2\n1.5\n1\n0.5\n0\n0\n1\n2\n3\n4\n5202224262830\nFIG. 5. Localised solutions of the quadratic Schr¨ odinger\nequation on the plane: the stationary soliton (a) and a\nbreather (b). Both figures were produced by direct numer-\nical simulations of equation (42) with b= 0. In panel (a), the\ninitial condition was taken in the form of the soliton (43) pe r-\nturbed by a random perturbation within 5% of the soliton’s\namplitude. After t= 100, the solution (shown in the panel)\nremains close to the soliton. In panel (b), the initial condi tion\nwas chosen as u= 1.6 exp(−r2). After an initial transient,\nthe solution settles to a localised oscillatory state shown in\nthe figure.\nD. Stable solitons in two dimensions\nWe close this section with a remark on the Landau-\nLifshitz-Gilbert-S/suppress lonczewski equation in two dimensions:\n∂M\n∂t=−M×∇2M−M×H−β(M·ˆz)M׈z\n−γM×M׈z+λM×∂M\n∂t.(41)\nHere∇2=∂2\n∂x2+∂2\n∂y2. Assuming that H0is only slightly\naboveγand that the anisotropy βand damping λare\nsmall, we consider a perturbation of the east-point uni-10\nform state (22). Following the asymptotic procedure out-\nlined in section IV A, the equation (41) is reducible in this\nlimit to a planar Schr¨ odinger equation:\niut=uxx+uyy+ 6u2−4u+b(u−u∗), (42)\nwhere\nb=2H0\nγβ/radicalbig\nH2\n0−γ2.\nLike its one-dimensional counterpart (25), equation\n(42) isPT-symmetric. The PT-operation can be cho-\nsen, for instance, in the form\nt→ −t, x→ −x, y→ −y, u→u∗.\nThe quadratic Schr¨ odinger equation (42) has a static\nradially-symmetric soliton solution,\nus(x,y) =R(r), (43)\nwhereR(r) is a nodeless (bell-shaped) solution of the\nboundary-value problem\nRrr+1\nrRr−4R+ 6R2= 0,\nRr(0) = 0,R(r)→0 asr→ ∞.\nPostponing the detailed stability analysis of the soliton\n(43) to future publications, we restrict ourselves to the\nsimplest case of isotropic ferromagnet, b= 0. A numeri-\ncal simulation of equation (42) with the initial condition\nin the form of the noise-perturbed soliton (43) indicates\nthat the soliton is stable against small perturbations.\n[See Fig 5(a).] On the other hand, generic localised initial\nconditions evolve into time-periodic breather-like states\n[Fig 5(b)]. This suggests that the quadratic Schr¨ odinger\nequation (42) [and hence the planar Landau-Lifshitz\nequation (41)] supports a broad class of stable stationary\nand oscillatory localised structures.\nVI. SOLITON EXCITATIONS OF\nLATITUDINAL STATE\nThe scaling transformation (29) takes the equations\n(28) to the nondimensional form\n(i+λ)ut−uxx−6u2=∓4iu−b(u+u∗). (44)\nAs in section V, b= 2B/γhere.\nIn what follows, we confine ourselves to the analysis of\nthe isotropic equations ( b= 0) as it is the only regime\nwhere we were able to obtain soliton solutions of (44). In\nthe isotropic case, the u= 0 solution of the top-sign equa-\ntion in (44) is stable and that of the bottom-sign equation\nunstable — regardless of whether λis zero or not. (This\nagrees with the stability properties of the north and south\nfixed-point solutions of the Landau-Lifshitz equation; see\nsection III B.) Hence we only keep the top-sign equation\nin what follows.A.sechmode\nLettingb= 0, the top-sign equation in (44) can be\nfurther transformed to\n(1−iλ)wt=wzz−4w+ 6w2, (45)\nwhere\nw(z,t) =−iu, z =eiπ/4x. (46)\nAn obvious static solution of the equation (45) is ws=\nsech2z; the corresponding solution of the original equa-\ntion (44) is\nus(x) =isech2/parenleftbig\neiπ\n4x/parenrightbig\n. (47)\nThe solution (47) decays to zero as x→ ±∞ and does\nnot have singularities on the real line. Similar to the\nsolution (31) over the equatorial background, we term\nthe solution (47) the sech soliton .\nTo classify the stability of the soliton (47), we linearise\nequation (45) about ws= sech2z. Assuming that the\nsmall perturbation depends on time as eµt, we obtain\nµ=−1 +iλ\n1 +λ2E, (48)\nwhereEis an eigenvalue of the P¨ oschl-Teller operator\nH=−d2\ndz2+ 4−12 sech2z.\nThe operator acts upon functions y(z) defined on the line\nz=eiπ/4ξ(−∞< ξ <∞) on the complex- zplane and\nsatisfying the boundary conditions y→0 asξ→ ±∞ .\nAs discussed in the previous section, the equation\nHy=EywithE=−5 has a solution y0= sech3z. The\nfunction sech3(eiπ/4ξ) is nonsingular for all −∞<ξ<∞\nand decays to zero as ξ→ ±∞ ; henceE0=−5 is a dis-\ncrete eigenvalue of the operator H. The corresponding\nexponentµin (48) has a positive real part regardless of\nλ. This implies that the sech soliton (47) is unstable\nirrespective of whether λis zero or not.\nB.sech-tanh modes\nApplying the transformation (46) to the solutions\nwT= 2sech2(2z)±2isech(2z) tanh(2z) (49)\nof the equation (45), we obtain a pair of localised solu-\ntions of the original equation (44):\nuT=∓2 sech(2eiπ/4x) tanh(2eiπ/4x) + 2isech2(2eiπ/4x).\n(50)\nBy analogy with solutions (36) over the equatorial back-\nground, we are referring to (50) as the sech-tanh modes .\nLinearising equation (45) about its stationary solutions\n(49) and assuming that the small perturbation depends11\non time as eµt, we obtain the following equation for the\nexponentµ:\nµ=−41 +iλ\n1 +λ2E.\nHereEis an eigenvalue of the Scarff-II operator:\nLy=Ey, (51a)\nL=−d2\ndZ2+ 1−6 sech2Z∓6isechZtanhZ, (51b)\nwithZ= 2z. The eigenvalue problem (51) is posed on\nthe line\nZ=eiπ/4ξ,−∞<ξ<∞ (52)\non the complex- Zplane, with the boundary conditions\ny→0 asξ→ ±∞ .\nThree solutions of the equation (51) are in (39)-(40).\nSinceyn(Z) (n= 0,1,2) are nonsingular and decay to\nzero asZtends to infinity in either direction along the\nline (52), these solutions are eigenfunctions of the opera-\ntorL— and the corresponding Enare eigenvalues. The\nexponentµ0pertaining to the eigenvalue E0=−5/4 has\na positive real part:\nµ0=−41 +iλ\n1 +λ2E0.\nConsequently, the sech-tanh modes (50) are unstable —\nno matter whether λis zero or not.\nC. Summary of one-dimensional solitons\nThe stability properties of six localised modes sup-\nported by the quadratic Ginsburg-Landau equations (30)\nand (44) are summarised in Table II. The Table includes\ntwosech solitons (the fundamental soliton (31) and its\nlatitudinal-background counterpart, equation (47)) and\nfoursech-tanh modes (the twisted modes (36) and their\nlatitudinal analogs (50)).\nNonlinear\nmodeover equatorial\nbackgroundover latitudinal\nbackground\n(withb= 0)\nsech stable if λ= 0\nunstable\nsoliton and−5\n2< b <3\n2\nsech-tanh exist ifb= 0;\nunstable\nmodes stable if λ= 0\nTABLE II. Stability of the stationary nonlinear modes in one\ndimension. Themiddlecolumnclassifies solutionsoftheequ a-\ntion(30)whiletheright-handcolumncorrespondstosoluti ons\nof (44).VII. CONCLUDING REMARKS\nWe have studied nonlinear structures associated with\nthe spin torque oscillator — an open system described\nby the Landau-Lifshitz-Gilbert-S/suppress lonczewski equation. In\nthe limit of zero damping ( λ= 0), this nonconservative\nsystem is found to be PT-symmetric. The nearly -PT\nsymmetric equation corresponds to small nonzero λ. In\nthis paper, we have considered both nearly-symmetric\nand nonsymmetric oscillators (small and moderate λ).\nThe spin torque oscillator has four stationary states of\nuniform magnetisation; they are described by four fixed\npoints on the unit M-sphere. Two of these states have\ntheir magnetisation vectors lying in the equatorial plane\nof the unit sphere while the other two correspond to\nfixed points in the northern and southern hemisphere, re-\nspectively. We have assumed that the external magnetic\nfieldH0has been tuned to values ǫ2-close to the bifurca-\ntion points of the “equatorial” and “latitudinal” uniform\nstatic states, and that the ferromagnet is only weakly\nanisotropic: β=O(ǫ). In that limit, small-amplitude lo-\ncalised perturbations of the uniform static states satisfy\nthe Ginsburg-Landau equations — equations (30) and\n(44), respectively.\nIf the damping coefficient λisO(ǫ) or smaller,\neach of the two Ginsburg-Landau reductions becomes a\nquadratic nonlinear Schr¨ odinger equation. Of the two\nSchr¨ odinger equations, the one corresponding to pertur-\nbations of the “equatorial” uniform static state turns out\nto bePT-symmetric. (Thus the asymptotic reduction of\nanearlyPT-symmetric Landau-Lifshitz system is exactly\nPT-symmetric.) This Schr¨ odinger equation proves to be\nquite remarkable. Indeed, despite both our Ginsburg-\nLandau reductions supporting soliton solutions, it is only\nin thePT-symmetric Schr¨ odinger limit that the solitons\nare found to be stable.\nThePT-symmetric Schr¨ odinger equation supports two\ntypes of stable solitons. The constant-phase solution (31)\nis stable in a band of β=O(ǫ) values, extending from the\neasy-axis to the easy-plane region. [The stability band is\ndemarcated by the inequality (35).] On the other hand, a\npair of stable solitons with the twisted phase, equations\n(36), are only supported by the nearly-isotropic ferro-\nmagnet:β=O(ǫ2) or smaller. In addition to stable\nstatic solitons, the PT-symmetric Schr¨ odinger equation\nexhibits stable breathers.\nIn the two-dimensional geometry, the Landau-Lifshitz\nequation for the spin torque oscillator admits an asymp-\ntotic reduction to a planar quadratic Schr¨ odinger equa-\ntion, equation (42). Like its one-dimensional counter-\npart, the PT-symmetric planar Schr¨ odinger equation has\nstable static and oscillatory soliton solutions.\nFinally, it is worth re-emphasising here that the\nPT-symmetric Schr¨ odinger equation is a reduction of\nthe whole family of nearly- PT symmetric Landau-\nLifshitz-Gilbert-S/suppress lonczewski equations with λ=O(ǫ) —\nand not just of its special case with λ= 0. Therefore\nour conclusion on the existence of stable solitons is12\napplicable to the physically relevant class of spin torque\noscillators with nonzero damping.\nACKNOWLEDGMENTS\nWe thank Boris Ivanov and Andrei Slavin for useful\ndiscussions. This project was supported by the NRF ofSouth Africa (grants No 105835, 120844 and 120467).\n[1] C.M. Bender and S. Boettcher, Phys. Rev. Lett. 805243\n(1998); C.M. Bender, S. Boettcher, and P.N. Meisinger,\nJ. Math. Phys. 402201 (1999); C.M. Bender, Contemp.\nPhys.46277 (2005); C.M. Bender, Rep. Prog. 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A 85063837 (2012); N.V. Alex-eeva, I.V. Barashenkov and Y.S. Kivshar, New J. Phys.\n19113032 (2017)."    },    {        "title": "1607.01307v1.Magnetic_moment_of_inertia_within_the_breathing_model.pdf",        "content": "Magnetic moment of inertia within the breathing model\nDanny Thonig,\u0003Manuel Pereiro, and Olle Eriksson\nDepartment of Physics and Astronomy, Material Theory, University Uppsala, S-75120 Uppsala, Sweden\n(Dated: June 20, 2021)\nAn essential property of magnetic devices is the relaxation rate in magnetic switching which\nstrongly depends on the energy dissipation and magnetic inertia of the magnetization dynamics.\nBoth parameters are commonly taken as a phenomenological entities. However very recently, a large\ne\u000bort has been dedicated to obtain Gilbert damping from \frst principles. In contrast, there is no\nab initio study that so far has reproduced measured data of magnetic inertia in magnetic materials.\nIn this letter, we present and elaborate on a theoretical model for calculating the magnetic moment\nof inertia based on the torque-torque correlation model. Particularly, the method has been applied\nto bulk bcc Fe, fcc Co and fcc Ni in the framework of the tight-binding approximation and the\nnumerical values are comparable with recent experimental measurements. The theoretical results\nelucidate the physical origin of the moment of inertia based on the electronic structure. Even though\nthe moment of inertia and damping are produced by the spin-orbit coupling, our analysis shows that\nthey are caused by undergo di\u000berent electronic structure mechanisms.\nPACS numbers: 75.10.-b,75.30.-m,75.40.Mg,75.78.-n,75.40.Gb\nThe research on magnetic materials with particular fo-\ncus on spintronics or magnonic applications became more\nand more intensi\fed, over the last decades [1, 2]. For\nthis purpose, \\good\" candidates are materials exhibiting\nthermally stable magnetic properties [3], energy e\u000ecient\nmagnetization dynamics [4, 5], as well as fast and stable\nmagnetic switching [6, 7]. Especially the latter can be\ninduced by i)an external magnetic \feld, ii)spin polar-\nized currents [8], iii)laser induced all-optical switching\n[9], or iv)electric \felds [10]. The aforementioned mag-\nnetic excitation methods allow switching of the magnetic\nmoment on sub-ps timescales.\nThe classical atomistic Landau-Lifshitz-Gilbert (LLG)\nequation [11, 12] provides a proper description of mag-\nnetic moment switching [13], but is derived within the\nadiabatic limit [14, 15]. This limit characterises the\nblurry boundary where the time scales of electrons and\natomic magnetic moments are separable [16] | usually\nbetween 10\u0000100 fs. In this time-scale, the applicabil-\nity of the atomistic LLG equation must be scrutinized\nin great detail. In particular, in its common formula-\ntion, it does not account for creation of magnetic inertia\n[17], compared to its classical mechanical counterpart of\na gyroscope. At short times, the rotation axis of the\ngyroscope do not coincide with the angular momentum\naxis due to a \\fast\" external force. This results in a\nsuperimposed precession around the angular-momentum\nand the gravity \feld axis; the gyroscope nutates. It is\nexpected for magnetisation dynamics that atomic mag-\nnetic moments behave in an analogous way on ultrafast\ntimescales [17, 18] (Fig. 1).\nConceptional thoughts in terms of \\magnetic mass\"\nof domain walls were already introduced theoretically by\nD oring [19] in the late 50's and evidence was found ex-\nperimentally by De Leeuw and Robertson [20]. More\nrecently, nutation was discovered on a single-atom mag-\nnetic moment trajectory in a Josephson junction [21{23]\nB\nprecession conenutation cone\nmFIG. 1. (Color online) Schematic \fgure of nutation in the\natomistic magnetic moment evolution. The magnetic moment\nm(red arrow) evolves around an e\u000bective magnetic \feld B\n(gray arrow) by a superposition of the precession around the\n\feld (bright blue line) and around the angular momentum\naxis (dark blue line). The resulting trajectory (gray line)\nshows an elongated cycloid.\ndue to angular momentum transfer caused by an elec-\ntron spin \rip. From micromagnetic Boltzman theory,\nCiornei et al. [18, 24] derived a term in the extended\nLLG equation that addresses \\magnetic mass\" scaled by\nthe moment of inertia tensor \u0013. This macroscopic model\nwas transferred to atomistic magnetization dynamics and\napplied to nanostructures by the authors of Ref. 17, and\nanalyzed analytically in Ref. 25 and Ref. 26. Even in the\ndynamics of Skyrmions, magnetic inertia was observed\nexperimentally [27].\nLike the Gilbert damping \u000b, the moment of inertia ten-\nsor\u0013have been considered as a parameter in theoretical\ninvestigations and postulated to be material speci\fc. Re-\ncently, the latter was experimentally examined by Li et\nal. [28] who measured the moment of inertia for Ni 79Fe21\nand Co \flms near room temperature with ferromagnetic\nresonance (FMR) in the high-frequency regime (aroundarXiv:1607.01307v1  [cond-mat.mtrl-sci]  5 Jul 20162\n200 GHz). At these high frequencies, an additional sti\u000b-\nening was observed that was quadratic in the probing fre-\nquency!and, consequently, proportional to the moment\nof inertia\u0013=\u0006\u000b\u0001\u001c. Here, the lifetime of the nutation \u001c\nwas determined to be in the range of \u001c= 0:12\u00000:47 ps,\ndepending not only on the selected material but also on\nits thickness. This result calls for a proper theoretical\ndescription and calculations based on ab-initio electronic\nstructure footings.\nA \frst model was already provided by Bhattacharjee\net al. [29], where the moment of inertia \u0013was derived\nin terms of Green's functions in the framework of the\nlinear response theory. However, neither \frst-principles\nelectronic structure-based numerical values nor a detailed\nphysical picture of the origin of the inertia and a poten-\ntial coupling to the electronic structure was reported in\nthis study. In this Letter, we derive a model for the\nmoment of inertia tensor based on the torque-torque cor-\nrelation formalism [30, 31]. We reveal the basic electron\nmechanisms for observing magnetic inertia by calculat-\ning numerical values for bulk itinerant magnets Fe, Co,\nand Ni with both the torque-torque correlation model\nand the linear response Green's function model [29]. In-\nterestingly, our study elucidate also the misconception\nabout the sign convention of the moment of inertia [32].\nThe moment of inertia \u0013is de\fned in a similar way\nas the Gilbert damping \u000bwithin the e\u000bective dissipation\n\feldBdiss[30, 33]. This ad hoc introduced \feld is ex-\npanded in terms of viscous damping \u000b@m=@tand magnetic\ninertia\u0013@2m=@t2in the relaxation time approach [32, 34]\n(see Supplementary Material). The o\u000b-equilibrium mag-\nnetic state induces excited states in the electronic struc-\nture due to spin-orbit coupling. Within the adiabatic\nlimit, the electrons equilibrate into the ground state at\ncertain time scales due to band transitions [35]. If this\nrelaxation time \u001cis close to the adiabatic limit, it will\nhave two implications for magnetism: i)magnetic mo-\nments respond in a inert fashion, due to formation of\nmagnetism, ii)the kinetic energy is proportional to mu2=2\nwith the velocity u=@m=@tand the \\mass\" m of mag-\nnetic moments, following equations of motion of classical\nNewtonian mechanics. The inertia forces the magnetic\nmoment to remain in their present state, represented in\nthe Kambersky model by \u000b=\u0000\u0013\u0001\u001c(Ref. 32 and 34);\ntheraison d'etre of inertia is to behave opposite to the\nGilbert damping.\nIn experiments, the Gilbert damping and the moment\nof inertia are measurable from the diagonal elements of\nthe magnetic response function \u001fvia ferromagnetic res-\nonance [31] (see Supplementary Material)\n\u000b=!2\n0\n!Mlim\n!!0=\u001f?\n!(1)\n\u0013=1\n2!2\n0\n!Mlim\n!!0@!<\u001f?\n!\u00001\n!0; (2)\nwhere!M=\rBand!0=\rB0are the frequencies re-lated to the internal e\u000bective and the external magnetic\n\feld, respectively. Thus, the moment of inertia \u0013is equal\nto the change of the FMR peak position, say the \frst\nderivative of the real part of \u001fwith respect to the prob-\ning frequency [29, 36]. Alternatively, rapid external \feld\nchanges induced by spin-polarized currents lead also to\nnutation of the macrospin [37].\nSetting\u001fonab-initio footings, we use the torque-\ntorque correlation model, as applied for the Gilbert\ndamping in Ref. 30 and 35. We obtain (see Supplemen-\ntary Material)\n\u000b\u0016\u0017=g\u0019\nmsX\nnmZ\nT\u0016\nnm(k)T\u0017\nnm(k)Wnmdk (3)\n\u0013\u0016\u0017=\u0000g~\nmsX\nnmZ\nT\u0016\nnm(k)T\u0017\nnm(k)Vnmdk; (4)\nwhere\u0016;\u0017 =x;y;z andmsis the size of the mag-\nnetic moment. The spin-orbit-torque matrix elements\nTnm=hn;kj[\u001b;Hsoc]jm;ki| related to the commuta-\ntor of the Pauli matrices \u001band the spin-orbit Hamilto-\nnian | create transitions between electron states jn;ki\nandjm;kiin bandsnandm. This mechanism is equal\nfor both, Gilbert damping and moment of inertia. Note\nthat the wave vector kis conserved, since we neglect non-\nuniform magnon creation with non-zero wave vector. The\ndi\u000berence between moment of inertia and damping comes\nfrom di\u000berent weighting mechanism Wnm;Vnm: for the\ndampingWnm=R\n\u0011(\")Ank(\")Amk(\")d\"where the elec-\ntron spectral functions are represented by Lorentzian's\nAnk(\") centred around the band energies \"nkand broad-\nened by interactions with the lattice, electron-electron\ninteractions or alloying. The width of the spectral func-\ntion \u0000 provides a phenomenological account for angular\nmomentum transfer to other reservoirs. For inertia, how-\never,Vnm=R\nf(\") (Ank(\")Bmk(\") +Bnk(\")Amk(\")) d\"\nwhereBmk(\") = 2(\"\u0000\"mk)((\"\u0000\"mk)2\u00003\u00002)=((\"\u0000\"mk)2+\u00002)3\n(see Supplementary Material). Here, f(\") and\u0011(\") are\nthe Fermi-Dirac distribution and the \frst derivative of it\nwith respect to \". Knowing the explicit form of Bmk, we\ncan reveal particular properties of the moment of inertia:\ni)for \u0000!0 (\u001c!1 ),Vnm=2=(\"nk\u0000\"mk)3. Sincen=m\nis not excluded, \u0013!\u00001 ; the perturbed electron system\nwill not relax back into the equilibrium. ii)In the limit\n\u0000!1 (\u001c!0), the electron system equilibrates imme-\ndiately into the ground state and, consequently, \u0013= 0.\nThese limiting properties are consistent with the expres-\nsion\u0013=\u0000\u000b\u0001\u001c. Eq. (4) also indicates that the time scale\nis dictated by ~and, consequently, on a femto-second\ntime scale.\nTo study these properties, we performed \frst-\nprinciples tight binding (TB) calculations [38] of the\ntorque-correlation model as described by Eq. (4) as well\nas for the Green's function model reported in Ref. 29.\nThe materials investigated in this letter are bcc Fe, fcc\nCo, and fcc Ni. Since our magnetic moment is \fxed3\n-1·10−3-5·10−405·10−41·10−3−ι(fs)\n10−110+0\nΓ (eV)Fe\nCo\nNiTorque\nGreen\n10−21α10−410−21Γ (eV)\nFIG. 2. (Color) Moment of inertia \u0013as a function of the band\nwidth \u0000 for bcc Fe (green dotes and lines), fcc Co (red dotes\nand lines), and fcc Ni (blue dotes and lines) and with two\ndi\u000berent methods: i)the torque-correlation method (\flled\ntriangles) and the ii)Greens function method [29](\flled cir-\ncles). The dotted gray lines indicating the zero level. The\ninsets show the calculated Gilbert damping \u000bas a function of\n\u0000. Lines are added to guide the eye. Notice the negative sign\nof the moment of inertia.\nin thezdirection, variations occur primarily in xory\nand, consequently, the e\u000bective torque matrix element is\nT\u0000=hn;kj[\u001b\u0000;Hsoc]jm;ki, where\u001b\u0000=\u001bx\u0000i\u001by. The\ncubic symmetry of the selected materials allows only di-\nagonal elements in both damping and moment of inertia\ntensor. The numerical calculations, as shown in Fig. 2,\ngive results that are consistent with the torque-torque\ncorrelation model predictions in both limits, \u0000 !0 and\n\u0000!1 . Note that the latter is only true if we assume\nthe validity of the adiabatic limit up to \u001c= 0. It should\nalso be noted that Eq. (4) is only valid in the adiabatic\nlimit (>10 fs). The strong dependency on \u0000 indicates,\nhowever, that the current model is not a parameter-free\napproach. Fortunately, the relevant parameters can be\nextracted from ab-initio methods: e.g., \u0000 is related ei-\nther to the electron-phonon self energy [39] or to electron\ncorrelations [40].\nThe approximation \u0013=\u0000\u000b\u0001\u001cderived by F ahnle et\nal. [32] from the Kambersk\u0013 y model is not valid for all\n\u0000. It holds for \u0000 <10 meV, where intraband transi-\ntions dominate for both damping and moment of inertia;\nbands with di\u000berent energies narrowly overlap. Here, the\nmoment of inertia decreases proportional to 1=\u00004up to a\ncertain minimum. Above the minimum and with an ap-\npropriate large band width \u0000, interband transitions hap-\npen so that the moment of inertia approaches zero for\nhigh values of \u0000. In this range, the relation \u0013=\u000b\u0001\u001c\nused by Ciornei et al [18] holds and softens the FMR res-\nonance frequency. Comparing qualitative the di\u000berence\n10−410−310−210−1−ι(fs)/α\n510+02510+12510+22\nτ(fs)\n5·10−310−22·10−23·10−2\nΓ (eV)−ι\nαFIG. 3. (Color online) Gilbert damping \u000b(red dashed line),\nmoment of inertia \u0013(blue dashed line), and the resulting nu-\ntation lifetime \u001c=\u0013=\u000b(black line) as a function of \u0000 in the\nintraband region for Fe bulk. Arrows indicating the ordinate\nbelonging of the data lines. Notice the negative sign of the\nmoment of inertia.\nbetween the itinerant magnets Fe, Co and Ni, we obtain\nsimilar features in \u0013and\u000bvs. \u0000, but the position of the\nminimum and the slope in the intraband region varies\nwith the elements: \u0013min= 5:9\u000110\u00003fs\u00001at \u0000 = 60 meV\nfor bcc Fe, \u0013min= 6:5\u000110\u00003fs\u00001at \u0000 = 50 meV for fcc\nCo, and\u0013min= 6:1\u000110\u00003fs\u00001at \u0000 = 80 meV for fcc Ni.\nThe crossing point of intra- and interband transitions for\nthe damping was already reported by Gilmore et al. [35]\nand Thonig et al. [41]. The same trends are also repro-\nduced by applying the Green's function formalism from\nBhattacharjee et al. [29] (see Fig. 2). Consequently, both\nmethods | torque-torque correlation and the linear re-\nsponse Green's function method | are equivalent as it\ncan also be demonstrated not only for the moment of\ninertia but also for the Gilbert damping \u000b(see Supple-\nmentary Material)[41]. In the torque-torque correlation\nmodel (4), the coupling \u0000 de\fnes the width of the en-\nergy window in which transitions Tnmtake place. The\nGreen function approach, however, provides a more ac-\ncurate description with respect to the ab initio results\nthan the torque-torque correlation approach. This may\nbe understood from the fact that a \fnite \u0000 broadens and\nslightly shifts maxima in the spectral function. In par-\nticular, shifted electronic states at energies around the\nFermi level causes di\u000berences in the minimum of \u0013in both\nmodels. Furthermore, the moment of inertia can be re-\nsolved by an orbital decomposition and, like the Gilbert\ndamping\u000b, scales quadratically with the spin-orbit cou-\npling\u0010, caused by the torque operator ^Tin Eq. (4). Thus,\none criteria for \fnding large moments of inertia is by hav-\ning materials with strong spin-orbit coupling.\nIn order to show the region of \u0000 where the approxi-\nmation\u0013=\u0000\u000b\u0001\u001cholds, we show in Fig. 3 calculated\nvalues of\u0013,\u000b, and the resulting nutation lifetime \u001cfor a\nselection of \u0000 that are below \u0013min. According to the data\nreported in Ref. 28, this is a suitable regime accessible4\nfor experiments. To achieve the room temperature mea-\nsured experimental values of \u001c= 0:12\u00000:47 ps, we have\nfurthermore to guarantee that \u0013 >> \u000b . An appropriate\nexperimental range is \u0000 \u00195\u000010 meV, which is realistic\nand caused, e.g., by the electron-phonon coupling. A nu-\ntation lifetime of \u001c\u00190:25\u00000:1 ps is revealed for these\nvalues of \u0000 (see Fig. 3), a value similar to that found in ex-\nperiment. The aforementioned electron-phonon coupling,\nhowever, is underestimated compared to the electron-\nphonon coupling from a Debye model (\u0000 \u001950 meV) [42].\nIn addition, e\u000bects on spin disorder and electron corre-\nlation are neglected, that could lead to uncertainties in\n\u0000 and hence discrepancies to the experiment. On the\nother hand, it is not excluded that other second order\nenergy dissipation terms, Bdiss, proportional to ( @e=@t)2\nwill also contribute [32] (see Supplementary material).\nThe derivation of the moment of inertia tensor from the\nKambersk\u0013 y model and our numerics corroborates that\nrecently observed properties of the Gilbert damping will\nbe also valid for the moment of inertia: i)the moment\nof inertia is temperature dependent [41, 43] and decays\nwith increasing phonon temperature, where the later usu-\nally increase the electron-phonon coupling \u0000 in certain\ntemperature intervals [42]; ii)the moment of inertia is\na tensor, however, o\u000b-diagonal elements for bulk mate-\nrials are negligible small; iii)it is non-local [36, 41, 44]\nand depends on the magnetic moment [45{47]. Note that\nthe sign change of the moment of inertia also e\u000bects the\ndynamics of the magnetic moments (see Supplementary\nMaterial).\nThe physical mechanism of magnetic moment of inertia\nbecomes understandable from an inspection of the elec-\ntron band structure (see Fig. 4 for fcc Co, as an example).\nThe model proposed here allows to reveal the inertia k-\nand band-index nresolved contributions (integrand of\nEq. (4)). Note that we analyse for simplicity and clarity\nonly one contribution, AnBm, in the expression for Vnm.\nAs Fig. 4 shows the contribution to Vnmis signi\fcant only\nfor speci\fc energy levels and speci\fc k-points. The \fg-\nure also shows a considerable anisotropy, in the sense that\nmagnetisations aligned along the z- or y-directions give\nsigni\fcantly di\u000berent contributions. Also, a closer in-\nspection shows that degenerate or even close energy levels\nnandm, which overlap due to the broadening of energy\nlevels, e.g. as caused by electron-phonon coupling, \u0000, ac-\ncelerate the relaxation of the electron-hole pairs caused\nby magnetic moment rotation combined with the spin\norbit coupling. This acceleration decrease the moment\nof inertia, since inertia is the tendency of staying in a\nconstant magnetic state. Our analysis also shows that\nthe moment of inertia is linked to the spin-polarization\nof the bands. Since, as mentioned, the inertia preserves\nthe angular momentum, it has largest contributions in\nthe electronic structure, where multiple electron bands\nwith the same spin-polarization are close to each other\n(cf. Fig. 4 c). However, some aspects of the inertia,\n-4-3-2-10E−EF(eV)\n-4-3-2-10E−EF(eV)\nι<0\nι>0\n-4-3-2-10E−EF(eV)\nΓ H N\nk(a−1\n0)(a)\n(b)\n(c)y\nz\nFIG. 4. (Color online) Moment of inertia in the electron band\nstructure for bulk fcc Co with the magnetic moment a) in y\ndirection and b) in zdirection. The color and the intensity\nindicates the sign and value of the inertia contribution (blue\n-\u0013 <0; red -\u0013 >0; yellow - \u0013\u00190). The dotted gray line\nis the Fermi energy and \u0000 is 0 :1 eV. c) Spin polarization of\nthe electronic band structure (blue - spin down; red - spin up;\nyellow - mixed states).\ne.g. being caused by band overlaps, is similar to the\nGilbert damping [48], although the moment of inertia is\na property that spans over the whole band structure and\nnot only over the Fermi-surface. Inertia is relevant in\nthe equation of motion [17, 35] only for \u001c&0:1 ps and\nparticularly for low dimensional systems. Nevertheless,\nin the literature there are measurements, as reported in\nRef. 37, where the inertia e\u000bects are present.\nIn summary, we have derived a theoretical model for\nthe magnetic moment of inertia based on the torque-\ntorque correlation model and provided \frst-principle\nproperties of the moment of inertia that are compared\nto the Gilbert damping. The Gilbert damping and the\nmoment of inertia are both proportional to the spin-\norbit coupling, however, the basic electron band struc-5\nture mechanisms for having inertia are shown to be dif-\nferent than those for the damping. We analyze details\nof the dispersion of electron energy states, and the fea-\ntures of a band structure that are important for having\na sizable magnetic inertia. We also demonstrate that\nthe torque correlation model provides identical results\nto those obtained from a Greens functions formulation.\nFurthermore, we provide numerical values of the moment\nof inertia that are comparable with recent experimen-\ntal measurements[28]. The calculated moment of inertia\nparameter can be included in atomistic spin-dynamics\ncodes, giving a large step forward in describing ultrafast,\nsub-ps processes.\nAcknowledgements The authors thank Jonas Frans-\nson and Yi Li for fruitful discussions. The support of\nthe Swedish Research Council (VR), eSSENCE and the\nKAW foundation (projects 2013.0020 and 2012.0031) are\nacknowledged. The computations were performed on re-\nsources provided by the Swedish National Infrastructure\nfor Computing (SNIC).\n\u0003danny.thonig@physics.uu.se\n[1] S. S. P. Parkin, J. X., C. Kaiser, A. Panchula, K. Roche,\nand M. Samant, Proceedings of the IEEE 91, 661 (2003).\n[2] Y. Xu and S. 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Appl.\nPhys. 103, 07D303 (2008)."    },    {        "title": "1503.01478v2.Critical_current_destabilizing_perpendicular_magnetization_by_the_spin_Hall_effect.pdf",        "content": "arXiv:1503.01478v2  [cond-mat.mes-hall]  1 Aug 2015Critical current destabilizing perpendicular magnetizat ion by the spin Hall effect\nTomohiro Taniguchi1, Seiji Mitani2, and Masamitsu Hayashi2\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba 305-8568, Japan\n2National Institute for Materials Science, Tsukuba 305-004 7, Japan\n(Dated: July 5, 2018)\nThe critical current needed to destabilize the magnetizati on of a perpendicular ferromagnet via\nthe spin Hall effect is studied. Both the dampinglike and field like torques associated with the spin\ncurrent generated by the spin Hall effect is included in the La ndau-Lifshitz-Gilbert equation to\nmodel the system. In the absence of the fieldlike torque, the c ritical current is independent of the\ndamping constant and is much larger than that of conventiona l spin torque switching of collinear\nmagnetic systems, as in magnetic tunnel junctions. With the fieldlike torque included, we find that\nthe critical current scales with the damping constant as α0(i.e., damping independent), α, and\nα1/2depending on the sign of the fieldlike torque and other parame ters such as the external field.\nNumerical and analytical results show that the critical cur rent can be significantly reduced when\nthe fieldlike torque possesses the appropriate sign, i.e. wh en the effective field associated with the\nfieldlike torque is pointing opposite to the spin direction o f the incoming electrons. These results\nprovideapathwaytoreducingthecurrentneededtoswitch ma gnetization usingthespin Hall effect.\nPACS numbers: 75.78.-n, 75.70.Tj, 75.76.+j, 75.40.Mg\nI. INTRODUCTION\nThe spin Hall effect1–3(SHE) in a nonmagnetic heavy\nmetal generates pure spin current flowing along the di-\nrection perpendicular to an electric current. The spin\ncurrent excites magnetization dynamics in a ferromagnet\nattached to the nonmagnetic heavy metal by the spin-\ntransfer effect4,5. There have been a number of exper-\nimental reports on magnetization switching and steady\nprecession induced by the spin Hall effect6–9. These dy-\nnamics have attracted great attention recently from the\nviewpoints ofboth fundamental physicsand practicalap-\nplications.\nAn important issue to be solved on the magnetization\ndynamics triggered by the spin Hall effect is the reduc-\ntion of the critical current density needed to destabilize\nthe magnetization from its equilibrium direction, which\ndetermines the current needed to switch the magneti-\nzation direction or to induce magnetization oscillation.\nThe reported critical current density for switching8,10–13\nor precession9is relatively high, typically larger than 107\nA/cm2. One of the reasons behind this may be related\nto the recently predicted damping constant independent\ncritical current when SHE is used14,15. This is in con-\ntrast to spin-transfer-induced magnetization switching in\na typical giant magnetoresistance (GMR) or magnetic\ntunnel junction (MTJ) device where the critical current\nis expected to be proportional to the Gilbert damping\nconstant α. Here the magnetization dynamics is excited\nas a result of the competition between the spin torque\nand the damping torque16. Since the damping constant\nfor typical ferromagnet in GMR or MTJ devices is rela-\ntively small ( α∼10−2−10−3)17,18, it can explain why\nthe critical current is larger for the SHE driven systems.\nThus in particular for device application purposes, it is\ncrucial to find experimental conditions in which the mag-netization dynamics can be excited with lower current.\nAnother factor that might contribute to the reduc-\ntion of the critical current is the presence of the field\nlike torque19. In the GMR/MTJ systems, both the con-\nventional spin torque, often referred to as the damp-\ninglike torque, and the fieldlike torque arise from the\nspin transfer between the conduction electrons and the\nmagnetization4,19–23. Due to the short relaxation length\nof the transverse spin of the conduction electrons24,25,\nthe damping like torque is typically larger than the field-\nlike torque. Indeed, the magnitude of the field like\ntorque experimentally found in GMR/MTJ systems has\nbeen reported to be much smaller than the damping like\ntorque26–29. Because of its smallness, the fieldlike torque\nhad notbeen consideredin estimatingthe criticalcurrent\nintheGMR/MTJsystems16,30–32,althoughitdoesplaya\nkeyrolein particularsystems33,34. In contrast, recentex-\nperiments found that the fieldlike torque associated with\nthe SHE is larger than the damping like torque35–40.\nThe physical origin of the large SHE-induced field-\nlike torque still remains unclear. Other possible sources\ncan be the Rashba effect36,41–44, bulk effect45, and the\nout of plane spin orbit torque46. Interestingly, the field\nlike torque has been reported to show a large angu-\nlar dependence36,37,47(the angle between the current\nand the magnetization), which cannot be explained by\nthe conventional formalism of spin-transfer torque in\nGMR/MTJsystems. Thefieldliketorqueactsasatorque\nduetoanexternalfieldandmodifiestheenergylandscape\nof the magnetization. As a result, a large fieldlike torque\ncan significantly influence the critical current. However,\nthe fieldlike torque had not been taken into account in\nconsidering the current needed to destabilize the magne-\ntization from its equilibrium direction and thus its role\nis still unclear.\nIn this paper, we study the critical current needed to2\ndestabilize a perpendicular ferromagnet by the spin Hall\neffect. The Landau-Lifshitz-Gilbert(LLG) equationwith\nthe dampinglike and fieldlike torques associated with the\nspin Hall effect is solved both numerically and analyti-\ncally. Wefindthatthecriticalcurrentcanbesignificantly\nreduced when the fieldlike torque possesses the appropri-\nate sign with respect to the dampinglike torque. With\nthe fieldlike torque included, the critical current scales\nwith the damping constant as α0(i.e., damping indepen-\ndent),α, andα1/2, depending on the sign of the field-\nlike torque and other parameters. Analytical formulas\nof such damping-dependent critical current are derived\n[Eqs. (19)-(21)], and they show good agreement with the\nnumerical calculations. From these results, we find con-\nditions in which the critical current can be significantly\nreduced compared to the damping-independent thresh-\nold, i.e., systems without the fieldlike torque.\nThe paper is organized as follows. In Sec. II, we\nschematically describe the system under consideration.\nWe discuss the definition of the critical current in Sec.\nIII. Section IV summarizes the dependences of the crit-\nical current on the direction of the damping constant,\nthe in-plane field, and the fieldlike torque obtained by\nthe numerical simulation. The analytical formulas of the\ncritical current and their comparison to the numerical\nsimulations are discussed in Sec. V. The condition at\nwhich damping-dependent critical current occurs is also\ndiscussed in this section. The conclusion follows in Sec.\nVI.\nII. SYSTEM DESCRIPTION\nThe system we consider is schematically shown in Fig.\n1, where an electric current flowing along the x-direction\ninjects a spin current into the ferromagnet by the spin\nHall effect. The magnetization dynamics in the ferro-\nmagnet is described by the LLG equation,\ndm\ndt=−γm×H+αm×dm\ndt\n−γHsm×(ey×m)−γβHsm×ey,(1)\nwhereγandαarethe gyromagneticratioandtheGilbert\ndamping constant, respectively. We assume that the\nmagnetization of the ferromagnet points along the film\nnormal (i.e., along the zaxis), and an external in-plane\nmagnetic field is applied along the xoryaxis. The total\nmagnetic field His given by\nH=HapplnH+HKmzez, (2)\nwhereHapplis the external field directed along the xor\nyaxis and HKis the uniaxial anisotropy field along the\nzaxis.nHandeiare unit vectors that dictate the di-\nrection of the uniaxial anisotropy field and the iaxis,\nrespectively. Here we call the external field along the x\nandydirections the longitudinal and transverse fields,\nrespectively. The third and fourth terms on the right-\nhand side of Eq. (1) are the damping like and fieldlikeHappl // y\nHappl // x\nm\ncurrentxz\ny\nFIG. 1. Schematic view of the spin-Hall system. The x\naxis is parallel to current, whereas the zaxis is normal to the\nfilm plane. The spin direction of the electrons entering the\nmagnetic layer via the spin Hall effect points along the + yor\n−ydirection.\ntorques associated with the spin Hall effect, respectively.\nThetorquestrength Hscanbeexpressedwiththecurrent\ndensityj, the spin Hall angle ϑ, the saturation magneti-\nzationM, and the thickness of the ferromagnet d, i.e.,\nHs=/planckover2pi1ϑj\n2eMd. (3)\nThe ratio of the fieldlike torque to the damping like\ntorque is represented by β. Recent experiments found\nthatβis positive and is larger than 135–40.\nThe magnetization dynamics described by the LLG\nequation can be regarded as a motion of a point particle\non a two-dimensional energy landscape. In the presence\nof the fieldlike torque, the energy map is determined by\nthe energy density given by34\nE=−M/integraldisplay\ndm·H−βMHsm·ey.(4)\nThen, the external field torque and the fieldlike torque,\nwhich are the first and fourth terms on the right-hand-\nside of Eq. (1), can be expressed as −γm×B, where the\neffective field Bis\nB=−∂E\n∂Mm. (5)\nThe initial state of the numerical simulation is chosen to\nbe the direction corresponding to the minimum of the\neffective energy density E. The explicit forms of the ini-\ntial state for the longitudinal and the transverse external\nfields are shown in Appendix A.\nWe emphasize for the latter discussion in Sec. V that,\nusing Eqs. (1), (4), and (5), the time change of the effec-\ntive energy density is described as\ndE\ndt=dEs\ndt+dEα\ndt. (6)3\nHere the first and second terms on the right-hand side\nare the rates of the work done by the spin Hall torque\nand the dissipation due to damping, respectively, which\nare explicitly given by\ndEs\ndt=γMHs[ey·B−(m·ey)(m·B)],(7)\ndEα\ndt=−αγM/bracketleftBig\nB2−(m·B)2/bracketrightBig\n. (8)\nThe sign of Eq. (7) depends on the current direction\nand the effective magnetic field, while that of Eq. (8) is\nalways negative.\nThe magnetic parameters used in this paper mimic the\nconditions achieved in CoFeB/MgO heterostructures48;\nM= 1500 emu/c.c., HK= 540 Oe, ϑ= 0.1,γ=\n1.76×107rad/(Oe s), and d= 1.0 nm. The value of\nβis varied from −2, 0, to 2. Note that we have used a\nreducedHK(Refs.8,49) in ordertoobtain criticalcurrents\nthat are the same order of magnitude with that obtained\nexperimentally. We confirmed that the following discus-\nsions are applicable for a large value of HK(∼1T).\nIII. DEFINITION OF CRITICAL CURRENT\nIn this section, we describe how we determine the crit-\nical current from the numerical simulations. In exper-\niments, the critical current is determined from the ob-\nservation of the magnetization reversal8,12,41,46,48–50. As\nmentioned in Sec. II, in this paper, the initial state for\ncalculation is chosen to be the minimum of the effective\nenergy density. Usually, there are two minimum points\nabove and below the xyplane because of the symmetry.\nThroughout this paper, the initial state is chosen to be\nthe minimum point above the xyplane, i.e., mz(0)>0,\nfor convention.” It should be noted that, once the mag-\nnetization arrives at the xyplane during the current ap-\nplication, it can move to the other hemisphere after the\ncurrent is turned off due to, for example, thermal fluc-\ntuation. Therefore, here we define the critical current as\nthe minimum current satisfying the condition\nlim\nt→∞mz(t)< ǫ, (9)\nwhere a small positive real number ǫis chosen to be\n0.001. The duration of the simulations is fixed to 5 µs,\nlong enough such that all the transient effects due to the\ncurrent application are relaxed. Figures 2(a) and 2(b)\nshow examples of the magnetization dynamics close to\nthe critical current, which are obtained from the numer-\nical simulation of Eq. (1). As shown, the magnetization\nstays near the initial state for j= 3.1×106A/cm2, while\nit moves to the xyplane for j= 3.2×106A/cm2. Thus,\nthe critical current is determined as 3 .2×106A/cm2in\nthis case.\nWe note that the choice of the definition of the criti-\ncal current has some arbitrariness. For comparison, weFIG. 2. Time evolution ofthe zcomponentof themagnetiza-\ntionmzin the presence of the transverse field of Happl= 200\nwith (a) j= 3.1×106A/cm2and (b)j= 3.2×106A/cm2.\nThe value of βis zero.\nshow numerically evaluated critical current with a differ-\nent definition in Appendix B. The main results of this\npaper, e.g., the dependence of the critical current on the\ndamping constant, are not affected by the definition.\nWe also point out that the critical current defined by\nEq. (9) focuses on the instability threshold, and does\nnot guarantee a deterministic reversal. For example,\nin the case of Fig. 2(b), the reversal becomes prob-\nabilistic because the magnetization, starting along + z,\nstops its dynamics at the xyplane and can move back\nto its original direction or rotate to a point along −z\nresulting in magnetization reversal. Such probabilistic\nreversal can be measured experimentally using transport\nmeasurements8,12,41,46,49,50or by studying nucleation of\nmagnetic domains via magnetic imaging48. On the other\nhand, it hasbeen reportedthat deterministicreversalcan\ntake place when a longitudinal in-plane field is applied\nalongside the current41,49. It is difficult to determine the\ncritical current analytically for the deterministic switch-\ning for all conditions since, as in the case of Fig. 2(b),\nthe magnetization often stops at the xyplane during the\ncurrent application. This occurs especially in the pres-\nence of the transverse magnetic field because all torques\nbecome zero at m=±eyand the dynamics stops. Here\nwe thus focus on the probabilistic reversal.4\nFIG. 3. Numerically evaluated mzatt= 5µs for (a)-(c) the longitudinal ( nH=ex) and (d)-(f) the transverse ( nH=ey)\nfields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0. The damping constant is α= 0.005. The color scale\nindicates the zcomponent of the magnetization ( mz) att= 5µs. The red/white boundary indicates the critical current fo r\nprobabilistic switching, whereas the red/blue boundary gi ves the critical current for deterministic switching.\nIV. NUMERICALLY ESTIMATED CRITICAL\nCURRENT\nIn this section, we show numerically evaluated critical\ncurrent for different conditions. We solve Eq. (1) and\napply Eq. (9) to determine the critical current. Figure\n3 shows the value of mzatt= 5µs in the presence of\n(a)-(c) the longitudinal ( nH=ex) and (d)-(f) the trans-\nverse (nH=ey) fields. The value of βis 0 for Figs.\n3(a) and 3(d), 2 .0 for Figs. 3(b) and 3(e), and −2.0 for\nFigs. 3(c) and 3(f), respectively. The damping constant\nisα= 0.005. The red/white boundary indicates the crit-\nical current for the probabilistic switching, whereas the\nred and blue ( mz=−1) boundary gives the critical cur-\nrent for the deterministic switching. Using these results\nand the definition of the critical current given by Eq. (9),\nand performing similar calculations for different values of\nα, wesummarizethedependenceofthecriticalcurrenton\nthe longitudinal and transverse magnetic fields in Fig. 4.\nThe damping constant is varied as the following in each\nplot:α= 0.005, 0.01, and 0 .02. The solid lines in Fig. 4\nrepresent the analytical formula derived in Sec. V.A. In the presence of longitudinal field\nIn the case of the longitudinal field and β= 0\nshown in Fig. 4(a), the critical current is damping-\nindependent. Such damping-independent critical current\nhas been reported previously for deterministic magneti-\nzation switching14,15. Similarly, in the case of the longi-\ntudinal field and negative β(β=−2.0) shown in Fig.\n4(c), the critical current is damping-independent. In\nthese cases, the magnitude of the critical current is rel-\natively high. In particular, near zero field, the critical\ncurrent exceeds ∼108A/cm2, which is close to the limit\nof experimentally accessible value. These results indicate\nthat the useofthe longitudinal field with zeroornegative\nβis ineffective for the reduction of the critical current.\nOn the other hand, when βis positive, the critical cur-\nrent depends on the damping constant, as shown in Fig.\n4(b). Note that positive βis reported for the torques\nassociated with the spin Hall effect or Rashba effect in\nthe heterostructures studied experimentally35–37,39. The\nmagnitude of the critical current, ∼10×106A/cm2, is\nrelatively small compared with the cases of zero or neg-\nativeβ. In this case, the use of a low damping material\nis effective to reduce the critical current. Interestingly,\nthe critical current is not proportional to the damping\nconstant, while that previously calculated for a GMR or\nMTJ system16is proportional to α. For example, the5\nlongitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-6040 60 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-5040 50 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02-30-1010 30 \ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=-2.0\n-50\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02(a) (b) (c)\n(d) (e) (f)longitudinal magnetic field (Oe)0 50 100 150 200β=-2.0\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02critical current density (10 6 A/cm 2)\n0\n-10050 \n-150100150\n-50\nFIG. 4. Numerically evaluated critical currents in the pres ence of (a)-(c) the longitudinal ( nH=ex) and (d)-(f) the transverse\n(nH=ey) fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0, respectively. The solid lines are analytically\nestimated critical current in Sec. V.\ncritical current at zero longitudinal field in Fig. 4(b) is\n12.3,17.2, and24 .0×106A/cm2forα= 0.005,0.01, and\n0.02, respectively. These values indicate that the critical\ncurrent is proportional to α1/2. In fact, the analytical\nformula derived in Sec. V shows that the critical current\nis proportional to α1/2for positive β[see Eq. (19)].\nTo summarize the case of the longitudinal field, the\nuse of a heterostructure with positive β, which is found\nexperimentally, has the possibility to reduce the critical\ncurrent if a ferromagnet with low damping constant is\nused. In this case, the critical current is proportional to\nα1/2, which has not been found in previous works.\nB. In the presence of transverse field\nIn the presence of the transverse field with β= 0,\nthe critical current shows a complex dependence on the\ndamping constant α, as shown in Fig. 4(d). When the\ncurrent and the transversefield areboth positive (or neg-\native), the critical current is proportional to the damping\nconstant αexcept near zero field. The numerically cal-\nculated critical current matches well with the analytical\nresult, Eq. (20), shown by the solid lines. In this case,\nthe use of the low damping material results in the reduc-\ntion of the critical current. On the other hand, when the\ncurrent and the transversefield possessthe opposite sign,\nthecriticalcurrentisdampingindependent. Moreover,in\nthis case, thecriticalcurrentisofthe orderof108A/cm2.\nThus, it is preferable to use the current and field having\nthe same sign for the reduction of the critical current. Itshould be noted that, in our definition, the same sign of\ncurrent and field corresponds to the case when the direc-\ntion ofincoming electrons’spin (due to the SHE) and the\ntransverse field are opposite to each other. The reason\nwhy the critical current becomes damping dependent in\nthis situation will be explained in Sec. V.\nWhenβis positive the critical current depends on the\ndamping constant for the whole range of the transverse\nfield, as shown in Fig. 4(e). The critical current is\nroughly proportional to α1/2, in particular, close to zero\nfield. The solid lines display the analytical formula, Eq.\n(21), and showgood agreementwith the numericalcalcu-\nlations. The damping dependence of the critical current\nbecomes complex when the magnitude of the transverse\nfield is increased [see Eq. (21)]. We note that the critical\ncurrentfor the positive βin Fig. 4(e) is smallerthan that\nforβ= 0 in Fig. 4(d) for the whole range of Happl.\nOn the other hand, when βis negative, the critical\ncurrent is almost independent of α, especially near zero\nfield. However, when the transverse field is increased,\nthere is a regime where the critical current depends on\nthe damping constant. Such transition of the critical\ncurrent with the transverse field is also predicted by the\nanalytical solution, Eq. (21).\nTosummarizethe caseofthe transversefield, the αde-\npendence of the critical current can be categorized into\nthe following: α0(damping independent), α,α1/2, or\nother complex behavior. As with the case of the longi-\ntudinal field, the use of a heterostructure with positive β\nallowsreductionofthe criticalcurrentwhen lowdamping\nferromagnet is used. Overall, the most efficient condition6\nto reduce the critical current is to use the transverse field\nwith heterostructures that possess low αand positive β.\nIn this case, the critical current is reduced to the order\nof 106A/cm2.\nV. ANALYTICAL FORMULA OF CRITICAL\nCURRENT\nIn this section, we derive the analytical formula of the\ncritical current from the linearized LLG equation51. The\ncomplex dependences ofthe critical currentonthe damp-\ning constant αdiscussed in Sec. IV are well explained by\nthe analytical formula. We also discuss the physical in-\nsight obtained from the analytical formulas.\nA. Derivation of the critical current\nTo derive the critical current, we consider the stable\ncondition of the magnetization near its equilibrium. It is\nconvenient to introduce a new coordinate XYZin which\ntheZaxis is parallel to the equilibrium direction. The\nrotationfromthe xyz-coordinatetothe XYZcoordinate\nis performed by the rotation matrix\nR=\ncosθ0−sinθ\n0 1 0\nsinθ0 cosθ\n\ncosϕsinϕ0\n−sinϕcosϕ0\n0 0 1\n,(10)\nwhere (θ,ϕ) are the polar and azimuth angles of the\nmagnetization at equilibrium. The equilibrium magne-\ntization direction under the longitudinal and transverse\nmagnetic field is given by Eqs. (A1) and (A2), respec-\ntively. Since we are interested in small excitation of the\nmagnetization around its equilibrium, we assume that\nthe components of the magnetization in the XYZcoor-\ndinate satisfy mZ≃1 and|mX|,|mY| ≪1. Then, the\nLLG equation is linearized as\n1\nγd\ndt/parenleftbigg\nmX\nmY/parenrightbigg\n+M/parenleftbigg\nmX\nmY/parenrightbigg\n=−Hs/parenleftbigg\ncosθsinϕ\ncosϕ/parenrightbigg\n,(11)\nwhere the components of the 2 ×2 matrix Mare\nM1,1=αBX−Hssinθsinϕ, (12)\nM1,2=BY, (13)\nM2,1=BX (14)\nM2,2=αBY−Hssinθsinϕ. (15)\nHere,BXandBYare defined as\nBX=Happlsinθcos(ϕ−ϕH)+βHssinθsinϕ+HKcos2θ,\n(16)BY=Happlsinθcos(ϕ−ϕH)+βHssinθsinϕ+HKcos2θ,\n(17)\nwhereϕHrepresents the direction of the external field\nwithin the xyplane:ϕH= 0 for the longitudinal field\nandπ/2 for the transverse field.\nThe solution of Eq. (11) is mX,mY∝\nexp{γ[±i/radicalbig\ndet[M]−(Tr[M]/2)2−Tr[M]/2]t}, where\ndet[M] and Tr[ M] are the determinant and trace of\nthe matrix M, respectively. The imaginary part of the\nexponent determines the oscillation frequency around\ntheZaxis, whereas the real part determines the time\nevolution of the oscillation amplitude. The critical\ncurrent is defined as the current at which the real part\nof the exponent is zero. Then, the condition Tr[ M] = 0\ngives\nα(BX+BY)−2Hssinθsinϕ= 0,(18)\nFor the longitudinal field, Eq. (18) gives\njLONG\nc=±2e√αMd\n/planckover2pi1ϑ/radicalBig\n2H2\nK−H2\nappl/radicalbig\nβ(2+αβ),(19)\nindicating that the critical current is roughly propor-\ntional to α1/2. This formula works for positive βonly52\nif we assume 0 <2+αβ≃2, which is satisfied for typical\nferromagnets. The critical current when the transverse\nfield is applied reads\njTRANS\nc=2αeMd\n/planckover2pi1ϑ(Happl/HK)HK/bracketleftBigg\n1−1\n2/parenleftbiggHappl\nHK/parenrightbigg2/bracketrightBigg\n,(20)\nwhenβ= 0, indicating that the critical current is pro-\nportional to α. The critical current for finite βis\njTRANS\nc=2eMd\n/planckover2pi1ϑ\n×−(1+αβ)Happl±/radicalBig\nH2\nappl+2αβ(2+αβ)H2\nK\nβ(2+αβ).\n(21)\nEquation (21) works for the whole range of |Happl|(<\nHK) for positive β, while it only works when |Happl|>\n2αβ(2 +αβ)HKfor negative β. For example, when\nβ=−2.0, this condition is satisfied when |Happl|>108\nOe forα= 0.005 and |Happl|>152 Oe for α= 0.01.\nHowever the condition is not satisfied for the present\nrange of Happlforα= 0.02. The solid lines in Fig. 4(f)\nshow where Equation (21) is applicable. The zero-field\nlimits of Eqs. (19) and (21) become identical,\nlim\nHappl→0jc=±2e√αMd\n/planckover2pi1ϑ√\n2HK/radicalbig\nβ(2+αβ),(22)\nindicating that the critical current near zero field is pro-\nportional to α1/2whenβ >0.7\nFIG. 5. Magnetization dynamics under the conditions of (a)\nnH=ey,Happl= 50 Oe, β= 0,α= 0.005, and j= 13.2×106\nA/cm2, and (b) nH=ex,Happl= 50 Oe, β= 0,α= 0.005,\nandj= 90×106A/cm2.\nB. Discussions\nThe solid lines in Fig. 4(b), 4(d), 4(e), and 4(f) show\nthe analytical formulas, Eqs. (19), (20), and (21). As\nevident, these formulas agree well with the numerical re-\nsults in the regions where the critical currents depend on\nthe dampingconstant. In this section, we discussthe rea-\nson why the critical current becomes damping dependent\nor damping independent depending on the field direction\nand the sign of β.\nIt is useful for the following discussion to first study\ntypical magnetization dynamics found in the numerical\ncalculations. Figure 5 shows the time evolution of the\nx,yandzcomponents of the magnetization when the\ncritical current depends on [Fig. 5(a)] or is independent\nof [Fig. 5(b)] the damping constant. For the former,\nthe instability is accompanied with a precession of the\nmagnetization. On the other hand, the latter shows that\nthe instability takes place without the precession.\nWe start with the case when the critical current be-\ncomes damping dependent. To provide an intuitive pic-\nture, we schematically show in Fig. 6(a) the torques ex-\nerted on the magnetization during one precession period\nwhen current is applied. The condition is the same with\nthat described in Fig. 5(a), i.e., the transverse magnetic\nfield is applied with β= 0. In Fig. 6(a), magnetization\nis shown by the large black arrow, while the directions\nof the spin Hall torque, the damping torque and the ex-ternal field torque are represented by the solid, dotted\nand dashed lines, respectively (the external field torque\nis tangent to the precession trajectory). As evident in\nFig. 5(a), the precession trajectory is tilted to the posi-\ntiveydirection due to the transversefield. Depending on\nthe direction of the magnetization the spin Hall torque\nhas a component parallel, antiparallel, or normal to the\ndamping torque. This means that the work done by the\nspin Hall torque, denoted by ∆ Esin Fig. 6 (a), is pos-\nitive, negative, or zero at these positions. This can be\nconfirmed numerically when we calculate the work done\nby the spin Hall torque using Eq. (7). For an infinites-\nimal time ∆ t, the work done by the spin Hall torque\nis equal to the rate of its work ( dEs/dt), given in Eq.\n(7), times ∆ t, i.e. ∆Es= (dEs/dt)∆t. The solid line\nin Fig. 6(b) shows an example of the calculated rate of\nthe work done by the spin Hall torque (solid line), dEs/dt\nin Eq. (7). As shown, dEs/dtis positive, negative, and\nzero, when the magnetization undergoes one precession\nperiod. Similarly, the energy dissipated by the damping\ntorque,dEα/dt, can be calculated using Eq. (8) and is\nshown by the dotted line in Fig. 6(b). The calculated\ndissipation due to damping over a precession period is\nalways negative. Details of how the rates, shown in Fig.\n6, are calculated are summarized in Appendix C.\nNote that the strength of the spin Hall torque for\n∆Es>0 is larger than that for ∆ Es<0 due to the an-\ngular dependence of the spin Hall torque, |m×(ey×m)|.\nAlthough it is difficult to see, thesolid line in Fig. 6(b) is\nslightly shifted upward. Thus the total energy supplied\nby the spin Hall torque during one precession, given by/contintegraltext\ndt(dEs/dt), does not average to zero and becomes posi-\ntive. When the current magnitude, |j|, is larger than |jc|\nin Eq. (20), the energy supplied by the spin Hall torque\novercomes the dissipation due to the damping and con-\nsequently the precession amplitude grows, which leads to\nthe magnetization instability shown in Fig. 5(a). The\nsame picture is applicable when both directions of field\nand current are reversed. For this condition, the insta-\nbility of the magnetization is induced by the competition\nbetween the spin Hall torque and the damping torque.\nTherefore, the critical current depends on the damping\nconstant α. When only the current direction is reversed\nin Figs. 6(a) and 6(b) (i.e., the sign of the magnetic field\nand current is opposite to each other), the sign of ∆ Esis\nreversed and thus the total energy supplied by the spin\nHall torque becomes negative. This means that the spin\nHall torque cannot overcome the damping torque to in-\nduce instability. Therefore, the critical current shown in\nEq. (20) only applies to the case when the sign of the\nfield and current is the same. As described in Sec. IV,\nthe same sign of the current and field in our definition\nmeans that the incoming electrons’ spin direction, due\nto the spin Hall effect, is opposite to the transverse field\ndirection.\nNext, we consider the case when the critical current is\ndamping independent. Figure 6 (c) schematically shows\nthe precession trajectory when the applied field points to8\nFIG. 6. (a) A schematic view of the precession trajectory\nin the presence of the applied field in the positive y-direction.\nThe solid and dotted arrows indicate the directions of the\nspin Hall torque and the damping torque, respectively. The\ndashed line, which is the tangent line to the precession tra-\njectory, shows the field torque. The damping torque always\ndissipates energy from the ferromagnet. On the other hand,\nthe spin Hall torque supplies energy (∆ Es>0) when its di-\nrection is anti-parallel to the damping torque, and dissipa tes\nenergy (∆ Es<0) when the direction is parallel to the damp-\ning torque. When the direction of the spin Hall torque is\northogonal to the damping torque, the spin Hall torque does\nnot change the energy (∆ Es= 0). (b) Typical temporal vari-\nation of the rates of the work done by the spin Hall torque,\nEq. (7), (solid) and the dissipation due to damping, Eq. (8)\n(dotted) in the presence of the transverse field. The time is\nnormalized by the period given by Eq. (C7). (c), (d) Similar\nfigures with the longitudinal field.\nthexdirection and β= 0. The corresponding rate of\nwork done by the spin Hall torque and the dissipation\nrate due to the damping torque are shown in Fig. 6 (d).\nSimilar to the previous case, ∆ Escan be positive, nega-\ntive, or zero during one precession period. However, the\ntotal workdoneby the spin Hall torque,/contintegraltext\ndt(dEs/dt), be-\ncomes zero in this case due to the symmetry of angular\ndependence of the spin Hall torque. This means that the\nspin Hall torque cannot compensate the damping torque,\nand thus, a steady precession assumed in the linearized\nLLG equation is not excited. This is evident in the nu-\nmerically calculated magnetization trajectory shown in\nFig. 5(b). For this case, the linearized LLG equation\ngives|jc| → ∞, indicating that the spin Hall torque can-\nnot destabilize the magnetization. The same picture is\nalsoapplicable, forexample, in the absenceofthe applied\nfield and β= 0.\nHowever, an alternative mechanism can cause destabi-\nlization of the magnetization. As schematically shown in\nFigs. 6(a) and 6(c), there is a component of the damping\nlike spin Hall torque that is orthogonal to the damping\ntorque when ∆ Es= 0. The spin Hall torque at this pointis parallel or antiparallel to the field torque depending on\nthe position of the magnetization. When the spin Hall\ntorqueissufficientlylargerthanthefieldtorque,themag-\nnetization moves from its equilibrium position even if the\ntotal energy supplied by the spin Hall torque is zero or\nnegative. This leads to an instability that occurs before\none precession finishes. In this case, it is expected that\nthe critical current is damping-independent because the\ninstability is induced as a competition between the spin\nHall torque and the field torque, not the damping torque.\nThe time evolution of the magnetization shown in Fig.\n5 (b) represents such instability. The work reported in\nRefs.14,49discusses a similar instability condition.\nThe above physical picture is also applicable in the\npresence of the fieldlike torque. The fieldlike torque,\nwhich acts like a torque due to the transversefield, modi-\nfies the equilibrium direction ofthe ferromagnetand thus\nthe precession trajectory. Consequently, the amount of\nenergy supplied by the spin Hall torque and the dissipa-\ntion due to damping is changed when the fieldlike torque\nis present. Depending on the sign of β, the amount of the\nwork done by the spin Hall torque increases or decreases\ncompared to the case with β= 0. In our definition, posi-\ntiveβcontributes to the increase of the supplied energy,\nresulting in the reduction of the critical current. The\ncomplex dependence of the critical current on αarises\nwhen the fieldlike torque is present.\nTo summarize the discussion, the critical current be-\ncomes damping dependent when the energy supplied by\nthe spin Hall torque during a precession around the equi-\nlibrium is positive. The condition that meets this criteria\ndepends on the relative direction of the spin Hall torque\nand the damping torque, as briefly discussed above. To\nderive an analytical formula that describes the condition\natwhichthe criticalcurrentbecomesdamping dependent\nis not an easy task except for some limited cases53.\nVI. CONCLUSION\nIn summary, we have studied the critical current\nneeded to destabilize a perpendicularly magnetized fer-\nromagnet by the spin Hall effect. The Landau-Lifshitz-\nGilbert (LLG) equation that includes both the damping-\nlike and fieldlike torques associated with the spin Hall\neffect is solved numerically and analytically. The criti-\ncal current is found to have different dependence on the\ndamping constant, i.e., the critical current scales with α0\n(damping-independent), α, andα1/2depending on the\nsign of the fieldlike torque. The analytical formulas of\nthe damping-dependent critical current, Eqs. (19), (20),\nand (21), are derived from the linearized LLG equation,\nwhich explain well the numerical results. We find that\nsystems with fieldlike torque having the appropriate sign\n(β >0 in our definition) are the most efficient way to re-\nduce the criticalcurrent. Fortypicalmaterialparameters\nfound in experiment, the critical current can be reduced\nto the order of 106A/cm2when ferromagnets with rea-9\nsonable parameters are used.\nACKNOWLEDGMENTS\nThe authorsacknowledgeT. Yorozu, Y. Shiota, and H.\nKubota in AIST for valuable discussion sthey had with\nus. This workwassupported by JSPS KAKENHIGrant-\nin-AidforYoungScientists(B),GrantNo. 25790044,and\nMEXT R & D Next-Generation Information Technology.\nAppendix A: Initial state of the numerical\nsimulation\nWe assume that the magnetization in the absence of\nthe applied field points to the positive zdirection. In\nthe presence of the field, the equilibrium direction moves\nfrom the zaxis to the xyplane. Let us denote the zenith\nand azimuth angles of the initial state m(t= 0) asθand\nϕ, i.e.,m(t= 0) = (sin θcosϕ,sinθsinϕ,cosθ). When\nthe applied field points to the x-direction ( nH=ex), the\ninitial state is\n/parenleftbigg\nθ\nϕ/parenrightbigg\nnH=ex=/parenleftBigg\nsin−1[/radicalBig\nH2\nappl+(βHs)2/HK]\ntan−1(βHs/Happl)/parenrightBigg\n,(A1)\nwhere the value of ϕis 0< ϕ < π/ 2 forHappl>0 and\nβHs>0,π/2< ϕ < π forHappl<0 andβHs>0,π <\nϕ <3π/2forHappl<0andβHs<0,and3π/2< ϕ <2π\nforHappl>0 andβHs<0. On the other hand, when\nthe applied field points to the y-direction ( nH=ey), the\ninitial state is\n/parenleftbigg\nθ\nϕ/parenrightbigg\nnH=ey=/parenleftbigg\nsin−1[(Happl+βHs)/HK]\nπ/2/parenrightbigg\n,(A2)\nwhere the range of the inverse sine function is −π/2≤\nsin−1x≤π/2. We note that the choice of the initial\nstate does not affect the evaluation of the critical cur-\nrent significantly, especially in the small field and current\nregimes.\nAppendix B: Numerically evaluated critical current\nwith different definition\nAs mentioned in Sec. III, the definition of the critical\ncurrent has arbitrariness. As an example, we show the\ntime evolution of mzunder the conditions of nH=ex,\nHappl=−30 Oe,β= 0, and j= 110×106A/cm2in\nFig. 7. In this case, the magnetization initially starts at\nmz= cos[sin−1(Happl/HK)]≃0.99, and finally moves to\na pointmz→0.12. Since the final state does not satisfy\nEq. (9), this current, j= 110×106A/cm2, should be\nregarded as the current smaller than the critical current\nin Sec. IV. However, from the analytical point of view,\nthis current can be regarded as the current larger than\nmagnetization 01\n-1 -0.50.5\nj=110×10 6 A/cm2\ntime (ns)0 2 4 6 8 10 Happl=-30 Oe\nFIG. 7. Time evolution of the zcomponent of the mag-\nnetization mzin the presence of the longitudinal field with\nHappl=−30 Oe,β= 0, and j= 110×106A/cm2. The\ndotted line is a guide showing mz= 0.\nthe critical current because the final state of the magne-\ntization is far away from the initial equilibrium.\nRegarding this point, we show the numerically eval-\nuated critical current with a different definition. The\nmagnetic state can be regarded as unstable when it fi-\nnally arrives at a point far away from the initial state54.\nThus, for example, one can define the critical current as\na minimum current satisfying\nlim\nt→∞|mz(t)−mz(0)|> δ, (B1)\nwhere a small positive real number δis chosen to be\n0.1 here. Figure 8 summarizes the numerically evalu-\nated critical current with the definition of Eq. (B1). The\nanalytical formulas, Eqs. (19)-(21), still fit well with the\nnumerical results. The absolute values of the damping-\ndependent critical current are slightly changed when the\ndefinition of the critical current is changed. This is be-\ncause Eq. (B1) is more easily satisfied than Eq. (9),\nand thus the critical current in Fig. 8 is smaller than\nthat shown in Fig. 4. However, the main results of this\npaper, such as the damping dependence of the critical\ncurrent, are not changed by changing the definition of\nthe critical current in the numerical simulations.\nAppendix C: Energy change during a precession\nAs described in Sec. V, the linearized LLG equation\nassumes a steady precession of the magnetization due to\nthe field torque when the current magnitude is close to\nthe critical current. This is because the spin Hall torque\ncompensates with the damping torque. Thus, Figs. 6(b)\nand 6(d) are obtained by substituting the solution of m\nprecessing a constant energy curve of Einto Eqs. (7) and\n(8).\nWhen the transverse field is applied and β= 0, i.e.,\nE=E, whereE=−M/integraltext\ndm·H, the precession trajec-\ntory on the constant energy curve of Eis given by55\nmx(E) = (r2−r3)sn(u,k)cn(u,k),(C1)10\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-6040 60 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=-2.0\n-50\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02(a) (b) (c)\n(d) (e) (f)longitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-4020 \n-5040 50 \nβ=2.0\n-20\n-100 -50 -150 -200: α=0.005 : α=0.01 : α=0.02-30-1010 30 \nlongitudinal magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\nlongitudinal magnetic field (Oe)0 50 100 150 200β=-2.0\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ntransverse magnetic field (Oe)critical current density (10 6 A/cm 2)\n0 50 100 150 2000\n-10050 \n-150100150\nβ=0.0\n-50\n-100 -50 -150 -200: α=0.005: α=0.01: α=0.02\ncritical current density (10 6 A/cm 2)\n0\n-10050 \n-150100150\n-50\nFIG. 8. Numerically evaluated critical currents with a diffe rent definition, Eq. (B1), in the presence of (a)-(c) the long itudinal\n(nH=ex) and (d)-(f) the transverse ( nH=ey) fields, where the value of βis (a), (d) 0 .0; (b), (e) 2 .0; and (c), (f) −2.0. The\nsolid lines are the analytically estimated critical curren t described in Sec. V.\nmy(E) =r3+(r2−r3)sn2(u,k),(C2)\nmz(E) =/radicalBig\n1−r2\n3−(r2\n2−r2\n3)sn2(u,k),(C3)\nwhereu=γ/radicalbig\nHtHK/2√r1−r3t, andrℓare given by\nr1(E) =−E\nMHappl, (C4)\nr2(E) =Happl\nHK+/radicalBigg\n1+/parenleftbiggHappl\nHK/parenrightbigg2\n+2E\nMHK,(C5)\nr3(E) =Happl\nHK−/radicalBigg\n1+/parenleftbiggHappl\nHK/parenrightbigg2\n+2E\nMHK.(C6)The modulus of Jacobi elliptic functions is k=/radicalbig\n(r2−r3)/(r1−r3). The precession period is\nτ(E) =2K(k)\nγ/radicalbig\nHapplHK/2√r1−r3,(C7)\nwhereK(k) is the first kind of complete elliptic inte-\ngral. The initial state is chosen to be my(0) =r3. Fig-\nure 6(b) is obtained by substituting Eqs. 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We find\nthat the standard Gilbert damping term naturally arises as the simplest leading-order symmetry-\nconsistentnon-conservativecontributionwithinthein-inframework. TheEFTiseasilygeneralized\nto scenarios with anisotropy and inhomogeneity. In particular, we find the classic Landau-Lifshitz\ndamping term emerges when isotropy is broken by a constant external background field. This\nprovides a first principle explanation for distinguishing the two types of damping dynamics that\nwere originally constructed phenomenologically. Furthermore, the EFT framework could also in-\ncorporate intrinsic anisotropy of the material in a straightforward way using the spurion method.\nFor systems with inhomogeneity such as nontrivial spin textures, we find that the leading order\nderivative correction yields the generalized Gilbert damping equations that were found in con-\ndensed matter literature. This shows that the EFT approach enables us to derive the form of\nhigher-derivative-order corrections in a systematic way. Lastly, using the phonon-magnon cou-\npling deduced in the magnetoelastic EFT, we are able to make a prediction for the generic form\nof the phononic contribution to the damping equation.arXiv:2312.13093v1  [hep-th]  20 Dec 2023Contents\n1 Introduction 1\n2 Review of the magnon EFT and Schwinger-Keldysh formalism 2\n2.1 Symmetry breaking in magnetoelastic systems 2\n2.2 The magnon EFT and the conservative equation of motion 4\n2.3 The Schwinger-Keldysh formalism 4\n3 Gilbert damping term from EFT 6\n3.1 Coupling at the leading order 6\n3.2 Gilbert damping 6\n4 More general materials 9\n4.1 Anisotropic materials 9\n4.2 Inhomogeneous materials 10\n5 Magnon damping term from phonons 11\n6 Conclusion and discussions 12\n1 Introduction\nIt has long been established that the methodology of coset construction serves as a powerful tool\nof relativistic effective field theories (EFTs) of Goldstone bosons (e.g. pions from spontaneously\nbroken approximate chiral symmetry) [2–5]. In recent years, many works have demonstrated\nthat its versatility is extendable to condensed matter systems where we are interested in the\nmacroscopic behavior which are usually the massless low energy excitations [6, 7]. Using this\napproach, a recent paper [1] constructed an EFT of magnetoelastic systems where the phonons\nand magnons are considered Goldstones associated with translations spontaneously broken by\nthe ground state location of the material (the lattice) and an SO(3)symmetry of the magnetic\nmoments by their ground state orientations. The EFT approach provides a systematic way to\nunderstand phonon-magnon interactions from first principles and predict the forms of higher-order\ncorrections which has not been done previously.\nWhile the paper was focused on conservative dynamics, there has also been extensive study\non the theoretical description of non-conservative dynamics of damped magnetic systems since\nthe seminal work of Landau, Lifshitz, and later Gilbert [8, 9]. However, to our knowledge, the\nprior works were mostly model-dependent phenomenological descriptions. It is therefore desirable\nto have a first principle derivation from a similar many-body EFT perspective.\nOn the other hand, the Schwinger-Keldysh formalism [10, 11] (and the related in-in formalism\n[12, 13]) has been known to describe the quantum field theory of open systems and hence fully\n– 1 –capable of describing dissipative effects. In recent years, its power has been successfully extended\nto the EFT framework of dissipative dynamics in astrophysics and black holes [14, 15, 17] that\nsystematically derives dissipative equations of motion. Therefore, it is natural to consider its\nutility in describing deriving damping equations in condensed matter EFTs.\nIn this paper, combining the power of the two techniques, we apply the Schwinger-Keldysh\nformalism to incorporate dissipative effects into the EFT of magnons to reproduce the known\nresults of magnetic damping. In section 2, we review the coset construction of the magnon EFT\nas well as the techniques in Schwinger-Keldysh formalism to be applied in this paper. Section 3\nderives the original Gilbert damping equation for homogeneous and isotropic materials. In section\n4, we move on to more general materials and recover the Landau-Lifshitz damping equation for\nanisotropic systems and generalized Gilbert damping for spatially inhomogeneous materials. In\nsection 5, we derive the damping terms originating from the magnon-phonon interaction.\nConventions: we use natural units where ℏ= 1. Unless specified, the uppercase Latin indices\nA, B, C . . . denote the full internal spin symmetry space which runs over 1,2,3while the lower\ncase ones a, b, c . . . in the begining alphabet index the broken subspace 1,2. Those in the middle\nalphabet i, j, k . . . run over the three spatial dimensions (to generalize to higher dimensions, the\ninternal spin symmetry will have to be modified accordingly).\n2 Review of the magnon EFT and Schwinger-Keldysh formalism\nIn this section, we first provide a self-contained review on the symmetries and the corresponding\ncoset construction of magnon-phonon EFT proposed by [1]. Furthermore, we summarize the\nSchwinger-Keldysh formalism which is the central tool for deriving the dissipative equations of\nmotion.\n2.1 Symmetry breaking in magnetoelastic systems\nWe follow the derivations in [1]. The symmetries under consideration are the spatial Galilean\ngroup (generated by translations Pi, rotations, Li, boost Ki) and internal symmetries (internal\ntranslations Ti, internal rotations Qi, spin rotations SA). The algebra of the generators is given\nby\n[Li, Kj] =iϵijkKk,[Li, Pj] =iϵijkPk, (2.1)\n[Ki, H] =−iPk,[Ki, Pj] =−iMδ ij, (2.2)\n[Qi, Tj] =iϵijkTk,[Qi, Qj] =iϵijkQk, (2.3)\n[SA, SB] =iϵijkKk,[Li, Lj] =iϵijkLk. (2.4)\nIn particular, the TiandQigenerators generate translations and rotations on the “comoving”\ncoordinates ϕI(x)(or the Lagrangian coordinates in continuum mechanics which simply gives an\ninitial labeling to the continuum)\nϕI(x)→ϕI(x) +aI, ϕI(x)→RI\nJϕJ(x), (2.5)\nandSAgenerates the internal rotation on the orientation of the spin\nNA→OA\nBNB, (2.6)\n– 2 –where O=eiχaSaand the Néel vector ⃗Nis the order parameter for the spin orientation.\nIn the ground state, the order parameters gain vacuum expectation values (VEVs) which we\nchoose to be D\n⃗ϕ(x)E\n=⃗ x,D\n⃗NE\n= ˆx3. (2.7)\nThe VEVs are not invariant under the transformations and hence spontaneously break some of\nthe symmetries\nUnbroken =\n\nH\nPi+Ti≡¯Pi\nLi+Qi\nS3\nM, Broken =\n\nKi\nTi\nQi\nS1, S2≡Sa. (2.8)\nThe parametrization of the ground state manifold which is simply the broken symmetry transfor-\nmations plus the unbroken translations is given by\nΩ =e−itHeixi¯PieiηiKieiπiTieiθiQieiχaSa, (2.9)\nwhere ηi,θi,χa, and πi=ϕi−xiare the corresponding Goldstone fields.\nFor any Goldstone fields ψicorresponding to the broken generators Xi, their covariant deriva-\ntives are the basic building blocks of the low energy EFT. They are systematically computed by\nthe coset construction using the Mauer-Cartan forms of the broken group\nΩ−1∂µΩ⊃\u0000\n∇µψi\u0001\nXi, (2.10)\nby computing the coefficients of the broken generator Xi. In addition, in the case that one broken\ngenerator X′appears in the commutation algebra of the other X’ with the unbroken translations\n\u0002¯P, X\u0003\n⊃X′, (2.11)\nit means that the two Goldstones are not independent and one of them can be eliminated. This\nis known as the inverse Higgs phenomenon.\nThe result of this exercise is that ηi,θiare eliminated and the only independent degrees of\nfreedom are the magnons χaand phonons πi, and at leading order in derivatives, they appear in\nthe following combinations:\n∇(iπj)= (D√\nDTDD−1)ij−δij, (2.12)\n∇tχa=1\n2ϵaBCn\nO−1h\n∂t−∂tπk(D−1)j\nk∂ji\nOo\nBC, (2.13)\n∇iχa=1\n2ϵaBC\u0000\nO−1∂iO\u0001\nBC, (2.14)\nwhere Dij=δij+∂iπj. [1] found the most general action for ferromagnetic material in the form\nL=c1\n2det (D)ϵabh\u0000\nO−1∂tO\u0001\nab−∂tπk\u0000\nD−1\u0001j\nk\u0000\nO−1∂jO\u0001\nabi\n(2.15)\n−1\n2Fij\n2(∇(iπj))∇iχa∇jχa−1\n2F3(∇(iπj))∇tχa∇tχa, (2.16)\nwhere the first term is similar to a Wess-Zumino-Witten (WZW) term that differs by a total\nderivative under the symmetry transformation.\n– 3 –2.2 The magnon EFT and the conservative equation of motion\nTo derive the equations of motion for magnons, it is more convenient to express the magnon fields\nin the nonlinear form\nˆn=R(χ)ˆx3= (sin θcosϕ,sinθsinϕ,cosθ), (2.17)\nwhere the two magnon fields are related to the angular fields by\nχ1=θsinϕ, χ 2=θcosϕ. (2.18)\nPhysically, this unit vector represents the direction of the magnetic moment. Under this repre-\nsentation, the pure magnon Lagrangian (in the absence of phonon excitations) becomes\nL →c2\n2ϵab\u0000\nO−1∂tO\u0001\nab+c6\n2(∂tˆn)2−c7\n2(∂iˆn)2, (2.19)\nwhere F3(0) = c6andFij\n2(0) = c7δij.\nThe dispersion relation for the quadratic Lagrangian has two solutions [1]\nω2\n+=\u0012c2\nc6\u00132\n+O(k2), ω2\n−=\u0012c7\nc2\u00132\nk4+O(k6). (2.20)\nFor ferromagnetic materials, where c2(c6c7)3/4, the first mode is gapped around the EFT cutoff\nscale, while the second has ω∼k2scaling and exits the EFT. In the long wavelength limit, we\nmay assign the scaling ∂1/2\nt∼∂ito the derivatives for ferromagnets.\nTo derive the equation of motion, we notice that an action of this form has a symmetry under\nthe infinitesimal spin rotation δˆn=⃗ ω׈n, where ⃗ ωis the constant infinitesimal parameter. It can\nbe shown that the Wess-Zumino-Witten term contributes to a total derivative ∂µ⃗Fµunder this\ntransformation. Using the equation for Noether current in Lagrangian mechanics\n⃗Jµ= ˆn×∂L\n∂∂µˆn−⃗Fµ, (2.21)\nwe find the conserved current\n⃗J0=−c2ˆn−c6∂tˆn׈n,⃗Ji=c7∂in׈n. (2.22)\nThe continuity equation ∂µ⃗Jµ= 0is then explicitly\nc2∂tˆn=−\u0000\nc6∂2\ntˆn−c7∇2ˆn\u0001\n׈n, (2.23)\nwhich is the equation of motion for Landau-Lifshitz model of magnetism.\n2.3 The Schwinger-Keldysh formalism\nThe appropriate formalism for non-conservative system is the so-called in-in or Schwinger-Keldysh\nformalism [10–13]. The basic idea is that there is an external sector Xthat the energy is dissipated\ninto since the total energy needs to be conserved. The external sector could evolve into any final\nstate which we do not observe, so all the dynamics are inclusive of the final states in the Hilbert\nspace of XX\nXout⟨Xin|. . .|Xout⟩⟨Xout|. . .|Xin⟩ ≡ ⟨. . .⟩in, (2.24)\n– 4 –and depends only on the initial state (hence the name in-in). We can generate an effective action\nfor the in-in observables via the Schwinger-Keldysh closed time path integral\nexp\u0014\niΓ[q; ˜q]\u0015\n=Z\ninitialDXD˜Xexp\u0014\niS[q, X]−iS[˜q,˜X]\u0015\n, (2.25)\nwhere we are integrating over an additional copy of variables ˜Xwhich corresponds to evolving\nback to the boundary conditions fixed at the initial time.\nThe equation of motion for the degrees of freedom in the observed sector qcan be derived\nfrom the action functional Γ[q; ˜q]by\nδ\nδqΓ[q; ˜q]\f\f\f\f\f\nq=˜q= 0. (2.26)\nAny external sector operator O(X)coupled to some operator in the observable sector F(q)by\nthe interaction termR\ndxO(X(x))F(q(x′))(where xis the corresponding spacetime coordinates)\nwould enter the equations of motion in terms of\n⟨O(X(x))⟩in=Z\ninitialDXD˜Xexp\u0014\niS[q, X]−iS[˜q,˜X]\u0015\nO(X), (2.27)\nwhere we have abbreviated ⟨O(X)⟩in≡ ⟨Xin|O(X)|Xin⟩. Just as in the perturbative quantum\nfield theory correlation functions calculated by Feynman propagators, this can be similarly calcu-\nlated using the Schwinger-Keldysh propagators\n⟨Oa(x)Ob(x′)⟩= \n⟨TO(x)O(x′)⟩ ⟨O (x′)O(x)⟩\n⟨O(x)O(x′)⟩ ⟨˜TO(x)O(x′)⟩!\n, (2.28)\nwhere Tand ˜Trepresent time and anti-time orderings. The sub-indices label the first and the\nsecond copy, which determines the relative time-ordering of the operators.\nExplicitly, the linear response gives\n⟨O(x)⟩=iZ\ndx′{⟨TO(x)O(x′)⟩ − ⟨O (x′)O(x)⟩}F\u0000\nq(x′)\u0001\n+O(F2). (2.29)\nor equivalently\n⟨O(x)⟩=Z\ndx′GR(x, x′)F\u0000\nx′\u0001\n, (2.30)\nwith the retarded Green’s function given by\nGR(x, x′) =iθ(t−t′)⟨[O(x),O(x′)]⟩\n=i(⟨TO(x)O(x′)⟩ − ⟨O (x′)O(x)⟩). (2.31)\nTherefore, the exact form of the damping term in the equation of motion would depend on the\ndetailed structure of these retarded response functions.\n– 5 –3 Gilbert damping term from EFT\n3.1 Coupling at the leading order\nThe composite operators Or(X)that encapsulate the external sector transform under arbitrary\nrepresentations (labeled by r), provided they form invariants of the unbroken SO(2)with the\nmagnon χaand derivatives. In order to achieve this, the operators have to be dressed with\nthe broken SO(3)/SO(2)subgroup parametrized by the Goldstones TR(χ)in the corresponding\nrepresentation\n˜Or(X)≡ Rr(χ)Or(X), (3.1)\nsuch that they transform covariantly under the unbroken subgroup [16].\nIn the long wavelength regime, the theory is organized by a spatial derivative expansion.\nIn fact, the simplest invariant operator at zeroth-order in the derivative expansion is the singlet\naligned along the ground state orientation ˆx3\nSint=Z\nd4x˜O3(X)≡Z\nd4xˆx3·˜⃗O(X), (3.2)\n(note that we are adopting manifestly relativistic notations for spacetime and energy-momentum\nfor convenience, albeit the system may or may not be relativistic). Equivalently, we may write\nSint=Z\nd4xˆn·⃗O(X) (3.3)\nwhere ˆn=O(χ)ˆx3as defined previously.\nAt the same order in this expansion, there could be more operators that can be added, such\nas when the operator is a two-index tensor operator O(2)and we may have combinations of the\nform ˆn· O(2)·ˆn. However, as will be explained in the next subsection, due to the constraint\nˆn2= 1, these will not lead to any new contributions, and from the perspective of the EFT, they\nare redundant operators. Hence, Eq. (3.3) is the only non-trivial operator at this order.\nFurthermore, we observe that the form of the operator restricts the possible external sector\nthat can couple in this way. For example, one can form such operators from fermions, where\n⃗O(ψ) =ψ†⃗ σψ+ψ⃗ σψ†, (3.4)\nwhere ⃗ σare the Pauli matrices acting as the intertwiner between the spinor and SO(3)spin space.\nOn the other hand, phonons cannot form operators in this form, and hence will not contribute in\nthis way.\n3.2 Gilbert damping\nThe undamped equation of motion is derived from the continuity equation\n∂µ⃗Jµ= 0 (3.5)\ncorresponding to the spin rotation transformation δˆn=⃗ ω׈n. To derive the non-conservative\nequation of motion, we need to find out how the additional term Eq. (3.3) affects the continuity\nequation.\n– 6 –The spin rotation transformation on ˆnalone is itself a valid symmetry for the pure magnon\naction Eq. (2.19), but the interaction term Eq. (3.3) is not invariant if we keep the external sector\nfixed. A standard trick of Noether theorem is that for an arbitrary symmetry transformation\nϕ(x)7→ϕ(x) +f(x)ϵ, if we promote the global symmetry variation parameter to be an arbitrary\nlocal variation ϵ(x), the total variation takes the form\nδS=−Z\nd4xJµ∂µϵ, (3.6)\nsuch that when ϵis a constant, invariance under the symmetry transformation is guaranteed\nδS= 0even off-shell (equation of motion is not satisfied). Integrate by parts, we find that\non-shell\nδS=Z\nd4x(∂µJµ)ϵ. (3.7)\nHowever, for an arbitrary ϵ(x), this is also the equation of motion, since\nδS=Z\nd4xδS\nδϕf(x)ϵ(x). (3.8)\nTherefore, the effect of an additional term in the action ∆Sis adding a term to the current\ndivergence\n∂µJµ= 0→∂µJµ+δ∆S\nδϕf(x) = 0 . (3.9)\nFor the spin rotations, the variation of the pure magnon EFT with local ω(x)is given by\nδS=Z\nd4x\u0010\n∂µ⃗Jµ\u0011\n·⃗ ω. (3.10)\nCorrespondingly, the addition of Eq. (3.3) leads to the modification\nδS=Z\nd4x\u0010\n∂µ⃗Jµ+ ˆn×⃗O(X)\u0011\n·⃗ ω. (3.11)\nThus, the (non-)continuity equation becomes\n∂µ⃗Jµ=−ˆn×⃗O(X). (3.12)\nWhen we focus on the measurements of the magnons, the effect of the external sector enters as\nan in-in expectation valueD\n⃗OE\nin. This may be evaluated via the in-in formalism, and the leading\norder contribution is given by\n⃗ ω·Z\nd4x∂µ⃗Jµ=−⃗ ω·Z\nd4xˆn×D\n⃗OE\nin=−⃗ ω·Z\nd4xˆn×Z\nd4x′\u0000\nGR·ˆn′\u0001\n, (3.13)\nwhere GR(t′, ⃗ x′;t, ⃗ x)is the retarded response function of the operator ⃗O.\nIn frequency space, we have GR(t, ⃗ x) =Rd3⃗kdω\n(2π)4e−iωt+⃗k·⃗ x˜GR(ω,⃗k)and furthermore, using the\nspectral representation (making the spin space indices explicit temporarily)\n˜GAB\nR(ω,⃗k) =Z∞\n−∞dω0\nπi\nω−ω0+iϵρAB(ω0,⃗k), (3.14)\n– 7 –we can separate the prefactor using the identity\ni\nω−ω0+iϵ=πδ(ω−ω0) +Pi\nω−ω0, (3.15)\ninto a δ-function and a principal part. The dissipative part is captured by the former\n˜GAB\nR,diss (ω,⃗k) =Z∞\n−∞dω0δ(ω−ω0)ρAB(ω0,⃗k) =ρAB(ω,⃗k). (3.16)\nTheindicesofthisspectralfunctionlivesinthespin SO(3)spaceand, forisotropicsystems, should\nbe built from invariant tensors δAB,ϵABC. However, the latter could not neither form a two-index\nobject nor respect parity invariance by itself, so the only symmetry-consistent possibility is\nρAB(ω,⃗k) =f(ω,|⃗k|2)δAB, (3.17)\nwith fassumed to be an analytic function of its arguments such that it has a smooth limit as\nωgoes to zero. Dissipative dynamics is antisymmetric under time reversal, so it should be odd\nunder the simultaneous transformation ω↔ −ωand(A, B)↔(B, A), meaning that the leading\norder contribution is given by\nρAB(ω,⃗k) =−iCωδAB(3.18)\nor in real spacetime\nGAB\nR,diss (t, ⃗ x) =C∂\n∂tδ(t)δ3(⃗ x)δAB. (3.19)\nCcould be understood as a Wilson coefficient in this non-conservative sector.\nFrom this, we arrive at the equation\n∂µ⃗Jµ=−Cˆn×∂\n∂tˆn. (3.20)\nCombining with the conservative part of the continuity equation Eq. (2.23), we find the Gilbert\ndamping equation\n∂\n∂t⃗ m=−γ ⃗ m×∂\n∂t⃗ m+. . . , (3.21)\nwhere γ=C/(c2ms),msˆn=⃗ m(for uniform materials), and the higher-order terms on the right-\nhand side of the original equation of motion Eq. (2.23) are contained in . . .which will be omitted\nin the following.\nWhen we have another singlet of the form ˆn·O(2)·ˆn, the effect of the extra ˆns is a replacement\nofthespectraldensity ρAB→ρACBDˆnCˆnD. ThetensorbasisstillconsistsofKroneckerdeltasince\nthe only structure that Levi-Civita tensors could contract to ˆnand would vanish automatically.\nTherefore, any additional structures will appear in the form of the inner product ˆn·ˆndue to\ncontractions with the Kronecker delta. Consequently, they do not lead to anything new due to\nthe normalization condition ˆn2= 1.\nWe see that using the in-in formalism, the form of the damping equation and the coeffi-\ncients are completely fixed by the principles of EFT: the symmetries, power counting, and Wilson\ncoefficients.\n– 8 –4 More general materials\nBy choosing the spectral function to depend only on invariant tensors and the frequency, we\nnaturally arrive at the Gilbert damping Eq. (3.21) which applies to isotropic and homogeneous\nsystems. However, in generic materials, we may be interested in situations with more general\nmaterials which for instance have non-trivial spin textures or highly anisotropic lattices and thus\ninhomogeneous or anisotropic. The advantage of the EFT framework is that these generalizations\ncan be systematically incorporated by including additional couplings. In this section, we explore\nseveral possibilities along these lines.\n4.1 Anisotropic materials\nFor homogeneous systems, one can still have anisotropy due to a background field. The retarded\nresponse function can then depend on the background. Given a homogeneous background vector\nfield⃗heff, the Levi-Civita tensor can now be incorporated into the response ρAB=DϵAB\nChC\neff.\nSince dissipative effects need to be antisymmetric under the simultaneous transformation A↔B,\nω↔ − ωandA↔Bantisymmetry is already included in the Levi-Civita tensor structure,\nthe response function has to be symmetric under ω↔ −ω. This means that the leading order\ncontribution is now independent of ω.1The corresponding response function is\nGA,B\nR,diss(t, ⃗ x) =DϵABChC\neffδ(t)δ3(⃗ x) (4.1)\nand gives rise to the conservation equation\n∂\n∂t⃗ m=−λ⃗ m×(⃗ m×⃗heff). (4.2)\nFor systems with conserved parity, the external field ⃗heffis an effective magnetic field in the sense\nof being parity odd to ensure an even-parity response function.\nThis damping equation involving an effective external background magnetic field is known\nas the Landau-Lifshitz damping equation [8]. We observe that from the EFT point of view, the\nLandau-Lifshitz and Gilbert damping terms are distinguished by symmetries.\nFor completeness, we note that in the literature there are generalizations to Gilbert damping\nby introducing anisotropic damping tensors. In field theoretic language, anisotropy corresponds\nto explicitly breaking of SO(3)and is thus straightforwardly realized in the EFT by a spurion\ncondensation. Instead of introducing a symmetry-breaking VEV to an explicit spurion operator\nin the action, one may assume the 2-point functions acquire a VEV and the most general resulting\ndissipative response function at LO is given by\nGA,B\nR,diss(ω) =SABω+AAB, (4.3)\nwhere SABandAABare general symmetric and antisymmetric tensors which are not SO(3)\ninvariant. Thegeneralizeddampingequationforananisotropicbuthomogeneousmagneticsystem\nthen become∂\n∂t⃗ m=−⃗ m×S·∂\n∂t⃗ m+⃗ m×A·⃗ m. (4.4)\n1This is analogous to the dissipative EFT of a spinning black hole in which case the role of this background is\nplayed by the direction of the spin vector [17].\n– 9 –The exact form of these anisotropy tensors can then be extracted from the microscopic details of\nthe given full theory.\n4.2 Inhomogeneous materials\nFor inhomogenous materials, e.g. configurations with background spin textures, we may have\ncontributions from higher orders in (spatial) derivative expansion. The simplest possible operator\nis given by the coupling\nSint=Z\nd4x∂iχa˜Oai(X). (4.5)\nIn terms of the orientation vector, this is equivalent to\nSint≈Z\nd4x(ˆn×∂iˆn)·⃗Oi(X), (4.6)\nat leading order in χ-field.\nAgain promoting the spin rotation transformation δˆn=⃗ ω׈nto a local parameter ⃗ ω(x), we\nfind an additional contribution to the current divergence\nδSint=Z\nd4x⃗ ω·(2∂iˆnˆn+ (ˆnˆn−δ)·∂i)·⃗Oi, (4.7)\nwhere we have used the fact that ˆn·∂iˆn=1\n2∂iˆn2= 0to simplify the expression. For the spectral\nfunction of the form\nρiA,jB(ω) =EωδijδAB, (4.8)\nthis leads to the damping term\nc2∂tˆn=E\u0010\n2∂iˆnˆn·(∂tˆn×∂iˆn) + (ˆnˆn−δ)·∂t\u0010\nˆn×⃗∇2ˆn\u0011\u0011\n=E\u0010\n2ˆn×(ˆn×∂iˆn) (ˆn×∂iˆn)·∂tˆn+ (ˆnˆn−δ)·∂t\u0010\nˆn×⃗∇2ˆn\u0011\u0011\n, (4.9)\nwhere we have used that ˆn×(ˆn×∂iˆn) =−∂iˆnon the first term. More compactly, this is\n∂\n∂t⃗ m=⃗ m×A·∂\n∂t⃗ m+E(ˆnˆn−δ)\nc2ms·∂t\u0010\n⃗ m×⃗∇2⃗ m\u0011\n, (4.10)\nwhere the first term contains the generalized damping tensor\nA=2E\nc2ms(⃗ m×∂i⃗ m) (⃗ m×∂i⃗ m). (4.11)\nThis corresponds to the generalized Gilbert damping in the presence of non-trivial spin textures\n(i.e. when ∇⃗ m̸= 0) [18].\nWe note that Eq. (4.6) is only the leading order derivative correction to the damping dynam-\nics. The EFT framework is capable of systematically generating higher derivative corrections. For\nexample, other types of inhomogeneity may be attributed to interactions in the lattice model [19]\nby\n⃗ mi×X\nijGij·⃗ mj, (4.12)\n– 10 –where the i, jindices label the lattice sites associated with the magnetic moments. In the contin-\nuum field theory, the lattice variables become ⃗ mi7→⃗ m(⃗ xi)and the tensorial structure becomes\nthe response function Gij7→GR,diss (t, ⃗ xi−⃗ xj), except that unlike the one in Eq. (3.19), it is\nnon-local (no longer proportional to δ(⃗ x)). In the simpler case that the long-range coupling falls\noff sufficiently quickly, these terms are traded for a series expansion\n⃗ m×X\nnAi1...in∂n\ni1...in∂t⃗ m, (4.13)\nfor some coefficient tensors Ai1...in. In terms of the action, this means the spectral functions are\nnow dependent on the wave vectors\nρAB(ω,⃗k) =X\nn˜Ai1...inki1. . . k inωδAB. (4.14)\n5 Magnon damping term from phonons\nFrom the magnetoelastic EFT, the generic magnon-phonon couplings are given by [1]\nLph=−1\n2Fij\n2(∇(iπj))∂iˆn·∂jˆn+1\n2ρ˜F3(∇(iπj))Dtˆn·Dtˆn, (5.1)\nwhere Dt≡∂t+vi∂iwith the velocity of the material given by vi=−∂tϕ(D−1)i\nj. The \"full\ntheory\" (technically the EFT at the next level of the hierarchy) action constrains the form of the\ncouplings between magnons the external sector to be\nLph=1\n2∂iˆn·∂jˆnOij\n2(π) +1\n2∂tˆn·∂tˆnO3(π) +1\n2∂tˆn·∂iˆnOi\n4(π), (5.2)\nwhere the last term arises from the linear-in- vcontribution in the expansion Dtˆn·Dtˆn.\nFor ferromagnets, the dispersion relation dictates the first term to be dominant. After inte-\ngrating out the external sector using the Schwinger-Keldysh methodR\nDπD˜π, we find its contri-\nbution to the in-in equation of motion is given by\nc2∂tˆn= ˆn×∂i\u0010\n∂jˆnD\nOij\n2E\u0011\n, (5.3)\nwhere the in-in expectation value is given by\nD\nOij\n2(x)E\n=Z\nd4x′Gij,kl\nR,2(x−x′)∂kˆn(x′)·∂lˆn(x′). (5.4)\nFor 4-index tensor structures under SO(3), there are two invariant tensors corresponding to the\nsymmetric-traceless and trace irreps. Therefore, one may write the leading-order dissipative con-\ntribution to the retarded response function as\nGij,kl\nR,2(x)≃\u00121\n2δijδklC2+δi(kδl)jD2\u0013\nδ3(⃗ x)∂tδ(t), (5.5)\nwhere C2andD2are the independent (Wilson) coefficients.\n– 11 –Substituting the results, we find the damping equation in a similar form\nc2∂tˆn=1\n2C2ˆn×∂i\u0000\n∂iˆn∂t\u0000\n∂jˆn·∂jˆn\u0001\u0001\n+D2ˆn×∂i\u0000\n∂jˆn∂t\u0000\n∂iˆn·∂jˆn\u0001\u0001\n=C2ˆn×∂i\u0000\n∂iˆn∂jˆn·∂j∂tˆn\u0001\n+D2ˆn×∂i\u0000\n∂jˆn∂iˆn·∂j∂tˆn\u0001\n+D2ˆn×∂j\u0000\n∂iˆn∂iˆn·∂j∂tˆn\u0001\n,(5.6)\nor more compactly\n∂\n∂t⃗ m=⃗ m×D·∂\n∂t⃗ m, (5.7)\nwhere the \"damping tensor\" Dis given by\nD=1\nc2msh\n∂i\u0000\nC2∂iˆn∂jˆn+D2∂jˆn∂iˆn\u0001\n+D2∂j\u0000\n∂iˆn∂iˆn\u0001\n+\u0000\nC2∂iˆn∂jˆn+D2∂jˆn∂iˆn\u0001\n∂i+D2∂iˆn∂iˆn∂ji\n∂j. (5.8)\nWe notice that the form of the couplings restricts the damping tensor to appear at higher-\norders in derivative expansions and hence they are expected to be small compared with contribu-\ntions from fermions (e.g. electrons) in the long wavelength limit. However, for insulating materials\nthat have electron-magnon coupling suppressed, we expect their effects to be more significant.\n6 Conclusion and discussions\nIn this paper, we used the in-in (Schwinger-Keldysh) formalism to generalize the recently con-\nstructed EFT of magnetoelasticity [1] to describe damped magnetic dynamics. We discover that\nthe Gilbert damping term naturally arises as the simplest symmetry consistent dissipative cor-\nrection within the in-in formalism. Systematic generalizations to anisotropic and inhomogeneous\nsetups also yield desired results such as the Landau-Lifshitz magnetic damping equation. More-\nover, we are able to predict the form of phononic contribution to the damping dynamics. Thus\nwe have shown that this is a useful framework to derive dissipative dynamics from first principles\nand to predict the forms of higher-order corrections in a systematic way.\nIt would be interesting to investigate the explicit full theory model of the “external sector”\nsuch as the fermionic fields in Eq. (3.4) and extract the relevant Wilson coefficients by matching\nthe response functions. In this way, we may gain better insights into what controls the damping\nparameter and give more predictive power to the EFT approach. It would also be interesting to\nmatch the relevant coefficients in Eq. (4.14) to obtain an EFT framework for a generalized class\nof models.\nFurthermore, various applications of the magnetoelastic EFT [20, 21] have appeared more\nrecently. It would be interesting to investigate the effects of adding dissipative terms into these\nproblem. There are also further developments in the technical aspects of such EFTs [22, 23]. It\nis natural to consider their implications on the non-conservative sector. We leave these problems\nfor future works.\nAcknowledgement\nThe author thanks Ira Rothstein for advising throughout the project and a careful reading of the\nmanuscript. The author also thanks Riccardo Penco for important discussions, Shashin Pavaskar\n– 12 –for other useful discussions, and Witold Skiba for comments on the draft. This work is partially\nsupported by the grants DE- FG02-04ER41338 and FG02- 06ER41449.\nReferences\n[1] S. Pavaskar, R. Penco, I. Z. Rothstein, An Effective Field Theory of Magneto-Elasticity , SciPost\nPhys.12.5.155 (2022), arXiv:2112.13873 [hep-th].\n[2] S. R. Coleman, J. Wess, and B. Zumino, Structure of phenomenological Lagrangians. 1. , Phys.Rev.\n177 2239 (1969).\n[3] J. Callan, Curtis G., S. R. Coleman, J. Wess, and B. Zumino, Structure of phenomenological\nLagrangians. 2. , Phys.Rev. 177 2247 (1969).\n[4] D. V. Volkov, Phenomenological Lagrangians , Fiz. Elem. Chast. Atom. Yadra 4 3 (1973).\n[5] V. I. Ogievetsky, Nonlinear Realizations of Internal and Space-time Symmetries , Proc. of. X-th\nWinter. School of Theoretical Physics in Karpacz, Vol. 1, Wroclaw 227 (1974) .\n[6] M. Baumgart et. al., Snowmass White Paper: Effective Field Theories for Condensed Matter\nSystems, in2021 Snowmass Summer Study. 10, 2022, arXiv:2210.03199[hep-ph].\n[7] T. Brauner et. al., Snowmass White Paper: Effective Field Theories for Condensed Matter Systems ,\nin2022 Snowmass Summer Study. 3, 2022„ arXiv:2203.10110[hep-th].\n[8] L. D. Landau and E. M. Lifshitz, Theory of the dispersion of magnetic permeability in ferromagnetic\nbodies, Phys. Z. Sowjetunion. 8, 153 (1935).\n[9] T. L. Gilbert, A phenomenological theory of damping in ferromagnetic materials , IEEE\nTransactions on Magnetics, vol. 40, no. 6, (2004).\n[10] J. Schwinger, Brownian Motion of a Quantum Oscillator , J. Math. Phys. 2 407 (1961).\n[11] L. V. Keldysh, Diagram technique for nonequilibrium processes , Zh. Eksp. Teor. Fiz. 47 1515 (1964).\n[12] C. R. Galley, The classical mechanics of non-conservative systems , Phys. Rev. Lett. 110, 174301\n(2013), arXiv:1210.2745 [gr-qc].\n[13] C. R. Galley, D. Tsang, and L. C. Stein, The principle of stationary nonconservative action for\nclassical mechanics and field theories (2014), arXiv:1412.3082 [math-ph].\n[14] S. Endlich, R. Penco, An effective field theory approach to tidal dynamics of spinning astrophysical\nsystems, Phys. Rev. D. 93.064021 (2016), arXiv:1510.08889 [gr-qc].\n[15] W. D. Goldberger and I. Z. Rothstein, Horizon radiation reaction forces , JHEP 10 026 (2020),\narXiv:2007.00731[hep-th].\n[16] L. V. Delacrétaz, S. Endlich, A. Monin, R. Penco, F. Riva, (Re-)Inventing the Relativistic Wheel:\nGravity, Cosets, and Spinning Objects , JHEP 11 (2014) 008, arXiv:1405.7384 [hep-th].\n[17] W. D. Goldberger, J. Li, and I. Z. Rothstein, Non-conservative effects on spinning black holes from\nworld-line effective field theory , JHEP 06 053 (2021) arXiv:2012.14869[hep-th].\n[18] S. Zhang, S. S.-L. Zhang, Generalization of the Landau-Lifshitz-Gilbert Equation for Conducting\nFerromagnets , Phys. Rev. Lett. 102, 086601 (2009).\n[19] S. Brinker, M. dos Santos Dias, S. Lounis, Generalization of the Landau-Lifshitz-Gilbert equation by\nmulti-body contributions to Gilbert damping for non-collinear magnets , J. Phys.: Condens. Matter\n34 285802 (2022), arXiv:2202.06154 [cond-mat.mtrl-sci].\n– 13 –[20] A. Esposito, S. Pavaskar, Optimal anti-ferromagnets for light dark matter detection (2022),\narXiv:2210.13516 [hep-ph].\n[21] S. Pavaskar, I. Z. Rothstein, The Dynamics of Line Defects and Their Sensitivity to the Lattice\nStructure (2022), arXiv:2212.10587 [hep-th].\n[22] A. Nicolis, I. Z. Rothstein, Apparent Fine Tunings for Field Theories with Broken Space-Time\nSymmetries (2022), arXiv:2212.08976 [hep-th].\n[23] C. O. Akyuz, G. Goon, R. Penco, The Schwinger-Keldysh Coset Construction (2023),\narXiv:2306.17232 [hep-th].\n– 14 –"    },    {        "title": "1207.2192v2.Thermal_vortex_dynamics_in_thin_circular_ferromagnetic_nanodisks.pdf",        "content": "arXiv:1207.2192v2  [cond-mat.stat-mech]  29 Aug 2012Thermal vortex dynamics in thin circular ferromagnetic nan odisks\nG. M. Wysin∗\nDepartment of Physics, Kansas State University, Manhattan , KS 66506-2601\nW. Figueiredo\nDepartamento de F´ ısica, Universidade Federal de Santa Cat arina, Florian´ opolis, Santa Catarina, Brazil\n(Dated: August 27, 2012)\nThe dynamics of gyrotropic vortex motion in a thin circular n anodisk of soft ferromagnetic mate-\nrial is considered. The demagnetization field is calculated using two-dimensional Green’s functions\nfor the thin film problem and fast Fourier transforms. At zero temperature, the dynamics of the\nLandau-Lifshitz-Gilbert equation is simulated using four th order Runge-Kutta integration. Pure\nvortex initial conditions at a desired position are obtaine d with a Lagrange multipliers constraint.\nThese methods give accurate estimates of the vortex restori ng force constant kFand gyrotropic\nfrequency, showing that the vortex core motion is described by the Thiele equation to very high\nprecision. At finite temperature, the second order Heun algo rithm is applied to the Langevin dy-\nnamical equation with thermal noise and damping. A spontane ous gyrotropic motion takes place\nwithout the application of an external magnetic field, drive n only by thermal fluctuations. The\nstatistics of the vortex radial position and rotational vel ocity are described with Boltzmann distri-\nbutions determined by kFand by a vortex gyrotropic mass mG=G2/kF, respectively, where Gis\nthe vortex gyrovector.\nPACS numbers: 75.75.-c, 85.70.Ay, 75.10.Hk, 75.40.Mg\nKeywords: magnetics, dipolar field, demagnetization, vort ex dynamics, nanoparticles.\nI. INTRODUCTION: VORTEX STATES IN\nTHIN NANOPARTICLES\nVortices in nanometer-sized thin magnetic particles1\nhave attracted a lot of attention, due to the possibili-\nties for application in resonators or oscillators, in detec-\ntors, as objects for data storage.2We consider the dy-\nnamic motion of an individual vortex in a thin circular\ndisk (radius Rand height L≪R) of soft ferromagnetic\nmaterial such as Permalloy-79 (Py) where vortices have\nbeen commonly studied.3,4In disks of appropriate size,\nthe single vortex state is very stable and of lower en-\nergy than a single-domain state.5Especially, we study\nthe effective force constant kFresponsible for the restor-\ning force on a vortex when it is displaced from the disk\ncenter,F=−kFX, whereXis the vortex core position\nrelative to the disk center. If the vortex is initially dis-\nplaced from the disk center, say, by a pulsed magnetic\nfield,6it oscillates in the gyrotropic mode,7at an angu-\nlar frequency ωG=kF/G, whereG=Gˆzis the vortex\ngyrovector, pointing perpendicular to the plane of the\ndisk. The vortex gyrotropic motion has been observed,\nforexample, byphotoemissionelectronmicroscopyusing\nx-rays.8Not only in small disks but in many easy-plane\nmagnetic models this type of vortex dynamics has been\nstudied for its interesting gyrotropic dynamics.9,10The\ngyrotropic mode is due to the translational mode11,12in\na whole spectrum of internal vibrations of a magnetic\nvortex.13\nTheforceconstantestimatedanalyticallybyGuslienko\net al.7using the two-vortices model14gave predictions\nof the gyrotropic frequencies obtained in micromagnetic\nsimulations, with reasonably good accord between thetwo. Here, we discuss direct numerical calculations of\nkFbased on static vortex energies, by using a Lagrange\nmultipliers technique to secure the vortex position Xat\na desired location,15and map out its potential within\nthe disk. We apply an adapted two-dimensional (2D)\nmicromagnetics approach for thin systems to calculate\nthe demagnetization field. As a result, the calculations\ncanbedirectlycomparedwiththetwo-vorticesprediction\nforkF.\nAt the same time, we give a corresponding study of\nthe vortex dynamics to calculate the gyrotropic frequen-\ncies. It is found that the staticresults forkFcombined\nwith the dynamics results forωGagree with the predic-\ntion of the Thiele dynamical equation,16,17ωG=kF/G,\nto very high precision. Similar to Ref. 7, kFis found\nto be close to linear in the aspect ratio L/Rif the disks\nare thin but not so thin that the vortex would be desta-\nbilized. Our values for kFare slightly less than those\nin the two-vortices model, as the numerical relaxation of\nthe vortex structure allows for more flexibility than an\nanalytic expression.\nAmicromagneticsstudy ofthis system18forfinite tem-\nperatureshowsevidencefora spontaneous gyrotropic vor-\ntex motion with a radius ofa couple ofnanometers, with-\nout the application of a magnetic field. The spontaneous\ngyrotropic motion occurs even if the vortex is initiated\nat the center of the disk in the simulations. It is clear\nthat thermal fluctuations should lead to a random dis-\nplacement of the vortex core away from the disk center,\nhowever, it is striking that the ordered gyrotropic rota-\ntion appears and even dominates over the thermal fluc-\ntuations. Here we confirm this effect, and also find that\na spin wave doublet19(of azimuthal quantum numbers\nm=±1) is excited together with the gyrotropic motion.2\nHaving at hand the force constant kF, we can analyze\nboth the dynamics and the statistics of the gyrotropic\nmotion induced by the temperature. The study of the\nfinite temperature dynamics is carried out using a mag-\nnetic Langevin equation that includes stochastic mag-\nnetic fields together with damping. We discuss the so-\nlution via the second order Heun method20,21applied to\nmagnetic systems. Further, we introduce a technique for\nestimating the location of the vortex core accurately in\nthe presence of fluctuations. Based on the behavior of\nkFwith disk geometry, we find it possible to predict the\nRMS displacement of the vortex core in equilibrium. By\nusing the collective coordinate Hamiltonian for the vor-\ntex, as derived from the Thiele equation, it is also pos-\nsible to determine the probability distributions for vor-\ntex radial displacement r=|X|and rotational velocity\nV=ωGr. It is interesting to see that the velocity dis-\ntribution,f(V), is of the Boltzmann form for a particle\nwith an effective mass given by mG=G2/kF, which is\nfound to depend only on the gyromagnetic ratio γ, the\nmagnetic permeability of free space µ0and the disk ra-\ndius.\nII. DISCRETE MODEL FOR THE CONTINUUM\nMAGNET\nWe determine the magnetic dynamics for a continuum\nmagnetic particle, but using a thin-film micromagnet-\nics approach,22defining appropriate dipoles at cells of\na two-dimensional grid. This is a modification of usual\nmicromagnetics23where a 3D grid is used. The parti-\ncle has a thickness Lalong thez-axis, and a circular\ncross-section of radius R. For thin film magnets it is rea-\nsonable to make the assumption that the magnetization\n/vectorM(r) does not depend on the coordinate zthrough the\nthickness. This is acceptable as long as the particle is\nvery thin. The demagnetization field tends to cause M\nto lie within the xyplane in most of the sample,24except\nfor the vortex core region. Even in the vortex core, how-\never, one should not expect large variations of Mwithz,\ndue to the dominance of the ferromagnetic exchange over\nthe dipolar interactions through short distances. In this\nsituation for verythin magnets, this 2Dapproachhas the\nobvious advantage of greater speed over 3D approaches,\nwithout sacrificing accuracy. It is somewhat like using a\nsingle layer of computation cells in 3D micromagnetics,\nwiththe cellheightlongerthanitstransversedimensions.\nTheenergyoftheoriginalcontinuumsystem, including\nexchange and magnetic field energy, can be expressed as\na volume integral,\nH=/integraldisplay\ndV/braceleftBig\nA∇/vector m·∇/vector m−µ0/bracketleftBig\n/vectorHext+1\n2/vectorHM/bracketrightBig\n·/vectorM/bracerightBig\n.\n(1)\nThe magnetization scaled by saturation magnetization\nMsis used to define the scaled magnetization /vector m=\n/vectorM/Ms, that enters in the exchange term, where Aisthe exchange stiffness (about 13 pJ/m for Permalloy).\nThe last term is the interaction with an externally gen-\nerated field, /vectorHext. The demagnetization energy involves\nthe demagnetization field /vectorHMthat is generated by /vectorM,\nand which is determined through a Poisson equation in-\nvolving the scalar magnetic potential Φ M,\n/vectorHM=−/vector∇ΦM,−∇2ΦM=ρM≡ −/vector∇·/vectorM.(2)\nThis is solved formally in three dimensions using a con-\nvolution with the 3D Green’s function:\nΦM(r) =/integraldisplay\nd3r′G3D(r−r′)ρM(r′),(3)\nG3D(r) =1\n4π|r|. (4)\nHowever, this is reduced to an effective 2D Green’s op-\nerator, appropriate for thin magnetic film problems, re-\nviewed below.\nA. The micromagnetics model\nThe micromagnetics25,26is set up to use nanometer-\nscaled cells in which to define coordinates /vector mias the av-\neraged scaled magnetization in that cell. The system is\ndivided into cells of size a×a×L(a×ais the cross\nsection in the xy-plane), rather than cubical cells. Each\ncellicontains a magnetic moment of fixed magnitude\nµ=La2Ms, whereMsis the saturation magnetization.\nThe direction of the (assumed uniform) magnetization\nin a cell is a unit vector, ˆ mi, whose dynamics is to be\nfound. The cells interact with neighboring cells via the\nexchange interaction, and with all other cells, due to the\ndemagnetization field, and also with any external field.\nFor the square grid of cells, the exchange energy is\nfound to be equivalent to\nHex=−2AL/summationdisplay\n(i,j)ˆmi·ˆmj (5)\nwhere the sum is over nearest neighbor cell pairs. The\nenergyscaleofexchangeistakenasthe basicenergyunit.\nThus it is convenient to define an effective exchange con-\nstant acting between the cells,\nJ= 2AL, (6)\nand for the computations, all other energies will be mea-\nsured in this unit. In addition, the saturation magneti-\nzation is a convenient unit for magnetic fields as well as\nfor/vectorM. So we define scaled fields,\n˜HM≡/vectorHM\nMs,˜Hext≡/vectorHext\nMs. (7)\nAs a result of this, the magnetic field interaction energy\nterms are scaled here as follows. For the demagnetiza-\ntion,\nHdemag=−Ja2\n2λ2ex/summationdisplay\ni˜HM\ni·ˆmi, (8)3\nand for the energy in the external field,\nHext=−Ja2\nλ2ex/summationdisplay\ni˜Hext\ni·ˆmi. (9)\nThese depend on the definition of the exchange length,\nλex=/radicalBigg\n2A\nµ0M2s, (10)\nthat gives a measure of the competition between ex-\nchange and dipolar forces. This means that the effective\n2D Hamiltonian can be written as\nH=−J\n\n/summationdisplay\n(i,j)ˆmi·ˆmj\n+a2\nλ2ex/summationdisplay\ni/parenleftBig\n˜Hext\ni+1\n2˜HM\ni/parenrightBig\n·ˆmi/bracerightBigg\n.(11)\nB. The demagnetization field /vectorHMin a thin film\nIt is important to calculate the demagnetization field\nefficiently and accurately, as it plays an important role\nin the dynamics, and is the most computational effort.\nAn approach for thin films, described by Huang22is used\nhere, where we need effective Green’s functions that act\nin 2D on the magnetization /vectorM(x,y). This is somewhat\ndifferent from that used in Refs. 15 and 27, where the\nin-plane part of /vectorHMwas calculated by first estimating\nthe magnetic charge density ρM. Here, it is preferred to\ncalculate/vectorHMdirectly from the field /vectorM, which has less\nsteps, and is found to result in extremely precise energy\nconservation in the absence of damping.\nBy applying an integration by parts, and throwing out\na surface term outside the magnet, the solution for the\nmagnetic potential is first written as an operation on /vectorM:\nΦM(r) =/integraldisplay\nd3r′/vector∇′G3D(r−r′)·/vectorM(r′).(12)\nOne can notice that this involves the propagator for the\ndipole potential, that is,\n/vector∇′G3D(r−r′) =r−r′\n4π|r−r′|3(13)\nis the function whose product with a source dipole at\nposition r′gives the magnetic potential at rdue to that\ndipole.\nToproceedfurther, itisusefultoconsiderthecontribu-\ntions to the vertical ( z) and horizontal ( xy) components\nof/vectorHMseparately. Consider a source cell centered at\n(x′,y′), and the vertical component of /vectorHMit generates,\ndue toM′\nz≡Mz(x′,y′), at an observer position ( x,y).\nThe usual procedure is to sumover the source point z′andaverageover the observer position z. One has the\ncontribution from this cell, of area dA′=dx′dy′,\ndΦM=dA′M′\nz\n4π/integraldisplayδ\n−δdz′(z−z′)\n[˜r2+(z−z′)2]3/2(14)\nwhereδ=L/2 and the notation ˜ r2= (x−x′)2+(y−y′)2\nis used. The integration gives\ndΦM=−dA′M′\nz\n4π/bracketleftBigg\n1/radicalbig\n˜r2+(z+δ)2−1/radicalbig\n˜r2+(z−δ)2/bracketrightBigg\n.\n(15)\nThis would also be obtained exactly the same if starting\nfrom the magnetic surface charge density. Then, its neg-\native gradient with respect to zgives the contribution to\nthe demagnetization field. If we also do the averaging\nover the observer position z, these two operations undo\neach other. The field averaged in the observer cell posi-\ntion is\n∝angb∇acketleftdHMz∝angb∇acket∇ight=−1\nL/integraldisplayδ\n−δdzd\ndzΦM=−1\nLdΦM/vextendsingle/vextendsingle/vextendsingle/vextendsingle+δ\n−δ.(16)\nEvaluation of the limits, and then including a sum over\nthesourcepoint r′= (x′,y′), showsthatthefieldisdeter-\nmined by convolution with an effective Green’s function\nin 2D,\nHMz(r) =/integraldisplay\nd2r′Gzz(r−r′)Mz(r′),(17)\nGzz(˜ r) =1\n2πL/parenleftbigg1√\n˜r2+L2−1\n˜r/parenrightbigg\n.(18)\nIntheseexpressions,it isunderstoodthatthe positions r,\nr′and the displacement between the two, ˜ r=r−r′, are\nnowtwo-dimensional. Theexpressionfor Gzzisdivergent\nat zero radius. However, it is a weak divergence that\ncan be regularized for the computation on the grid, by\naveragingoverthe cell area. For removingthe divergence\nat ˜r= 0, averaging over a circle of area equal to the cell\nareaa2replaces the value of Gzz(0) by the longitudinal\ndemagnetization factor Nzfor cylinder of length Land\nradiusro=a/√π. So we set\nGzz(0) =∝angb∇acketleftGzz∝angb∇acket∇ighto=−Nz=−1\nL/parenleftBig\nL+ro−/radicalbig\nL2+r2o/parenrightBig\n.\n(19)\nThe ”o” subscript refers to averaging over the circle of\nradiusro. See Ref. 15 for more details. Note that Gzz\nis always negative; it correctly gives the demagnetization\nfield opposite to the magnetization /vectorMwhich generated\n/vectorHM.\nFor the in-plane components of /vectorHM, a similar proce-\ndure can be followed. Due to symmetry considerations,\nonlyMxandMycancontribute. Onecanstartbyfinding\nthe magnetic potential,\ndΦM=dA′\n4π/integraldisplayδ\n−δdz′(x−x′)M′\nx+(y−y′)M′\ny\n[˜r2+(z−z′)2]3/2.(20)4\nThe integration over the source vertical coordinate z′\ngives\ndΦM=dA′\n4π˜r2/bracketleftbig\n(x−x′)M′\nx+(y−y′)M′\ny/bracketrightbig\n×/bracketleftBigg\nz+δ/radicalbig\n˜r2+(z+δ)2−z−δ/radicalbig\n˜r2+(z−δ)2/bracketrightBigg\n.(21)\nThe averaging over the observer point zcan be carried\nout, and gives,\n∝angb∇acketleftdΦM∝angb∇acket∇ight=1\nL/integraldisplayδ\n−δdz dΦM(z) =√\n˜ r2+L2−|˜ r|\n2πL˜ r2\n×/bracketleftbig\n(x−x′)M′\nx+(y−y′)M′\ny/bracketrightbig\ndA′.(22)\nFinally, the in-plane gradient leads to the in-plane de-\nmagnetization components. Including also the zcompo-\nnents, the demagnetization field averaged in the observer\ncell is obtained from\n/vectorHM\nα(r) =/integraldisplay\nd2r′/summationdisplay\nβ=x,y,zGαβ(r−r′)Mβ(r′).(23)\nTheelementsoftheGreenfunctionneededherearefound\nto be\nGxx(˜ r) =L\n2π˜r4/parenleftbigg˜x2\n√\n˜r2+L2−˜y2\n√\n˜r2+L2+ ˜r/parenrightbigg\n,(24)\nGxy(˜ r) =L\n2π˜r42√\n˜r2+L2+ ˜r√\n˜r2+L2+ ˜r˜x˜y√\n˜r2+L2.(25)\nThe element Gyyis obtained from Gxxby swapping x\nandyindices, and Gyx=Gxy. One can verify that these\nmatrix elements go over to those for the far-field of a\npoint dipole, in the limit ˜ r→ ∞.\nThese transverse elements of Galso are not defined at\nzeroradius, becauseanimplicit assumptioninthederiva-\ntion is that the observation point is outside of the source\ncell. Thereneedstobeaninternaldemagnetizationeffect\nwithin a cell even for a transverse magnetization such as\nMx∝negationslash= 0 orMy∝negationslash= 0. For long thin cells with L≫a,\nthis internal transverse demagnetization factor would be\napproximately Nx=Ny≈1\n2. As a better alternative,\nwe setGxy(0) = 0, and replace Gxx(0) andGyy(0) with\nthe transverse demagnetization factor of a cylinder with\ncross-sectional radius ro=a/√π,\nGxx(0) =Gyy(0) =−Nx=1\n2L/parenleftBig/radicalbig\nL2+r2o−ro/parenrightBig\n.(26)\nIn this way, the internal demagnetization components of\nthe computation cells satisfy the requirement Nx+Ny+\nNz= 1, while making Gxx(0) andGyy(0) consistent with\nthe regularization done for Gzz(0).\nThe above results show that /vectorHMis found by convo-\nlution of the 2D Green’s operator, as a matrix, with /vectorM.\nThecalculationcanbemadefasterbyusingafastFourier\ntransform(FFT) approach,28whichreplacestheconvolu-\ntion in real space with multiplication in reciprocal space.Of course, the simplest FFT approach requires a grid\nwith a size like 2n×2n, wherenis an integer. Our 2D\nsystem is a circle of radius R=Na(Nis the size in\ninteger grid units). For the FFT approach to work, so\nthat the system being simulated is a single copy of the\ncircle with no periodic interactions with the images, one\ncan choosethe smallest nsuch that 2n≥2N. By making\nthe FFT grid at least twice as large as the circle to be\nstudied, the wrap-aroundproblem, due to the periodicity\nof Fourier transforms, is avoided in the evaluation of the\nconvolution. The FFT of the Green’s matrix, which is\nstatic, is done only once at the start of the calculation.\nDuring every time step of the integrations, however, the\nFFT of the magnetization field components must be car-\nried out, for every stage at which the demagnetization\nfield is required. Of course, the inverse FFTs to come\nback to/vectorHMare needed as well in every stage of the time\nintegrator.\nIII. THE DYNAMICS AND UNITS\nA. Zero temperature\nThe zero-temperature undamped dynamics of the sys-\ntem is determined by a torque equation, for each cell of\nthe micromagnetics system,\nd/vector µi\ndt=γ/vector µi×/vectorBi. (27)\nHere/vectorBiis the local magnetic induction acting on the\nithcell,γis the electronic gyromagnetic ratio, and the\ndipole moment of the cell is /vector µi=La2Msˆmi. The local\nmagnetic induction can be defined supposing an energy\n−/vector µi·/vectorBifor each dipole, with\n/vectorBi=−δH\nδ/vector µi=−1\nµδH\nδˆµi=J\nLa2Ms/vectorbi,\n/vectorbi≡/summationdisplay\nj=z(i)ˆmj+a2\nλ2ex/parenleftBig\n˜Hext\ni+˜HM\ni/parenrightBig\n.(28)\nThe sum over jcontains only sites z(i) that are near-\nest neighbors of site i. This dimensionless induction /vectorbi\nused in the simulations is converted to real units by the\nfollowing unit of magnetic induction,\nB0≡J\nLa2Ms=2A\na2Ms=λ2\nex\na2µ0Ms.(29)\nForcomputations, thedynamicsiswrittenintermsofthe\ndimensionless fields, also scaling the time appropriately:\ndˆmi\ndτ= ˆmi×/vectorbi, τ=γB0t. (30)\nThismeansthattheunitoftimeinthesimulationsis t0=\n(γB0)−1. For Permalloy with A= 13 pJ/m, Ms= 860\nkA/m, one has λex≈5.3 nm. In our simulations we put5\nthetransverseedgeofthecellsas a= 2.0nm. Thenusing\nthe gyromagneticratio, γ=e/me≈1.76×1011T−1s−1,\nthe computation units are based on µ0Ms= 1.08 T and\nB0≈7.59 T. This large value for B0is the scale of the\nlocal magnetic induction due to the exchange interaction\nbetween the cells. The time unit is then t0≈0.75 ps;\na frequency unit is f0=γB0= 1.336 THz. We may\ndisplay frequency results, however, in units ofµ0\n4πγMs≈\n15.1 GHz for Permalloy, as this expression is equivalent\ntoγMsinCGSunits. Forthedisksizesusedhere, typical\nperiods of the vortex gyrotropic motion are around τG∼\n4000, which then corresponds to dimensionless frequency\nν= 1/τG∼2.5×10−4, and hence, physical frequency\nf=νf0∼0.3 GHz.\nIn some cases we also need to include Landau-Gilbert\ndamping, with some dimensionless strength α. Then this\nis included into the dynamics with the usual modifica-\ntion,\ndˆmi\ndτ= ˆmi×/vectorbi−αˆmi×/parenleftBig\nˆmi×/vectorbi/parenrightBig\n.(31)\nThe zero temperature dynamics was integrated numer-\nically for this equation, using a standard fourth-order\nRunge-Kutta (RK4) scheme. Typically, a time step of\n∆τ= 0.04 was found sufficient to insure the correct en-\nergy conserving dynamics (when α= 0) and result in\ntotal energy conserved to better than 12 digits of preci-\nsion over5.0×105time steps in a system with as many as\n4000 cells. To get this high precision, however, it is nec-\nessary to always evaluate the full demagnetization field\nat all four intermediate stages of the individual Runge-\nKutta time steps.\nB. Finite temperature: Langevin dynamics\nFornon-zerotemperature, thedynamicsisinvestigated\nhere using a Langevin approach. This requires including\nboth a damping term and a stochastic torque in the dy-\nnamics; together they represent the interaction with a\nheat bath. The size of the stochastic torques is related\nto the temperature and the damping constant, such that\nthe system reaches thermal equilibrium.\nIt is reasonable to think of the dynamics depending on\nstochastic magnetic inductions /vectorbs, in addition to the de-\nterministic fields /vectorbifrom the Hamiltonian dynamics. For\nthe discussion here, suppose we consider the dynamics\nof one computation cell, and suppress the iindex. The\ndynamical equation for that cell’s ˆ m, including both the\ndeterministic and random fields, is\ndˆm\ndτ= ˆm×/parenleftBig\n/vectorb+/vectorbs/parenrightBig\n−αˆm×/bracketleftBig\nˆm×/parenleftBig\n/vectorb+/vectorbs/parenrightBig/bracketrightBig\n.(32)\nThe first term is the free motion and the second term is\nthe damping. Alternatively, the dynamics can be viewed\nas that due to the superposition of the deterministic ef-\nfects (due to /vectorb) and stochastic effects (due to /vectorbs).For a given temperature T, the stochastic fields estab-\nlish thermal equilibrium, provided the time correlations\nsatisfy the fluctuation-dissipation (FD) theorem,\n∝angb∇acketleftbλ\ns(τ)bλ′\ns(τ′)∝angb∇acket∇ight= 2αTδλλ′δ(τ−τ′).(33)\nδλλ′is the Kronecker delta and the indices λ,λ′refer to\nany of the Cartesian coordinates; δ(τ−τ′) is a Dirac\ndelta function. The dimensionless temperature Tis the\nthermal energy scaled by the energy unit J,\nT ≡kT\nJ=kT\n2AL, (34)\nwherekis Boltzmann’s constant. The fluctuation-\ndissipation theorem expresses how the power in the ther-\nmal fluctuations is carried in the random magnetic fields.\nIn terms of the physical units, the relation is\nγµ∝angb∇acketleftBλ\ns(t)Bλ′\ns(t′)∝angb∇acket∇ight= 2αkTδ λλ′δ(t−t′).(35)\nwhereµ=La2Msis the magnetic dipole moment per\ncomputation cell.\nC. Time evolution with second order Heun (H2)\nmethod\nThe Langevin equation (32) is a first-order differential\nequation that is linear in multiplicative noise. If y=\ny(τ) represents the full state of the system (a vector of\ndimension 3 N, whereNis the number of cells), then the\ndynamics follows an equation of the form\ndy\ndτ=f[τ,y(τ)]+fs[τ,y(τ)]·bs(τ).(36)\nThe vector function fis the deterministic time deriva-\ntive and the vector function fsdetermines the stochastic\ndynamics;bsrepresents the whole stochastic field of the\nsystem. An efficient method for integrating this type of\nequation forward in time is the second order Heun (H2)\nmethod.20,21That is in the family of predictor-corrector\nschemes and is rather stable. It involves an Euler step as\nthe predictor stage, and a corrector stage that is equiv-\nalent to the trapezoid rule. Some details of the method\naresummarizedhere, toindicatehowthestochasticfields\nare included, and to show why it is used rather than the\nfourth order Runge-Kutta method (the latter seems dif-\nficult to adapt to the stochastic fields).\nWe use the notation yn≡y(τn) to show the values at\ntimesτn=n∆τ, according to the choice of some inte-\ngration time step ∆ τ. Integrating Eq. (36) over one time\nstep gives the Euler predictor estimate for y(τn+∆τ):\n˜yn+1=yn+f(τn,yn)∆τ+fs(τn,yn)·(σswn).(37)\nThelastfactor, σswn,isintroducedtorepresentthetime-\nintegral of the stochastic magnetic inductions. σsis a\nvariance and wnrepresents a vector of 3 Nrandom num-\nbers, oneforeachCartesiancomponentateachsiteofthe6\ngrid. Consider, say, the result of integrating the equation\nof motion for just one component for one site:\n/integraldisplayτn+∆τ\nτndτ bx\ns(τ)−→σswx\nn. (38)\nThe physical variance σsneeded for this to work cor-\nrectly, must be determined by the FD theorem. For this\nindividual component at one site, the squared variance is\nσ2\ns=/angbracketleftBigg/parenleftBigg/integraldisplayτn+∆τ\nτndτ bx\ns(τ)/parenrightBigg2/angbracketrightBigg\n=/integraldisplayτn+∆τ\nτndτ/integraldisplayτn+∆τ\nτndτ′∝angb∇acketleftbx\ns(τ)bx\ns(τ′)∝angb∇acket∇ight.(39)\nNow applying the FD theorem to this gives the required\nvariance of the random fields, that depends on the time\nstep being used:\nσs=√\n2αT∆τ. (40)\nThis means that individual stochastic field components\nbλ\ns(τ), integrated over one time step, are replaced by ran-\ndom numbers of zero mean with variance σs, as used\nabove.\nForthecorrectorstage,thepoints ynand ˜yn+1areused\nto get better estimates of the slope of the solution. Then\ntheir average is used in the trapezoid corrector stage:\nyn+1=yn+1\n2[f(τn,yn)+f(τn+1,˜yn+1)]∆τ(41)\n+1\n2[fs(τn,yn)+fs(τn+1,˜yn+1)]·(σswn).\nThe error is of order O((∆τ)3), hence it is a second or-\nder scheme. Note that the same vector of 3 Nrandom\nnumberswnused in the predictor stage are re-used in\nthe corrector stage, because it is the evolution over the\nsame time interval.\nIn the coding for computations, one does not use the\nexplicit form of the functions fandfs. Rather, at each\ncell, firstonecancalculatethedeterministiceffectivefield\n/vectorbibased on the present state of the system. Its effect in\nthe dynamics will be actually proportional to its product\nwith the time step, i.e., it gives a contribution ∆ˆ mi∝\n/vectorbi∆τ. Of course, the stochastic change in this same site\nwillbeproportionaltothestochasticeffectivefield, which\nis someσs/vector wifor that site, where /vector wi= (wx\ni,wy\ni,wz\ni). So\nthe total change at this site is linearly determined by a\ncombination,\n∆ˆmi∝/vector gi, /vector gi≡/vectorbi∆τ+σs/vector wi. (42)\nAn effective field combination /vector giacts in this way both\nduring the predictor and the corrector stages. In either\nstage, a dynamic change in a site is given by a simple\nrelation,\n∆ˆmi= ˆmi×[/vector gi−α(ˆmi×/vector gi)]. (43)Of course, the predictor stage uses the last configuration\nof the whole system to determine all the /vectorbi, while the\ncorrector finds the needed /vectorbibased on the predicted po-\nsitions. And, the corrector actually does the average of\n∆ˆmifrom the Euler stage and the second estimate from\nthe corrector stage. The same random numbers wnused\nin the predictor stage are used again in the corrector, for\na chosen time step.\nThe integration requires a long sequence of quasi-\nrandom numbers wn. It is important that the simulation\ntime does not surpass the period of the random num-\nbers. We used the generator mzran13 due to Marsaglia\nand Zaman,29implemented in the C-language for long\nintegers. This generator is very simple and fast and has\na period of about 2125, and is based on a combination of\ntwo separate generators with periods of 232and 295.\nIV. VORTEX STATE PROPERTIES AND\nZERO-TEMPERATURE DYNAMICS\nThe dynamics at zero temperature, calculated with\nRK4, was used to check basic vortex dynamic proper-\nties such as the stability and gyrotropic mode frequency.\nWe also used the Langevin dynamics calculated with sec-\nond order Heun method to include finite temperature to\nsee the primary thermal effects for some specific vortex\ninitial configurations. For some of these studies, it is ex-\ntremely beneficial to produce a well-formed initial vortex\nstate in some desired location without the presence of\nspin waves.\nAn initial vortex state is prepared first in a planar con-\nfiguration of positive vorticity q= +1, namely, in-plane\nmagnetization angle φ= tan−1my/mxgiven by\nφ(x,y) =qtan−1x−x0\ny−y0. (44)\n(The negative vorticity state q=−1 is destabilized by\nthe demagnetization field, so there is no reason to con-\nsider it.) This is the profile of a vortex centered at\nposition (x0,y0). The out-of-plane component here is\nmz= 0, however, the stable vortex state has a nonzero\nout-of-plane component close to mz=±1 at the vor-\ntex core (polarization p=±1). This stable vortex state\nwas reached by the local spin alignment procedure11for\na vortex at the constrained position ( x0,y0), described\nin Ref. 15. Briefly, that is a procedure where each ˆ mi\nis aligned along its local induction /vectorbi, and the process\nis iterated until convergence. The constraint is applied\nas extra fictitious fields included with the Lagrange mul-\ntiplier technique, that force the desired vortex starting\nposition. This procedure helps to remove any spin waves\nthat would otherwise be generated starting from any ar-\nbitrary initial state. This state would be a perfect static\nstate if generated in the center of the disk. When gen-\nerated off-center, the dynamics associated with its mo-\ntion still is able to produce some spin waves. A cleaner7\nvortex motion can be generated if there is a weak damp-\ning applied ( α= 0.02) over some initial time interval\n(τ≈1000). After that, the system can be let to evolve\nin energy-conserving dynamics, if needed.\nThis relaxed vortex state develops either positive or\nnegative out-of-plane component, including some small\nrandomness in the initial state before the relaxation. If\nmz≈+1 (−1) in the vortex core region, the vortex has\npositive (negative) polarization and a positive (negative)\ngyrovector G=Gz, defined from\nG= 2πQm0\nγˆz, Q≡qp. (45)\nγis the electron gyromagnetic ratio and m0=µ/a2=\nLMsis the magnetic dipole moment per unit area. The\nintegerQ=±1 defines the quantized topological charge\nthat determines the twoalloweddiscrete values of the gy-\nrovector. To a good degree of precision, the vortex states\nstudied here obey a dynamics for the vortex velocity V\ndescribedbyaThieleequation,16,17ignoringanyintrinsic\nvortex mass11or damping effects,\nF+G×V= 0. (46)\nThis equation comes from an analysis of the Hamilto-\nnian dynamics of a magnetic system,9,30in which the\nvortex excitation profile preserves its shape but moves\nwith some collective coordinate center position X(t),\nwithV(t) =˙X(t). The force Fis the gradient of the\npotential experienced by the vortex. The force points\ntowards the nanodisk center, and can be approximated\nby some harmonic potential with force constant kF, for\na vortex at distance rfrom the center,\nF=−kFrˆr. (47)\nHence, the presence of the gyrovector leads to the well-\nknown gyrotropic (or uniform circular) motion. Solving\nfor the vortex velocity results in\nV=ˆz×F\nG=−γkFr\n2πQLM sˆφ. (48)\nGincludes the sign of the gyrovector (vector Gpoints\nperpendicular to the plane of the disk, and it has only a\nzcomponent). Thus, thevorticesgeneratedwith positive\n(negative) gyrovector move clockwise (counterclockwise)\nin thexyplane. Furthermore, the angular frequency of\nthis gyrotropic motion is given by a related equation,\nωG=V\nr=−kF\nG=−γkF\n2πQLM s. (49)\nThe force constant has been estimated theoretically from\ntherigidvortexapproximation31andfromthetwo-vortex\nmodel.7Below, we determine kFnumerically from re-\nlaxed vortex states15(a flexible vortex). The frequency\nin Eq. (49) applies to the stable vortex states. If the\ndisk is too thin, the vortex could be unstable; this pro-\nduces an outward force F, and results in the gyrotropic-20 -10 0 10 20\nxc (nm)-20-1001020 yc (nm)R=30 nm, L=5 nm, α=0.02, T=0\nFIG. 1: Vortex motion with damping, at zero temperature.\nThis is clockwise motion for a vortex with positive (+ˆ z) gy-\nrovector, starting from the dot on the x-axis. The vortex\nperforms gyrotropic motion of decreasing radius and increa s-\ning frequency as it moves towards the disk center, r= (0,0).\nmotion in the “wrong” direction. Thus it is easy to iden-\ntify whether a vortex is stable or unstable from a short\nintegration of its dynamics.\nIn the time and frequency units applied in the sim-\nulations, the dimensionless gyrotropic frequency Ω Gis\nobtained from\nΩG=ωGt0=ωG\nγB0=−kFa2\n4πLAQ.(50)\nThe negative sign shows that vortices with a negative\ngyrovector ( Q=−1) have a counterclockwise rotational\nmotion; the opposite sense holds for positive gyrovector.\nThe force constant kFincreases with thickness Lbut de-\ncreases with disk radius R. Therefore, in the simulation\ntime units, the gyrotropic frequency could depend pri-\nmarily on their ratio, L/R.\nFor detection of the vortex motion, one method is to\nmeasure the spatially averaged magnetization,\n∝angb∇acketleft/vector m∝angb∇acket∇ight=1\nN/summationdisplay\ni/vector mi. (51)\nThis is a useful measure of vortex gyrotropic motion, es-\npecially for experiments, where it may not be possible to\nobserve the rapidly changing instantaneous vortex core\nposition. However, ∝angb∇acketleft/vector m∝angb∇acket∇ightcan show rotational oscillations\neven when no vortex is present. Thus, we need instead a\nmeasure of the vortex core position based on the location\nof the vorticity charge center.\nThe vorticity center position rvis the point around\nwhich the in-plane magnetization components give a di-\nvergent curl. That is, a continuum magnetization field of\na vortex located at position rv, with in-plane angle φ(r),\nwould be expected to have the curl,\n/vector∇×/vector∇φ(r) = 2πˆzδ(r−rv). (52)8\n0 10000 20000 30000 40000 50000 60000τ-0.6-0.4-0.200.20.40.6\nxc/R\n<my>R=30 nm, L=5 nm,  α=0.02, T=0\nFIG. 2: (Color online) For the vortex motion in Figure 1,\nthe phase relationship between perpendicular components o f\nposition and in-plane magnetization.\nWhen used on the discrete grid of cells, the vorticity cen-\nter falls between the four nearest neighbor grid cells that\nhave a net 2 πcirculation in φ. However, this discretely\ndefined position always jumps in increments of the cell\nsizea, hence, it cannot be used directly. Instead, we use\nan average position weighted by the squared mz\nicompo-\nnents, of only those cells nearthe vorticity center:\nrc=/summationtext\n|ri−rv|<4λex(mz\ni)2ri/summationtext\n|ri−rv|<4λex(mz\ni)2. (53)\nTheriare the cell positions and the sum is restricted to\nthose cells within four exchange lengths of the vorticity\ncenter. The center of the nanodisk is the origin, ( x,y) =\n(0,0). Including this cutoff in the sums helps to reduce\nthe contributions from other oscillations in the system\n(i.e., spin waves)that arenot directly associatedwith the\nvortexposition. Byweightingwith( mz\ni)2, the position rc\nisabletochangesmoothlyasthevortexmoves,especially\natT= 0, in contrast to the discrete vorticity center rv.\nIt is a reasonable estimate of the mean location of out-\nof-plane magnetization energy of the vortex, i.e., close to\nthe vortex core position. The mz-weighted position rc\nand the vorticity center rvare usually within one lattice\nconstant. Thismeasureissupplementedbyobservingthe\nactual magnetization field when there is any doubt about\nthe presence or stability of the vortex.\nA. Gyrotropic frequencies in circular disks\nCalculationswerecarriedoutforcirculardisksofthick-\nness5.0nm, 10nm and 20nm ( L= 2.5a,5a,10a,all with\na= 2.0 nm) for radii 30 nm, 60 nm, 90 nm and 120 nm.\nThe stability of the vortex state is easily checked for a\ngivengeometry,bystartingfromarelaxedvortexatsome\nradius near half the radius of the disk. Including a weak0 10000 20000 30000τ-50510152025xc(nm)\nR=120 nmR=60 nmR=30 nmL=10 nm,  T=0 K \nall with x0= 4 nm\nFIG. 3: (Color online) Typical motions of the vortex core\ncoordinate xc(τ) at zero temperature, for circular disks of\nthickness L= 10 nm with different radii (shifted vertically\nfromxc= 0 for clarity). The damping α= 0.02 was turned\noffattime τ= 1000. Periods were calculated from theenergy-\nconserving motion after τ >1000. The motion of yc(τ) is\nsimilar but shifted a quarter of a period.\ndampingα= 0.02, it isnecessaryonly torun ashortsim-\nulation of the dynamics and observe whether the vortex\nmoves in the direction given by the Thiele equation,16\nEq. (48).\nFor example, with R= 30 nm,L= 5.0 nm, a vor-\ntex was initially relaxed at a position ( x0,y0) = (16,0)\nnm, and then the dynamics was started, including damp-\ningα= 0.02 in the RK4 method. In this case the vor-\ntex is very stable and spirals into the center of the disk,\nsee Figures 1 and 2. The instantaneous vortex displace-\nment on one axis, scaled by disk radius, takes approxi-\nmatelythesamemagnitudeastheperpendicularin-plane\ncomponent of ∝angb∇acketleftˆm∝angb∇acket∇ight, such asxc/Rand∝angb∇acketleftmy∝angb∇acket∇ightin Figure 2.\nAnother feature is that the period of rotation becomes\nless as the vortex moves inward. The first few peri-\nods are ∆τ= 6000,3140,2580,but the later revolutions\nhave an average period τG≈2020 (1.51 ns, frequency\nfG= 1/τG= 0.661 GHz for Py).\nOther similar dynamics calculations were done at vari-\nous disk sizes, but turning off the damping α= 0.02 after\nτ= 1000, see Figure 3. This initial damped motion is\nused to remove spin waves that might be generated when\nthe vortex is initially released, after being relaxed at a\ndesired starting position. Once the damping is turned\noff, the dynamics is energy conserving. Because we are\nlater interested in small movements near the disk center,\ntheinitial positionwastakenas( x0,y0) = (2a,0), usinga\nlattice constant a= 2.0 nm. These simulations result in\nvery smooth circular motion of the vortex center rc(Fig.\n3), from which very precise estimates of the gyrotropic\nperiodτGwere determined by following the motion for\ntypically five to ten periods. The resulting frequencies\nfG, in units ofµ0\n4πγMs, are shown versus aspect ratio9\n0 0.1 0.2 0.3 0.4 0.5\nL/R00.050.10.15fG(µ0__\n4π γMs)R=15a\nR=30a\nR=45a\nR=60a\n0 0.1 0.2 0.3 0.4 0.500.511.52\nfG (GHz, Py)slope = 0.28\nFIG. 4: (Color online) Zero-temperature vortex gyrotropic\nfrequency fGfor various diskradii R, versus aspect ratio L/R.\n[For Permalloy,µ0\n4πγMs≈15.1 GHz]. The computation cell\nsize isa= 2.0 nm. The vortex state is unstable below a\nminimum disk thickness, as expected due to the diminished\nrestoring forces from the reduced edge area. The dashed line\nshows the result [Eq. (59)] from using the linear approxima-\ntion in Eq. (55) for kF.\n0 0.1 0.2 0.3 0.4 0.5\nL/R00.020.040.060.080.1kF / L (A/a2)slope = 0.25R=15a\nR=30a\nR=45a\nR=60a\nR=90a\nFIG. 5: (Color online) Vortex force constant kFscaled by\ndisk thickness, versus disk aspect ratio. These were obtain ed\nby assuming a parabolic potential for vortex motion within\nthe disk. The dashed line indicates that the slope of this\nrelationship is close to 1 /4 for some range of parameters, Eq.\n(55), for disks of adequate thickness. Cell edge is a= 2.0 nm.\nL/Rin Figure 4. The scale is also given there for the\nparameters of Permalloy, for whichµ0\n4πγMs≈15.1 GHz.\nOnecan note the obviousfeature, that the gyrotropicfre-\nquency goes to zero at some minimum thickness needed\nfor vortex stability.B. Relation to force constant kF\nThevortexrestoringforceconstants kFwereestimated\nbasedonlyonstaticenergyconsiderations. We compared\nthe total system energy with the displaced vortex, U(x),\ntakingx= 2a, with the energy for the vortex at the disk\ncenter,U(0). It is known that the vortex potential is\nclose to parabolic, as long as the vortex displacement is\nsmall compared to the disk radius.15The force constant\nis then estimated simply by solving\nU(x) =U(0)+1\n2kFx2. (54)\nThe energies applied in this equation are those obtained\nafter the vortex is relaxed by the Lagrange-constrained\nmethod. These calculations are relatively fast because\nthere is no need to run the dynamics. The raw force\nconstants were obtained for a wide variety of disk sizes.\nGenerally, we find that kFincreases faster than linearly\nwith disk thickness Land decreases with disk radius R.\nIt is expected that the force constant should scale\nsomewhat with the aspect ratio, L/R. Further, the\nThiele equation suggests that the ratio kF/Lis most rel-\nevant in determining ωG[see Eq. (49)]. Therefore, we\nshowkF/LversusL/Rin Figure 5, which presents a re-\nlationship somewhat close to linear, with a slope near\n1/4. Thus we can write as a rough approximation (far\nenough from the critical disk thickness for vortex stabil-\nity),\nkF≈1\n4L2\nRA\na2=λ2\nex\n8a2µ0M2\nsL2\nR= 0.878µ0M2\nsL2\nR.(55)\nThe last form, obtained by applying the definition of ex-\nchange length, is preferred because the vortex restoring\nforce ultimately is due to the demagnetization fields gen-\nerated byMs.\nOnecancheckwhethertheseforceconstantsareconsis-\ntent with the gyrotropic frequencies found in the dynam-\nics. If the Thiele equation applies to this motion, then\nthe gyrotropic frequencies must be linearly proportional\ntokF/L, [Equations (49) and (50)]. Therefore we have\nplotted the dimensionless frequency Ω GversuskF/Lin\nFigure 6. For the wide variety of disk sizes studied, all\npoints in this plot fall on a single line of unit slope, ex-\nactly consistent with the Thiele equation. This shows\nthat the calculations of the dynamics over fairly long\ntimes (many periods) are completely consistent with the\nforce constants found only from static energy consider-\nations. It further implies that we can safely use static\nenergy calculations to predictdynamic properties. This\nis based on the assumption of an isotropic parabolic po-\ntential in which the vortex moves. There may be some\nlimitation to this idea, however, only because the poten-\ntial will deviate from parabolic for larger displacements\nfrom the disk center.\nThese results are consistent with the two-vortices\nmodel applied by Guslienko et al.7With the boundary10\n0 0.04 0.08 0.12\nkF/L (A/a2)00.040.080.124πΩG\nslope = 1R=15a\nR=30a\nR=45a\nR=60a\nFIG. 6: (Color online) The dimensionless gyrotropic freque n-\ncies (found from dynamics) versus force constant scaled by\ndisk thickness. The dashed line of unit slope is Eq. (50). Thi s\nverifies the dynamics of the Thiele equation, and shows the\ncomplete consistency between the static energetics and the\ndynamics. Cell edge is a= 2.0 nm.\nparameterξ= 2/3 and the initial susceptibility at small\naspect ratio being χ(0)−1≈9.98L/R, their result (con-\nverted to SI units by factorµ0\n4π) is approximately\nkF=πLµ0\n4πM2\nsξ2χ(0)−1≈1.109µ0M2\nsL2\nR.(56)\nOur results have a somewhat weaker potential, which is\nto be expected because the numerical simulations allow\nfor a wider range of possible deformations of the vortex\nstructure than is possible in an analytic approximation.\nIn addition, ournumericalresults include the destabiliza-\ntion of the vortex at sufficiently small L/R, hence, it is\nimpossible to fit anystraight line for kF/Lvs.L/Rdown\nto arbitrarily small aspect ratio, see Figure 5.\nWe showed above that the gyrotropic frequencies νG\nare exactly linearly proportionalto kF/L, hence, this im-\nplies that the frequencies also scale close to linearly with\nL/R. Combining our fit of kFwith relation (50) then\nshows that roughly, the dimensionless angular frequency\nmagnitude is\nΩG≈1\n16πL\nR≈0.0199L\nR. (57)\nIn physical units, this is\nωG=γB0ΩG≈0.140γµ0MsL\nR. (58)\nThen the frequency comes out\nfG=ωG\n2π≈0.280/parenleftBigµ0\n4πγMs/parenrightBigL\nR. (59)\nThe dashed line in Figure 4 shows Eq. (59) compared\nwith data from various disk sizes. These frequencies aresmaller than those in the rigid vortex model,31and only\nslightly smaller than those for the two-vortices model.7\nHowever, this result fits quite well with the experimental\ndata presented in Ref. 8 by also using the higher value\nfor the gyromagnetic ratio, γ= 1.85×1011s−1T−1,\nin conjunction with saturation magnetization still at the\nvalueMs= 860 kA/m. The calculation here can be\nconsidered as that for a more flexible vortex. The mag-\nnetization at the edge of the disk adjusts itself to try to\nfollow the boundary. The magnetization can also adjust\nitself, to a lesser extent, in the vortex core region. These\neffects lead to lower force constants and therefore lower\ngyrotropic frequencies.\nThese results show that the adapted 2D methods ap-\nplied here give reliable results, consistent with experi-\nmentandwiththetwo-vorticesanalyticcalculationofthe\ngyrotropic frequencies. We note that the smaller value of\ncell constant used here ( a= 2.0 nm) is important for the\nsimulation to correctly describe the magnetization dy-\nnamics in the vortex core. Of course, this then imposes\na limitation on the system size that can be studied.\nThese results confirm the basic dynamic properties,\nthat the vortex resonance frequency ωGdiminishes with\nincreasing dot radius, and increases with increasing dot\nthickness. A wider dot has a weaker spring constant kF\nin its potential, U(r) =U(0)+1\n2kFr2, leading to the re-\nduction of its resonance frequency. Similarly, in a thicker\ndot, the greaterareaat the edge produces a largerrestor-\ning force, leading to a higher resonance frequency.\nV. THERMAL EFFECTS IN VORTEX\nDYNAMICS IN CIRCULAR DISKS\nInthefollowingpart, theeffectsofthermalfluctuations\non the vortex dynamics are considered. We consider two\nbasic situations left to evolve in time via Langevin dy-\nnamics: (1) A vortex started off-center, and (2) a vortex\nstarted at the minimum energy position, the center of\nthe disk. In the latter case, the question is whether ther-\nmal fluctuations alone are sufficient to initiate gyrotropic\nmotion. If so, we can also study its frequency and range\nof motion. In all simulations we used cell size a= 2.0 nm\nand damping parameter α= 0.02 .\nA. Vortex initially off-center\nFor the same system used above [ R= 30 nm,L= 5.0\nnm], the same initial condition was used, with vortex\nat (x0,y0) = (16,0) nm, but a finite temperature cor-\nresponding to Permalloy at 300 K was considered. The\ndynamics was solved now by the H2 scheme. The scaled\ntemperature depends on the thickness Lof the disk and\nthe exchange stiffness Aof the material. The energy\nunit here is J= 2AL= 130 zJ, while 300 K corre-\nsponds tokT= 4.14 zJ, so the scaled temperature is\nT=kT/J= 0.032. Thex-component of the vortex po-11\n0 10000 20000 30000 40000 50000 60000τ-16-12-8-40481216 xc(nm)R=30 nm, L=5 nm\nT=300K (kT/J=0.032)x0=16nm\n0 10000 20000 30000 40000 50000 60000τ-0.4-0.200.20.4 <my>R=30 nm, L=5 nm\nT=300K (kT/J=0.032)(x0=16nm)\nFIG. 7: Vortex motion in Py at room temperature (300 K),\nstarting from an initial displacement of 16 nm from the disk\ncenter. The y-componentofaverage magnetization inthe disk\nis correlated to the x-component of the vortex position.\nsition versus time is shown in Figure 7. In this case, the\nvortex still spirals towards the center of the disk, how-\never, thermal fluctuations remain present in the motion\neven at time τ= 60000 ( ≈90 ns), 25 revolutions later.\nThe range of the motion there remains close to ±6 nm.\nThe time dependence of ∝angb∇acketleftmy∝angb∇acket∇ight(most closely relatedto xc)\nisalsoshowninFigure7; italsoshowsaneffectpersisting\nat the 25% levelout to τ= 60000. Note that at zerotem-\nperature, the time-scale for relaxation (Figure 2) was on\nthe order of τ∼20000. This shows that thermal forces\napparently are able to maintain the gyrotropic motion\nto very long times. The average period of the motion is\nτG≈2278 (1.705 ns, frequency f= 1/τG= 0.586 GHz\nfor Py), showing that the temperature also softened the\npotential experienced by the vortex.0 10000 20000 30000 40000 50000 60000τ-6-4-202468xc(nm)R=30 nm, L=5 nm\nT=300K (kT/J=0.032)\nx0=y0=0spontaneous motion\nFIG. 8: Spontaneous gyrotropic vortex motion in Py due to\nthermal fluctuations at 300 K, starting from a vortex at the\ncenter of the disk.\n0 0.0005 0.001 0.0015ν05101520<|mx(ν)|2>x0=y0=0R=30 nm, L=5 nm\nT=300K (kT/J=0.032)FFT2\nFIG. 9: Thermal power spectrum of the in-plane magnetiza-\ntionfluctuations duetospontaneousgyrotropic vortexmoti on\nin Py at 300 K, for the motion in Figure 8.\nB. Vortex initially at disk center\nThe same system is used [ R= 30 nm,L= 5.0 nm],\nbut this time the vortex was initiated at the center of\nthe disk, (x0,y0) = (0,0). At zero temperature, such an\ninitial state is static. Instead, the dynamics correspond-\ning to Py at 300 K was considered (scaled temperature\nT= 0.032). Any thermal fluctuations can move the vor-\ntex core off-center, and if that happens, gyrotropic mo-\ntion can initiate spontaneously. This indeed happens,\nas can be seen in the vortex core position rc(τ) plotted\nin Figure 8. It needs to be stressed that these vortex\nmotions of the order of ±4 nm, and magnetization fluc-\ntuations on the order of ±15%, occur without the ap-\nplication of any external magnetic field. The motion is\nsufficiently coherent that it can be followed for dozens12\n0 10000 20000 30000 40000 50000 60000τ-4-2024xc(nm)R=120 nm, L=20 nm, 300K (kT/J=0.008)\nFIG. 10: Spontaneous gyrotropic vortex motion, due to ther-\nmal fluctuations, in a 20 nm thick Py disk at 300 K, with\nthe vortex starting at the center of the disk. The natural pe-\nriodic motion executes 32 revolutions in this time sequence ,\nwith period τG≈1870.\nof rotations. The gyrotropic motion was followed out to\ntwice the time shown in the plots. An average over 24\nrotations results in a period τG= 2250, corresponding to\n1.68 ns or a frequency f= 0.594 GHz. To verify this, we\nalso show the power spectrum of the in-plane magnetiza-\ntion oscillationsin Figure 9. This was obtained by taking\ntime FFTs of ∝angb∇acketleftmx(τ)∝angb∇acket∇ightof length 256 points at different\nstarting times in the data out to τ= 120000 and aver-\naging their absolute squares. The middle peak in Figure\n9 falls at dimensionless frequency ν≈4.52×10−4, cor-\nresponding to physical frequency f=ν/t0= 0.600 GHz,\nconsistent with the estimate from counting oscillations.\nThere is some structure in the FFT, possibly the beating\nbetween three different primary frequencies, that causes\nthe amplitude of the oscillations to wax and wane.\nThe spontaneous gyrotropic vortex motion takes place\nforawiderangeofsystemsizesthatweretested. Another\nexample is givenfor a largersystem [ R= 120nm,L= 20\nnm] in Figure 10, where the vortex core displacement is\ndisplayed. An interesting feature is apparent. The gy-\nrotropic motion loses its phase coherence at times, lead-\ning randomly to brief intervals of dramatically changed\namplitude. This is only one example; in other time se-\nquences for other system sizes, this behavior is particu-\nlarly intermittent and random. For the same simulation,\nFigure 11 also shows both components of vortex core po-\nsition and both components of the average in-plane mag-\nnetization, zoomed in to show details at earlier times.\nHere one can see the quarter-period phase difference be-\ntweenxandycomponents for the vortex position as well\nas for the magnetization. In addition, the magnetiza-\ntion exhibits a high-frequency oscillation with a period\nof about ∆τ≈125 on top of the gyrotropic oscillations.\nThis can be expected to be spin wave excitations that\nareexcited thermally togetherwith the vortexgyrotropic0 2000 4000 6000 8000 10000τ-4-2024rc(nm)xcyc\n0 2000 4000 6000 8000 10000τ-0.02-0.0100.010.02< m ><mx><my>\nFIG.11: (Color online) Forthespontaneous gyrotropic vort ex\nmotion in Figure 10, [ R= 120 nm, L= 20 nm Py disk at 300\nK], details of the motion at earlier times. The vortex starte d\nat the center of the disk. There is a high-frequency spin wave\noscillation apparent in the magnetization dynamics, excit ed\ntogether with the gyrotropic motion.\nmotion.\nTo confirm the identity of these spin wave oscillations,\nwealsoshowinFigure12thepowerspectruminnetmag-\nnetization component mx, from a longer simulation out\ntotimeτ= 2.5×105. Theverticalscalehasbeenzoomed\nin to bring out the appearance of a doublet with frequen-\ncies of 9.3 GHz and 11.4 GHz, for Permalloy parameters,\nwhile the gyrotropic frequency is only 0.71 GHz. A spin\nwave doublet with azimuthal quantum numbers m=±1\n(wavefunction varying as ψ∼eimφaround the disk cen-\nter) has been discussed in Ref. 19. The doublet is pre-\ndicted to have a splitting32of ∆f=f2−f1= 3.5fG\nand an averaged frequency33of¯f= 1.8/parenleftbigµ0\n4πγMs/parenrightbig/radicalBig\nL\nR.\nFor the situation here, these formulas predict ∆ f= 2.5\nGHz and ¯f= 11.1 GHz, while the observed doublet has\n∆f= 2.1 GHz and ¯f= 10.3 GHz. Although slightly\nsofter, these are of the right orders of magnitude and are\nconsistentwiththethetheoreticalpredictionforthisdou-13\n0 0.002 0.004 0.006 0.008 0.01 0.012ν00.010.020.030.040.05<|mx(ν)|2>R=120 nm, L=20 nm\nT=300 K  (kT/J=0.008) amplitude ~ 1.0\nFIG. 12: (Color online) The thermally averaged power spec-\ntrum in one component of the magnetization (squared FFT)\nfor the vortex motion in Figure 10. The low frequency gy-\nrotropic mode dominates strongly over a much weaker dou-\nblet at high frequency. For Permalloy parameters ( f=\n1336GHz ×ν), the gyrotropic frequency is fG= 0.71 GHz\nwhile the components of the doublet lie at f1= 9.3 GHz and\nf2= 11.4 GHz.\nblet. This lowest doublet relates to the presence of spin\nwaves propagating azimuthally around the disk, in the\npresence of the vortex. The splitting can be attributed\nto the breaking of symmetry for the two directions of\npropagation,duetothe presenceoftheout-of-planemag-\nnetization at the vortex core. Based on these results and\nresults at other disk sizes, we then note that the primary\ndeviation from a smooth gyrotropic motion is due to the\nthermal excitation of this doublet on top of the vortex\nmagnetization.\nC. Analysis of thermal vortex motion in circular\nnanodisks\nThe spontaneous vortex motion at 300 K takes place\nwithout the application of any externally generated mag-\nnetic field. Only the thermal energyis responsible for the\nmotion. Indeed, boththefrequencyandamplitudeofthis\nspontaneous gyrotropic motion is determined directly by\nthe temperature. Here we give some analysis and sug-\ngest where this motion might be most easily observed\nexperimentally.\nFor some smaller disks with R= 30 nm, and for some\nlarger disks, with R= 120 nm, Figures 13 and 14 ex-\nhibit the typical time dependence of the vortex coordi-\nnatexc(τ), for Permalloy systems at 300 K. The vortex\nwasinitially relaxedat the center ofthe disk ( x=y= 0).\nAs seen for the systems studied above, the gyrotropic\nmotion is spontaneous, and furthermore, takes place at\na lower frequency for thinner disks. In addition, there\nis a dependence of the amplitude of the motion on the0 10000 20000 30000 40000 50000 60000τ-50510152025xc(nm)\nL=5 nmL=10 nmL=20 nmR=30 nm,  T=300K \nFIG. 13: (Color online) Typical spontaneous fluctuations of\nthe vortex core x-coordinate for 30 nm radius Py disks with\nvarious thicknesses, at 300 K. The vortex was initiated at th e\ndisk center. Curves are shifted vertically from xc= 0 for\nclarity.\n0 10000 20000 30000 40000 50000 60000τ-50510152025xc(nm)\nL=5 nmL=10 nmL=20 nm R=120 nm,  T=300K \nFIG. 14: (Color online) Typical spontaneous fluctuations of\nthe vortex core x-coordinate for 120 nm radius Py disks with\nvarious thicknesses, at 300 K. The vortex was initiated at th e\ndisk center. Curves are shifted vertically from xc= 0 for\nclarity.\ndisk thickness. The amplitude is observed to be larger\nfor thinner disks. Also it is apparent that generally the\namplitude is larger for the larger radius disks. This is\nsomewhat difficult to analyze precisely, due to the lim-\nited time sequences that can be obtained during a rea-\nsonable computation time. However, from knowledge of\nthe force constants kFand their dependence on the disk\ngeometry, the RMS range of the vortex core motion can\nbe predicted.\nThe statistical mechanics of the vortex core position\nX= (X(t),Y(t)) and velocity V=˙Xcan be obtained\nfromthe effective Hamiltonianassociatedwith the Thiele\nequation. The Thiele equation is mathematically equiv-14\nalent to the equation of motion for a massless charge\nein a uniform magnetic field B, witheB=−G, and\nalso affected by some other force F. We can start from\na Lagrangian that leads to the Thiele equation, using\nthe symmetric gauge for the effective vector potential,\nand including a circularly symmetric parabolic potential\n(harmonic approximation),\nL(X,˙X) =−1\n2G(X˙Y−Y˙X)−1\n2kF(X2+Y2).(60)\nThe first term on the RHS is equivalent to eV·A, with\nvector potential A=1\n2B×Xin a magnetic problem;\nthere is no usual kinetic energy term like1\n2mV2, be-\ncause the intrinsic mass is considered zero here. Only\nthez-component of the gyrovector is present, G≡Gz=\n2πpqm0γ−1. Then the components of the Thiele equa-\ntion are recovered from the Euler-Lagrange variations,\n∂L\n∂X−d\ndt∂L\n∂˙X=−kFX−G˙Y= 0, (61)\n∂L\n∂Y−d\ndt∂L\n∂˙Y=−kFY+G˙X= 0. (62)\nThe Lagrangian is written equivalently as\nL(X,V) =−1\n2(G×X)·V−1\n2kFX2.(63)\nThis leads to the canonical momentum,\nP=∂L\n∂V=−1\n2G×X= (G\n2Y,−G\n2X).(64)\nThis allows the transformation to the collective coordi-\nnate Hamiltonian, H(X,P). Following the usual pre-\nscription, we have\nH(X,P) =P·˙X−L=1\n2kFX2=1\n2kF/parenleftbig\nX2+Y2/parenrightbig\n.(65)\nNote that the derivation of the Hamiltonian does not\ndepend on the choice ofthe gaugefor the gyrovector(i.e.,\nforits effective magneticfield). In Ref.34, it isshownthat\nthe Landau gauge leads to the same result for H, but\nwhereP=GYis found to be the momentum conjugate\ntoX.\nTechnically this is all that is needed to analyze the\nstatistics of the vortex position. By being purely po-\ntential energy, however, this Hamiltonian needs careful\ntreatment. Its variation via the Hamiltonian equations\nof motion does not lead back to the correct dynamics,\ni.e., it does not give the Thiele equation. One can see\nthat the difficulty is due to the fact that the position and\ncanonical momentum coordinates are redundant, since\nPx=1\n2GYandPy=−1\n2GX. Even so, all of these\nshouldbe consideredlinearlyindependent mechanicalco-\nordinates, and all should appear in Hto give the correct\ndynamics (gyrotropicmotion does not conserve XnorP,\nso both should appear in H). For that to work out, H\nmust be expressed so that there are both potential and\nkinetic energy terms. (A similar care is needed even inthe Landau gauge, where GYmust be identified by and\nreplaced as the momentum Pconjugate to X.) We can\nsplit out half of the potential energy and redefine it in\nterms of P2as a kinetic energy,\nH(X,P) =1\n4kFX2+1\n4kF/parenleftbigg2P\nG/parenrightbigg2\n.(66)\nOne can easily demonstrate that the correct dynamic\nequations result only by allocating exactly half of the en-\nergy as kinetic energy and half as potential energy. This\nthen leads to the Hamilton dynamic equations for oscil-\nlations along the two perpendicular axes. For example,\nalongxthere is\n˙X=∂H\n∂Px=2kFPx\nG2, (67)\n˙Px=−∂H\n∂X=−1\n2kFX. (68)\nThese give a second order equation for simple harmonic\nmotion (SHO),\n¨X=−k2\nF\nG2X. (69)\nThe other variations with respect to YandPylead to\nthe same dynamics for Y. However, note that the Thiele\nequation is recovered from these dynamics only by in-\ncluding the connection (64) that defines the canonical\nmomentum in terms of the position.\nIt isclearthat the Hamiltonian(66) is the sameasthat\nfor a two-dimensionalsimple harmonicoscillatorwith co-\nordinate Xand momentum P. For that oscillator, the\neffective spring constant is kSHO=1\n2kF, and the corre-\nsponding effective mass is mSHO=G2\n2kF. It is interesting\nto see that these lead back to the natural frequency of\ngyrotropic motion [or see Eq. (69)],\nωG=ωSHO=/radicalbigg\nkSHO\nmSHO=kF\nG. (70)\nOf course, as Gis proportional to the disk thickness via\nthe factorm0=LMs, andkFdepends on both Rand\nL, then this contains the various geometrical effects, es-\npecially those associated with the vortex force constant.\nIn consideration of the classical statistical mechanics,\nthe important fact here is that the Hamiltonian (65) has\na dynamics due to only two coordinates ( X,Y) appear-\ningquadratically. Althoughthe dynamicequationsfor ˙X\nand˙YmustcomefromtheHamiltonian(66)oftheequiv-\nalent 2D SHO, the phase space of the Thiele dynamics is\nmore restricted, due to relation (64) between PandX.\nThis forces the Thiele phase space to be only two dimen-\nsional; this does not depend on the choice of the gauge.\nAs an example of that reduction of the phase space, ellip-\ntical motions are present for the 2D SHO, while the zero-\ntemperatureThieledynamicshasonlycircularorbits. As\nweareconsideringthermalequilibrium, each independent15\n051015202530\nkF-1 (a/A)0246810< (r/a)2 >300 K, R=30 nm\n300 K, R=60 nm\n300 K, R=120 nm\n150 K, R=30 nm\n150 K, R=60 nm\n150 K, R=120 nm 300 K\n150 K\nFIG. 15: (Color online) Average squared displacement of\nthe vortex core from the disk center, versus reciprocal forc e\nconstant. The points come from simulations out to time\nτ= 2.5×105; the solid lines are the predictions from the\nequipartition theorem, Eq. 72, using the parameters for Py.\nquadratic coordinate receives an average thermal energy\nof1\n2kT. This gives the connection needed to predict the\naverage RMS vortex displacement from the disk center.\nSpecifically, for each vortex core coordinate,\n∝angb∇acketleft1\n2kFX2∝angb∇acket∇ight=∝angb∇acketleft1\n2kFY2∝angb∇acket∇ight=1\n2kT. (71)\nThen the average squared displacement of the vortex\nfrom the disk center should be\n∝angb∇acketleftr2∝angb∇acket∇ight=∝angb∇acketleftX2+Y2∝angb∇acket∇ight=r2\nrms=2kT\nkF.(72)\nTheseshowthatthe averagethermalenergyinthe vortex\nmotion must be\n∝angb∇acketleftH(X,P)∝angb∇acket∇ight=kT. (73)\nTherefore,wecancheckthat theserelationsactuallyhold\nin the simulations. The average squared displacement\nshould be proportional to the reciprocal of the force con-\nstant, with the same proportionality factor (twice the\ntemperature) when disks of different geometries are con-\nsidered. Some results for the average squared displace-\nments versusreciprocalforce constantin different geome-\ntriesaregiveninFigure15. Theresultsdependonthebe-\nhavior of the force constant with disk geometry, showing\nthe importance of static calculations for understanding\nthe statistical dynamics behavior. The simulation data\nhavea generaltrend consistent with Eq. 72, but there are\nlarge fluctuations due to the finite time sequences used,\nwhich is more of a problem for the systems with small\nkF.\nWe can further substantiate the statistical behavior of\nthe vortex core, by calculating the probability distribu-\ntionp(r) of its distance r=√\nX2+Y2from the disk0 1 2 3 4 5\nr/a00.511.5p(r)L=20 nm\nL=10 nm\nL=5 nmR=30 nm, T=300 K\n0 1 2 3\nr/a00.511.522.5p(r)L=20 nm\nL=10 nm\nL=5 nmR=30 nm, T=150 K\nFIG. 16: (Color online) Probability distributions in Py dis ks\nof radius 30 nm at temperatures 300 K and 150 K, for the\nradial position rof the vortex, measured from the disk center,\nin units of the cell size, a= 2 nm. Solid curves are the\ntheoretical expression (74) based on a Boltzmann distribut ion\nusing the static force constants; points are from simulatio ns\nout to time τ= 2.5×105.\ncenter. Assuming that its position is governed by Boltz-\nmann statistics for Hamiltonian (65), the normalized dis-\ntribution from p(r)dr∝2πrdre−βHis predicted to be\np(r) =βkFre−1\n2βkFr2, (74)\nwhereβ= (kT)−1is the inverse temperature. This dis-\ntribution also has some particular distinctive points that\nare relatively easy to check. For instance, the distribu-\ntion has a peak at the point of maximum probability, at\nthe radius\nrmax=/radicalbigg\nkT\nkF=rrms√\n2. (75)\nIn addition, the value of the function at this point is\npmax=p(rmax) =e−1/2\nrmax. (76)16\n0 12 3 4 56\nr/a00.20.40.60.811.2p(r)L=20 nm\nL=10 nm\nL=5 nmR=120 nm, T=300 K\n0 1 2 3 4\nr/a00.511.5p(r)L=20 nm\nL=10 nm\nL=5 nmR=120 nm, T=150 K\nFIG. 17: (Color online) Probability distributions for vort ex\nradial position in Py disks of radius 120 nm, as explained in\nFigure 16.\nWe have found that the vortex core position satisfies\nthis distribution reasonably well, while the vortex is\nundergoing the spontaneously generated gyrotropic mo-\ntion. There is a certain difficulty to verify this, be-\ncause very long time sequences (we used final time τ=\n250000) are needed so that many gyrotropic revolutions\nare performed. During the motion, at times there are\nrather large fluctuations in the amplitude of the motion.\nThe motion varies between time intervals of smooth gy-\nrotropic motion of large amplitude and other time inter-\nvals where the motion seems to be impeded, and is of\nmuch smaller amplitude. Even so, we were able to take\nthese long sequences and produce histograms of the vor-\ntex radial position to compare with the predicted proba-\nbility distribution. An example for R= 30 nm is given in\nFigure 16. The temperatures are defined here by apply-\ning the material parameters for Permalloy (that is, 300\nK corresponds to kT= 0.1592Aa, where the exchange\nstiffness for Py is A= 13 pJ/m and cell size a= 2.0 nm\nwas used in all simulations). The data (points) are com-\npared with the prediction of equation (74) (solid curves),for different disk thicknesses. For these smaller systems,\nthe agreement is quite good between the simulations and\nthe theoretical expression, Eq. (74).\nThe distributions were also found in simulations for\nlarger radius, see Figure 17 for the distribution at R=\n120 nm. In this case, the errors are considerably greater.\nThis is due primarily to the larger gyrotropic period.\nOver the sampling time interval to τ= 2.5×105, there\nare less periods being sampled. The system has a some-\nwhat erratic behavior, in that the orbital radius of the\nvortex motion seems to switch suddenly between differ-\nent values, as already mentioned. As a result, at this\nsystem size a greater time interval is needed to obtain a\nsample that could be considered in thermal equilibrium,\nwith well defined averages.\nFor thinner disks, the number of revolutions in the\ngiven time interval is lesser, which means the thinner\ndisks may also require longer time sequences to give the\nsame relative errors. Of course, the thinner (thicker)\ndisks have a weaker (stronger) force constant, leading\nto the greater (lesser) amplitude spontaneous motions.\nThis is clearly exhibited in the probability distributions.\nAlthough these aspects may be difficult to verify exper-\nimentally, the results do indeed point to much stronger\nspontaneous gyrotropic fluctuations for very thin mag-\nnetic disks. In the cases where these motions were of\ngreater amplitude, there may start to appear deviations\nfrom the distribution in (74), simply because the larger\namplitude vortex motions cause the vortex to move out\nof the region where the potential is parabolic.\nVI. DISCUSSION AND CONCLUSIONS\nThe calculations here give a precise description of\nthe magnetostatics and dynamics for thin-film nanomag-\nnets, especially in the situations where a single vortex is\npresent. The continuum problem for some finite thick-\nnessLhas been mapped onto an equivalent 2D prob-\nlem, i.e., the modified micromagnetics adapted here. For\nhigh aspect ratios, L≪2R, the shape anisotropy is very\nstrong, and this 2D system is a very good approximation\nof the full 3D problem, because it leads to the physical\nsituation where the magnetization has little dependence\nonzand is predominately planar, except in the vortex\ncore.\nAt zero temperature, we have been able to test this\napproach and compare with the predictions for vortex\ngyrotropic motion based on the Thiele equation. This\ncomparisonismadepossibleherebecausethevortexforce\nconstantskFcan be calculated from the energetics of a\nvortex with a constrained position . The application of\nthe Lagrange undetermined multipliers technique15for\nenforcing a desired static vortex position Xhas been es-\nsential in the determination of kF. In addition, that re-\nlaxation procedure also is of great utility for initiating a\nvortex at some radius while removing most of the initial\nspin wave like oscillations that would otherwise be gen-17\nerated when the time dynamics is started. As a result,\nwe have been able to determine the zero temperature gy-\nrotropic frequencies for the motion of the vortex core,\nX(t), to fairly high precision. The confirmation of the\napplicability of the Thiele equation to the T= 0 dynam-\nics of vortex velocity Vis impressive, as demonstrated\nin the straight line fit for gyrotropic frequency ωGversus\nscaled force constant kF/Lin Figure 6. This shows the\ncomplete consistency between the staticscalculations of\nthe force constants and the dynamics calculations of the\nfrequencies, when interpreted via the Thiele equation.\nAt larger disk radii, the gyrotropic frequency is found\nto be close to linear in the aspect ratio, L/R, see Eq. 58.\nThe frequencies are also close to those found in the two-\nvortices model and micromagnetics calculations carried\nout in Ref. 7. The differences from those results may be\nduetothefactthatwehaveusedthecellparameter ahalf\nof what was used in Ref. 7. This is important, because\nthe cell parameter should be sufficiently less than the\nexchange length for results to be reliable. Otherwise, if\nais too large, the details of the energetics and dynamics\nin the vortex core cannot be correctly represented.\nAtT >0, the Langevin dynamics shows some surpris-\ning behavior that was reported earlier in Ref. 18, even\nwhen the vortex is initiated at the center of a nanodisk.\nThe thermal fluctuations are indeed sufficiently strong to\nproduce a spontaneous motion of the vortex core, with-\nout the application of any external field, which is not a\nsimple random walk. Instead, the gyrotropic nature of\nthe motion is still present, and in fact, persistent vor-\ntex rotation is the dominant feature of the motion. The\nthermal fluctuations can be viewed as a perturbation on\ntop of the gyrotropic motion, however, it is the tempera-\nture that determines the expected squared radius of the\norbit. The orbital radius is very well described from the\nstatistical mechanics of the vortex collective coordinate\nHamiltonian (65), that possesses only the potential en-\nergy associated with the vortex force constant.\nIntegrations of the dynamics over very long times\n(equivalent to hundreds ofvortex revolutions)showsthat\nthe statistics of the vortex position follows the simple\nBoltzmann distribution in Eq. (74). The averagesquared\nvortex displacement from the origin, r2\nrms, scales linearly\nin the temperature divided onlyby the force constant\nkF. This is in contrast to the vortex gyrotropic frequen-\ncies, which depend on kF/L. Thus, the results for force\nconstant indirectly predict the expected position fluctu-\nations. However, very long time sequences are needed to\nsee this average behavior; over some short time intervals\nthere can be large variations in the instantaneous vortex\norbital radius. The largest spontaneous vortex position\nfluctuations will be possible in thin dots of larger radius,\nwhere the force constants are weakest. Even so, this is a\nsmall effect (RMS radii on the order of several nanome-ters), and it may be difficult to observe experimentally.\nAs an example based only on the calculated force con-\nstants, a magnetic dot of radius R= 180 nm and thick-\nnessL= 20 nm has kF≈0.29A/a. For Py at 300 K,\nthis gives the estimate rrms≈2.1 nm. If the thickness\nis reduced to 10 nm, then kF≈0.080A/aand the RMS\norbital radius increases to rrms≈4.0 nm. Even though\nthese are rather small, the distributions p(r) are rather\nwide and therefore at times one can expect even larger\nvortex gyrotropic oscillations.\nFinally we note that the thermal distribution of the\nvortex rotational velocity is connected to the radial dis-\ntributionp(r), because the Hamilton equations (67) im-\nply\nV=/vector ωG×X, /vector ωG=−kF\nGˆz. (77)\nThus,wecantransformmagnitudeswith V=ωGr. Then\nthe RMS rotational velocity is\nVrms=|ωG|rrms=√2kFkT\nG, (78)\nwhich varies proportional to√kF/L. This is connected\nto a Boltzmann distribution for the probability f(V)dV\nof vortex speed Vin some interval of width dV, where\nf(V) =p(V/ωG)\nωG=βmGVe−1\n2βmGV2.(79)\nThis involves a gyrotropic effective mass mG,\nmG≡G2\nkF≈(2π)2\n0.878R\nµ0γ2, (80)\ndetermined both by the vortex force constant and by the\ndisk thickness contained in the definition of G. For small\naspectratio,however,thethicknesscancelsandthismass\nis proportional to the disk radius alone. At R= 100\nnm, the mass is about 1 .2×10−22kg, independent of\nthe material. Although f(V) has a mathematical form\nidentical to that for p(r), it leads to another interesting\ninterpretation of the vortex dynamics in equilibrium.\nAcknowledgments\nG. M. Wysin acknowledges the financial support of\nFAPEMIG grant BPV-00046-11 and the hospitality of\nUniversidade Federal de Vi¸ cosa, Minas Gerais, Brazil,\nand of Universidade Federal de Santa Catarina, Flo-\nrian´ opolis,Brazil, wherethis workwascarriedout during\nsabbatical leave. W. Figueiredo acknowledges the finan-\ncial support of CNPq (Brazil).\n∗Electronic address: wysin@phys.ksu.edu;\nURL:http://www.phys.ksu.edu/personal/wysin1N.A. Usov and S.E. Peschany, J. Mag. Magn. Mater. 118,18\n290 (1993).\n2K.Yu. Guslienko, K.-S. Lee and S.-K. Kim, Phys. Rev.\nLett.100, 027203 (2008).\n3R.P. Cowburn, D.K. Koltsov, A.O.Adeyeye, M.E. Welland\nand D.M. Tricker, Phys. Rev. Lett. 83, 1042 (1999).\n4M. Schneider, H. Hoffmann and J. Zweck, Appl. Phys.\nLett.77, 2909 (2000).\n5J. Raabe, R. Pulwey, S. Sattler, T. Schweinbock, J. Zweck\nand D. Weiss, J. Appl. Phys. 88, 4437 (2000).\n6J.P. Park, P. Eames, D.M. Engebretson, J. Berezovsky and\nP.A. Crowell, Phys. Rev. B 67, 020403 (2003).\n7K.Yu. 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Galkin, Low Temp.\nPhys.36, 747 (2010)."    },    {        "title": "1111.4295v2.Charge_and_Spin_Transport_in_Magnetic_Tunnel_Junctions__Microscopic_Theory.pdf",        "content": "arXiv:1111.4295v2  [cond-mat.mes-hall]  8 Jun 2012JournalofthePhysicalSocietyofJapan FULLPAPERS\nCharge and Spin Transport in MagneticTunnel Junctions: Mic roscopic\nTheory\nDaisukeMiura∗and AkimasaSakuma\nDepartmentofAppliedPhysics,Tohoku University\nSendai980-8579\nWe study the charge and spin currents passing through a magne tic tunnel junction (MTJ) on\nthe basis of a tight-binding model. The currents are evaluat ed perturbatively with respect to\nthetunnelHamiltonian.Thechargecurrenthastheform A[M1(t)×˙M1(t)]·M2+B˙M1(t)·M2,\nwhereM1(t) andM2denote the directions of the magnetization in the free layer and fixed\nlayer,respectively.Theconstant Avanisheswhenoneorbothlayers areinsulators,whilethe\nconstantBdisappearswhenbothlayersareinsulatorsorthesameferro magnets.Thefirstterm\nintheexpressionforchargecurrentrepresentsdissipatio ndrivenbytheeffectiveelectricfield\ninducedbythedynamicmagnetization.Inaddition,fromani nvestigationofthespincurrent,\nwe obtain the microscopic expression for the enhanced Gilbe rt damping constant ∆α. We\nshowthat∆αisproportionaltothetunnelconductanceand dependsonthe biasvoltage.\nKEYWORDS: spintronics, magnetictunneljunction,spin cur rent, spindynamics\n1. Introduction\nMagnetictunnel junctions(MTJs), which consist of a thin tu nnel barrier sandwiched be-\ntween two ferromagnetic layers,1–5are promising for their use in magnetic random access\nmemory (MRAM).6However, the primary disadvantage of conventional MRAM des igns,\nwhich employ a current-induced field to write data, is that th e writing current increases\nwiththedevicedensity.Thus,therehas been considerablei nterestin exploitingspin-transfer\ntorque(STT)7,8instead.9–13InsuchanSTTMRAMdevice,thecriticalcurrentisproportio nal\nto the product of the volume and the Gilbert damping constant αof the free layer, making\nlowαan importantcriterionforelectrodematerials.\nTothisend,severalstudieshaveexploredthedynamicsandt hedistributionofthemagne-\ntizationsin STT MRAMby usingtheLandau–Lifshitz–Gilbert (LLG)equation withan STT\nterm.14–18However,othertorques(spintorques)alsoactonthedynami cmagnetizationinthe\n∗E-mailaddress:dmiura@solid.apph.tohoku.ac.jp\n1/13J.Phys. Soc. Jpn. FULLPAPERS\nfreelayer,whichforminreactiontotheoutwardflowofspins fromthelayer:Mizukami etal.\nexperimentally showed that αincreases with the thickness of the nonmagnetic metal (NM)\nlayer in NM/Py/NM films, and that this enhancement continues up to thickness es of several\nhundrednanometers.19Theirexperimentsupportstheimportanceofspintorquesin themag-\nnetizationdynamicsofmesoscopicdevicessuch asSTTMRAMs .Further, thisexperimental\nfindingwassupportedimmediatelybyTserkovnyak etal.’s20,21theoryofspinpumpingbased\nonscatteringtheory,withadditionaltheoreticalconfirma tionbyUmetsu etal.onthebasisof\ntheKuboformula.22,23\nSeveral studies have also investigated charge transport in the presence of magnetization\ndynamicsin magneticmultilayers.It is known that dynamicm agnetizations inducean e ffec-\ntiveelectromagneticfield.24,25Oheet al.simulated the effectiveelectric field induced by the\nmotion of the magnetic vortex core in a magnetic disk,26and the field was observed experi-\nmentally.27Furthermore, Zhang et al.phenomenologicallyderived the LLG equation having\nthe STT term induced by this e ffective electric field.28And Moriyama et al.observed the dc\nvoltage across generated by the precession of the magnetiza tion in an Al/AlOx/Ni80Fe20/Cu\ntunnel junction.29The origins of this voltage have been discussed from a theore tical stand-\npoint(scatteringtheory).30–32Inaddition,chargeandspincurrentsinferromagnetswithm ag-\nnetizations that slowly vary in space and time have been stud ied microscopically.33–35These\nstudies employed the s-d model in continuous space and treat ed the perturbation within the\nframeworkoftheKeldysh–Greenfunction.36,37\nSimilarly,ouraimistodescribethechargeandspintranspo rtinMTJsinthepresenceofa\nvoltageacrossthebarrierandthedynamicalmagnetization inthefreelayer.Thissituationjust\ncorrespondstoanSTTMRAMcellduringthewritingstage.Int hispaper,wemicroscopically\ndescribe the charge and spin currents passing through an MTJ . However, in contrast with\nprevious works that relied on models in continuous space, we calculate the currents on the\nbasisofatight-bindingscheme.Thismakesiteasiertoacco untforthepropertiesofmaterials\nand the space dependence of the magnetization in magnetic mu ltilayers, such as MTJs, with\nstronglyinhomogeneousmagneticstructures.Inthecalcul ations,weconsiderthevoltageand\nthe dynamics of the magnetization in Berry’s adiabatic appr oximation under the assumption\nthattheeffectiveexchangefieldislargerthanthevoltageanddynamics .Ourmodelshowsthat\nthe charge current induced by the dynamical magnetization h as the form A[ML(t)×˙ML(t)]·\nMR+B˙ML(t)·MR, whereML(t) andMRdenote the directions of the magnetization in the\nfreelayerandfixedlayer,respectively.Thefirsttermtends totheformgivenbyTserkovnyak\net al.,31which expressedthedc current dueto theprecession of ML(t)aboutMRas aspecial\n2/13J.Phys. Soc. Jpn. FULLPAPERS\ni jtij \n... L RTLR \n... ... ML(t)\nMR\n... \nLeft hand side \nlayer Right hand side \nlayer \nTunnel barrier \nFig. 1. Schematicofone-dimensionalmagnetictunneljunction. TLRis thetunnelingamplitudeand tijrepre-\nsents the hoppingmatrix between sites iandjlocated at either side of the interface. ML(t) andMRdenote the\ndirectionsoftheeffectiveexchangefieldsforthe left(L)andright(R)handside layer,respectively.\ncase; in this sense, our result is a generalization of their w ork. Furthermore, from the results\nconcerningspintransport,wesuccessfullyderivetheenha ncedGilbertdampingandpropose\namicroscopicexpressionforit.\n2. Model and Formalism\n2.1 Model Hamiltonian\nWe consider the motion of electrons in an e ffective exchange field. Furthermore, assume\nthat the ferromagnetic layer on the left-hand side (LHS) of t he MTJ is the free layer; that\nis, the direction of the field at time tin this layer, ML(t), rotates time-dependently (see Fig.\n1). Thus, the direction of the field on the right-hand side (RH S) (fixed layer), MR, is time-\nindependent. Note that we ignore the inner structure of the t unnel barrier and account for\nits properties via the simple tunnel amplitude TLRbetween sites L and R, which denote the\nsurfaceson theLHSandRHS, respectively.In thismodel,the totalHamiltonianfortheMTJ\nisthesumoftheonedimensionaltight-bindingHamiltonian sintheferromagneticlayers,\nHL(t) :=/summationdisplay\ni,j∈LHSc†\ni/bracketleftBig\n−tijˆ1−δijJLML(t)·ˆσ/bracketrightBig\ncj, (1)\nHR:=/summationdisplay\ni,j∈RHSc†\ni/bracketleftBig\n−tijˆ1−δijJRMR·ˆσ/bracketrightBig\ncj, (2)\nandthetunnelHamiltonian,\nHT:=−TLRc†\nLcR+H.c., (3)\n3/13J.Phys. Soc. Jpn. FULLPAPERS\nwherec†\niσ(ciσ) is an operator that creates (annihilates) the σspin electron at site i, andtijis\nthehopping integral between sites iandj. The constant JL(JR) represents the strength of the\ninteractionbetweenthespinofanelectronandthee ffectiveexchangefieldontheLHS(RHS)\nlayer;and ˆσisthePaulimatrix,wherehat ‘ˆ’denotesa2 ×2 matrixinspin-space.\n2.2 Adiabaticapproximation\nAssuming JL≫/planckover2pi1|dML(t)/dt|,weadopt Berry’s adiabaticapproximation38forHL(t):\nci(t)≃ˆUL(t)eiγ(t)ˆσzdifori∈LHS, (4)\nHL(t)→Had\nL:=/summationdisplay\ni,j∈LHSd†\ni/bracketleftBig\n−tijˆ1−δijJLˆσz/bracketrightBig\ndj, (5)\nwhereci(t)isintheHeisenbergrepresentationwithrespectto HL(t),ˆUL(t)isarotationmatrix\nsatisfyingtheequation ˆU†\nL(t)ML(t)·ˆσˆUL(t)=ˆσz, andγ(t)is Berry’s phasedefined by\nγ(t) :=i/integraldisplay\ndt/bracketleftBigg\nˆU†\nL(t)dˆUL(t)\ndt/bracketrightBigg\n↑↑. (6)\nWiththeapproximation(4), wereplace HTwith\nHad\nT(t) :=−TLRd†\nLe−iγ(t)ˆσzˆU†\nL(t)ˆURdR+H.c., (7)\nwhereˆURis a rotation matrix satisfying the equation ˆU†\nRMR·ˆσˆUR=ˆσz, anddi:=\nˆU†\nRcifori∈RHS. Finally, our total Hamiltonian is H(t) :=Had\nL+HR+Had\nT(t), where\nHR=/summationtext\ni,j∈RHSd†\ni/bracketleftBig\n−tijˆ1−δijJRˆσz/bracketrightBig\ndj. Thus, a nonequilibrium statistical average of the form/angbracketleftBig\ndiσ(t)d†\ni′σ′(t′)/angbracketrightBig\ncan bederivedperturbativelywithrespectto Had\nT(t)usingtheKeldysh–Green\nfunctiontechnique.\n2.3 Chargeand spincurrents\nThechargecurrent Ie(t)andspincurrent Is(t)passingthroughtheMTJaredefined by\nIe(t) :=2ℜi\n/planckover2pi1TRL/angbracketleftBig\nd†\nR(t)ˆU†\nRˆUL(t)eiγ(t)ˆσzdL(t)/angbracketrightBig\n[1/s], (8)\nIs(t) :=2ℜi\n/planckover2pi1TRL/angbracketleftBig\nd†\nR(t)ˆU†\nRˆσˆUL(t)eiγ(t)ˆσzdL(t)/angbracketrightBig\n[1/s], (9)\nwhere∝angbracketleft···∝angbracketrightdenotesastatisticalaveragein H(t).36,37\nBy introducingthelesserfunction,\n/bracketleftBigˆG<\nLR(t,t′)/bracketrightBig\nσσ′:=i\n/planckover2pi1/angbracketleftBig/bracketleftBig\nd†\nR(t′)ˆU†\nR/bracketrightBig\nσ′/bracketleftBigˆUL(t)eiγ(t)ˆσzdL(t)/bracketrightBig\nσ/angbracketrightBig\n,\neqs.(8)and (9)can bewrittenin theform\nIe(t)=2ℜTRLtrˆG<\nLR(t,t), (10)\n4/13J.Phys. Soc. Jpn. FULLPAPERS\nIs(t)=2ℜTRLtr ˆσˆG<\nLR(t,t). (11)\nInthefirst orderin Had\nT(t),wehave\nˆG<\nLR(t,t)≃−TLR/integraldisplay\ndt′ˆUL(t)eiγ(t)ˆσzˆgL(t−t′)e−iγ(t)ˆσzˆU†\nL(t)ˆA(t,t′)ˆURˆgR(t′−t)ˆU†\nR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<\n,(12)\nwhere<denotesthelessercomponentofKeldysh–Greenfunctions,36,37and\nˆA(t,t′) :=ˆUL(t)eiγ(t)ˆσze−iγ(t′)ˆσzˆU†\nL(t′). (13)\nMoreover,weintroducetheunperturbedKeldysh–Greenfunc tionsdefined by\n/bracketleftbigˆgL(t)/bracketrightbig\nσσ′:=−i\n/planckover2pi1/angbracketleftBig\nTdLσ(t)d†\nLσ′/angbracketrightBig\n0=−i\n/planckover2pi1/angbracketleftBig\nTdLσ(t)d†\nLσ/angbracketrightBig\n0δσσ′, (14)\n/bracketleftbigˆgR(t)/bracketrightbig\nσσ′:=−i\n/planckover2pi1/angbracketleftBig\nTdRσ(t)d†\nRσ′/angbracketrightBig\n0=−i\n/planckover2pi1/angbracketleftBig\nTdRσ(t)d†\nRσ/angbracketrightBig\n0δσσ′, (15)\nwhere T is the time-ordering operator on the Keldysh contour , and∝angbracketleft···∝angbracketright0denotes an equilib-\nrium statistical average in Had\nL+HR. Since ˆgL(t) is the diagonal matrix in spin-space, ˆ gL(t)\nandBerry’s phasefactors commute.Thus, ˆG<\nLR(t,t)reduces to\nˆG<\nLR(t,t)=−TLR/integraldisplaydE\n2π/planckover2pi1/integraldisplaydE′\n2π/planckover2pi1e−iE′t//planckover2pi1ˆU†\nL(t)ˆgL(E)ˆUL(t)ˆA(t,E′)ˆU†\nRˆgR(E−E′)ˆUR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<\n,(16)\nwhere we employ the Fourier transform of a function f(t) with respect to t, defined by the\nrelation\nf(E) :=/integraldisplay\ndteiEt//planckover2pi1f(t). (17)\nE′ineq.(16)representstheenergythatanelectronobtainsfr omthedynamicsofthemagne-\ntization;weconsideritinthefirst order:\nˆG<\nLR(t,t)≃−TLR/integraldisplaydE\n2π/planckover2pi1ˆU†\nL(t)ˆgL(E)ˆUL(t)ˆU†\nRˆgR(E)ˆUR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<\n−TLR/integraldisplaydE\n2π/planckover2pi1ˆU†\nL(t)ˆgL(E)ˆUL(t)/planckover2pi1\nidˆA(t,t′)\ndt′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet′=tˆU†\nRdˆgR(E)\ndEˆUR/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle<\n. (18)\nThen,usingtherelations\n/planckover2pi1\nidˆA(t,t′)\ndt′/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglet′=t=/planckover2pi1\n2ˆσ·ML(t)×dML(t)\ndt, (19)\nˆU†\nL(t)ˆgL(E)ˆUL(t)=ˆ1¯gL(E)+ˆσ·ML(t)∆gL(E), (20)\nˆU†\nRˆgR(E)ˆUR=ˆ1¯gR(E)+ˆσ·MR∆gR(E), (21)\nwhere\n¯gL(R)(E) :=1\n2tr ˆgL(R)(E)=/bracketleftbigˆgL(R)(E)/bracketrightbig\n↑↑+/bracketleftbigˆgL(R)(E)/bracketrightbig\n↓↓\n2, (22)\n5/13J.Phys. Soc. Jpn. FULLPAPERS\n∆gL(R)(E) :=1\n2tr ˆσzˆgL(R)(E)=/bracketleftbigˆgL(R)(E)/bracketrightbig\n↑↑−/bracketleftbigˆgL(R)(E)/bracketrightbig\n↓↓\n2, (23)\nwecan decompose ˆG<\nLR(t,t)intotwoterms:\nˆG<\nLR(t,t)=ˆ1G<\nLR(t,t)+ˆσ·G<\nLR(t,t). (24)\nHerewedefine\nG<\nLR(t,t) :=−TLR/integraldisplaydE\n2π/planckover2pi1/bracketleftBigg\n¯gL(E)¯gR(E)+ML(t)·MR∆gL(E)∆gR(E)\n+/planckover2pi1\n2ML(t)×dML(t)\ndt·MR¯gL(E)d∆gR(E)\ndE−i/planckover2pi1\n2dML(t)\ndt·MR∆gL(E)d∆gR(E)\ndE/bracketrightBigg<\n,\n(25)\nG<\nLR(t,t) :=−TLR/integraldisplaydE\n2π/planckover2pi1/bracketleftBigg\nML(t)∆gL(E)¯gR(E)+MR¯gL(E)∆gR(E)+iML(t)×MR∆gL(E)∆gR(E)\n+/planckover2pi1\n2ML(t)×dML(t)\ndt¯gL(E)d¯gR(E)\ndE−i/planckover2pi1\n2dML(t)\ndt∆gL(E)d¯gR(E)\ndE\n+i/planckover2pi1\n2/braceleftBigg\nML(t)×dML(t)\ndt/bracerightBigg\n×MR¯gL(E)d∆gR(E)\ndE+/planckover2pi1\n2dML(t)\ndt×MR∆gL(E)d∆gR(E)\ndE/bracketrightBigg<\n.\n(26)\nIe(t)andIs(T)are expressedinterms of G<\nLR(t,t)andG<\nLR(t,t)as follows:\nIe(t)=4ℜTRLG<\nLR(t,t), (27)\nIs(t)=4ℜTRLG<\nLR(t,t). (28)\nFinally,takingthelessercomponents,weobtainthefollow ingin thelow-temperaturelimit:\nIe(t)=2π|TLR|2/braceleftBigg\n¯ρL(µ)∆ρR(µ)ML(t)×dML(t)\ndt·MR\n−/integraldisplayµ\ndE/bracketleftBigg\n∆ρL(E)d∆χR(E)\ndE−d∆χL(E)\ndE∆ρR(E)/bracketrightBiggdML(t)\ndt·MR/bracerightBigg\n, (29)\nIs(t)=4π|TLR|2\n/planckover2pi1/integraldisplayµ\ndE/bracketleftbig∆ρL(E)∆χR(E)+∆χL(E)∆ρR(E)/bracketrightbigML(t)×MR\n+4π|TLR|2\n/planckover2pi1/braceleftBigg\n¯ρL(µ)¯ρR(µ)/planckover2pi1\n2ML(t)×dML(t)\ndt\n−/integraldisplayµ\ndE/bracketleftBigg\n∆ρL(E)d¯χR(E)\ndE−d∆χL(E)\ndE¯ρR(E)/bracketrightBigg/planckover2pi1\n2dML(t)\ndt\n+/integraldisplayµ\ndE/bracketleftBigg\n¯ρL(E)d∆χR(E)\ndE−d¯χL(E)\ndE∆ρR(E)/bracketrightBigg/planckover2pi1\n2/bracketleftBigg\nML(t)×dML(t)\ndt/bracketrightBigg\n×MR\n+∆ρL(µ)∆ρR(µ)/planckover2pi1\n2dML(t)\ndt×MR/bracerightBigg\n, (30)\n6/13J.Phys. Soc. Jpn. FULLPAPERS\nHereµis the chemical potential of the system. ¯ ρL(R)(E) and∆ρL(R)(E) are the spin-averaged\nlocal density of states (DOS) and the spin polarization of th e local DOS, respectively, at the\nLHS(RHS) layersurface, defined by\n¯ρL(R)(E) :=−1\nπℑ¯gr\nL(R)(E), (31)\n∆ρL(R)(E) :=−1\nπℑ∆gr\nL(R)(E), (32)\nwheregr’s are retarded Green’s functions from the calculations tak ing the lesser component.\nFurthermore,theχ’s aredefined as thereal parts oftheretarded Green’sfuncti ons,\n¯χL(R)(E) :=1\nπℜ¯gr\nL(R)(E), (33)\n∆χL(R)(E) :=1\nπℜ∆gr\nL(R)(E). (34)\n3. DiscussionandSummary\n3.1 Chargecurrent\nThe form A[ML(t)×˙ML(t)]·MR+B˙ML(t)·MRof the charge current (eq. (29)) driven\nby the magnetizationdynamics is consistent with previous w orks; The first term tends to the\nform given by Tserkovnyak et al.31in the special case where ML(t) precesses around MR,\nas discussed in§3.3. And Xiao et al.30have derived the same form for the charge current\npassing through the MTJ on the basis of scattering theory in t he continuum space, whereas,\nwecalculatethecurrentonthebasisofthetight-bindingmo del.Newinsightswhichwefound\nout in this study are as follows.If the electronic structure in the two layers of the MTJ is the\nsame [i.e.,∆ρL(E)=∆ρR(E) and∆χL(E)=∆χR(E)] or both layers are insulators, we have\nB=0. However, if either layer is metallic, finite Bshould be measured because the real\npartoftheretarded Green’s functionremainsfinite,which r eflects virtualtransitionsthrough\nforbiddenbands.Theterm[ ML(t)×˙ML(t)]·MRineq.(29)representsthechargecurrentdriven\nbytheeffectiveelectricfield(i.e.,thespinelectricfield),asment ionedin§3.3.Inotherwords,\ntheeffectiveelectrochemicalpotentialofthefreelayerischang ed bythedynamicsof ML(t),\nandtheresultantdi fferenceinelectrochemicalpotentialsbetweenthetwolayer smanifestsas\na bias voltage.31This situation may be realized when a barrier exists between the electrode\nand lead, or when the di ffusion constant of the free layer is small enough to maintain t he\nchanged effective chemical potential. Otherwise, this charge current will flow back to the\nreservoirconnected to thefree layerwithouttunnelingthr oughthebarrier oftheMTJ.\n7/13J.Phys. Soc. Jpn. FULLPAPERS\n3.2 Spincurrent\nThe term ML(t)×MRin eq. (30) represents the static e ffective Heisenberg coupling be-\ntweenML(t)andMR.Thatis,theequationofmotionfor ML(t)describedbythisspincurrent\ncorresponds to the equationdML(t)\ndt=Jeff\n/planckover2pi1|SL(t)|ML(t)×MR[SL(t) is defined by eq. (42)]. This\naffordsaHeisenberg couplingenergy of −JeffML(t)·MR, where\nJeff:=−4π|TLR|2/integraldisplayµ\ndE/bracketleftbig∆ρL(E)∆χR(E)+∆χL(E)∆ρR(E)/bracketrightbig(35)\n=1\nπ/integraldisplayµ\ndEℑGr↑\nLR(E)∆R(E)Gr↓\nRL(E)∆L(E), (36)\nGrσ\nij(E) :=gr\niσ(E)Tijgr\njσ(E), (37)\n∆i(E) :=gr\ni↑(E)−1−gr\ni↓(E)−1. (38)\n∆i(E) describes the exchange splitting at site i, and this result agrees with the expression\npresentedby Liechtenstein et al.39\nLet usconsidertheterm ML×dML(t)\ndtineq. (30):\n2π|TLR|2¯ρL(µ)¯ρR(µ)ML(t)×dML(t)\ndt=/planckover2pi1\n2e2¯ΓML(t)×dML(t)\ndt, (39)\nwheree>0is theelementary charge and ¯ΓisthetunnelconductanceoftheMTJ,\n¯Γ:=4π|TLR|2e2\n/planckover2pi1¯ρL(µ)¯ρR(µ). (40)\nThis term describes the spin pumping in the MTJ and a ffords the following microscopic\nexpressionfortheenhanced Gilbertdampingconstant:\n∆α=/planckover2pi1\n2e2¯Γ\n|SL(t)|, (41)\nwhereSL(t)isthetotalspinpolarizationoftheelectrons intheLHSla yer,\nSL(t) :=2/summationdisplay\ni∈LHS/integraldisplayµ\ndE∆ρi(E)ML(t). (42)\nEquation (41) agrees with the corrected Gilbert damping con stant derived by Zhang et al.28\nphenomenologically after considering the e ffect of the spin electric field induced by the dy-\nnamic magnetization. In addition, in the present formulati on, from the fact that ∆αvanishes\nif one ignores Berry’s phase (6),40it follows that one of the origins of spin pumping is the\nspinelectric field. Asaconsequenceofthis, ∆αis proportionalto theconductance ¯Γ.\nThesizedependenceof ∆αcan bedescribed as follows:\n∆α∝1\nλ, (43)\nwhereλisthicknessofthefreelayer,because |SL(t)|isroughlyproportionaltothevolumeof\n8/13J.Phys. Soc. Jpn. FULLPAPERS\nthefree layer, and ¯Γtothecross-sectionalarea ofthebarrier.\n3.3 Analysisofeffectivefield\nFor a more transparent physical interpretation of the curre nts, we rewrite eqs. (29) and\n(30)as follows:\n−eIe(t)=/summationdisplay\nσ=±1/bracketleftBig\nΓR\nσε1\nσ(t)+γL\nσε2\nσ(t)/bracketrightBig\n·MR, (44)\n−eIs(t)=/bracketleftBiggeJeff\n/planckover2pi1ML(t)−∆Γ/planckover2pi1\n2edML(t)\ndt/bracketrightBigg\n×MR\n+/summationdisplay\nσ=±1σ/braceleftBig\nΓR\nσε1\nσ(t)+/bracketleftBig\nγL\nσε2\nσ(t)·MR/bracketrightBig\nML(t)−/bracketleftBig\nγR\nσ+ML(t)·MRγL\nσ/bracketrightBig\nε2\nσ(t)/bracerightBig\n,(45)\nwherethe“conductances” aredefined by\nΓR\nσ:=2π|TLR|2e2¯ρL(µ)ρRσ(µ)\n/planckover2pi1, (46)\n∆Γ:=4π|TLR|2e2∆ρL(µ)∆ρR(µ)\n/planckover2pi1, (47)\nγL\nσ:=−2π|TLR|2e2\n/planckover2pi1/integraldisplayµ\ndE/bracketleftBigg\nρLσ(E)d∆χR(E)\ndE−dχLσ(E)\ndE∆ρR(E)/bracketrightBigg\n, (48)\nγR\nσ:=−2π|TLR|2e2\n/planckover2pi1/integraldisplayµ\ndE/bracketleftBigg\nρRσ(E)d∆χL(E)\ndE−dχRσ(E)\ndE∆ρL(E)/bracketrightBigg\n, (49)\nandtheeffectivedrivingfields can bedefined by\nε1\nσ(t) :=−σ/planckover2pi1\n2eML(t)×dML(t)\ndt, (50)\nε2\nσ(t) :=−σ/planckover2pi1\n2edML(t)\ndt. (51)\nThe conductances represented by a capital letter denote the “Fermi surface terms,” whereas\nthoserepresentedbyasmallletterdenotethe“Fermiseater ms.”Thespin-dependente ffective\nvoltageε1\nσ(t)·MRin eq. (44) just corresponds to the spin electric field betwee n the layers.\nTo compare the expressionsobtained in continuous space and in discrete space, let us define\nthe correspondences M(r,t) :=ML(t) andM(r+∆r,t) :=MR, where∆rdenotes the barrier\nthickness.Thenwefind ε1\nσ(t)·MR≃∆ri/parenleftBig\n−σ/planckover2pi1\n2e/parenrightBig∂M(r,t)\n∂t×∂M(r,t)\n∂xi·M(r,t),whichiswell-knownas\nthespinelectricfield.When ML(t)steadilyprecessaboutthedirectionof MRwithaconstant\nconeangleθand aconstantfrequency ω,thevoltageis time-independent:\nε1\nσ(t)·MR=−σ/planckover2pi1ω\n2esin2θ, (52)\nThisaffords an estimate/planckover2pi1ω/2e∼20µV at 10 GHz. The Fermi sea term in eq. (44) vanishes\nin this case. This result is in good agreement with that of Xia oet al.30and Tserkovnyak et\n9/13J.Phys. Soc. Jpn. FULLPAPERS\nal.31Notethatin general theFermisea termiscertainly theaccur rent.\nNext, let us consider the spin current (45). The terms includ ingΓR\nσε1\nσ(t)+/bracketleftBig\nγL\nσε2\nσ(t)·\nMR/bracketrightBig\nML(t) describe the spin transport due to the spin σcomponent of the charge current.\nBy consideringε2\nσ(t) as a driving force, we can interpret the term/bracketleftBig\nγR\nσ+ML(t)·MRγL\nσ/bracketrightBig\nε2\nσ(t)\nasthe“tunnelingmagnetoresistance(TMR) e ffect”inspintransport.\n3.4 Effects ofbiasvoltage\nFinally, we consider the charge and spin transport in the pre sence of a bias voltage V(t)\nacross the MTJ. In Berry’s adiabatic approximation under th e assumption JL≫e|V(t)|, the\neffectsofV(t)can beincludedbyreplacing eq. (4)with\nci(t)≃e−ie\n/planckover2pi1/integraltext\ndtV(t)ˆUL(t)eiγ(t)ˆσzdifori∈LHS. (53)\nInthefirst orderindV(t)\ndt, theeffectiveexchangeconstantand conductancesdi fferas follows:\nJeff→Jeff+(γL\n↑−γL\n↓)/planckover2pi1\neV(t)+∆Γ/planckover2pi12\n2ed\ndµln/bracketleftBigg∆ρL(µ)\n∆ρR(µ)/bracketrightBiggdV(t)\ndt, (54)\nΓR\nσ→ΓR\nσ1−d\ndµln/bracketleftBigg¯ρL(µ)\nρRσ(µ)/bracketrightBigg\neV(t)−/integraldisplayµ\ndE¯ρL(E)d3χRσ(E)\ndE3−d3¯χL(E)\ndE3ρRσ(E)\n¯ρL(µ)ρRσ(µ)e/planckover2pi1\n2dV(t)\ndt,(55)\n∆Γ→∆Γ1−d\ndµln/bracketleftBigg∆ρL(µ)\n∆ρR(µ)/bracketrightBigg\neV(t)−/integraldisplayµ\ndE∆ρL(E)d3∆χR(E)\ndE3−d3∆χL(E)\ndE3∆ρR(E)\n∆ρL(µ)∆ρR(µ)e/planckover2pi1\n2dV(t)\ndt,\n(56)\nγL\nσ→γL\nσ−2π|TLR|2e2\n/planckover2pi1/integraldisplayµ\ndE/bracketleftBigg\nρLσ(E)d2∆χR(E)\ndE2+d2χLσ(E)\ndE2∆ρR(E)/bracketrightBigg\neV(t)\n+2π|TLR|2e2\n/planckover2pi1/bracketleftBiggdρLσ(µ)\ndµd∆ρR(µ)\ndµ−d2ρLσ(µ)\ndµ2∆ρR(µ)−ρLσ(µ)d2∆ρR(µ)\ndµ2/bracketrightBigge/planckover2pi1\n2dV(t)\ndt,(57)\nγR\nσ→γR\nσ−2π|TLR|2e2\n/planckover2pi1/integraldisplayµ\ndE/bracketleftBigg\nρRσ(E)d2∆χL(E)\ndE2+d2χRσ(E)\ndE2∆ρL(E)/bracketrightBigg\neV(t)\n+2π|TLR|2e2\n/planckover2pi1/bracketleftBiggdρRσ(µ)\ndµd∆ρL(µ)\ndµ−d2ρRσ(µ)\ndµ2∆ρL(µ)−ρRσ(µ)d2∆ρL(µ)\ndµ2/bracketrightBigge/planckover2pi1\n2dV(t)\ndt.(58)\nIn addition,atermdescribingtheTMRe ffect,\n1\ne/bracketleftBig¯Γ+∆ΓML(t)·MR/bracketrightBig\nV(t)+1\n−e/bracketleftbig¯γ+∆γML(t)·MR/bracketrightbig/planckover2pi1\n2dV(t)\ndt(59)\nappears inthechargecurrent, where\n¯γ:=4π|TLR|2e2\n/planckover2pi1/integraldisplayµ\ndE/bracketleftBigg\n¯ρL(E)d2¯χR(E)\ndE2+d2¯χL(E)\ndE2¯ρR(E)/bracketrightBigg\n,\n∆γ:=4π|TLR|2e2\n/planckover2pi1/integraldisplayµ\ndE/bracketleftBigg\n∆ρL(E)d2∆χR(E)\ndE2+d2∆χL(E)\ndE2∆ρR(E)/bracketrightBigg\n.\n10/13J.Phys. Soc. Jpn. FULLPAPERS\nForthespincurrent, aterm describingtheSTT e ffect,\n1\ne/bracketleftBigg\n(ΓL\n↑−ΓL\n↓)V(t)−(γR\n↑+γR\n↓)/planckover2pi1\n2edV(t)\ndt/bracketrightBigg\nML(t)+1\ne/bracketleftBigg\n(ΓR\n↑−ΓR\n↓)V(t)+(γL\n↑+γL\n↓)/planckover2pi1\n2edV(t)\ndt/bracketrightBigg\nMR\nisadded, where\nΓL\nσ:=2π|TLR|2e2ρLσ(µ)¯ρR(µ)\n/planckover2pi1. (60)\nThenfortheGilbertdamping,since ¯Γ=ΓR\n↑+ΓR\n↓,∆αchanges as follows:\n∆α→∆α1−d\ndµln/braceleftBigg¯ρL(µ)\n¯ρR(µ)/bracerightBigg\neV(t)−/integraldisplayµ\ndE¯ρL(E)d3¯χR(E)\ndE3−d3¯χL(E)\ndE3¯ρR(E)\n¯ρL(µ)¯ρR(µ)e/planckover2pi1\n2dV(t)\ndt.(61)\nThisresultindicatesthat when writingdatato an STT MRAM ce ll,thedampingofthemag-\nnetization dynamics is influenced by not only the spin pumpin g but also the bias voltage.\nHowever, the effect of the bias voltage on ∆αvanishes when both electrodes have the same\nelectronicstructure.\nIn summary, we derived, at the microscopic level, the charge and spin currents passing\nthrough an MTJ in response to arbitrary motion of the magneti zation in the free layer. The\nchargecurrentconsistsofbothFermisurfaceandFermiseat erms.TheFermisurfacetermis\ndriven by the spin electric field and manifests as a dc current for steady precession of ML(t)\nin the direction of MR, whereas the Fermi sea term is due to virtual transitions and essen-\ntially manifests as the ac current. With regard to spin trans port, we focused particularly on\ntheenhancedGilbertdamping(orthespinpumpinge ffect)andthusobtainedthemicroscopic\nexpression for the enhanced Gilbert damping constant ∆α=/planckover2pi1\n2e2¯Γ\n|SL(t)|. Under a bias voltage,\nthe DOSs of the two layers in the MTJ are shifted. Thus, the bia s voltage changes the ef-\nfective exchange constant and the conductances, thus produ cing modulation of ∆α. All the\nconductances consist of the tunneling amplitude TLRand the local DOS on the surfaces of\nthe layers; the real part of a retarded Green’s function can b e obtained from the imaginary\npart (namely, the local DOS) via the Kramers–Kronig relatio nship. 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Box 3049, 67663 Kaiserslautern, Germany\n(Dated: June 4, 2018)\nWe present a microscopic calculation of magnetization damping for a magnetic \\toy model.\"\nThe magnetic system consists of itinerant carriers coupled antiferromagnetically to a dispersionless\nband of localized spins, and the magnetization damping is due to coupling of the itinerant carriers\nto a phonon bath in the presence of spin-orbit coupling. Using a mean-\feld approximation for\nthe kinetic exchange model and assuming the spin-orbit coupling to be of the Rashba form, we\nderive Boltzmann scattering integrals for the distributions and spin coherences in the case of an\nantiferromagnetic exchange splitting, including a careful analysis of the connection between lifetime\nbroadening and the magnetic gap. For the Elliott-Yafet type itinerant spin dynamics we extract\ndephasing and magnetization times T1andT2from initial conditions corresponding to a tilt of the\nmagnetization vector, and draw a comparison to phenomenological equations such as the Landau-\nLifshitz (LL) or the Gilbert damping. We also analyze magnetization precession and damping for\nthis system including an anisotropy \feld and \fnd a carrier mediated dephasing of the localized spin\nvia the mean-\feld coupling.\nPACS numbers: 75.78.-n, 72.25.Rb, 76.20.+q\nI. INTRODUCTION\nThere are two widely-known phenomenological ap-\nproaches to describe the damping of a precessing mag-\nnetization in an excited ferromagnet: one introduced\noriginally by Landau and Lifshitz1and one introduced\nby Gilbert,2which are applied to a variety of prob-\nlems3involving the damping of precessing magnetic mo-\nments. Magnetization damping contributions and its in-\nverse processes, i.e., spin torques, in particular in thin\n\flms and nanostructures, are an extremely active \feld,\nwhere currently the focus is on the determination of novel\nphysical processes/mechanisms. Apart from these ques-\ntions there is still a debate whether the Landau-Lifshitz\nor the Gilbert damping is the correct one for \\intrin-\nsic\" damping, i.e., neglecting interlayer coupling, inter-\nface contributions, domain structures and/or eddy cur-\nrents. This intrinsic damping is believed to be caused\nby a combination of spin-orbit coupling and scattering\nmechanisms such as exchange scattering between s and d\nelectrons and/or electron-phonon scattering.4{6Without\nreference to the microscopic mechanism, di\u000berent macro-\nscopic analyses, based, for example, on irreversible ther-\nmodynamics or near equilibrium Langevin theory, prefer\none or the other description.7,8However, material param-\neters of typical ferromagnetic heterostructures are such\nthat one is usually \frmly in the small damping regime so\nthat several ferromagnetic resonance (FMR) experiments\nwere not able to detect a noticeable di\u000berence between\nLandau Lifshitz and Gilbert magnetization damping. A\nrecent analysis that related the Gilbert term directly to\nthe spin-orbit interaction arising from the Dirac equa-\ntion does not seem to have conclusively solved this dis-\ncussion.9\nThe dephasing term in the Landau-Lifshitz form isalso used in models based on classical spins coupled\nto a bath, which have been successfully applied to\nout-of-equilibrium magnetization dynamics and magnetic\nswitching scenarios.10The most fundamental of these\nare the stochastic Landau-Lifshitz equations,10{13from\nwhich the Landau-Lifshitz Bloch equations,14,15can be\nderived via a Fokker-Planck equation.\nQuantum-mechanical treatments of the equilibrium\nmagnetization in bulk ferromagnets at \fnite temper-\natures are extremely involved. The calculation of\nnon-equilibrium magnetization phenomena and damp-\ning for quantum spin systems in more than one dimen-\nsion, which include both magnetism and carrier-phonon\nand/or carrier-impurity interactions, at present have to\nemploy simpli\fed models. For instance, there have been\nmicroscopic calculations of Gilbert damping parameters\nbased on Kohn-Sham wave functions for metallic ferro-\nmagnets16,17and Kohn-Luttinger p-dHamiltonians for\nmagnetic semiconductors.18While the former approach\nuses spin density-functional theory, the latter approach\ntreats the anti-ferromagnetic kinetic-exchange coupling\nbetween itinerant p-like holes and localized magnetic\nmoments originating from impurity d-electrons within a\nmean-\feld theory. In both cases, a constant spin and\nband-independent lifetime for the itinerant carriers is\nused as an input, and a Gilbert damping constant is ex-\ntracted by comparing the quantum mechanical result for\n!!0 with the classical formulation. There have also\nbeen investigations, which extract the Gilbert damping\nfor magnetic semiconductors from a microscopic calcula-\ntion of carrier dynamics including Boltzmann-type scat-\ntering integrals.19,20Such a kinetic approach, which is of\na similar type as the one we present in this paper, avoids\nthe introduction of electronic lifetimes because the scat-\ntering is calculated dynamically.arXiv:1405.2347v1  [cond-mat.mtrl-sci]  9 May 20142\nThe present paper takes up the question how the spin\ndynamics in the framework of the macroscopic Gilbert\nor Landau-Lifshitz damping compare to a microscopic\nmodel of relaxation processes in the framework of a rel-\natively simple model. We analyze a mean-\feld kinetic\nexchange model including spin-orbit coupling for the itin-\nerant carriers. Thus the magnetic mean-\feld dynamics is\ncombined with a microscopic description of damping pro-\nvided by the electron-phonon coupling. This interaction\ntransfers energy and angular momentum from the itin-\nerant carriers to the lattice. The electron-phonon scat-\ntering is responsible both for the lifetimes of the itiner-\nant carriers and the magnetization dephasing. The lat-\nter occurs because of spin-orbit coupling in the states\nthat are connected by electron-phonon scattering. To be\nmore speci\fc, we choose an anti-ferromagnetic coupling\nat the mean-\feld level between itinerant electrons and\na dispersion-less band of localized spins for the magnetic\nsystem. To keep the analysis simple we use as a model for\nthe spin-orbit coupled itinerant carrier states a two-band\nRashba model. As such it is a single-band version of the\nmulti-band Hamiltonians used for III-Mn-V ferromag-\nnetic semiconductors.18,21{24The model analyzed here\nalso captures some properties of two-sublattice ferrimag-\nnets, which are nowadays investigated because of their\nmagnetic switching dynamics.25,26The present paper is\nset apart from studies of spin dynamics in similar mod-\nels with more complicated itinerant band structures19,20\nby a detailed comparison of the phenomenological damp-\ning expressions with a microscopic calculation as well as\na careful analysis of the restrictions placed by the size\nof the magnetic gap on the single-particle broadening in\nBoltzmann scattering.\nThis paper is organized as follows. As an extended\nintroduction, we review in Sec. II some basic facts con-\ncerning the Landau-Lifshitz and Gilbert damping terms\non the one hand and the Bloch equations on the other.\nIn Sec. III we point out how these di\u000berent descriptions\nare related in special cases. We then introduce a micro-\nscopic model for the dephasing due to electron-phonon\ninteraction in Sec. IV, and present numerical solutions\nfor two di\u000berent scenarios in Secs. V and VI. The \frst\nscenario is the dephasing between two spin subsystems\n(Sec. V), and the second scenario is a relaxation process\nof the magnetization toward an easy-axis (Sec. VI). A\nbrief conclusion is given at the end.\nII. PHENOMENOLOGIC DESCRIPTIONS OF\nDEPHASING AND RELAXATION\nWe summarize here some results pertaining to a single-\ndomain ferromagnet, and set up our notation. In equilib-\nrium we assume the magnetization to be oriented along\nits easy axis or a magnetic \feld ~H, which we take to\nbe thezaxis in the following. If the magnetization\nis tilted out of equilibrium, it starts to precess. As\nillustrated in Fig. 1 one distinguishes the longitudinal\nFIG. 1. Illustration of non-equilibrium spin-dynamics in pres-\nence of a magnetic \feld without relaxation (a) and within\nrelaxation (b).\ncomponent Mk, inzdirection, and the transverse part\nM?\u0011q\nM2\u0000M2\nk, precessing in the x-yplane with the\nLarmor frequency !L.\nIn connection with the interaction processes that re-\nturn the system to equilibrium, the decay of the trans-\nverse component is called dephasing. There are three\nphenomenological equations used to describe spin de-\nphasing processes:\n1. The Bloch(-Bloembergen) equations27,28\n@\n@tMk(t) =\u0000Mk(t)\u0000Meq\nT1(1)\n@\n@tM?(t) =\u0000M?(t)\nT2(2)\ndescribe an exponential decay towards the equilib-\nrium magnetization Meqinzdirection. The trans-\nverse component decays with a time constant T2,\nwhereas the longitudinal component approaches its\nequilibrium amplitude with T1. These time con-\nstants may be \ft independently to experimental\nresults or microscopic calculations.\n2. Landau-Lifshitz damping1with parameter \u0015\n@\n@t~M(t) =\u0000\r~M\u0002~H\u0000\u0015~M\nM\u0002\u0000~M\u0002~H\u0001\n(3)\nwhere\ris the gyromagnetic ratio. The \frst term\nmodels the precession with a frequency !L=\rj~Hj,\nwhereas the second term is solely responsible for\ndamping.\n3. Gilbert damping2with the dimensionless Gilbert\ndamping parameter \u000b\n@\n@t~M(t) =\u0000\rG~M\u0002~H+\u000b\u0010~M\nM\u0002@t~M\u0011\n(4)\nIt is generally accepted that \u000bis independent of\nthe static magnetic \felds ~Hsuch as anisotropy\n\felds,18,29and thus depends only on the material\nand the microscopic interaction processes.3\nThe Landau-Lifshitz and Gilbert forms of damping are\nmathematically equivalent2,7,30with\n\u000b=\u0015\n\r(5)\n\rG=\r(1 +\u000b2) (6)\nbut there are important di\u000berences. In particular, an in-\ncrease of\u000blowers the precession frequency in the dynam-\nics with Gilbert damping, while the damping parameter\n\u0015in the Landau-Lifshitz equation has no impact on the\nprecession. In contrast to the Bloch equations, Landau-\nLifshitz and Gilbert spin-dynamics always conserve the\nlengthj~Mjof the magnetization vector.\nAn argument by Pines and Slichter,31shows that there\nare two di\u000berent regimes for Bloch-type spin dynamics\ndepending on the relation between the Larmor period and\nthe correlation time. As long as the correlation time is\nmuch longer than the Larmor period, the system \\knows\"\nthe direction of the \feld during the scattering process.\nStated di\u000berently, the scattering process \\sees\" the mag-\nnetic gap in the bandstructure. Thus, transverse and\nlongitudinal spin components are distinguishable and the\nBloch decay times T1andT2can di\u000ber. If the correlation\ntime is considerably shorter than the Larmor period, this\ndistinction is not possible, with the consequence that T1\nmust be equal to T2. Within the microscopic approach,\npresented in Sec. IV D, this consideration shows up again,\nalbeit for the energy conserving \u000efunctions resulting from\na Markov approximation.\nThe regime of short correlation times has already been\ninvestigated in the framework of a microscopic calcula-\ntion by Wu and coworkers.32They analyze the case of\na moderate external magnetic \feld applied to a non-\nmagnetic n-type GaAs quantum well and include di\u000ber-\nent scattering mechanisms (electron-electron Coulomb,\nelectron-phonon, electron-impurity). They argue that\nthe momentum relaxation rate is the crucial time scale\nin this scenario, which turns out to be much larger than\nthe Larmor frequency. Their numerical results con\frm\nthe identity T1=T2expected from the Pines-Slichter\nargument.\nIII. RELATION BETWEEN\nLANDAU-LIFSHITZ, GILBERT AND BLOCH\nWe highlight here a connection between the Bloch\nequations (1, 2) and the Landau-Lifshitz equation (3).\nTo this end we assume a small initial tilt of the mag-\nnetization and describe the subsequent dynamics of the\nmagnetization in the form\n~M(t) =0\n@\u000eM?(t) cos(!Lt)\n\u000eM?(t) sin(!Lt)\nMeq\u0000\u000eMk(t)1\nA (7)\nwhere\u000eM?and\u000eMjjdescribe deviations from equilib-\nrium. Putting this into eq. (3) one gets a coupled set ofequations.\n@\n@t\u000eM?(t) =\u0000\u0015HMeq\u0000\u000eMk(t)\nj~M(t)j\u000eM?(t) (8)\n@\n@t\u000eMk(t) =\u0000\u0015H1\nj~M(t)j\u000eM2\n?(t) (9)\nEq. (8) is simpli\fed for a small deviation from equilib-\nrium, i.e.,\u000eM(t)\u001cMeqandj~M(t)j\u0019Meq:\n\u000eM?(t) =Cexp(\u0000\u0015Ht) (10)\n\u000eMk(t) =C2\n2Meqexp(\u00002\u0015Ht) (11)\nwhereCis an integration constant. For small excitations\nthe deviations decay exponentially and Bloch decay times\nT1andT2result, which are related by\n2T1=T2=1\n\u0015H: (12)\nOnly this ratio of the Bloch times is compatible with a\nconstant length of the magnetization vector at low exci-\ntations. By combining Eqs. (12) and (5) one can connect\nthe Gilbert parameter \u000band the dephasing time T2\n\u000b=1\nT2!L: (13)\nIf the conditions for the above approximations apply, the\nGilbert damping parameter \u000bcan be determined by \ft-\nting the dephasing time T2and the Larmor frequency !L\nto computed or measured spin dynamics. This dimen-\nsionless quantity is well suited to compare the dephasing\nthat results from di\u000berent relaxation processes.\nFigure 2 shows the typical magnetization dynamics\nthat results from (3), i.e., Landau-Lifshitz damping. As\nan illustration of a small excitation we choose in Fig. 2(a)\nan angle of 10\u000efor the initial tilt of the magnetization,\nwhich results in an exponential decay with 2 T1=T2.\nFrom the form of Eq. (3) it is clear that this behavior\npersists even for large !Land\u0015. Obviously the Landau-\nLifshitz and Gilbert damping terms describe a scenario\nwith relatively long correlation times (i.e., small scat-\ntering rates), because only in this regime both decay\ntimes can di\u000ber. The microscopic formalism in Sec. IV\nworks in the same regime and will be compared with\nthe phenomenological results. For an excitation angle\nof 90\u000e, the Landau-Lifshitz dynamics shown in Fig. 2(b)\nbecome non-exponential, so that no well-de\fned Bloch\ndecay times T1,T2exist.\nIV. MICROSCOPIC MODEL\nIn this section we describe a microscopic model that in-\ncludes magnetism at the mean-\feld level, spin-orbit cou-\npling as well as the microscopic coupling to a phonon\nbath treated at the level of Boltzmann scattering inte-\ngrals. We then compare the microscopic dynamics to4\n0 5000.51δM⊥/Meq\ntime (ps)0 5000.51\ntime (ps)δM/bardbl/Meq0 5000.010.02δM/bardbl/Meq\ntime (ps)0 5000.10.2\ntime (ps)δM⊥/Meq\n  \nT1= 5.02 ps(a)\n(b)T2= 10.04 ps\nFIG. 2. Dynamics of \u000eM?and\u000eMkcomputed using to\nLandau-Lifshitz damping ( !L= 1 ps\u00001,H= 106A\nm\u0019\n1:26\u0001104Oe,\u0015= 10\u00007m\nA ps). (a) An angle of 10\u000eleads to\nexponential an exponential decay with well de\fned T1andT2\ntimes. (b). For an angle of 90\u000e, the decay (solid line) is not\nexponential as comparison with the exponential \ft (dashed\nline) clearly shows.\nthe Bloch equations (1), (2), as well as the Landau-\nLifshitz (3) and Gilbert damping terms (4). The mag-\nnetic properties of the model are de\fned by an anti-\nferromagnetic coupling between localized magnetic im-\npurities and itinerant carriers. As a prototypical spin-\norbit coupling we consider an e\u000bectively two-dimensional\nmodel with a Rashba spin-orbit coupling. The reason\nfor the choice of a model with a two-dimensional wave\nvector space is not an investigation of magnetization dy-\nnamics with reduced dimensionality, but rather a reduc-\ntion in the dimension of the integrals that have to be\nsolved numerically in the Boltzmann scattering terms.\nSince we treat the exchange between the localized and\nitinerant states in a mean-\feld approximation, our two-\ndimensional model still has a \\magnetic ground state\"\nand presents a framework, for which qualitatively dif-\nferent approaches can be compared. We do not aim at\nquantitative predictions for, say, magnetic semiconduc-\ntors or ferrimagnets with two sublattices. Finally, we\ninclude a standard interaction hamiltonian between the\nitinerant carriers and acoustic phonons. The correspond-\ning hamiltonian reads\n^H=^Hmf+^Hso+^He\u0000ph+^Haniso: (14)\nOnly in Sec. VI an additional \feld ^Haniso is included,\nwhich is intended to model a small anisotropy.A. Exchange interaction between itinerant carriers\nand localized spins\nThe \\magnetic part\" of the model is described by the\nHamiltonian\n^Hmf=X\n~k\u0016~2k2\n2m\u0003^cy\n~k\u0016^c~k\u0016+J^~ s\u0001^~S: (15)\nwhich we consider in the mean-\feld limit. The \frst term\nrepresents itinerant carriers with a k-dependent disper-\nsion relation. In the following we assume s-like wave\nfunctions and parabolic energy dispersions. The e\u000bective\nmass is chosen to be m\u0003= 0:5me, wheremeis the free\nelectron mass, and the ^ c(y)\n~k\u0016operators create and annihi-\nlate carriers in the state j~k;\u0016iwhere\u0016labels the itinerant\nbands, as shown in Fig. 3(a).\nThe second term describes the coupling between itiner-\nant spins~ sand localized spins ~Svia an antiferromagnetic\nexchange interaction\n^~ s=1\n2X\n~kX\n\u0016\u00160h~k;\u00160j^~ \u001bj~k;\u0016i^cy\n~k\u0016^c~k\u00160 (16)\n^~S=1\n2X\n\u0017\u00170h\u00170j^~ \u001bj\u0017iX\n~K^Cy\n~K\u0017^C~K\u00170 (17)\nHere, we have assumed that the wave functions of the lo-\ncalized spins form dispersionless bands, i.e., we have im-\nplicitly introduced a virtual-crystal approximation. Due\nto the assumption of strong localization there is no or-\nbital overlap between these electrons, which are therefore\nconsidered to have momentum independent eigenstates\nj\u0017iand a \rat dispersion, as illustrated in Fig. 3(a). The\ncomponents of the vector ^~ \u001bare the Pauli matrices ^ \u001biwith\ni=x;y;z , and ^C(y)\n~K\u0017are the creation and annihilation op-\nerators for a localized spin state.\nWe do notinclude interactions among localized or itin-\nerant spins, such as exchange scattering. For simplicity,\nwe assume both itinerant and localized electrons to have\na spin 1=2 and therefore \u0016and\u0017to run over two spin-\nprojection quantum numbers \u00061=2. In the following we\nchosse an antiferromagnetic ( J > 0) exchange constant\nJ= 500 meV, which leads to the schematic band struc-\nture shown in Fig. 3(b).\nIn the mean \feld approximation used here, the itiner-\nant carriers feel an e\u000bective magnetic \feld ^Hloc\n~Hloc=\u0000J\u0016B\u0016\ng~S (18)\ncaused by localized moments and vice versa. Here \u0016B\nis the Bohr magneton and g= 2 is the g-factor of the\nelectron. The permeability \u0016is assumed to be the vac-\nuum permeability \u00160. This time-dependent magnetic\n\feld~Hloc(t) de\fnes the preferred direction in the itiner-\nant sub-system and therefore determines the longitudinal\nand transverse component of the itinerant spin at each\ntime.5\nr#k (a) (b) E(k) \nk \n\u0010\nk \n\u000e\u0010,k\n\u000e,kE(k) \nEF \nFIG. 3. Sketch of the band-structure with localized (\rat\ndispersions) and itinerant (parabolic dispersions) electrons.\nAbove the Curie-Temperature TCthe spin-eigenstates are de-\ngenerate (a), whereas below TCa gap between the spin states\nexists.\nB. Rashba spin-orbit interaction\nThe Rashba spin-orbit coupling is given by the Hamil-\ntonian\n^Hso=\u000bR(^\u001bxky\u0000^\u001bykx) (19)\nA Rashba coe\u000ecient of \u000bR= 10 meV nm typical for semi-\nconductors is chosen in the following calculations. This\nvalue, which is close to the experimental one for the\nInSb/InAlSb material system,33is small compared to the\nexchange interactions, but it allows the exchange of an-\ngular momentum with the lattice.\nC. Coherent dynamics\nFrom the above contributions (15) and (19) to the\nHamiltonian we derive the equations of motion contain-\ning the coherent dynamics due to the exchange interac-\ntion and Rashba spin-orbit coupling as well as the inco-\nherent electron-phonon scattering. We \frst focus on the\ncoherent contributions. In principle, one has the choice\nto work in a basis with a \fxed spin-quantization axis or\nto use single-particle states that diagonalize the mean-\n\feld (plus Rashba) Hamiltonian. Since we intend to use\na Boltzmann scattering integral in Sec. IV D we need to\napply a Markov approximation, which only works if one\ndeals with diagonalized eigenenergies. In our case this is\nthe single-particle basis that diagonalizes the entire one-\nparticle contribution of the Hamiltonian ^Hmf+^Hso. In\nmatrix representation this one-particle contribution for\nthe itinerant carriers reads:\n^Hmf+^Hso= \n~2k2\n2m\u0003+ \u0001loc\nz(\u0001loc\n++R~k)\u0003\n\u0001loc\n++R~k~2k2\n2m\u0003\u0000\u0001loc\nz!\n(20)\nwhere we have de\fned \u0001loc\ni=J1\n2h^SiiandR~k=\n\u0000i\u000bRkexp(i'k) with'k= arctan(ky=kx). The eigenen-\nergies are\n\u000f\u0006\n~k=~2k2\n2m\u0003\u0007q\nj\u0001loczj2+jR~k+ \u0001loc\n+j2: (21)and the eigenstates\nj~k;+i=\u0012\n1\n\u0018~k\u0013\n;j~k;\u0000i=\u0012\u0000\u0018\u0003\n~k\n1\u0013\n(22)\nwhere\n\u0018~k=\u0001loc\n++R~k\n\u0001locz+q\nj~\u0001locj2+jR~kj2(23)\nIn this basis the coherent part of the equation of mo-\ntion for the itinerant density matrix \u001a\u0016\u00160\n~k\u0011 h^cy\n~k\u0016^c~k\u00160i\nreads\n@\n@t\u001a\u0016\u00160\n~k\f\f\f\ncoh=i\n~\u0000\n\u000f\u0016\n~k\u0000\u000f\u00160\n~k\u0001\n\u001a\u0016\u00160\n~k: (24)\nNo mean-\feld or Rashba terms appear explicitly in these\nequations of motion since their contributions are now hid-\nden in the time-dependent eigenstates and eigenenergies.\nSince we are interested in dephasing and precessional\ndynamics, we assume a comparatively small spin-orbit\ncoupling, that can dissipate angular momentum into the\nlattice, but does not have a decisive e\u000bect on the band-\nstructure. Therefore we use the spin-mixing only in the\ntransition matrix elements of the electron-phonon scat-\nteringM~k0\u00160\n~k\u0016(31). For all other purposes we set R~k= 0.\nIn particular, the energy-dispersion \u000f\u0006\n~kis assumed to be\nuna\u000bected by the spin-orbit interaction and therefore it\nis spherically symmetric.\nWith this approximation the itinerant eigenstates are\nalways exactly aligned with the e\u000bective \feld of the local-\nized moments ~Hloc(t). Since this e\u000bective \feld changes\nwith time, the diagonalization and a transformation of\nthe spin-density matrix in \\spin space\" has to be re-\npeated at each time-step. This e\u000bort makes it easier\nto identify the longitudinal and transverse spin compo-\nnents with the elements of the single-particle density\nmatrix: The o\u000b-diagonal entries of the density matrix\n\u001a\u0006\u0007\n~k, which precess with the k-independent Larmor fre-\nquency!L= 2\u0001loc=~, always describe the dynamics of\nthe transverse spin-component. The longitudinal compo-\nnent, which does not precess, is represented by the diag-\nonal entries \u001a\u0006\u0006\n~k. Since both components change their\nspatial orientation continuously, we call this the rotating\nframe. The components of the spin vector in the rotating\nframe are\nh^ski=1\n2X\n~k\u0000\n\u001a++\n~k\u0000\u001a\u0000\u0000\n~k\u0001\n(25)\nh^s?i=X\n~k\f\f\u001a+\u0000\n~k\f\f (26)\nThe components in the \fxed frame are obtained from\nEq. (16)\nh^~ si=1\n2X\n~kX\n\u0016\u00160h~k;\u00160j^~ \u001bj~k;\u0016i\u001a\u0016\u00160\n~k(27)6\nIn this form, the time-dependent states carry the infor-\nmation how the spatial components are described by the\ndensity matrix at each time step. No time-independent\n\\longitudinal\" and \\transverse\" directions can be identi-\n\fed in the \fxed frame.\nIn a similar fashion, the diagonalized single-particle\nstates of the localized spin system are obtained. The\neigenenergies are\nE\u0006=\u0007\f\f~\u0001itin\f\f (28)\nwhere \u0001itin\ni=J1\n2h^siiis the localized energy shift caused\nby the itinerant spin component si. The eigenstates are\nagain always aligned with the itinerant magnetic mo-\nment. In this basis the equation of motion of the localized\nspin-density matrix \u001a\u0017\u00170\nloc\u0011P\n~Kh^Cy\n~K\u0017^C~K\u00170iis simply\n@\n@t\u001a\u0017\u00170\nloc=i\n~(E\u0017\u0000E\u00170)\u001a\u0017\u00170\nloc (29)\nand does not contain explicit exchange contributions.\nEqs. (25), (26), and (27) apply in turn to the components\nhSkiandhS?iof the localized spin and its spin-density\nmatrix\u001a\u0017\u00170\nloc.\nD. Electron-phonon Boltzmann scattering with\nspin splitting\nRelaxation is introduced into the model by the interac-\ntion of the itinerant carriers with a phonon bath, which\nplays the role of an energy and angular momentum sink\nfor these carriers. Our goal here is to present a derivation\nof the Boltzmann scattering contributions using stan-\ndard methods, see, e.g., Refs. 34 and 36. However, we\nemphasize that describing interaction as a Boltzmann-\nlike instantaneous, energy conserving scattering process\nis limited by the existence of the magnetic gap. Since we\nkeep the spin mixing due to Rashba spin-orbit coupling\nonly in the Boltzmann scattering integrals, the resulting\ndynamical equations describe an Elliott-Yafet type spin\nrelaxation.\nThe electron-phonon interaction Hamiltonian reads34\n^He\u0000ph=X\n~ q~!ph\nq^by\n~ q^b~ q\n+X\n~k~k0X\n\u0016\u00160\u0000\nM~k0\u00160\n~k\u0016^cy\n~k\u0016^b~k\u0000~k0^c~k0\u00160+ h.c.\u0001(30)\nwhere ^b(y)\n~ qare the bosonic operators, that create or an-\nnihilate acoustic phonons with momentum ~ qand linear\ndispersion!ph(q) =cphj~ qj. The sound velocity is taken\nto becph= 40 nm/ps and we use an e\u000bectively two-\ndimensional transition matrix element35\nM~k0\u00160\n~k\u0016=Dq\nj~k\u0000~k0jh~k;\u0016j~k0;\u00160i (31)\nwhere the deformation potential is chosen to be D=\n60 meVnm1=2. The scalar-product between the initialstatej~k0;\u00160iand the \fnal state j~k;\u0016iof an electronic\ntransition takes the spin-mixing due to Rashba spin-orbit\ncoupling into account.\nThe derivation of Boltzmann scattering integrals for\nthe itinerant spin-density matrix (24) leads to a memory\nintegral of the following shape\n@\n@t\u001aj(t)\f\f\f\ninc=1\n~X\nj0Zt\n\u00001ei(\u0001Ejj0+i\r)(t\u0000t0)Fjj0[\u001a(t0)]dt0;\n(32)\nregardless whether one uses Green's function36or\nequation-of-motion techniques.34Since we go through a\nstandard derivation here, we highlight only the impor-\ntant parts for the present case and do not write the equa-\ntions out completely. In particular, for scattering process\nj0=j\u00160;~k0i!j=j\u0016;~ki, we useFjj0[\u001a(t0)] as an abbre-\nviation for a product of dynamical electronic spin-density\nmatrix elements \u001a, evaluated at time t0<t, and equilib-\nrium phononic distributions. The corresponding energy\ndi\u000berence is denoted by \u0001 Ejj0=Ej\u0000Ej0\u0006~!ph(j~k\u0000~k0j),\nand\rdescribes the decay of the exponential function due\nto dissipation and/or higher order correlation functions.\nIn general, the integral has to be evaluated numerically\nand contains memory e\u000bects. To apply the Markov ap-\nproximation one needs to compare two time scales: the\n\\memory depth\" 1 =\r, i.e., the time scale on which the\nexp[\u0000\r(t\u0000t0)] factor essentially cuts o\u000b the integral, and\nthe typical time scale on which the Fterm changes. In\nthis paper we deal with relaxation processes not too far\naway from equilibrium, so that the typical time scale of\nthe components of the spin-density matrix contained in\nFis set by the Bloch times T1andT2. We can thus\napproximate Fjj0[\u001a(t0)] byFjj0[\u001a(t)], for all transitions\nlabeled byjandj0, if the memory depth is shorter than\nthe Bloch time(s), or\n\r\u001d1\nT1: (33)\nProvided condition (33) holds, the integral (32) can\nbe done using the Markov approximation Fjj0[\u001a(t0)]'\nFjj0[\u001a(t)]\n@\n@t\u001aj(t)\f\f\f\nincoh=i\n~X\nj0Fjj0[\u001a(t)]1\n\u0001Ejj0+i~\r:(34)\nAs it is customary, we neglect in the following the real\npart of the complex energy denominator, which results in\nshifts of the single-particle energies. While these shifts\nmay play an important role in non-Markovian problems\nwith discrete energy levels37, the imaginary parts yield\nthe relaxation contributions that are important for the\npresent paper\n@\n@t\u001aj(t)\f\f\f\nincoh=X\nj0Fjj0[\u001a(t)]~\r\n(\u0001Ejj0)2+ (~\r)2(35)\nAll transitions are thus weighted by a Lorentzian peaked\nat resonant transitions (\u0001 Ejj0= 0) with a broadening7\nof~\rthat may be interpreted as an energy uncertainty.\nFor relaxation processes in a system with a spin-splitting\n(due to internal \felds and/or spin-orbit coupling), this\nbroadening must not be so large as to blur the distinc-\ntion between the split bands. Consequently, only if the\nbroadening \ris smaller than the magnetic splitting, i.e,,\nif\r\u001c!L, it is possible to distinguish between longi-\ntudinal and transverse components of the spin-density\nmatrix. With Eq. (33) the inequality \r\u001c!Lyields the\ncondition\n!L\u001d1\nT1(36)\nfor the Larmor frequencies and Bloch times, for which\nit is permissible to replace the Lorentzian by an energy\nconserving \u000efunction\n~\r\n(\u0001Ejj0)2+ (~\r)2\r!0\u0000!\u0019\u000e(\u0001Ejj0): (37)This reduces the numerical e\u000bort very considerably, be-\ncause it allows one to eliminate an integration from the\nscattering term, and the energy conserving \u000efunction is\ntherefore often used without explicitly checking its valid-\nity.\nThe considerations leading to the connection between\nEqs. (36) and (37) are a microscopic version of an argu-\nment due to Pines and Slichter,31according to which T1\nandT2can di\u000ber only for correlation times that long in\ncomparison to a Larmor period. The microscopic Boltz-\nmann scattering terms, which contain the energy con-\nserving\u000efunctions and will be used in the following,\ndo not apply in a regime outside of condition (36). If\n!L'1=T1, a \fnite broadening \rhas to be taken into\naccount. Together the full equation of motion in the\nregime (36) for the itinerant-carrier spin density matrix\nthus reads\n@\n@t\u001a\u0016\u00160\n~k=i\n~\u0000\n\u000f\u0016\nk\u0000\u000f\u00160\nk\u0001\n\u001a\u0016\u00160\n~k\n+\u0019\n~X\nk0X\n\u00161\u00162\u00163M~k\u0016\n~k0\u00161M~k0\u00162\n~k\u00163\u000e\u0000\n\u0001E~k0\u00162~k\u00163\u0001h\u0000\n1 +Nph\nj~k0\u0000~kj\u0001\n\u001a\u00161\u00162\n~k0\u0000\n\u000e\u00163\u00160\u0000\u001a\u00163\u00160\n~k\u0001\n\u0000Nph\nj~k0\u0000~kj\u001a\u00163\u00160\n~k\u0000\n\u000e\u00161\u00162\u0000\u001a\u00161\u00162\n~k0\u0001i\n\u0000\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k\u0016\n~k0\u00161M~k0\u00162\n~k\u00163\u000e\u0000\n\u0001E~k\u00163~k0\u00162\u0001h\u0000\n1 +Nph\nj~k\u0000~k0j\u0001\n\u001a\u00163\u00160\n~k\u0000\n\u000e\u00161\u00162\u0000\u001a\u00161\u00162\n~k0\u0001\n\u0000Nph\nj~k\u0000~k0j\u001a\u00161\u00162\n~k0\u0000\n\u000e\u00163\u00160\u0000\u001a\u00163\u00160\n~k\u0001i\n+\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k0\u00161\n~k\u00160M~k\u00163\n~k0\u00162\u000e\u0000\n\u0001E~k0\u00162~k\u00163\u0001h\u0000\n1 +Nph\nj~k0\u0000~kj\u0001\n\u001a\u00162\u00161\n~k0\u0000\n\u000e\u0016\u00163\u0000\u001a\u0016\u00163\n~k\u0001\n\u0000Nph\nj~k0\u0000~kj\u001a\u0016\u00163\n~k\u0000\n\u000e\u00162\u00161\u0000\u001a\u00162\u00161\n~k0\u0001i\n\u0000\u0019\n~X\n~k0X\n\u00161\u00162\u00163M~k0\u00161\n~k\u00160Mk\u00163\n~k0\u00162\u000e\u0000\n\u0001E~k\u00163~k0\u00162\u0001h\u0000\n1 +Nph\nj~k\u0000~k0j\u0001\n\u001a\u0016\u00163\n~k\u0000\n\u000e\u00162\u00161\u0000\u001a\u00162\u00161\n~k0\u0001\n\u0000Nph\nj~k\u0000~k0j\u001a\u00162\u00161\n~k0\u0000\n\u000e\u0016\u00163\u0000\u001a\u0016\u00163\n~k\u0001i\n(38)\nHere, \u0001E~k\u0016~k0\u00160=\u000f\u0016\nk\u0000\u000f\u00160\nk0\u0000~!ph\nj~k\u0000~k0j, andNph\nqis the\noccupation function of a thermalized phonon bath, given\nby a Bose-Einstein distribution\nNph\nq=1\ne\f~!ph\u00001(39)\nwhere\f= 1=(kBTph). The numerical results for the mi-\ncroscopic dynamics in the following sections are obtained\nby numerically solving the equations of motion (29) and\n(38). In the numerical calculations the spin-density ma-\ntrix is transformed to the single-particle basis of the in-\nstantaneous, diagonalized eigenstates (22).\nV. DEPHASING BETWEEN LOCALIZED AND\nITINERANT SPINS: NUMERICAL RESULTS\nSince we are interested in this paper in a comparison\nof the model described above with Landau-Lifshitz and\nGilbert damping, we investigate magnetization dynam-\nics with an initial spin-density matrix that correspondsto a tilting of the spins out of their equilibrium posi-\ntion without changing the kinetic energy of the carriers,\nbecause we need initial conditions that lead to generic\nmagnetization dephasing without carrier heating and the\ncorresponding demagnetization dynamics.\nA. Initial state and spin dynamics\nThus we take as the equilibrium initial state the\nsteady-state that is reached for the coupled spins inter-\nacting with the phonon bath at low temperature ( Tph=\n1 K), as shown in Fig. 4(a). In this equilibrium state\nthe spin density matrix is characterized by shifted Fermi\nfunctions for the distributions \u001a\u0016\u0016\n~k=f(\u000f\u0016\nk\u0000EF;Tph)\nand vanishing coherences \u001a+\u0000\n~k= 0. In particular, the\nsteady-state calculation determines the equilibrium mag-\nnetic gap.\nWe then change the itinerant density matrix to that\ncorresponding to an itinerant spin tilted by \f= 10\u000e\nout of equilibrium, see Fig. 4(b). This initial condition8\nFIG. 4. (a) Localized spin h~Siand itinerant spin h~ siin ther-\nmal equilibrium. (b) Itinerant spins tilted out of equilibrium\nby an angle \f.\n0 1 2 300.51\nk(nm−1)Occupations  \n0 1 2 3−0.500.5\nk(nm−1)Coherences\n  ρ|k|− −\nρ|k|+ +\nℜ(ρ|k|− +)\nℑ(ρ|k|− +)\nFIG. 5. Initial spin density-matrix (occupations and coher-\nences) for itinerant electrons corresponding to a tilt \f= 50\u000e.\nThe deviation from equilibrium occurs only between the Fermi\nwave-vectors of the \\+\" and \\ \u0000\" bands.\nachieves a tilting of the spins without heating and avoids\ngeneric de- and remagnetization dynamics. Microscop-\nically the tilted spin corresponds to the spin density-\nmatrix shown in Fig. 5. The perturbation for distribu-\ntions and coherences exists only between the two Fermi\nwave vectors for the \u0016= + and\u0016=\u0000bands. For smaller\ntilt angles the deviation is much less pronounced.\nFrom this initial condition, both spins start to precess\naround the instantaneous direction, along which the ex-\nchange interaction tries to align them. This direction\nis determined for the itinerant carriers by the localized\nspins, and vice versa. A return back into equilibrium re-\nquires the scattering of itinerant electrons with phonons.\nIf we switch o\u000b spin mixing, the dynamics shown in\nFig. 6(a) result: No angular momentum is exchanged\nwith the phonon bath, the excited system cannot relax\ninto equilibrium and the precession goes on inde\fnitely.\nFig. 6(b) shows the same result including spin-mixed itin-\nerant states. Now angular momentum can be transferred\nfrom the itinerant sub-system into the lattice and the to-\ntal spinh~Si+h~ sichanges. In the presence of spin-orbit\ncoupling, electron-phonon scattering, which is by itself\nspin-diagonal, can return the spin system into equilib-\nrium, characterized by aligned spins and vanishing trans-\nverse components. Since we consider here a small Rashba\n(a) (b) \n5&:P; 5&:P; \nO&:P; O&:P; FIG. 6. Non-equilibrium dynamics of localized (red) and itin-\nerant (blue) spins including electron-phonon scattering with-\nout spin-mixing (a) and within spin-mixing (b).\n0 1 2 30.120.130.14\ntime (ps)|s|0 1 2 3−0.0400.04\ntime (ps)sx\n0 1 2 3−0.0400.04\ntime (ps)sy\n0 1 2 3−0.14−0.13−0.12\ntime (ps)sz\nFIG. 7. Relaxation dynamics of itinerant spins in the \fxed\nframe.\ncoupling, the interaction with the phonon bath removes\nenergy much faster than angular momentum. The car-\nrier temperature therefore stays practically equal to the\nphonon temperature Tphduring the entire relaxation pro-\ncess, and no heat-induced demagnetization processes oc-\ncur. The \fnal magnetization is, however, not necessarily\noriented in the zdirection of the \fxed frame, because\nin the results discussed in the this section there is no\nexternal \feld or anisotropy to induce such an alignment.\nWe plot the resulting dynamics of the itinerant spins\nduring the dephasing process in Fig. 7, which shows that9\n00.511.522.53−0.133−0.132−0.131\ntime(ps)s/bardbl\n00.511.522.5300.010.020.03\ntime(ps)s⊥\nFIG. 8. Relaxation dynamics of itinerant spins in the rotating\nframe. An exponential \ft determines the Bloch decay times\ntoT1= 0:5 ps andT2= 1:0 ps.\nall itinerant spatial components precess, and no spatially\n\fxed component can be considered to be longitudinal.\nFirst, the localized spin turns away from the zdirec-\ntion due to the tilted itinerant spin and subsequently\nboth spin-systems precess around each other because the\nquantization axis of each system changes continuously\ndue to the mutual interaction. During the entire relax-\nation process the absolute value of the itinerant spin is\nconserved to better than 1%. Fig. 8 shows the same\ndynamics in the rotating frame, where the longitudinal\nand transverse dynamics can be seen clearly. Both itin-\nerant components show exponential dynamics, which are\ntherefore well described by decay times T1andT2. If well-\nde\fned decay times exist, one expects a ratio T2=T1= 2\nas long as the length of the spin is conserved. The \ft for\nour numerical results indeed gives 2 T1'T2.\nFigure 9 plots the T1andT2values extracted from the\ndynamics as a function of the strength of the electron-\nphonon coupling, or deformation potential, D. The de-\npendence of the decay times on Dcan be \ft extremely\nwell by a 1=D2relation, which demonstrates the propor-\ntionality of the Bloch decay times T1;2/1=D2. Further,\nthe ratioT2=T1stays equal to 2 for all coupling strengths.\nAs discussed in Sec. IV D about the Markov approxima-\ntion, our microscopic description cannot reach regimes\nwhereT1andT2are indistinguishable and therefore\nequal. However, these results show that for small tilting\nangles, not even a pronounced electron-phonon coupling\nleads to a noticeable deviation from the T2= 2T1be-\nhavior. Because this relation between T1andT2holds,\nthe dynamics in Fig. 8 can be equally well described by\nan Landau-Lifshitz or Gilbert damping term. By \ftting\nthe dephasing time T2\u00191 ps and the Larmor-frequency\n!L\u0019281 ps\u00001, equation (13) yields the corresponding\nGilbert damping parameter \u000biso\u00193:6\u000110\u00003.\n204060801001201401.822.2\nD(meV√nm)T2/T1\n  2040608010012014010−2100102\nD(meV√nm)T1,2(ps)\n  FIG. 9. Top: Longitudinal (black squares) and the transverse\n(blue circles) Bloch decay times vs. electron-phonon coupling\nstrengthD. Two \ft curves /1=D2show thatT1;2/1=D2.\nBottom:The ratio of both decay times remains almost con-\nstantT2=T1\u00192 over the entire range of coupling.\n369  \n0 0.5 1279279.5280280.5281\n1/T2(ps−1)ω(ps−1)\n  \nLL\nGilbert\nMicroscopic\nFIG. 10. Precession frequency with respect to the damping\nstrength in terms of the decay rate 1 =T2.\nB. Precession-frequency shift due to dephasing\nIn the Landau-Lifshitz equation the contributions de-\nscribing, respectively, the precession and the damping are\ncompletely independent, so that the damping constant \u0015\nhas no impact on the precession. By contrast, an increase\nof\u000bin the Gilbert equation does not only increase the\ndephasing rate, it lowers the precession frequency as well.\nIn this section we investigate the change of the preces-\nsion frequency in the microscopic calculation and com-\npare it with the macroscopic descriptions. To this end,\nwe use the o\u000b-diagonal components of the itinerant den-\nsity matrix \u001a+\u0000(t) =P\n~k\u001a+\u0000\n~k(t), which describe the dy-\nnamics of the transverse spin components in the rotating\nframe. The modulus of its Fourier transform j\u001a+\u0000(!)j\nshows a distinct peak, which is exactly at the precession\nfrequency.\nTo compare the precession frequency for di\u000berent\ndamping parameters, we use the dependence on T2, be-\ncause all dephasing parameters can be related to T2for10\nsmall excitations. Fig. 10 plots the Larmor frequency vs.\nthe transverse relaxation rate 1 =T2. The precession pa-\nrameters of the Landau-Lifshitz and the Gilbert equation\nare chosen such that the Larmor frequency in the un-\ndamped limiting case is equal to that of the microscopic\nsimulation. In order to stay within the bounds set by con-\ndition (36), we do not extend the plot in Fig. 10 to higher\ndephasing rates. Fig. 10 shows that the microscopic cal-\nculation yields a reduction of the precession frequency\nwith the damping rate. Although the Gilbert dynamics\nalso show such a reduction, it occurs only at shorter T2.\nAs mentioned above, for the Landau-Lifshitz damping,\nthe frequency is independent of the damping parame-\nter. Even though the change of precession frequency in\nthe microscopic calculation is small, the Landau-Lifshitz\ndamping completely fails to include this e\u000bect. While\nGilbert damping does show a reduction of precession fre-\nquency, it is not at all close to the microscopic calcula-\ntion on the frequency scale considered here. Both phe-\nnomenological damping expressions thus do not repro-\nduce the dependence of the precession frequency on T2.\nEven though the numerical di\u000berences are small, these\ndi\u000berences already occur in the small-excitation regime,\nand may perhaps be detectable.\nC. Dephasing at larger excitation angles\nThe phenomenological Landau-Lifshitz and Gilbert\ndamping contributions describe an exponential decay\nonly for small excitation angles, as studied in the pre-\nvious section. In this section we investigate the e\u000bect of\nlarger excitation angles ( >10\u000e) on the spin dynamics in\nthe microscopic calculation. Apart from this the initial\ncondition of the dynamics is the same as before, in par-\nticular, the itinerant spin is tilted such that the absolute\nvalue of the spin is unchanged.\nFigure 11 shows the time development of the skand\ns?components of the itinerant spin in the rotating frame\nfor an initial tilt angle \f= 140\u000e. While the transverse\ncomponent s?in the rotating frame can be well described\nby an exponential decay, the longitudinal component sk\nshows a di\u000berent behavior. It initially decreases with a\ntime constant of less than 1 ps, but does not reach its\nequilibrium value. Instead, the eventual return to equi-\nlibrium takes place on a much longer timescale, during\nwhich the s?component is already vanishingly small.\nThe long-time dynamics are therefore purely collinear.\nFor the short-time dynamics, the transverse component\ncan be \ft well by an exponential decay, even for large ex-\ncitation angles. This behavior is di\u000berent from Landau-\nLifshitz and Gilbert dynamics, cf. Fig. 2, which both ex-\nhibit non-exponential decay of the transverse spin com-\nponent.\nIn Fig. 12 the dependence of T2on the excitation an-\ngle is shown. From small \fup to almost 180\u000e, the decay\ntime decreases by more than 50%. This dependence is\nexclusively due to the \\excitation condition,\" which in-\n0 1 2 3 4−0.100.1\ntime (ps)s/bardbl\n0 1 2 3 400.050.1\ntime (ps)s⊥FIG. 11. Dynamics of the longitudinal and transverse itiner-\nant spin components in the rotating frame (solid lines) for a\ntilt angle of \f= 140\u000e, together with exponential \fts toward\nequilibrium (dashed lines). The longitudinal equilibrium po-\nlarization is shown as a dotted line.\nvolves only spin degrees of freedom (\\tilt angle\"), but no\nchange of temperature. Although one can \ft such a T2\ntime to the transverse decay, the overall behavior with\nits two stages is, in our view, qualitatively di\u000berent from\nthe typical Bloch relaxation/dephasing picture.\nTo highlight the similarities and di\u000berences from the\nBloch relaxation/dephasing we plot in Fig. 13 the mod-\nulus of the itinerant spin vector j~ sjin the rotating\nframe, whose transverse and longitudinal components\nwere shown in Fig. 11. Over the 2 ps, during which the\ntransverse spin in the rotating frame essentially decays,\nthe modulus of the spin vector undergoes a fast initial\ndecrease and a partial recovery. The initial length of ~ s\nis recovered only over a much larger time scale of several\nhundred picoseconds (not shown). Thus the dynamics\ncan be seen to di\u000ber from a Landau-Lifshitz or Gilbert-\nlike scenario because the spin does not precess toward\nequilibrium with a constant length. Additionally they\ndi\u000ber from Bloch-like dynamics because there is a com-\nbination of the fast and slow dynamics that cannot be\ndescribed by a single set of T1andT2times. We stress\nthat the microscopic dynamics at larger excitation angles\nshow a precessional motion of the magnetization with-\nout heating and a slow remagnetization. This scenario is\nsomewhat in between typical small angle-relaxation, for\nwhich the modulus of the magnetization is constant and\nwhich is well described by Gilbert and Landau-Lifshitz\ndamping, and collinear de/remagnetization dynamics.\nVI. EFFECT OF ANISOTROPY\nSo far we have been concerned with the question\nhow phenomenological equations describe dephasing pro-\ncesses between itinerant and localized spins, where the11\n0 50 100 1500.40.60.81\nβ(◦)T2(ps)\nFIG. 12.T2time extracted from exponential \ft to s?dynam-\nics in rotating frame for di\u000berent initial tilting angles \f.\n0 0.5 1 1.5 20.040.060.080.10.120.14\ntime (ps)|s|\n  \n10°\n50°\n90°\n140°\nFIG. 13. Dynamics of the modulus j~ sjof the itinerant spin\nfor di\u000berent initial tilt angles \f. Note the slightly di\u000berent\ntime scale compared to Fig. 11.\nmagnetic properties of the system were determined by a\nmean-\feld exchange interaction only. Oftentimes, phe-\nnomenological models of spin dynamics are used to de-\nscribe dephasing processes toward an \\easy axis\" deter-\nmined by anisotropy \felds.29\nIn order to capture in a simple fashion the e\u000bects of\nanisotropy on the spin dynamics in our model, we sim-\nply assume the existence of an e\u000bective anisotropy \feld\n~Haniso, which enters the Hamiltonian via\n^Haniso =\u0000g\u0016B\u0016^~ s\u0001~Haniso (40)\nand only acts on the itinerant carriers. Its strength is\nassumed to be small in comparison to the \feld of the\nlocalized moments ~Hloc. This additional \feld ~Haniso has\nto be taken into account in the diagonalization of the\ncoherent dynamics as well, see section IV C.\nFor the investigation of the dynamics with anisotropy,\nwe choose a slightly di\u000berent initial condition, which is\nshown in Fig. 14. In thermal equilibrium, both spins\nare now aligned, with opposite directions, along the\nanisotropy \feld ~Haniso, which is assumed to point in the\nzdirection. At t= 0 they are both rigidly tilted by an\n5&:P; \nO&:P; U T \nV E *_lgqm FIG. 14. Dynamics of the localized spin ~Sand itinerant spin\n~ s. Att= 0, the equilibrium con\fguration of both spins is\ntilted (\f= 10\u000e) with respect to an anisotropy \feld ~Haniso.\nThe anisotropy \feld is only experienced by the itinerant sub-\nsystem.\n01002003004005006000.490.4950.5\ntime(ps)Sz(t)\n010020030040050060000.050.1\ntime(ps)/radicalBig\nS2x(t)+S2y(t)\n  \nFIG. 15. Relaxation dynamics of the localized spin toward the\nanisotropy direction for longitudinal component Szand the\ntransverse componentp\nS2x+S2y. An exponential \ft yields\nBloch decay times of Taniso\n1 = 67:8 ps andTaniso\n2 = 134:0 ps.\nangle\f= 10\u000ewith respect to the anisotropy \feld.\nFigure 14 shows the time evolution of both spins in the\n\fxed frame, with zaxis in the direction of the anisotropy\n\feld for the same material parameters as in the previous\nsections and an anisotropy \feld ~Haniso =\u0000108A\nm\u0001~ ez.\nThe dynamics of the entire spin-system are somewhat\ndi\u000berent now, as the itinerant spin precesses around the\ncombined \feld of the anisotropy and the localized mo-\nments. The localized spin precesses around the itinerant\nspin, whose direction keeps changing as well.\nFigure 15 contains the dynamics of the components\nof the localized spin in the rotating frame. Both com-\nponents show an exponential behavior that allows us to\nextract well de\fned Bloch-times Taniso\n1 andTaniso\n2. Again\nwe \fnd the ratio of 2 Taniso\n1\u0019Taniso\n2, because the abso-\nlute value of the localized spin does not change, as it is\nnot coupled to the phonon bath.\nIn Fig. 16 the Larmor-frequency !aniso\nL, which is the\nprecession frequency due to the anisotropy \feld, and the\nBloch decay times Taniso\n2 are plotted vs. the strength of\nthe anisotropy \feld ~Haniso. The Gilbert damping pa-12\n0 5 10 150510\nHaniso(107A/m)ωaniso\nL (ps−1)\n0 5 10 1505001000\nHaniso(107A/m)Taniso\n2 (ps)\n0 5 10 1501020\nHaniso(107A/m)αaniso (10−4)\nFIG. 16. Larmor frequency !aniso\nL and Bloch decay time Taniso\n2\nextracted from the spin dynamics vs. anisotropy \feld Haniso,\nas well as the corresponding damping parameter \u000baniso.\nrameter\u000baniso for the dephasing dynamics computed via\nEq. (13) is also presented in this \fgure.\nThe plot reveals a decrease of the dephasing time Taniso\n2\nand a almost linear increase of the Larmor frequency\n!aniso\nL with the strength of the anisotropy \feld Haniso.\nThe Gilbert damping parameter \u000baniso shows only a neg-\nligible dependence on the anisotropy \feld Haniso. This\ncon\frms the statement that, in contrast to the dephas-\ning rates, the Gilbert damping parameter is independent\nof the applied magnetic \feld. In the investigated range\nwe \fnd an almost constant value of \u000baniso'9\u000210\u00004.\nThe Gilbert damping parameter \u000baniso for the de-\nphasing toward the anisotropy \feld is about 4 times\nsmaller than \u000biso, which describes the dephasing between\nboth spins. This disparity in the damping e\u000eciency\n(\u000baniso< \u000b iso) is obviously due to a fundamental di\u000ber-\nence in the dephasing mechanism. In the anisotropy case\nthe localized spin dephases toward the zdirection with-\nout being involved in scattering processes with itinerant\ncarriers or phonons. The dynamics of the localized spins\nis purely precessional due to the time-dependent mag-\nnetic moment of the itinerant carriers ~Hitin(t). Thus,\nonly this varying magnetic \feld, that turns out to be\nslightly tilted against the localized spins during the en-\ntire relaxation causes the dephasing, in presence of the\ncoupling between itinerant carriers and a phonon bath,\nwhich acts as a sink for energy and angular momentum.\nThe relaxation of the localized moments thus occurs only\nindirectly as a carrier-meditated relaxation via their cou-\npling to the time dependent mean-\feld of the itinerant\nspin.\nNext, we investigate the dependence of the Gilbert pa-\nrameter\u000baniso on the bath coupling. Fig. 17 shows that\n0 50 100 150 20000.0040.0080.012\nD(meV√nm)αaniso\n  FIG. 17. Damping parameter \u000baniso vs. coupling constant D\n(black diamonds). The red line is a quadratic \ft, indicative\nof\u000baniso/D2.\n\u000baniso increases quadratically with the electron-phonon\ncoupling strength D.\nSince Fig. 9 establishes that the spin-dephasing rate\n1=T2for the fast dynamics discussed in the previous sec-\ntions, is proportional to D2, we \fnd\u000baniso/1=T2. We\nbrie\ry compare these trends to two earlier calculations\nof Gilbert damping that employ p-dmodels and assume\nphenomenological Bloch-type rates 1 =T2for the dephas-\ning of the itinerant hole spins toward the \feld of the\nlocalized moments. In contrast to the present paper, the\nlocalized spins experience the anisotropy \felds. Chovan\nand Perakis38derive a Gilbert equation for the dephasing\nof the localized spins toward the anisotropy axis, assum-\ning that the hole spin follows the \feld ~Hlocof the localized\nspins almost adiabatically. Tserkovnyak et al.39extract\na Gilbert parameter from spin susceptibilities. The re-\nsulting dependence of the Gilbert parameter \u000baniso on\n1=T2in both approaches is in qualitative accordance and\nexhibits two di\u000berent regimes. In the the low spin-\rip\nregime, where 1 =T2is small in comparison to the p-dex-\nchange interaction a linear increase of \u000baniso with 1=T2\nis found, as is the case in our calculations with micro-\nscopic dephasing terms. If the relaxation rate is larger\nthan thep-ddynamics,\u000baniso decreases again. Due to\nthe restriction (36) of the Boltzmann scattering integral\nto low spin-\rip rates, the present Markovian calculations\ncannot be pushed into this regime.\nEven though the anisotropy \feld ~Haniso is not cou-\npled to the localized spin ~Sdirectly, both spins precess\naround the zdirection with frequency !aniso\nL. In analogy\nto Sec. V B we study now the in\ruence of the damping\nprocess on the precession of the localized spin around\nthe anisotropy axis and compare it to the behavior of\nLandau-Lifshitz and Gilbert dynamics. Fig. 18 reveals a\nsimilar behavior of the precession frequency as a function\nof the damping rate 1 =Taniso\n2 as in the isotropic case. The\nmicroscopic calculation predicts a distinct drop of the\nLarmor frequency !aniso\nL for a range of dephasing rates\nwhere the precession frequency is unchanged according\nto the Gilbert and Landau-Lifshitz damping models. Al-\nthough Gilbert damping eventually leads to a change in\nprecession frequency for larger damping, this result shows\na qualitative di\u000berence between the microscopic and the13\n0 0.02 0.04 0.06 0.087.127.167.27.24\n1/Taniso\n2(ps−1)ωaniso(ps−1)\n  \nGilbert\nLL\nMicroscopic\nFIG. 18. Precession frequency of the localized spin around\nthe anisotropy \feld vs. Bloch decay time 1 =Taniso\n2.\nphenomenological calculations.\nVII. CONCLUSION AND OUTLOOK\nIn this paper, we investigated a microscopic descrip-\ntion of dephasing processes due to spin-orbit coupling\nand electron-phonon scattering in a mean-\feld kinetic\nexchange model. We \frst analyzed how spin-dependent\ncarrier dynamics can be described by Boltzmann scat-\ntering integrals, which leads to Elliott-Yafet type relax-\nation processes. This is only possible for dephasing rates\nsmall compared to the Larmor frequency, see Eq. (36).\nThe microscopic calculation always yielded Bloch times\n2T1=T2for low excitation angles as it should be due\nto the conservation of the absolute value of the mag-\nnetization. A small decrease of the e\u000bective precession\nfrequency occurs with increasing damping rate, which is\na fundamental di\u000berence to the Landau-Lifshitz descrip-\ntion and exceeds the change predicted by the Gilbert\nequation in this regime.We modeled two dephasing scenarios. First, a relax-\nation process between both spin sub systems was studied.\nHere, the di\u000berent spins precess around the mean-\feld of\nthe other system. In particular, for large excitation an-\ngles we found a decrease of the magnetization during the\nprecessional motion without heating and a slow remag-\nnetization. This scenario is somewhat in between typi-\ncal small angle-relaxation, for which the modulus of the\nmagnetization is constant and which is well described\nby Gilbert and Landau-Lifshitz damping, and collinear\nde/remagnetization dynamics. Also, we \fnd important\ndeviations from a pure Bloch-like behavior.\nThe second scenario deals with the relaxation of the\nmagnetization toward a magnetic anisotropy \feld expe-\nrienced by the itinerant carrier spins for small excitation\nangles. The resulting Gilbert parameter \u000baniso is inde-\npendent of the static anisotropy \feld. The relaxation of\nthe localized moments occurs only indirectly as a carrier-\nmeditated relaxation via their coupling to the time de-\npendent mean-\feld of the itinerant spin.\nTo draw a meaningful comparison with Landau-\nLifshitz and Gilbert dynamics we restricted ourselves\nthroughout the entire paper to a regime where the elec-\ntronic temperature is equal to the lattice temperature Tph\nat all times. In general our microscopic theory is also ca-\npable of modeling heat induced de- and remagnetization\nprocesses. We intend to compare microscopic simulations\nof hot electron dynamics in this model, including scat-\ntering processes between both types of spin, with phe-\nnomenological approaches such as the Landau-Lifshitz-\nBloch (LLB) equation or the self-consistent Bloch equa-\ntion (SCB)40.\nWe \fnally mention that we derived relation (13) con-\nnecting the Bloch dephasing time T2and the Gilbert\ndamping parameter \u000b. 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Phys. 113, 163911 (2013)."    },    {        "title": "1508.07290v2.Control_of_magnetic_relaxation_by_electric_field_induced_ferroelectric_phase_transition_and_inhomogeneous_domain_switching.pdf",        "content": "Control of magnetic relaxation by electric-\feld-induced ferroelectric phase transition\nand inhomogeneous domain switching\nTianxiang Nan ,1Satoru Emori ,1Bin Peng ,2Xinjun Wang ,1Zhongqiang Hu ,1Li Xie ,1\nYuan Gao,1Hwaider Lin,1Jie Jiao,3Haosu Luo,3David Budil,4John G. Jones,5Brandon\nM. Howe,5Gail J. Brown,5Ming Liu,2,a)and Nian Sun1,b)\n1)Department of Electrical and Computer Engineering, Northeastern University,\nBoston, Massachusetts 02115,USA\n2)Electronic Materials Research Laboratory, Xi'an Jiaotong University,\nXi'an 710049, China\n3)Shanghai Institute of Ceramics, Chinese Academy of Sciences, Shanghai 201800,\nChina\n4)Department of Chemistry, Northeastern University, Boston, Massachusetts 02115,\nUSA\n5)Materials and Manufacturing Directorate, Air Force Research Laboratory,\nWright-Patterson AFB, Ohio 45433, USA\n(Dated: June 24, 2021)\nElectric-\feld modulation of magnetism in strain-mediated multiferroic heterostruc-\ntures is considered a promising scheme for enabling memory and magnetic microwave\ndevices with ultralow power consumption. However, it is not well understood how\nelectric-\feld-induced strain in\ruences magnetic relaxation, an important physical\nprocess for device applications. Here we investigate resonant magnetization dynam-\nics in ferromagnet/ferrolectric multiferroic heterostructures, FeGaB/PMN-PT and\nNiFe/PMN-PT, in two distinct strain states provided by electric-\feld-induced ferro-\nelectric phase transition. The strain not only modi\fes magnetic anisotropy but also\nmagnetic relaxation. In FeGaB/PMN-PT, we observe a nearly two-fold change in\nintrinsic Gilbert damping by electric \feld, which is attributed to strain-induced tun-\ning of spin-orbit coupling. By contrast, a small but measurable change in extrinsic\nlinewidth broadening is attributed to inhomogeneous ferroelastic domain switching\nduring the phase transition of the PMN-PT substrate.\na)Electronic mail: mingliu@mail.xjtu.edu.cn\nb)Electronic mail: n.sun@neu.edu\n1arXiv:1508.07290v2  [cond-mat.mtrl-sci]  8 Sep 2015Electrical manipulation of the magnetization state is essential for improving the scala-\nbility and power e\u000eciency of magnetic random access memory (MRAM)1{5. A particularly\npromising scheme relies on an electric \feld to assist or induce magnetization switching with\nminimal power dissipation4,6,7. Multiferroic magnetoelectric materials with coupled mag-\nnetization and electric polarization o\u000ber possibilities for electric-\feld-driven magnetization\nswitching at room temperature8{14. Such magnetoelectric e\u000bects have been demonstrated\nwith strain-15{19, charge-20{23and exchange bias mediated coupling mechanisms24{27. For\nexample, non-volatile magnetization switching with remarkable modulation of magnetic\nanisotropy was realized using electric-\feld-induced piezo-strain at the interface between\nferromagnetic and ferroelectric phases28{31.\nOn the other hand, a better understanding of the processes responsible for magnetic\nrelaxation, especially at various strain states, is required for electric-\feld-assisted MRAM\nor tunable magnetic microwave devices. Recent studies suggest that electric-\feld-induced\nchanges of magnetic relaxation are correlated to the piezo-strain state or e\u000bective magnetic\nanisotropy32{36. A similar modulation of magnetic relaxation has also been observed in a\ncharge-mediated magnetoelectric heterostructure with ultra-thin ferromagnets37. In general,\nthe contributions to magnetic relaxation include intrinsic Gilbert damping due to spin-orbit\ncoupling and extrinsic linewidth broadening due to inhomogeneity in the ferromagnet. So far,\nthe understanding of how a piezo-strain modi\fes these intrinsic and extrinsic contributions\nhas been limited33.\nIn this work, we quantify electric-\feld-induced modi\fcations of both intrinsic Gilbert\ndamping and inhomogeneous linewidth broadening in two ferromagnet/ferroelectric hetere-\nostructures: Fe 7Ga2B1/Pb(Mg 1=3Nb2=3)O3-PbTiO 3(FeGaB/PMN-PT) with a strong strain-\nmediated magnetoelectric (magnetostrictive) coupling and Ni 80Fe20/Pb(Mg 1=3Nb2=3)O3-\nPbTiO 3with a negligible magnetoelectric coupling. The rhombohedral (011) oriented\nPMN-PT substrate provides two distinct strain states through an electric-\feld-induced\nphase transformation38,39. We conduct ferromagnetic resonance (FMR) measurements at\nseveral applied electric \feld values to disentangle the intrinsic and extrinsic contributions\nto magnetic relaxation. FeGaB/PMN-PT exhibits pronounced electric-\feld-induced mod-\ni\fcations of the resonance \feld and intrinsic Gilbert damping, whereas these parameters\nremain mostly unchanged for NiFe/PMN-PT. These \fndings show that magnetic relaxation\ncan be tuned through a strain-mediated modi\fcation of spin-orbit coupling in a highly mag-\n2netostrictive ferromagnet. We also observe in both multiferroic hetereostructures a small\nelectric-\feld-induced change in extrinsic linewidth broadening, which we attribute to the\nferroelectric domain state in the PMN-PT substrate.\n30-nm thick \flms of FeGaB and NiFe were sputter-deposited on (011) oriented PMN-PT\nsingle crystal substrates bu\u000bered with 5-nm thick Ta seed layers. The FeGaB thin \flm was\nco-sputtered from Fe 80Ga20(DC sputtered) and B (RF sputtered) targets. Both FeGaB and\nNiFe \flms were capped with 2 nm of Al to prevent oxidation. All \flms were deposited in 3\nmTorr Ar atmosphere with a base pressure \u00141\u000210\u00007Torr. The thicknesses of deposited\n\flms were calibrated by X-ray re\rectivity.\nThe amorphous FeGaB thin \flm was selected for its high saturation magnetostriction\ncoe\u000ecient of up to 70 ppm40and large magnetoelectric e\u000bect when interfaced with ferro-\nelectric materials19. NiFe was chosen as the control sample with near zero magnetostriction;\nthe thickness of 30 nm is far above the thickness regime that shows high surface magne-\ntostricion41. Fig. 1 shows magnetic hysteresis loops of FeGaB/PMN-PT and NiFe/PMN-PT,\nmeasured by vibrating sample magnetometry with an in-plane magnetic \feld applied along\nthe [100] direction of PMN-PT. An electric \feld was applied in the thickness direction of\nthe PMN-PT substrate. Due to the anisotropic piezeoelectric coe\u000ecient of PMN-PT, an\nin-plane compressive strain is induced along the [100] direction, which results in uniaxial\nmagnetic anisotropy along the same axis. In FeGaB/PMN-PT the electric \feld ( E= 8\nkV/cm) increases the saturation \feld by \u001940 mT, whereas only a small change is observed\nin NiFe/PMN-PT, con\frming the signi\fcantly di\u000berent strengths of strain-mediated mag-\nnetoelectric coupling for the two multiferroic heterostructures.\nBoth ferromagnetic thin \flms exhibit comparatively narrow resonant linewidths, allowing\nfor sensitive detection of the electric-\feld modi\fcation of spin relaxation. Electric-\feld\ndependent FMR spectra of FeGaB/PMN-PT and NiFe/PMN-PT were measured using a\nBruker EMX electron paramagnetic resonance (EPR) spectrometer with a TE 102cavity\noperated at a microwave frequency of 9.5 GHz. The external magnetic \feld was applied\nalong the [100] direction of the PMN-PT single crystal. These spectra, shown in Fig. 2(a),\n(b) were \ftted to the derivative of a modi\fed Lorentzian function42to extract the resonance\n\feldHFMR and resonance linewidth W. In FeGaB/PMN-PT, upon applying E= 2 kV/cm\nalong the thickness direction of PMN-PT, a slight increase of HFMR by 10 mT is observed.\nA larger shift of 35 mT in HFMR is induced at E= 8 kV/cm. In comparison, NiFe/PMN-PT\n3exhibits a much smaller HFMR shift of 1.5 mT at E= 8 kV/cm, as shown in Fig. 2(b).\nThe shift of HFMR in FeGaB/PMN-PT and NiFe/PMN-PT as a function of E is summa-\nrized in Fig. 2(c) and (d). Both samples show hysteric behavior that follows the piezo-strain\ncurve of PMN-PT (inset of Fig. 2(c)) measured with a photonic sensor. This can be under-\nstood by the strain-mediated magnetoelectric coupling with the electric-\feld-induced change\nof magnetic anisotropy \feld (\u0001H k) expressed by\n\u0001Hk=3\u0015(\u001b100\u0000\u001b0\u000011)\n\u00160Ms(1)\nwhere\u001b100and\u001b0\u000011are the in-plane piezo-stress, \u0015andMsare the magnetostriction con-\nstant and the saturation magnetization respectively. Considering an in-plane compressive\nstrain along the [100] direction and a positive magnetostriction coe\u000ecient of both FeGaB\nand NiFe, a decrease of the magnetic anisotropy \feld Hkis expected with a positive electric\n\feld. The drop of Hkresults in an increase of HFMR described by the Kittel equation,\nf=2\u0019=\r\u00160q\n(HFMR +Hk)(HFMR +Hk+Meff) (2)\nwhere\r=2\u0019=28 GHz/T and Meffis the e\u000bective magnetization. At E < 4 kV/cm,HFMR\nincreases linearly, which corresponds to the linear region of piezoelectric e\u000bect of PMN-\nPT with a uniaxial compressive piezo-strain along [100] direction. The sudden change of\nHFMR atE= 4 kV/cm is attributed to the rhombohedral-to-orthorhombic (R-O) phase\ntransition of PMN-PT substrate39. The PMN-PT substrate reverts to the rhombohedral\nphase upon decreasing the electric \feld. Therefore, the R-O phase transformation with a\nlarge uniaxial in-plane strain induces two stable and reversible magnetic states at E= 0\nand 8 kV/cm. This provides a reliable platform for studying magnetization dynamics in a\ncontrolled manner with the applied electric \feld.\nThe peak-to-peak FMR linewidth Wof FeGaB/PMN-PT and NiFe/PMN-PT, extracted\nfrom the same FMR measurements in Fig. 2, also exhibits a strong dependence on the\napplied electric \feld as shown in Fig. 3. For FeGaB/PMN-PT, Wremains unchanged\nwithin experimental uncertainty at E <4 kV/cm and abruptly increases from \u00194.6 mT\nto\u00195.6 mT across the R-O phase transition. By removing the applied electric \feld, W\ndecreases to the original value with the reversal to the rhombohedral phase. Comparing\nFig. 2(c) and 3(a), it is evident that the observed electric-\feld-induced changes in HFMR\nandWin FeGaB/PMN-PT are correlated, consistent with recent studies34{36. The change\n4inWindicates a modulation in spin-orbit coupling in the ferromagnet; considering that\nspin-orbit coupling governs the intrinsic Gilbert damping, it is reasonable that we observe\nsimultaneous modi\fcation of WandHFMR by strain in the magnetostrictive FeGaB \flm.\nGiven the same sign of the magnetostriction coe\u000ecient for FeGaB and NiFe40,43, one\nwould expect to also observe a small increase in Wwith increasing electric \feld across the\nR-O phase transition in NiFe/PMN-PT. However, NiFe/PMN-PT exhibits a decrease in\nWacross the phase transition. This observation indicates that the piezo-strain modi\fes a\ndi\u000berent magnetic relaxation contribution in NiFe.\nThe FMR linewidth Wconsists of the intrinsic Gilbert damping contribution (parameter-\nized by the damping constant \u000b) and the frequency-independent inhomogeneous linewidth\nbroadening W0:\nW=W0+4\u0019\u000bp\n3\rf (3)\nwherefis the microwave excitation frequency. According to Eq. 3 , \u000bandW0can be\ndetermined simply by measuring the frequency dependence of W. For this purpose, we\nused a home-built broadband FMR system44with a nominal microwave power of -5 dBm\nandf= 6-19 GHz. Just as in the single-frequency measurement using the EPR system\n(Fig. 2 and Fig. 3), the external magnetic \feld was applied along the [100] direction of\nthe PMN-PT substrate. By \ftting the frequency dependence of HFMR to Eq. 1 (Fig. 4(a),\n(b)), we obtain anisotropy \feld shift \u0001 Hk\u001946 mT for FeGaB/PMN-PT and \u0001 Hk\u0019\n1 mT for NiFe/PMN-PT across the R-O phase transition, in agreement with the single-\nfrequency FMR measurement (Fig. 2), while \u00160Meffremains unchanged. Fig. 4(c) and (d)\nplotWas a function of the frequency for FeGaB/PMN-PT and NiFe/PMN-PT, respectively.\nFrom the slope of the linear \ft (Eq. 3), we \fnd that \u000bof FeGaB/PMN-PT increases from\n(0:6\u00060:01)\u000210\u00002atE= 0 to (1:06\u00060:02)\u000210\u00002atE= 8 kV/cm, whereas \u000bis unchanged\nat (1:29\u00060:16)\u000210\u00002for NiFe/PMN-PT within experimental uncertainty( \u000b=(1:27\u00060:2)\u0002\n10\u00002atE= 8 kV/cm). The large change in \u000bfor FeGaB and negligible change for NiFe\nsuggest a strong correlation between magnetostriction and the intrinsic Gilbert damping\nmechanism. In particular, a large in-plane uniaxial strain generated by the R-O phase\ntransformation induces an additional anisotropy \feld in FeGaB that enhances the dephasing\nof the magnetization precession43.\nHowever, both FeGaB/PMN-PT and NiFe/PMN-PT show a decreased W0upon apply-\ningE= 8 kV/cm. This could be related to the ferroelectric domain state in the PMN-PT\n5substrate that signi\fcantly a\u000bects the homogeneity of the magnetic \flm on top. The po-\nlarization domain phase images with various applied voltages are shown in Fig. 5 by using\na piezo-force microscope. For the unpoled state at, as shown in Fig. 5 (a), the polariza-\ntion state of PMN-PT surface is inhomogenous, with polarization vectors oriented randomly\nalong the eight body diagonals of the pseudocubic cell. By applying a voltage of 30 V\nwithin the gated area (dashed outline in Fig. 5 (b),(d)), the ferroelectric state becomes sat-\nurated within this area with all the polarization vectors pointing upward. This uniformly\npolarized state alters the surface topology the PMN-PT substrate31, thereby reducing the\ninhomogeneous linewidth broadening W0of the ferromagnetic \flm.\nWe also measured frequency-dependent FMR spectra with an external magnetic \feld\napplied along the [0 \u001611] direction to examine the anisotropy of magnetic relaxation. For\nFeGaB/PMN-PT, \u000bandW0are close to the [100] con\fguration at E= 0. AtE= 8 kV/cm,\nwe observed a non-linear relation between Wandf, which might have resulted from a highly\nnon-uniform magnetization state at low \felds due to the large electric-\feld-induced Hk45,46.\nTo extract \u000breliably in this case, we would need to conduct FMR measurements at higher\nfrequencies. For NiFe/PMN-PT, \u000band the electric-\feld dependence of W0are identical for\nthe [0 \u001611] and [100] directions. The parameters quanti\fed in this study are summarized in\nTable I.\nIn summary, we have quanti\fed electric-\feld-induced modi\fcations of magnetic anisotropy\nand magnetic relaxation contributions, namely intrinsic Gilbert damping and inhomogeneous\nlinewidth broadening, in multiferroic heterostructures. A large modi\fcation of intrinsic\ndamping arises from strain-induced tuning of spin-orbit coupling in the ferromagnet and is\ncorrelated with the magnitude of magnetostriction. A small change in the extrinsic linewidth\ncontribution is attained by controlling the ferroelectric domain states in the substrate. These\n\fndings are not only of technology importance for the application on low-power MRAM and\nmagnetic microwave devices, but also permit investigation of the structural dependence of\nspin-orbit-derived phenomena in magnetic thin \flms.\n6Table I. Parameters extracted from broadband FMR at 2 di\u000berent electric \felds\nFeGaB/PMN-PT NiFe/PMN-PT\nE(kV/cm) 0 8 0 8\n4\u0019Meff(T) 1:48\u00060:01 1:46\u00060:01 0:96\u00060:04 0:96\u00060:04\nHk(mT)[100] 5:8\u00060:5\u000041:3\u00060:3 1:67\u00060:2 0:27\u00060:2\n\u000b(10\u00002)[100] 0:6\u00060:01 1:06\u00060:02 1:29\u00060:16 1:27\u00060:2\nW0(mT)[100] 2:4\u00060:05 1:8\u00060:07 0:66\u00060:06 0:35\u00060:07\nHk(mT)[0\u001611] 3:24\u00060:4a1:54\u00060:2 3:1\u00060:3\n\u000b(10\u00002)[0\u001611] 0:6\u00060:02 1 :21\u00060:12 1:29\u00060:15\nW0(mT)[0\u001611] 2:8\u00060:05 5 :9\u00060:08 2:9\u00060:05\naNot able to obtain due to the frequency constraint and the \feld-dragging e\u000bect at measured low\nfrequencies.\nThis work was supported by the Air Force Research Laboratory through contract FA8650-\n14-C-5706 and in part by FA8650-14-C-5705, the W.M. 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Rev. B 69, 184417 (2004).\n46K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M. Farle, U. Von H orsten, H. Wende,\nW. Keune, J. Rocker, S. S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, and Z. Frait,\nPhys. Rev. B 76, 104416 (2007).\n10-80 -60 -40 -20 020 40 60 80-1.0-0.500.51.0\nM/MS\nμ0H (mT)0 kV/cm\n8 kV/cm\n-7.5 -5 -2.5 0 2.5 5 7.50 kV/cm\n8 kV/cm(a) (b)\nμ0H (mT)FeGaB/PMN-PT NiFe/PMN-PTFigure 1. (a,b) Electric-\feld dependent magnetic hysteresis loops with the magnetic \feld applied\nalong the [100] direction for FeGaB/PMN-PT (a) and NiFe/PMN-PT (b).\n40 60 80 100 120 140-1.5-1.0-0.500.51.01.5\n 0 kV/cm\n 2 kV/cm\n 8 kV/cmdP/dH (a.u.)\n-0.3-0.2-0.100.10.20.3\n 0 kV/cm\n 8 kV/cm\n60708090100110\n  μ0HFMR(mT)\n0 2 4 6 8\nElectric field (kV/cm)(a) (b)\n(c)40 60 80 100 120 140\nμ0H (mT) μ0H (mT)FeGaB/PMN-PT NiFe/PMN-PT\nFeGaB/PMN-PT\n0 2 4 6 8100100.5101101.5\nElectric field (kV/cm)(d)\nNiFe/PMN-PT\n10 120 2 4 6 8|Strain|(%) \n00.10.2\nFigure 2. (a,b) FMR (\fxed at 9.5 GHz) spectra at various electric \felds with the magnetic \feld\napplied along the [100] direction for FeGaB/PMN-PT (a) and NiFe/PMN-PT (b). (c,d) Resonance\n\feld HFMR as a function of the applied electric \feld for FeGaB/PMN-PT (a) and NiFe/PMN-PT\n(b). Inset of (c) shows the piezo-strain as a function of electric \feld for PMN-PT substrate along\nthe [100] direction.\n114.44.85.25.66\nW (mT)\n4.44.64.85\n0 2 4 6 8\nElectric field (kV/cm)4.2\n0 2 4 6 8\nElectric field (kV/cm)(a) (b)FeGaB/PMN-PT NiFe/PMN-PTFigure 3. (a,b) Resonance linewidth W at 9.5 GHz with the magnetic \feld applied along the [100]\ndirection as a function of the applied electric \feld for FeGaB/PMN-PT (a) and NiFe/PMN-PT\n(b).\n050100150200250300\n  μ0HFMR(mT)\n4 8 12 16 20\nf (GHz) 0 kV/cm\n 8 kV/cm\n 0 kV/cm\n 8 kV/cm(a)\n(b)\n0 5 10 15 200246810\nW (mT)\nf (GHz)0 5 10 15 20\nf (GHz)(c) (d)\n 0 kV/cm\n 8 kV/cm 0 kV/cm\n 8 kV/cmFeGaB/PMN-PT\nNiFe/PMN-PT\nFeGaB/PMN-PT NiFe/PMN-PT4 8 12 16 20\nf (GHz)13 14 12150165180\nFigure 4. (a,b) Frequency f as a function of resonance \feld HFMR at di\u000berent electric \felds for\nFeGaB/PMN-PT (a) and NiFe/PMN-PT (b). (c,d) Linewidth W as a function of frequency f at\ndi\u000berent electric \felds for FeGaB/PMN-PT (c) and NiFe/PMN-PT (d). The magnetic \feld was\napplied along the [100] direction.\n125um 5um5um 5um(a) (b)\n(c) (d)0V 30V\n0V 30VFigure 5. (a,b) The out-of-plane vertical PFM (VPFM) phase images upon applying di\u000berent\nvoltages to the square area outlined by a red dashed line. (c,d) Corresponding amplitude images\nat di\u000berent voltage biases.\n13"    },    {        "title": "1008.0674v1.Determination_of_the_spin_flip_time_in_ferromagnetic_SrRuO3_from_time_resolved_Kerr_measurements.pdf",        "content": "arXiv:1008.0674v1  [cond-mat.mtrl-sci]  3 Aug 2010Determinationofthe spin-flip timeinferromagnetic SrRuO 3from time-resolved Kerr\nmeasurements\nC.L.S.Kantner,1,2M.C.Langner,1,2W.Siemons,3J.L.Blok,4G.Koster,4A.J.H.M.Rijnders,4R.Ramesh,1,3andJ.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, Universi ty of California, Berkeley, CA 94720\n4MESA+Institute for Nanotechnology, University of Twente, 7500 A E Enschede, The Netherlands\n(Dated: December 6, 2018)\nWereport time-resolvedKerr effectmeasurements of magnet izationdynamics inferromagnetic SrRuO 3. We\nobserve that the demagnetization timeslows substantially at temperatures within15K of theCurie temperature,\nwhichis∼150K. We analyze the data witha phenomenological model that relates the demagnetization timeto\nthe spinfliptime. Inagreement withour observations the mod el yields a demagnetization timethat is inversely\nproportional toT-T c. Wealsomake adirectcomparisonofthespinfliprateandtheG ilbertdampingcoefficient\nshowing thattheir ratioveryclose tok BTc,indicating a common originfor these phenomena.\nI: Introduction\nThereisincreasinginterestincontrollingmagnetisminfe r-\nromagnets. Of particular interest are the related question s of\nhowquicklyandbywhatmechanismthemagnetizationcanbe\nchanged by external perturbations. In addition to advancin g\nour basic understanding of magnetism, exploring the speed\nwithwhichthemagneticstatecanbechangediscrucialtoap-\nplications such as ultrafast laser-writing techniques. De spite\nits relevance, the time scale and mechanisms underlying de-\nmagnetizationarenotwell understoodata microscopicleve l.\nBeforeBeaurepaireetal.’spioneeringworkonlaser-excit ed\nNi in 1996,it was thoughtthat spins wouldtake nanoseconds\ntorotate,withdemagnetizationresultingfromtheweakint er-\nactionofspinswiththelattice. TheexperimentsonNishowe d\nthat this was not the case and that demagnetizationcould oc-\ncur on time scales significantly less than 1 ps1. Since then\ndemagnetization is usually attributed to Elliott-Yafet me cha-\nnism, in which the rate of electron spin flips is proportional\nto the momentum scattering rate. Recently Koopmans et al.\nhave demonstrated that electron-phononor electron-impur ity\nscattering can be responsible for the wide range of demag-\nnetization time scales observed in different materials2. Also\nrecentlyit hasbeenproposedthat electron-electronscatt ering\nshould be included as well as a source of Elliott-Yafet spin\nflipping,andconsequently,demagnetization3. AlthoughRef.3\nspecifically refers to interband scattering at high energie s, it\nis plausible that intrabandelectron scattering can lead to spin\nmemorylossaswell.\nTime-resolved magneto-optical Kerr effect (TRMOKE)\nmeasurementshavebeendemonstratedtobeausefulprobeof\nultrafast laser-induceddemagnetization1. In this paper we re-\nportTRMOKEmeasurementsonthinfilmsofSRO/STO(111)\nbetween 5 and 165K. Below about 80 K we observe damped\nferromagneticresonance (FMR), from which we determine a\nGilbert damping parameter consistent with earlier measure -\nments on SrTiO 3with (001) orientation6. As the the Curie\ntemperature ( ∼150K) is approached the demagnetization\ntime slows significantly, as has been observed in other mag-\nnetic systems4. The slowing dynamics have been attributed\nto critical slowing down, due to the similarities between th e\ntemperature dependencies of the demagnetization time andthe relaxation time5. In this paper we develop an analytical\nexpression relating the demagnetization time to the spin-fl ip\ntimenearthe Curietemperature. Thisprovidesa newmethod\nof measuring the spin-flip time, which is essential to under-\nstandingthedynamicsoflaser-induceddemagnetization.\nII: SampleGrowthandCharacterization\nSRO thin films were grown via pulsed laser deposition at\n700◦C in 0.3 mbar of oxygenand argon(1:1) on TiO 2termi-\nnated STO(111)7. A pressed pellet of SRO was used for the\ntargetmaterial and the energyon the targetwas kept constan t\nat 2.1 J/cm2. High-pressure reflection high-energy electron\ndiffraction (RHEED) was used to monitor the growth speed\nand crystallinity of the SRO film in situ. RHEED patterns\nandatomicforcemicroscopyimagingconfirmedthepresence\nof smooth surfaces consisting of atomically flat terraces se p-\narated by a single unit cell step (2.2 ˚Ain the [111] direction).\nX-ray diffraction indicated fully epitaxial films and x-ray re-\nflectometry was used to verify film thickness. Bulk magneti-\nzationmeasurementsusingaSQUIDmagnetometerindicated\na Curie temperature,T c, of∼155K.Electrical transportmea-\nsurementswere performedin the Vander Pauwconfiguration\nandshowtheresidualresistanceratiotobeabout10forthes e\nfilms.\nIII: ExperimentalMethods\nIntheTRMOKEtechniqueamagneticsampleisexcitedby\ntheabsorptionofapumpbeam,resultinginachangeofpolar-\nizationangle, ∆ΘK(t),ofatimedelayedprobebeam. Theul-\ntrashortpulsesfroma Ti:Sapphlaser are used to achievesub -\npicosecondtime resolution. Near normalincidence,as in th is\nexperiment, ∆ΘKis proportional to the ˆzcomponent of the\nperturbedmagnetization, ∆Mz.∆ΘKis measured via a bal-\nanceddetectionscheme. Foradditionalsensitivity,thede riva-\ntiveof∆ΘKt)withrespecttotimeismeasuredbylockinginto\nthefrequencyofasmallamplitude( ∼500fs)fastscanningde-\nlay line in the probe beam path as time is stepped throughon\nanotherdelayline.\nIV.1: ExperimentalResults: Low Temperature\nFig. 1 shows the time derivative of ∆ΘKfor an 18.5nm\nSRO/STO(111)sample forthe 16psfollowingexcitationbya\npump beam, for temperatures between 5 and 85K. Clear fer-\nromagnetic resonance (FMR) oscillations are present, gene r-2\nFIG.1. DerivativeofthechangeinKerrrotationasafunctio noftime\ndelay followingpulsed photoexcitation, for 5 <T<85 K\nated bya suddenshift in easy axisdirectionuponthermal ex-\ncitation by a pump beam6. This motion is described by the\nLandau-Lifshitz-Gilbertequationwith thefrequencyofos cil-\nlation proportional to the strength of the magnetocrystall ine\nanisotropy field, and the damping described by dimension-\nless phenomenological parameter, α. The motion appears as\na decaying oscillation to TRMOKE. The orientation of the\nanisotropyfield, closer to in-planewith the sample surface in\nSRO/STO(111) than in SRO/STO(001), makes these oscilla-\ntions more prominent when observed with the polar Kerr ge-\nometrycomparedtopreviousmeasurements.\nAttempting to model the time derivative of ∆ΘKwith a\ndampedcosine revealsthat it cannot be fit by such a function\nfor t<2ps. The feature at short times in Fig. 1 contains\nhigherfrequencycomponents,whereastheoscillationswhi ch\nbecome clear after 2 ps are at a single frequency. A com-\nparison of the amplitude of the first peak (at t ∼.5 ps) with\nthe amplitude of the subsequent oscillations (defined as the\ndifference between d ∆ΘK/dt at the peak at ∼3.5 ps and the\ndip at∼5.5 ps), is shown as a functionof temperaturein Fig.\n2. The constant offset between the two amplitudes indicates\nthat d∆ΘK/dt is comprised of a superposition of a tempera-\ntureindependent,short-livedcomponentwiththelongerli ved\ndampedoscillations.\nFitting the oscillatory portion of the signal to a damped\ncosine, the temperature dependencies of the amplitude, fre -\nquency,anddampingparameterarefound,asshowninFig. 3.\nComparingtheseparametersforSRO/STO(111)topreviously\npublished work on SRO/STO(001), the frequencyis found to\nbe somewhat smaller and to change more with temperature.\nOf particular interest is α, which is also smaller in this ori-\nentation of SRO, consistent with the more pronounced FMR\noscillations. Strikingly, in both orientations there is a d ip in\nαaround45K,whichisrelativelystrongerinSRO/STO(111).\nThis further strengthens the link between αand the anoma-\nlous hall conductivity, speculated in that paper, through n ear\ndegeneraciesin thebandstructure6.\nIV.2: ExperimentalResults: HighTemeperatureFIG. 2. Comparing amplitudes of the short time feature and th e fer-\nromagnetic resonance oscillations\nBy taking the time derivative of ∆ΘK, the FMR oscilla-\ntionscanbe followeduntilthey disappearat elevatedtempe r-\natures,atwhichpointitbecomessimplertolookat ∆ΘKthan\nits time derivative. Fig. 4 shows ∆ΘKas a function of time\nfor the first 38 ps after excitation by the pump laser, for tem-\nperatures between 120K and 165K. A property of a second\norder phase transitions is that the derivative of the order p a-\nrameter divergesnear the transition temperature. The peak in\nmagnitudeof ∆ΘKin figure 4, shown in figure5, can be un-\nderstood as the result of the derivative of magnetization wi th\nrespect to temperature becoming steeper near the Curie tem-\nperature. A strongtemperaturedependenceofthedemagneti -\nzationtime, τM, is seen,with τMsignificantlyenhancednear\n150K,consistentwithpreviousreportsonSRO4,6.\n∆ΘK(t) in Fig. 4, normalized by the largest value of\n∆ΘK(t) in the first 38 ps, can be fit with the following func-\ntion:\nfort <0∆ΘK(t)\n∆Θmax(t)= 0\nfort >0∆ΘK(t)\n∆Θmax(t)=C−Ae−t/τM(1)\nwhere the decay time is τM. The resulting τMis plotted\nas a functionof temperaturein Fig. 6. Notably, τMincreases\nby a factor of 10 from 135K to 150K. Taking the fit value of\nTc= 148.8K, as will be discussed later, τMis plotted log-\nlog as function of reduced temperature, tR= (Tc−T)/Tc.\nTheresult looksapproximatelylinear,indicatinga powerl aw\ndependenceof τMonthereducedtemperature.\nV: Discussion ofResults:\nEfforts to explain demagnetization have been largely phe-\nnomenological thus far, understandably, given the dauntin g\nchallenge of a full microscopic model. Beaurepaire et al. in -\ntroduced the three temperature model (3TM) to describe de-\nmagnetization resulting from the interactions of the elect ron,\nphonon,andspinbaths1. In3TMthedynamicsaredetermined3\nFIG. 3. Temperature dependence of (a) Amplitude of oscillat ions,\n(b) FMRfrequency, and, (c)damping parameter\nFIG.4. ChangeinKerrrotationasafunctionoftimedelayfol lowing\npulsed photoexcitation, for 120 <T<165 KFIG. 5. Magnitude of change in Kerr rotation at 38ps as a funct ion\nof temperature\nFIG.6. Demagnetization timeat hightemperature\nby the specific heats of each bath as well as the coupling\nconstants between them. Demagnetization can generally be\ndescribed with the appropriate choice of coupling constant s,\nproviding a guide into the microscopic mechanism. Koop-\nmans et al. also offer a phenomenological description of de-\nmagnetization considering three baths, but one that follow s\nspin in additionto heat8. Spinis treated asa two state system\nwith energylevels separatedby an exchangegapand Fermi’s\ngoldenrule is usedto relate demagnetizationto electronsc at-\nteringwhichflipsaspin. Equationsforcouplingconstantsa re\nderived based on parameters such as the density of states of\nelectrons, phonons, and spins, the electron-phononscatte ring\nrate,andthe probabilityofspinflipat a scatteringevent.\nIn the following we attempt to understand the behavior of\nthe demagnetization time near T cwith an approach based on\nthe two spin state model. A general relationship between the\nlaser-induced τMand the spin flip time, τsf, can be derived\nnear the transition temperature based on the concept of de-\ntailed balance9. In equilibrium,the ratio of the probabilityof4\nFIG. 7. Log-log plot of demagnetization time as a function of re-\nduced temperature\naspinflippingfrommajoritytominoritytothereverseofthi s\nprocess is the Boltzmann factor, e−∆ex/kT, where∆exis the\nexchange energy gap. The time derivative of the number of\nmajorityandminorityelectronscanthenbewritten:\n˙Nmaj=−˙Nmin=Nmin\nτsf−Nmaj\nτsfe−∆ex/kBT(2)\nWhenthesampleisthermallyexcitedbyapumpbeam,the\nelectron temperature is increased by δTe. The rate of change\nofspinsisthenalteredinthefollowingway:\n˙Nmaj=−˙Nmin=Nmin\nτsf−Nmaj\nτsfe−∆ex/kB(T+δTe)(3)\nThe demagnetization time is related to the total change in\nspin,∆S,frominitialtofinaltemperature,where,setting /planckover2pi1=1,\nSis definedby:\nS= 1/2(Nmaj−Nmin)/Ntotal (4)\nAssumingthat ∆S,asa functionoftime,canbewritten:\n∆S(t) = [S(Tf)−S(Ti)](1−e−t/τM)(5)\nthedemagnetizationtimecanbewrittenas:\nτM=∆S\n˙S(0)(6)\nwhere˙S(0)is the initial change in the time derivative of the\nspin.\nThe total change in spin can be calculated by taking the\nderivative of Swith respect to T, and multiplying by ∆Teq,\ntheincreaseintemperatureonceelectrons,phonons,andsp inshavecomeintothermalequilibriumwitheachother. S(T)and\n∆Scanbe written:\nS(T) =−1\n2tanh/parenleftbigg∆\n2kT/parenrightbigg\n(7)\nand:\n∆S=dS\ndT/vextendsingle/vextendsingle/vextendsingle\nT=T0∆Teq=−∆ex\n4kBT2\n0/bracketleftbigg\nT0∆′\nex\n∆ex−1/bracketrightbigg\n∆Teq\n(8)\nwhere we have relied on the fact that near the transition tem-\nperature, ∆ex≪kBTand made the approximation that\nδTe≪Tforlowlaserpower. Inthelastequation T0∆′\nex\n∆ex≫1\nnearTc,so onlythefirst termwill beconsidered.\nThequantity ˙S(0),where˙S= 1/2(˙Nmaj−˙Nmin)/Ntotal,\ncan be found by taking the derivative of ˙S(0)with respect to\nTe, since immediately after excitation the electron tempera-\nturehasincreased,butthespin temperature, T,hasnot.\n˙S(0) =d˙S\ndTe/vextendsingle/vextendsingle/vextendsingle\nT=T0∆Teq=Nmaj\nN0τsf∆ex\nkBT∆Teq(9)\nNear the Curie temperature Nmaj∼Nmin∼1\n2Ntotal.\nUsingthisapproximationsandequation(6), wefind:\nτM=/parenleftbigg∆′\nex\n∆ex/parenrightbiggTcτsf\n2(10)\nwhere∆′\nexisthederivativeof ∆exwithrespecttotemperature\nand∆ex∼(Tc−T)β, whereβis the critical exponent of\nthe order parameter. Taking the derivative, we find ∆′\nex∼\n−β(Tc−T)β−1,andthuscanwrite\nτM=βτsf\n2/parenleftbiggTc\nTc−T/parenrightbigg\n(11)\nTherefore τMis predicted to scale as 1/(Tc−T)near the\ntransition temperature. A fit of T c∼148.8K is found for the\ndatainFig 6.\nNote that detailed balance suggests that the demagnetiza-\ntion time scales as 1/tRnear the transition temperature re-\ngardlessoftheunderlyingmechanismofthedemagnetizatio n.\nAdditionally,the critical exponentfoundis independento fβ.\nIt should also be noted that the current situation, where the\nsample has been excited by a laser, is distinct from critical\nbehavior as typically considered. In general, divergent ti me\nscales are linked to divergent length scales, but here excit a-\ntions of various length scales are not being excited. Instea d\nthelengthscale is alwayseffectivelyinfinite, havingbeen de-\nterminedby the laser spot size. τsfis plotted as a functionof\ntemperature for the mean field value of β= 1/2, which has\nbeenshowntobesuitableforSRO10,inFig. 8. τsfisrevealed\nto be approximately 200 fs and nearly constant as a function\noftemperature.5\nFIG.8. Spinfliptime at hightemperature\nPrevious reports of conductivity in SRO give a scattering\ntime of∼20 fs near the transition temperature11. A compar-\nison of the spin flip time with the scattering time implies a\nprobabilityof0.1thatascatteringeventsresultsinaspin flip.\nThoughelectron-phononinteractionsare the most commonly\nconsidered source of demagnetization, as mentioned previ-\nously, Eliot Yafet-like electron-electron coulomb scatte ring\ncanalsoresultindemagnetization3. Thisisespeciallytruefor\nmaterials with strong spin orbit coupling, such as SRO. Ad-\nditionally in SRO the interaction with the crystal field mean s\nthat total spin is not conserved[Goodenough], so every elec -\ntroninteractioncanperturbthespinstate.\nHaving found a relationship between the demagnetization\ntime and the spin flip time we would like to explore the rela-\ntionship between these parameters and the damping param-\neter,α. Intuitively, the damping parameter should be pro-\nportional to the spin flip scattering rate, or inversely prop or-\ntionaltothe spinflip scatteringtime: α∼1/τsf. Elliot-Yafet\ntype scattering dissipates energy from motion described by\nthe LLG equation by disrupting the coherent, collective pre -\ncession of spins. Spins that have had their angular momen-\ntum changed through electron collisions must be pulled back\ninto the precession through the exchange interaction, repr e-senting a transfer of energy away from the precessional mo-\ntion. These collision-mediatedspin-orbitcouplingeffec ts are\nthought to be the primary source of Gilbert-type damping in\nferromagnets12. Again, this should be particularly true in a\nferromagnetwith strongspinorbitcoupling.\nCombining the spin flip time and the damping parameter\nwith Planck’s constant reveals an energy scale, E, given by\ntheconditionthat:\n1\nα∼E\n/planckover2pi1τsf (12)\nNoting that the valuesfor αandτsffoundin figures 3 and\n7, respectively, are approximately constant as a function o f\ntemperature, this energy scale for SRO is ∼7 meV. The fun-\ndamental energy scales applicable to the magnetic system in\nSRO aretheFermienergy,the exchangeenergy,andthecriti-\ncaltemperature,thelasttwoofwhichareinterdependent. T he\nFermi energy is orders of magnitude larger than 7 meV, but\nthe energy associated with the critical temperature, kBTc∼\n13 meV, is of the same order. This suggests an underlying\nconnectionbetween the critical temperature(and thus the e x-\nchangeenergy),Gilbertdamping,andspinflip scattering.\nA relationshipsimilar to equation(12) hasbeenfoundpre-\nviously between τM(rather than τsf) andαby Koopmanset\nal. at lowtemperature:\nτM=1\n4/planckover2pi1\nkBTc1\nα(13)\nApplying this equation to SRO at 5K yields τm∼30fs,\nwhich is unphysical since it is below the total scattering ra te\nof∼100fsatlowtemperature11. Whetherthefundamentalre-\nlationshipisbetweentransitiontemperatureandthedemag ne-\ntizationtimeorthespin-flipscatteringtimeremainsaques tion\nfora microscopicmodeltoresolve.\nACKNOWLEDGMENTS\nThis research is supported by the US Department of En-\nergy, Office of Science under contract number DE-AC02-\n05CH1123.\n1E.Beaurepaire etal.,Phys. Rev. Lett. 76, 4250 (1996).\n2B.Koopmans et al.,Nat.Mater. 9,259 (2009).\n3M. Krauss et al.,Phys.Rev. B 80, 180407 (2009).\n4T. Ogasawara et al.,Phys.Rev. Lett. 94, 087202 (2005).\n5T. Kiseet al.,Phys.Rev. Lett. 85, 1986 (2000).\n6M. Langner etal.,Phys. Rev. Lett. 102, 177601 (2009).\n7G.Koster etal.,Appl. Phys.Lett. 73, 2920 (1998).\n8B.Koopmans et al.,Phys.Rev. Lett. 95, 267207 (2005).\n9N.Metropolis et al.,J.Chem. Phys. 21, 1087 (1953).\n10D.Kim et al.,Phys.Rev. B 67, 100406 (2003).11J.S.Dodge et al.,Phys.Rev. Lett. 85, 4932 (2000).\n12B. Heinrich, in Ultrathin Magnetic Structures III (Springer-\nVerlag,Berlin,Germany, 2005).\n13J.Fabian, and S.Das Sarma, Phys.Rev. Lett. 81, 5624 (1998).\n14P.Monod, andF.Beuneu, Phys.Rev. B 19, 911 (1979).\n15X.Wanget al.,Phys.Rev. B. 74, 195118 (2006).\n16D.L.Mills and S.M.Rezende, in Spin dynamics in confined mag-\nnetic structures II , edited by B. Hillebrands and K. Ounadjela\n(Springer-Verlag,Berlin,Germany, 2003).\n17M. Pickelet al.,Phys.Rev. Lett. 101, 066402 (2008)."    },    {        "title": "1303.4922v1.Spin_pumping_and_Enhanced_Gilbert_Damping_in_Thin_Magnetic_Insulator_Films.pdf",        "content": "arXiv:1303.4922v1  [cond-mat.mes-hall]  20 Mar 2013Spin-pumping and Enhanced Gilbert Damping in Thin Magnetic Insulator Films\nAndr´ e Kapelrud and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nPrecessing magnetization in a thin film magnetic insulator p umps spins into adjacent metals;\nhowever, this phenomenon is not quantitatively understood . We present a theory for the dependence\nof spin-pumpingon the transverse mode number and in-plane w ave vector. For long-wavelength spin\nwaves, the enhanced Gilbert damping for the transverse mode volume waves is twice that of the\nmacrospin mode, and for surface modes, the enhancement can b e ten or more times stronger. Spin-\npumping is negligible for short-wavelength exchange spin w aves. We corroborate our analytical\ntheory with numerical calculations in agreement with recen t experimental results.\nPACS numbers: 76.50.+g, 75.30.Ds, 75.70.-i, 75.76.+j, 75. 78.-n\nMetallic spintronics have been tremendously success-\nful in creating devices that both fulfill significant market\nneeds and challenge our understanding of spin transport\nin materials. Topics that are currently of great interest\nare spin transfer and spin-pumping [1–3], spin Hall ef-\nfects [4], and combinations thereof for use in non-volatile\nmemory, oscillator circuits, and spin wave logic devices.\nA recent experimental demonstration that spin transfer\nand spin-pumping can be as effective in magnetic insula-\ntors as in metallic ferromagnetic systems was surprising\nand has initiated a new field of inquiry [5].\nIn magnetic insulators, no moving charges are present,\nand in some cases, the dissipative losses associated with\nthe magnetization dynamics are exceptionally low. Nev-\nertheless, when a magnetic insulator is placed in con-\ntact with a normal metal, magnetization dynamics in-\nducespin-pumping,whichinturncausesangularmomen-\ntum to be dumped to the metal’s itinerant electron sys-\ntem. Duetothisnon-localinteraction, themagnetization\nlosses become enhanced. Careful experimental investiga-\ntions of spin-pumping and the associated enhanced mag-\nnetization dissipation were recently performed, demon-\nstrating that the dynamic coupling between the magne-\ntization dynamics in magnetic insulators and spin cur-\nrentsinadjacentnormalmetalsisstrong. Importantly,in\nmagnetic insulators, an exceptionally low intrinsic damp-\ning combined with good material control has enabled the\nstudy of spin-pumping for a much larger range of wave\nvectorsthan has previously been obtained in metallic fer-\nromagnets [5–14].\nIn thin film ferromagnets, the magnetization dynamics\nare strongly affected by the long-range dipolar interac-\ntion, which has both static and spatiotemporal contribu-\ntions. This yields different types of spin waves. When\nthe in-plane wavelength is comparable to the film thick-\nness or greater, the long-range dipolar interaction causes\nthe separation of the spin-wave modes into three classes\ndepending on the relative orientation of the applied ex-\nternal field, in relation to the film normal, and the spin-\nwave propagation direction [15–20]. These spin waves\nare classified according to their dispersion and transverse\nmagnetization distribution as forward volume magneto-static spin waves (FVMSWs) when the external field is\nout-of-plane, backward volume magnetostatic spin waves\n(BVMSWs) when the external field is in-plane and along\nthe direction of propagation, and magnetostatic surface\nwaves (MSSWs) when the external field is in-plane but\nperpendicular to the direction of propagation. In volume\nwaves, the magnetic excitation is spatially distributed\nacross the entire film, while surface modes are localized\nnear one of the surfaces. “Backward” waves have a fre-\nquencydispersionwithanegativegroupvelocityforsome\nwavelengths. While these spin waves have been studied\nin great detail overthe last decades, the effect of an adja-\ncent normal metal on these waves has only recently been\ninvestigated.\nExperimentally, it has been observed that spin-\npumping differs for FVMSWs, BVMSWs and MSSWs\nand that it depends on the spin-wave wavelength[6, 8, 9,\n12–14]. Recent experiments [8] have demonstrated that\nthe magnetization dissipation is larger for surface spin\nwaves in which the excitation amplitude is localized at\nthe magnetic insulator-normal metal interface. To uti-\nlize spin-pumping from thin film magnetic insulatorsinto\nadjacent normal metals, a coherent theoretical picture of\nthese experimental findings must be developed and un-\nderstood, which is the aim of our work.\nIn this Letter, we present a theory for energy dissipa-\ntionfromspin-waveexcitationsinaferromagneticinsula-\ntor (FI) thin film via spin-pumping when the ferromag-\nnetic insulator layer is in contact with a normal metal\n(NM). To this end, consider a thin film magnetic insu-\nlator of thickness Lon an insulating substrate with a\nnormal metal capping (see Figure 1). We consider a nor-\nmal metal such as Pt at equilibrium, where there is rapid\nspin relaxation and no back-flow of spin currents to the\nmagnetic insulator. The normal metal is then a perfect\nspin sink and remains in equilibrium even though spins\nare pumped into it.\nThe magnetization dynamics are described by the\nLandau-Lifshitz-Gilbert (LLG) equation [21] with a\ntorque originating from the FI/NM interfacial spin-2\npumping [22]\n˙M=−γM×Heff+α\nMSM×˙M+τsp,(1)\nwhereαis the Gilbert damping coefficient, MSis the\nsaturation magnetization, γis the gyromagnetic ratio,\nHeffis the effective field including the external field, ex-\nchange energy, surface anisotropy energy, and static and\ndynamic demagnetization fields.\nSpin-pumping throughinterfaces between magneticin-\nsulators and normal metals gives rise to a spin-pumping\ninduced torque that is described as [2]\nτsp=γ/planckover2pi12\n2e2M2\nSg⊥δ/parenleftBig\nξ−L\n2/parenrightBig\nM×˙M,(2)\nwhereg⊥is the transverse spin (“mixing”) conductance\nper unit area at the FI/NM interface. We disregard the\nimaginary part of the mixing conductance because this\npart has been found to be small at FI/NM interfaces\n[12]. In addition, the imaginary part is qualitatively less\nimportant and only renormalizes the gyromagnetic ratio.\nAssuming only uniform magnetic excitations,\n“macrospin” excitations, the effect of spin-pumping\non the magnetization dissipation is well known [2, 3].\nSpin-pumping leads to enhanced Gilbert damping,\nα→α+∆αmacro, which is proportional to the FI/NM\ncross section because more spin current is then pumpedout, but inversely proportional to the volume of the\nferromagnet that controls the total magnetic moment:\n∆αmacro=γ/planckover2pi1\n4πLMSh\ne2g⊥. (3)\nThus, the enhanced Gilbert damping due to spin-\npumping is inversely proportional to the film thickness\nLand is important for thin film ferromagnets. However,\na “macrospin” excitation, or the FMR mode, is only one\nout of many types of magnetic excitations in thin films.\nThe effect of spin-pumping on the other modes is not\nknown, and we provide the first analytical results for this\nimportant question, which is further supported and com-\nplemented by numerical calculations.\nWe consider weak magnetic excitations around a ho-\nmogenous magnetic ground state pointing along the di-\nrection of the internal field Hi=Hiˆz, which is the com-\nbination of the external applied field and the static de-\nmagnetizing field [19]. We may then expand M=MSˆz+\nmQ,xy(ξ)ei(ωt−Qζ), wheremQ,xy·ˆz= 0,|mQ,xy| ≪MS,\nandQis the in-plane wave number in the ζ-direction.\nFollowing the linearization approach of the LLG equa-\ntion (1) as in Ref. [19], we arrive at a two-dimensional\nintegro-differential equation of the dynamic magnetiza-\ntion (in the xy-plane) in the film’s transverse coordinate\nξ:\n/bracketleftbigg\niω\nωM/parenleftbigg\nα−1\n1α/parenrightbigg\n+11/parenleftbiggωH\nωM+8πγ2A\nω2\nM/bracketleftbigg\nQ2−d2\ndξ2/bracketrightbigg\n+iαω\nωM/parenrightbigg/bracketrightbigg\nmQ,xy(ξ) =/integraldisplayL\n2\n−L\n2dξ′/hatwideGxy(ξ−ξ′)mQ,xy(ξ′),(4)\nwhereωis the spin-wave eigenfrequency, Ais the ex-\nchange stiffness, ωH=γHi,ωM= 4πγMS, and/hatwideGxyis\nthe dipole-dipole field interaction tensor, which fulfills\nthe boundary conditions resulting from Maxwell’s equa-\ntions (see [23]).\nThe eigensystem must be supplemented by boundary\nconditions that account for spin-pumping and surface\nanisotropy. These boundary conditions are obtained by\nintegrating Eq. (1) over the interface [24] and expanding\nto the lowest order in the dynamic magnetization. When\nan out-of-plane easy axis surface anisotropy is present,\nthe boundary conditions are\n/parenleftbigg\nL∂\n∂ξ+iωχ+LKs\nAcos(2θ)/parenrightbigg\nmQ,x(ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nξ=L\n2= 0,(5a)\n/parenleftbigg\nL∂\n∂ξ+iωχ+LKs\nAcos2(θ)/parenrightbigg\nmQ,y(ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nξ=L\n2= 0,(5b)\nwhereKsis the surface anisotropy energy with units\nerg·cm−2andχ=L/planckover2pi12g⊥\n4Ae2is a parameter relating the ex-change stiffness and the spin mixing conductance ([ χ] =\ns). The boundary condition at the magnetic insulator–\nsubstrate interface at ξ=−L/2 is similar to Eq. (5), but\nsimpler because χ→0 andKs→0 at that interface.\nA mathematical challenge induced by spin-pumping\narisesbecausethesecondterminthe linearizedboundary\ncondition(5)isproportionaltotheeigenvalue ωsuchthat\nthe eigenfunctions cannot simply be expanded in the set\nofeigenfunctionsobtainedwhenthereisnospin-pumping\nor dipolar interaction. Instead, we follow an alternative\nanalytical route for small and large wave vectors. Fur-\nthermore, we numericallydetermine the eigenmodeswith\na custom-tailored technique, where we discretize the dif-\nferential equation (4), include the spin-pumping bound-\nary conditions (5), and transform the resulting equations\ninto an eigenvalue problem in ω[25].\nLet us now outline how we obtain analytical results for\nsmallQL≪1 and large QL≫1 wave vectors. First,\nwe consider the case of vanishing surface anisotropy and\ncompute the renormalization of the Gilbert damping for3\n(a)L/Slash12\n/MinusL/Slash12NM\nFI\nSUBΞ\n(b)\nFIG. 1. a) A thin film magnetic insulator of thickness L\nin its coordinate system; ξis the normal axis, the infinite\nηζ-plane is coplanar with the interfaces, and the spin waves\npropagate along the ζ-axis. The internal field and saturation\nmagnetization are along the z-axis. The y-axis is always kept\nin-plane, and the x-axis is selected such that the x-,y- and\nz-axes form a right-handed coordinate system. b) A cross-\nsection showing the material stack.\nthe resulting modes. Next, we demonstrate that the sur-\nface anisotropycreates a surface wavewith a comparably\nlarge enhancement of the Gilbert component.\nWhenQL≪1, the convolution integral on the right-\nhand side of Eq. (4) only contains the homogeneous de-\nmagnetization field. The magnetization is then a trans-\nverse standing wave mQ,xy/parenleftbig\neikξ+e−ikξ+φ/parenrightbig\n,wherekis\na transverse wave number, φis a phase determined by\nthe BC at the lower interface, and the two-dimensional\ncoefficient vector mQ,xyallows for elliptical polarization\nin thexy-plane.\nBy employing exchange-only boundary conditions [24]\nat the lower interface and using Eq. (5) with Ks= 0 on\nthe upper interface, the transverse wave number kis de-\ntermined by kLtankL=iωχ. Together with the bulk\ndispersion relation ω=ω(k), calculated from Eq. (4),\nthisexpressionallowsustocalculatethe magneticexcita-\ntion dispersion relation parameterized by the film thick-\nness, the Gilbert damping α, and the transverse conduc-\ntanceg⊥.\nWhen spin-pumping is weak, ωχis small, and the so-\nlutions of the transcendental equation can be expanded\naround the solutions obtained when there is no spin-\npumping, kL=nπ, where nis an integer. When\nn∝negationslash= 0, we expand to first order in kLand obtain\nkL≈nπ+iωχ/(nπ). When n= 0, we must perform\na second-order expansion in terms of kLaround 0, which\nresults in ( kL)2≈iωχ. Using these relations in turn\nto eliminate kfrom the bulk dispersion relation while\nmaintaining our linear approximation in small terms and\nsolving for ω, we obtain complex eigenvalues, where the\nimaginary part is proportional to a renormalized Gilbert\ndamping parameter, α∗=α+ ∆α. When n= 0, our\nresults agree with the spin-pumping-enhanced Gilbert\ndamping of the macrospin (FMR) mode derived in [2](see Eq. (3)), ∆ α0= ∆αmacro. Whenn∝negationslash= 0, we compute\n∆αn= 2∆αmacro. (6)\nThese new results indicate that allhigher transverse vol-\nume modes have an enhanced magnetization dissipation\nthat is twice that of the macrospin mode. Thus, coun-\nterintuitively, with the exception of the macrospin mode,\nincreasingly higher-order standing-wave transverse spin-\nwave modes have precisely the same enhanced Gilbert\ndamping.\nNext, let us discuss spin-pumping for surface waves\ninduced by the presence of surface anisotropy. When\nKs∝negationslash= 0, the lowest volume excitation mode develops into\na spatially localized surface wave. Expanding the ex-\npression for the localized wave to the highest order in\nLKs/A, we determine after some algebra that the result-\ning enhancement of the Gilbert damping is\n∆αn=0=γ/planckover2pi1Ks\n4πMsAh\ne2g⊥ωH\nωM/bracketleftbiggωH\nωM+1\n2−K2\ns\n4πM2sA/bracketrightbigg−1\n.\n(7)\nComparing Eqs. (7) and (6), we see that for large sur-\nface anisotropy LKs/A≫1, the spin-pumping-induced\nenhanced Gilbert damping is independent of L. This re-\nsult occurs because a large surface anisotropy induces\na surface wave with a decay length A/Ks, which re-\nplaces the actual physical thickness Las the effective\nthickness of the magnetic excitations, i.e., for surface\nwavesL→A/Ksin the expression for the enhanced\nGilbert damping of Eq. (3). This replacement implies\nthat the enhanced Gilbert damping is much larger for\nsurface waves because the effective magnetic volume de-\ncreases. For typical values of AandKs, we obtain an\neffective length A/Ks∼10nm. Compared with the film\nthicknesses used in recent experiments, this value corre-\nsponds to a tenfold or greater increase in the enhance-\nment of the Gilbert damping. In contrast, for the volume\nmodes (n∝negationslash= 0), we note from Eq. (5) that the dynamic\nmagnetization will decrease at the FI/NM interface due\nto the surface anisotropy; hence, ∆ αdecreases compared\nwith the results of Eq. (6).\nFinally, we can also demonstrate that for large wave\nvectorsQL≫1, the excitation energymostly arisesfrom\nthe in-plane (longitudinal) magnetization texture gradi-\nent. Consequently, spin-pumping, which pumps energy\nout of the magnetic system due to the transverse gradi-\nent of the magnetization texture, is much less effective\nand decays as 1 /(QL)2with respect to Eq. (3).\nTo complement our analytical study, we numerically\ncomputed the eigenfrequencies ωn(Q). The energy is de-\ntermined by the real part of ωn(Q), whileImωn(Q) de-\nterminesthe dissipationrateandhencethespin-pumping\ncontribution. Recent experiments [6, 11, 13, 14] on\ncontrolling and optimizing the ferrimagnetic insulator\nyttrium-iron-garnet (YIG) have estimated that the mix-\ningconductancesofbothYIG—AuandYIG—Ptbilayers4\n10/Minus610/Minus40.01 1 100QL12345/CΑpDeltΑΑ/LParen110/Minus4/RParen1\n/MinusL/Slash120 L/Slash12\nΞ/LBracketBar1m/RArroΩ\n/LParen1Ξ/RParen1/RBracketBar1\nFIG. 2. ∆ αversus wave vector for the MSSW geometry ( θ=\nφ=π/2) for thefour lowest eigenvalues. Inset: Magnitudesof\neigenvectors (in arbitrary units) across the film at QL= 1.5.\nare in the range of g⊥h/e2∼0.02–3.43·1015cm−2. We\nuseg⊥h/e2= 1.2·1014cm−2from Ref. [6] in this work.\nAll of our results can be linearly re-scaled with other val-\nues of the mixing conductance. In the following section,\nwe also use A= 2.9·10−8erg/cm,Ks= 0.05erg/cm2,\nL= 100nm, 4 πMS= 1750G, and α= 3·10−4.\nTo distinguish the spin-pumping contribution∆ αfrom\nthe magnetization dissipation due to intrinsic Gilbert\ndamping α, we first compute the eigenvalues, ωd, with\nintrinsic Gilbert damping, α∝negationslash= 0, and no spin-pumping,\ng⊥= 0. Second, we compute the eigenvalues ωspwith\ndissipation arising from spin-pumping only, α= 0 and\ng⊥∝negationslash= 0. Because Imωd∝α, we define a measure of\nthe spin-pumping-induced effective Gilbert damping as\n∆α=αImωsp/Imωd.\nWefirstconsiderthe caseofnosurfaceanisotropy. Fig-\nure 2showsthe spin-pumping-enhancedGilbert damping\n∆αas a function of the product of the in-plane wave vec-\ntor and the film thickness QLin the MSSW geometry.\nIn the long-wavelength limit, QL≪1, the numerical re-\nsult agreespreciselywithouranalyticalresultsofEq.(6).\nThe enhanced Gilbert damping of all higher transverse\nmodes is exactly twice that of the macrospin mode. In\nthe dipole-exchange regime, for intermediate values of\nQL, the dipolar interaction causes a small asymmetry in\nthe eigenvectors for positive and negative eigenfrequen-\ncies because modes traveling in opposite directions have\ndifferent magnitudes of precession near the FI/NM in-\nterface [26], and spin-pumping from these modes there-\nfore differ. This phenomenon also explains why the en-\nhanced damping, ∆ α, splits into different branches in\nthis regime, as shown in Fig. 2. For exchange spin waves,\nQL≫1, the exchange interaction dominates the dipo-\nlar interaction and removes mode asymmetries. We also\nsee that ∆ α→0 for large QL, in accordance with our\nanalytical theory.\nFigure 3 shows ∆ αfor the BVMSW geometry. The\neight first modes are presented; however, as no substan-\ntial asymmetry exists between eigenmodes traveling in10/Minus610/Minus40.01 1 100QL123456/CΑpDeltΑΑ/LParen110/Minus4/RParen1\n0.1 1 100.00.51.01.52.02.5Re/LBrace1Ω/Slash1ΩM/RBrace1\n/MinusL/Slash12 0 L/Slash12\nΞ/LBracketBar1m/RArroΩ\n/LParen1Ξ/RParen1/RBracketBar1\nFIG. 3. ∆ αversus wave vector for the BVMSW geometry\n(θ=π/2 andφ= 0). Left inset: Magnitude of eigenvectors\n(in arbitrary units) across the film when QL= 1.5. Right\ninset: The real part of the dispersion relation for the same\nmodes.\n10/Minus40.001 0.01 0.1 1 10 100QL510152025/CΑpDeltΑΑ/LParen110/Minus4/RParen1\n/MinusL/Slash12 0 L/Slash12\nΞ/LBracketBar1m/RArroΩ\n/LParen1Ξ/RParen1/RBracketBar1\nFIG. 4. ∆ αversus wave vector for the MSSW geometry\n(θ=φ=π/2) with surface anisotropy added at the inter-\nface. Inset: Magnitudes of eigenvectors (in arbitrary unit s)\nacross the film.\ndifferent directions, the modes have the same pairwise\nrenormalization of α. This symmetry occurs because the\ndirection of the internal field coincides with the direction\nof propagation. As in the previous case, the dipolar in-\nteraction causes a slight shift in the eigenvectors in the\nintermediate QLregime, thereby altering ∆ αfrom that\nof Eq. (6).\nFigure 4 shows ∆ αfor the MSSW geometry but with\nsurface anisotropy at the FI/NM interface. As expected\nfrom our analytical results, surface anisotropy induces\ntwo localized surface modes with a ten-fold larger en-\nhancement of∆ αcomparedwith the volume modes. The\nhorizontal dashed line in Figure 4 indicates the analyti-\ncal result for the enhanced Gilbert damping of the n∝negationslash= 0\nmodes when Ks= 0. For the volume modes, it is clear\nthattheeigenvectorshavealowermagnitudeclosertothe\nFI/NM interfaceandthat ∆ αis lowercomparedwith the\ncase ofKs= 0, which is consistent with our analytical\nanalysis.\nOur results also agree with recent experiments.\nSandweg et al.[8] found that spin-pumping is signifi-5\ncantly higher for surface spin waves compared with vol-\nume spin-wave modes. In addition, in Ref. [9], exchange\nwaves were observed to be less efficient at pumping spins\nthan dipolar spin waves, which is consistent with our re-\nsults. Furthermore, our results are consistent with the\ntheoretical finding that spin-transfer torques preferen-\ntially excite surface spin waves with a critical current\ninversely proportional to the penetration depth [27].\nIn conclusion, we have analyzed how spin-pumping\ncauses a wave-vector-dependent enhancement of the\nGilbert damping in thin magnetic insulators in con-\ntact with normal metals. In the long-wavelength limit,\nour analytical results demonstrate that the enhancement\nof the Gilbert damping for all higher-order volumetric\nmodes is twice as large as that of a macrospin excita-\ntion. Importantly, surface anisotropy-pinnedmodes have\na Gilbert renormalization that is significantly and lin-\nearly enhanced by the ratio LKs/A.\nA. Kapelrud would like to thank G. E. W. Bauer for\nhis hospitality at TU Delft. This work was supported by\nEU-ICT-7 contract No. 257159 “MACALO”.\n[1] A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11,\n372 (2012).\n[2] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer,\nPhys. Rev. Lett. 88, 117601 (2002).\n[3] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[4] T. Jungwirth, J. Wunderlich, and K. Olejnik, Nat.\nMater.11, 382 (2012).\n[5] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando,\nK. Takanasahi, S. Maekawa, and E. Saitoh, Nature 464,\n262 (2010).\n[6] B. Heinrich, C. Burrowes, E. Montoya, B. Kar-\ndasz, E. Girt, Y.-Y. Song, Y. Sun, and M. Wu,\nPhys. Rev. Lett. 107, 066604 (2011).\n[7] C. Burrowes, B. Heinrich, B. Kardasz, E. A. Mon-\ntoya, E. Girt, Y. Sun, Y.-Y. Song, and M. Wu,\nAppl. Phys. Lett. 100, 092403 (2012).\n[8] C. W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, and\nB. Hillebrands, Appl. Phys. Lett. 97(2010).\n[9] C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A.\nSerga, V. I. Vasyuchka, M. B. Jungfleisch, E. Saitoh, and\nB. Hillebrands, Phys. Rev. Lett. 106, 216601 (2011).\n[10] L. H. Vilela-Leao, A. A. C. Salvador, and S. M. Rezende,\nAppl. Phys. Lett. 99, 102505 (2011).\n[11] S. M. Rezende, R. L. Rodriguez-Suarez, M. M. Soares,\nL. H. Vilela-Leao, D. L. Dominguez, and A. Azevedo,\nAppl. Phys. Lett. 102, 012402 (2013).\n[12] X. Jia, K. Liu, X. K, and G. E. W. B. Bauer, EPL 96,\n17005 (2011).\n[13] M. B. Jungfleisch, V. Lauer, R. Neb, A. V. Chu-\nmak, and B. Hillebrands, ArXiv e-prints (2013),\narXiv:1302.6697 [cond-mat.mes-hall].\n[14] Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Taka-\nhashi, T. An, Y. Fujikawa, and E. Saitoh, ArXiv e-prints(2013), arXiv:1302.7091 [cond-mat.mes-hall].\n[15] J. R. Eshbach and R. W. Damon,\nPhys. Rev. 118, 1208 (1960).\n[16] R. Damon and J. Eshbach,\nJ. Phys. Chem. Solids 19, 308 (1961).\n[17] H. Puszkarski, IEEE Trans. Magn. 9, 22 (1973).\n[18] R. E. D. Wames and T. Wolfram,\nJ. Appl. Phys. 41, 987 (1970).\n[19] B. A. Kalinikos and A. N. Slavin,\nJ. Phys. C 19, 7013 (1986).\n[20] A. Serga, A. Chumak, and B. Hillebrands, J. Phys. D\n43, 264002 (2010).\n[21] T. Gilbert, Phys. Rev. 100, 1243 (1955).\n[22] Gaussian (cgs) units are employed throughout.\n[23] B. A. Kalinikos, Sov. Phys. J. 24, 719 (1981).\n[24] G. Rado and J. Weertman, J. Phys. Chem. Solids 11,\n315 (1959).\n[25] A. Kapelrud and A. Brataas, Unpublished.\n[26] Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, and A. Slavin,\nPhys. Rev. Lett. 107, 146602 (2011).\n[27] J. Xiao and G. E. W. Bauer,\nPhys. Rev. Lett. 108, 217204 (2012)."    },    {        "title": "1307.2648v2.Scaling_of_spin_Hall_angle_in_3d__4d_and_5d_metals_from_Y3Fe5O12_metal_spin_pumping.pdf",        "content": "1 \n Scaling of spin Hall angle in 3 d, 4d and 5d metals  from  Y3Fe5O12/metal spin \npumping  \nH. L. Wang†, C. H. Du†, Y . Pu,  R. Adur, P. C. Hammel*, F. Y . Yang*  \nDepartment of Physics, The Ohio State University, Columbus, OH 43210 ,USA  \n†These authors made equal contributions to this work  \n*E-mails:fyyang@physics.osu.edu; hammel@physics.osu.edu  \n \nAbstract  \nWe have investigated spin pumping  from Y3Fe5O12 thin films into Cu, Ag, Ta, W, Pt and Au  \nwith varying  spin-orbit coupling strengths . From measurements of Gilbert damping \nenhance ment and inverse spin Hall  signals  spanning  three orders of magnitude , we determine \nthe spin Hall angles and interfacial spin mixing conductance s for the six metals . For noble \nmetals Cu, Ag and Au  (same d-electron counts) , the spin Hall angles vary as  Z4 (Z: atomic \nnumber ), corroborating the role of spin -orbit coupling . In contrast, amongst the four 5 d \nmetals, the variation of the spin Hall angle is dominated by the sensitivity of the d-orbital \nmoment to the d-electron count, confirming theoretical predictions.  \n \nPACS: 75.47.Lx, 76.50.+g, 75.70.Ak, 61.05.cp  \n  2 \n Spin pumping of p ure spin c urrent s from a  ferromagnet ( FM) into a nonmagnetic \nmaterial ( NM) provide s a promising route  toward energy -efficient  spintronic  devices . The \ninverse spin Hall effect (ISHE) in FM/Pt bilayer systems [1-13] is the most widely used tool \nfor detecting s pin currents generated  by either ferromagnetic resonance ( FMR ) or a thermal \ngradient. The intense interest in spin pumping emphasizes the pressing need for quantitative \nunderstanding of ISHE in normal metals other than Pt  [10]. To date, spin Hall angles  (SH) \nhave been measured  for several metals and alloys by spin Hall or ISHE measurements, \nmostly using metallic FMs [ 14]. Due to current shunting of the metallic FMs and potential \nconfounding effects  of anisotropic magnetoresistance ( AMR ) or anomalous Hall effect \n(AHE ), the reported values of SH vary significantly , sometimes by more than one order of \nmagnitude  for the same materials [14]. Here we report a systematic study of FMR spin \npumping from  insulating  Y3Fe5O12 (YIG) epitaxial thin films grown by sputtering [15-22] \ninto six normal metal s, Cu, Ag, Ta, W, Pt and Au , that span a wide range in two key \nparameters : a factor of ~50 in spin-orbit coupling strength [ 23] and over two orders of \nmagnitude variation in spin diffusion length  (SD) [4, 24-26]. Due to their  weak spin-orbit \ncoupling and relatively long spin diffusion length s, Cu and Ag present a significant challenge \nfor ISHE detection of spin pumping . ISHE voltages  (VISHE) exceeding 5 mV are  generated  in \nour YIG/Ta and YIG /W bilayer s and here we report  ~1 V spin pumping signals  in YIG/Cu \nand YIG /Ag bilayer s. The recently rep orted proximity effect in Pt [ 9, 13 ] should lead to at most \nV-level  contribution to  the measured  VISHE, which is negligible compared with the observed \nmV-level VISHE in our YIG/Pt bilayer and should not be a factor in other five metals.  The large 3 \n dynamic range that this sensitivity provides  and the insulating nature of YIG films  enable \nquantitative determination of spin mixing conductance s across the YIG/metal interfa ces [5, \n12] and spin Hall angle s of these 3d, 4d and 5 d metals .  \nWe characterize the structural quality of our epitaxial YIG films deposited on \n(111) -oriented Gd3Ga5O12 (GGG) substrates  using  off-axis ultrahigh vacuum  (UHV)  \nsputtering  [15-22] by high-resolution x -ray diffraction (XRD).  A representative θ-2θ scan of a \n20-nm YIG film  shown in Fig. 1 a demonstrates phase  purity and clear  Laue oscillations , \nindicat ing high uniformity of the film . We find an out-of-plane lattice constant of the YIG \nfilm, c = 12.393 Å , very close to the bulk value of 12.376 Å . The XRD rocking curve in the  \nleft inset to Fig. 1a give s a full  width  at half maximum (FWHM) of  only 0.0092, near the \nresolution limit of our high-resolution XRD,  demonstrating the excellent crystalline quality  of \nthe YIG film . The atomic force microscopy (AFM) image in the right inset to Fig. 1a shows a \nsmooth surface with a roughness of 0.15 nm.  Figure 1b shows a representative FMR \nderivative absorption spectrum f or a 20-nm YIG film  used in this study  taken by a Bruker \nEPR spectrometer in a cavity at a radio -frequency (rf)  f = 9.65 GHz and an input microwave \npower Prf = 0.2 mW with a magnetic field H applied in the film plane. The peak -to-peak \nlinewidth ( H) obtained from the spectrum is 1 1.7 Oe and an effective saturation \nmagnetization 4π𝑀eff = 1786  ± 36 Oe is extracted  from fitt ing the angular dependence of \nresona nce field  [27]. Due to the small  magnetic  anisotropy of YIG, the satura tion \nmagnetization  4𝑀s can be approximated at 1786  Gauss  which agrees well with the value \nreported for single crystal YIG , indicating the high magnetic quality of our YIG films  [28]. In 4 \n this letter , all six YIG/metal bilayers are made from the same 20 -nm YIG film characterized \nin Figs. 1b and 1c.  \nOur s pin pumping measurements are carried out in the center of the EPR cavity  on the \nsix YIG/ metal  bilayer s at room temperature  (approximate dimension s of 1.0 mm   5 mm). \nThe thickness of  the metal layers is 5 nm for  Ag, Ta, W, Pt, Au and 10 nm for Cu, and all are \nmade by UHV off-axis sputtering . Resistivity  () measurements confirm that the Ta and W \nfilms are -phase  [29, 30] (high resistivity , see Table I) . During the spin pumping \nmeasurements, a  DC magnetic field H is applied in the xz-plane and the ISHE voltage is \nmeasured across the ~5-mm long metal  layer along the y-axis, as illustrated in Fig. 1c. At \nresonan ce, the precessing YIG magnetization  (M) transfers angular momentum  to the \nconduction  electrons in the normal metal. The resulting  pure spin current  Js is injected into \nthe metal layer along the z-axis with spin polarization 𝜎 parallel to M and then converted  to a \ncharge current  Jc  SHJs𝜎 by the ISHE  via the spin-orbit interaction .  \nFigure  2 show s VISHE vs. H spectra of the six YIG/metal  bilayers  at θH = 90 and 270 \n(both with in -plane field)  at Prf = 200  mW. For YIG/Ta and YIG /W bilayer s, |VISHE| exceeds  5 \nmV (1 mV/mm) . For YIG /Pt, YIG/Au and YIG /Ag, VISHE = 2.10 mV, 72.6 V and 1. 49 V, \nrespectively . Due to the opposi te sign s of SH, Pt, Au and Ag  give positive VISHE while Ta and \nW give negative VISHE at θH = 90. This agrees with  the predicted signs of SH [31, 32] of the \nmetals . When  H is reversed from  θH = 90 to 270, all the VISHE signal s change sign  as \nexpected from the ISHE . The rf-power dependenc ies of VISHE are shown in the upper insets to \nFigs. 2a -2f at θH = 90, each of  which  shows a linear dependence, indicat ing that the 5 \n observed spin pumping signals are in the linear regime.  Furthermore, the large spin pumping \nsignals provided by the YIG films e nable the observation of  VISHE = 0.99 V in the YIG/Cu \nbilayer  (Fig. 2f) . Due to the much weak er spin-orbit coupling  [24] in Cu compared to th ose \n5d metals , there is no previous report of ISHE detection of spin pumping in FM/Cu structure s. \nThis first observation of VISHE in YIG/Cu enables the determination of spin Hall angle  in Cu . \nWhen  H is rotated from in -plane to out -of-plane , M remains essentially  parallel to H at \nall angles  since  the FMR resonance field  Hres (between 2500 and  5000 Oe ) always exceeds  \n4Meff (1786  Oe). The lower insets to Figs. 2a-2f show the angular dependenc ies of the \nnormalized  VISHE for the six bilayer s; all clearly exhibit the expected sinusoidal shape  (VISHE \n Js𝜎  JsM  JsH  sinθH), confirming that the observed ISHE voltage s arise from FMR \nspin pumping. Since YIG is insulating we can rule out artifacts due to thermoelectric or \nmagnetoelectric effects, such as AMR  or AHE , enabling more straightforward  measurement \nof the inverse spin H all effect than using  metallic FMs.  \nFigure s 3a-3f show the FMR derivative absorption spectr a (f = 9.65 GHz) of the \n20-nm YIG film s before and after the deposition of the metal s. The FMR linewi dths are \nclearly enhanced  in YIG/metal bilayers relative to the bare YIG films . The linewidth \nenhancement  [10, 11] is a consequence of FMR spin pumping: the coupling that transfer s \nangular momentum from YIG to the metal adds to the damping of the precessing YIG \nmagnetization , thus increas ing the  linewidth.  In order to accurately determine  the \nenhance ment of  Gilbert damping, we measured the frequency dependenc ies of the linewidth \nof a bare YIG film and the six YIG/ metal  bilayers  using  a microstrip transmission line . In all 6 \n cases the linewidth increases linear ly with frequency (Fig.  3g). The Gilbert damping  constant \n can be  obtained using [ 33], \nΔ𝐻=Δ𝐻inh+4𝜋𝛼𝑓\n√3𝛾,            (1) \nwhere Hinh is the inhomog eneous broadening  and  is the gyromagnetic ratio . Table I shows \nthe damping enhancement  sp due to spin pumping : sp=YIG/NM−YIG, where YIG/NM \nand YIG = (9.1 ± 0.6)  10-4 are obtained from the least-squares fits in  Fig. 3g .  \nThe observed ISHE  voltages depend on several materials parameters  [4, 11],  \n𝑉ISHE=−𝑒𝜃SH\n𝜎N𝑡N+𝜎F𝑡F𝜆SDtanh(𝑡N\n2𝜆SD)𝑔↑↓𝑓𝐿𝑃(𝛾ℎrf\n2𝛼𝜔)2\n,    (2) \nwhere e is the electron  charge , N (F) is the conductivity of the NM (FM), tN (tF) is the \nthickness  of the NM (FM) layer, 𝑔↑↓ is the interfacial  spin mixing  conductance ,  = 2f is \nthe FMR frequency , L is the sample length,  and hrf = 0.25 Oe [34] in our FMR cavity at Prf = \n200 mW. The factor P arises from  the ellipticity of the magnetization  precession  [10],  \n𝑃=2𝜔[𝛾4𝜋𝑀s+√(𝛾4𝜋𝑀s)2+4𝜔2]\n(𝛾4𝜋𝑀s)2+4𝜔2= 1.21       (3) \nfor all the  FMR and spin pumping  measurements. The spin mixing conductance can be  \ndetermined from the damp ing enhancement  [10-12], \n𝑔↑↓=4𝜋𝑀s𝑡F\n𝑔𝜇B(YIG/NM−YIG)         (4) \nwhere 𝑔 and 𝜇B are the  Landé  𝑔 factor  and Bohr magneton , respectively.  \nAlthough  the reported spin diffusion length var ies from a few nm to a few hundred  \nnm across the range of metals we have measured , the term 𝜆SDtanh(𝑡N\n2𝜆SD) in Eq.  (2) is rather \ninsensitive to the value of SD for a given tN (e.g., 5 nm ) due to the limitation of film \nthickness ; for example, 𝜆SDtanh(𝑡N\n2𝜆SD) = 1.70  nm for SD = 2 nm and 2.50  nm for SD =  7 \n [10]. In this calculation, w e assume 𝜆SD = 10 nm for Pt  [4], 2 nm for W and Ta [25], 60 nm \nfor Au  [24], 700 nm for Ag  [26] and 500  nm for Cu  [24]. Electrical conduction in YIG can be \nneglected.  From  Eqs. (2)-(4), 𝑔↑↓ can be obtained from the  Gilbert damping  enhancement  \nand SH can be calculated for the six metals  (Table I) [ 35]. Consequently , the spin current \ndensity  Js can be estimated  using  [11], \n𝐽s=𝑡N𝜎N+𝑡F𝜎F\n𝜃SH𝜆SDtanh(𝑡N\n2𝜆SD)𝑉ISHE\n𝐿.          (5) \nThe power of inverse spin Hall effect as a probe of spin pumping calls for a \nquantitative understanding to enable more precise and detailed experiments.  Spin Hall angles  \nhave been measured in several normal metals  by spin Hall or ISHE measurements, mostly \nusing metallic FMs [10, 14, 32, 35]. Due to the impact of AMR or AHE in electrical ly \nconducting  metallic FMs in the heterostructures , and the variation of sample quality among \ndifferent groups, the reported values of SH vary significantly for the same materials, in some \ncases, by more  than one order of magnitude [ 14]. Here, we  report  measurements of the spin \nHall angle  for various 3d, 4d and 5 d metals  using  Eq. (2) from  the large ISHE signals and, \nindependently, spin -pumping enhancement of Gilbert damping  (to obtain 𝑔↑↓ and ) of the \ninsulating YIG thin film . This set of experimental data can be compared to discern trends and \nuncover the roles of various materials parameters  in spin -orbit coupling , including the atomic \nnumber as well as d-electron co unt in transition metals [ 23, 31]. We first show in Fig. 4a the \nlinear dependence of SH on Z4 for Cu, Ag and Au , reflecting the key role of atomic number \nin spin-orbit coupling  [23] and  spin Hall physics  in metals having a particular  d-electron \nconfiguration . We note that  the spin Hall angles of the four 5 d metals do not vary as Z4 at all , 8 \n indicating that  the d-orbital  filling plays the dominant  role [ 31]. Figure 4b shows SH vs. Z for \nTa, W, Pt and Au , which matches  well with the theoretical calculations by Tanaka et al.  [31], \nincluding the sign change and relative magnitude of spin Hall effect  in 5d metals . These two \nresults highlight and distinguish the roles of  atomic number  and d-orbital filling  in spin Hall \nphysics in transition metals , and clarify their relative importance . \nIn conclusion , FMR spin pumping measurements on YIG/NM bilayers  give mV-level \nISHE voltages in YIG /Pt, YIG/Ta  and YIG/W  bilayer s and robust spin pumping signals in \nYIG/Cu  and YIG/Ag . YIG/NM interfacial spin mixing conductance s are determined by the \nenhanced Gilbert damping  which are measured by frequency dependence  of FMR linewi dth \nbefore and after the deposition  of metals.  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ISHE voltages  at f = 9.65 GHz and Prf = 200 mW , FMR linewidth changes  at f = \n9.65 GHz , Gilbert damping enhancement  due to spin pumping sp=YIG/NM−YIG \n(YIG= 9.1 ± 0.6  10-4) and resistivity of the six YIG/metal bilayers , and the calculated \ninterfacial spin mixing conductance, spin Hall angle, and spin current density for each metal . \n \nBilayer  VISHE H \nchange  sp  ( m) 𝑔↑↓(m-2) SH 𝐽𝑠 (A/m2) \nYIG/Pt  2.10 \nmV 24.3 Oe  (3.6 ± 0.3) \n 10-3 4.8  10-7 (6.9 ± 0.6) \n 1018 0.10 ± 0.01 (2.0 ± 0.2) \n 107 \nYIG/Ta  -5.10 \nmV 16.5 Oe (2.8 ± 0.2) \n 10-3 2.9 10-6 (5.4 ± 0.5) \n 1018 -0.069 ± 0.006 (1.6 ± 0.2) \n 107 \nYIG/W  -5.26 \nmV 12.3 Oe (2.4 ± 0.2) \n 10-3 1.810-6 (4.5 ± 0.4) \n 1018 -0.14 ± 0.01 (1.4 ± 0.1) \n 107 \nYIG/Au  72.6 \nV 5.50 Oe (1.4 ± 0.1) \n 10-3 4.9 10-8 (2.7 ± 0.2) \n 1018 0.084 ± 0.007 (7.6 ± 0.7) \n 106 \nYIG/Ag  1.49 \nV 1.30 Oe (2.7 ± 0.2) \n 10-4 6.6  10-8 (5.2 ± 0.5) \n 1017 0.0068 ± 0.0007 (1.5 ± 0.1) \n 106 \nYIG/Cu  0.99 \nV 3.70 Oe (8.1 ± 0.6) \n 10-4 6.3 10-8 (1.6 ± 0.1) \n 1018 0.0032 ± 0.0003 (4.6 ± 0.4) \n 106 \n \n  14 \n Figure Captions:  \nFigure 1.  (a) Semi -log θ-2θ XRD scan of a 20-nm thick YIG film near the YIG (444) peak  \n(blue line) , which exhibits clear Laue oscillations corresponding to the film thickness. Left \ninset: rocking curve of the YIG film  measured at 2θ = 50.639 for the first satellite peak  \n(green arrow)  to the left of the main YIG  (444) peak  gives a FWHM of 0. 0092. Righ t inset: \nAFM image of the YIG film with a roughness of 0.15 nm.  (b) A representative \nroom -temperature FMR derivative spectrum of a 20 -nm YIG film with an in -plane field  at Prf \n= 0.2 mW, which gives a peak -to-peak linewidth of 11.7 Oe. (c) Schematic of experimental \nsetup for ISHE voltage measurements.  \nFigure 2 .  VISHE vs. H spectra of (a) YIG/ Pt, (b) YIG/Ta, (c) YIG/W, (d) YIG/Au, (e) \nYIG/Ag, and (f) YIG/Cu  bilayers  at θH = 90(red) and 270 (blue)  using  Prf = 200mW . Top \ninsets: rf-power dependencies of the corresponding  VISHE at θH = 90. Bottom insets: a ngular \ndependenc ies (θH) of VISHE normalized by the magnitude of VISHE at θH = 90, where t he \ngreen  curves are sin θH for Pt, Au, Ag, Cu, and -sinθH for Ta and W.  \nFigure 3 .  FMR derivative absorption spectr a of the 20-nm YIG films before  (blue)  and \nafter (red)  the deposition at f = 9.65 GHz of (a) Pt, (b) Ta, (c) W, (d) Au, (e) Ag, and (f) Cu. (g) \nFrequency dependence of peak -to-peak FMR linewi dth of a bare YIG film and the six \nYIG/ metal  bilayers . \nFigure 4.  (a) Spin Hall angles as a function of Z4 for Cu, Ag and Au , reflecting the Z4 \ndependence of SH for noble metals with the same d-orbital filling . (b) Spin Hall angles for 5d \ntransition metals Ta,  W, Pt and Au, of which b oth the signs and relative magnitudes agree \nwell with the theoretical predictions  in Ref. 3 1.   15 \n  \n \nFigure 1.  \n  \n100102104\n49 50 51 52 53Intensity (c/s)\n2 (deg)(a)\nGGG(444)YIG(444)\n25.6 25.65\n (deg)FWHM=\n0.0092o\n2400 2600 2800-2-1012dIFMR/dH (a.u.)\nH (Oe)H = 11.7 Oe\nPrf = 0.2 mW(b)\n(c)\nroughness: \n0.15 nm16 \n  \n \nFigure 2.  \n  \n-2000-1000010002000VISHE (V)\n(a)YIG/Pt\nH = 90o\nH = 270o010002000\n0100 200Prf (mW)\n-6000-4000-20000200040006000VISHE (V)\n(b)YIG/Ta\nH = 90oH = 270o\n-6000-4000-20000\n0100 200Prf (mW)\n-6000-4000-20000200040006000\n-150 -100 -50 0 50 100\nH - Hres (Oe)VISHE (V)\n(c)YIG/W\nH = 90oH = 270o\n-6000-4000-20000\n0100 200Prf (mW)-80-4004080VISHE (V)\n(d)YIG/Au\nH = 90o\nH = 270o0204060\n0100 200Prf (mW)\n-1.5-1-0.500.511.5\n(e)YIG/Ag\nH = 90o\nH = 270oVISHE (V)00.511.5\n0100 200Prf (mW)\n-1-0.500.51\n-150 -100 -50 0 50 100\nH - Hres (Oe)VISHE (V)\n(f)YIG/Cu\nH = 90o\nH = 270o00.51\n0100 200Prf (mW)-101\n0180 360\nH (deg)\n-101\n0180 360\nH (deg)\n-101\n0180 360H (deg)-101\n0180 360\nH (deg)\n-101\n0180 360H (deg)\n-101\n0180 360H (deg)17 \n  \n \nFigure 3 . \n  \n-2-1012\n(a)YIG\nYIG/Pt\n-2-1012dIFMR/dH (a.u.)\n(b)YIG\nYIG/Ta\n-2-1012\n-60-40-20 02040\nH - Hres (Oe)(c)YIG\nYIG/W(d)YIG\nYIG/Au\n(e)YIG\nYIG/Ag\n-40-20 0204060\nH -Hres (Oe)(f)YIG\nYIG/Cu\n01020304050\n0 5 10 15 20H (Oe)\nf (GHz)YIG/Pt\nYIG/Ta\nYIG/W\nYIG/Au\nYIG/Cu\nYIG/Ag\nYIG\n(g)18 \n  \n \nFigure 4. \n \n00.040.08\n01x1072x1073x1074x107SH\nZ 4(a)\nCuAgAu\n-0.100.1\n73747576777879\nZSH\nTa\nWPtAu (b)"    },    {        "title": "1911.09400v1.Low_damping_and_microstructural_perfection_of_sub_40nm_thin_yttrium_iron_garnet_films_grown_by_liquid_phase_epitaxy.pdf",        "content": " 1 Low damping and microstructural perfection of sub-4 0nm-thin \nyttrium iron garnet films grown by liquid phase epi taxy \n \nCarsten Dubs, 1* Oleksii Surzhenko, 1 Ronny Thomas, 2 Julia Osten, 2 Tobias Schneider, 2 Kilian Lenz, 2 \nJörg Grenzer, 2 René Hübner, 2 Elke Wendler 3 \n \n1 INNOVENT e.V. Technologieentwicklung, Prüssingstr.  27B, 07745 Jena, Germany \n2 Institute of Ion Beam Physics and Materials Resear ch, Helmholtz-Zentrum Dresden-Rossendorf, \nBautzner Landstr. 400, 01328 Dresden, Germany \n3 Institut für Festkörperphysik, Friedrich-Schiller- Universität Jena, Helmholtzweg 3, 07743 Jena, \nGermany \n* Correspondence: cd@innovent-jena.de  \n \nThe field of magnon spintronics is experiencing an increasing interest in the development of \nsolutions for spin-wave-based data transport and pr ocessing technologies that are complementary or \nalternative to modern CMOS architectures. Nanometer -thin yttrium iron garnet (YIG) films have \nbeen the gold standard for insulator-based spintron ics to date, but a potential process technology tha t \ncan deliver perfect, homogeneous large-diameter fil ms is still lacking. We report that liquid phase \nepitaxy (LPE) enables the deposition of nanometer-t hin YIG films with low ferromagnetic \nresonance losses and consistently high magnetic qua lity down to a thickness of 20 nm. The obtained \nepitaxial films are characterized by an ideal stoic hiometry and perfect film lattices, which show \nneither significant compositional strain nor geomet ric mosaicity, but sharp interfaces. Their \nmagneto-static and dynamic behavior is similar to t hat of single crystalline bulk YIG. We found, \nthat the Gilbert damping coefficient α is independent of the film thickness and close to 1 × 10 -4, and \nthat together with an inhomogeneous peak-to-peak li newidth broadening of ∆H0||  = 0.4 G, these \nvalues are among the lowest ever reported for YIG f ilms with a thickness smaller than 40 nm. These \nresults suggest, that nanometer-thin LPE films can be used to fabricate nano- and micro-scaled \ncircuits with the required quality for magnonic dev ices. The LPE technique is easily scalable to YIG \nsample diameters of several inches. \n \n \nI. INTRODUCTION  \n \nYttrium iron garnet (Y 3Fe 5O12 ; YIG) in the micrometer thickness range is the mat erial of choice in \nradio-frequency (RF) engineering for decades (see, e.g., Refs. [1-5]). Especially the lowest spin \nwave loss of all known magnetic materials and the f act, that it is a dielectric are of decisive \nimportance. Since one has learned how to grow YIG f ilms in the nanometer thickness range, there \nhas been a renaissance of this material, as its mag netic and microwave properties are in particular \ndemand in many areas of modern physics.  \nA growing field of application for magnetic garnets  is (i) magnonics, which deals with future \npotential devices for data transfer and processing using spin waves [1,6-9]. The significant thickness  \nreduction achieved today allows reducing the circui t sizes from classical millimeter dimensions [1] \ndown to 50 nm [10-12] . Another important field is (ii) spintronics: By in creasing the YIG surface-\nto-volume ratio as much as possible (while keeping its magnetic properties), physical phenomena, \nsuch as the inverse spin Hall effect [13], spin-tra nsfer torque [14], and the spin Seebeck effect [15]  \n(generated by a spin angular momentum transfer at t he interfaces between YIG and a nonmagnetic \nmetallic conductor layer) become much more efficien t [7,16-29].  Also (iii) the field of terahertz \nphysics, which uses ultrafast spin dynamics to cont rol ultrafast magnetism, for example for potential  2 terahertz spintronic devices [30,31,32], and (iv) t he field of low-temperature physics, which deals \nwith magnetization dynamics at cryogenic temperatur es [33 ] for prospective quantum computer \nsystems, are possible fields of applications for na nometer-thin iron garnet films. \nThere are several different techniques to grow YIG on different substrates. (i) Pulsed laser \ndeposition (PLD) is an excellent technique for fabr icating small samples of nanometer-thin YIG \nfilms with narrow ferromagnetic resonance (FMR) lin ewidths [17,19,21,22,28,34-36] whereas its \nup-scaling to larger sample dimensions of several i nches is challenging. (ii) Magnetron sputtered \nYIG usually yields wider FMR linewidths, and inhomo geneous line broadening is frequently \nobserved [37-40]. (iii) For large-scale, low-cost c hemical solution techniques, such as spin coating, \nstrongly broadened FMR linewidths and increased Gil bert damping parameters were reported \n[41,42]. (iv) Liquid phase epitaxy (LPE) from high- temperature solutions (flux melts), is a well-\nestablished technique. Since nucleation and crystal  growth take place under almost thermodynamic \nequilibrium conditions, this guarantees high qualit y with respect to narrow absolute FMR linewidths \nand a small Gilbert damping coefficient [43-45] at the same time, making LPE comparable or \nsuperior to the other growth techniques. In additio n, LPE allows YIG to be deposited in the required \nquality on 3- or 4-inch wafers [46]. This is import ant for possible applications mentioned above. \nSo far, classical LPE was applied to grow micromete r-thick samples used for magneto-static \nmicrowave devices [47,48] or for magneto-optical im aging systems [49]. The typical shortcomings \nof the LPE technology making thin-film growth so di fficult lie in the fact, that, due to high growth \nrates, nanometer-thin films were technologically di fficult to access. The etch-back processes in high-\ntemperature solutions or interdiffusion processes a t the substrate/film interface at high temperatures  \nusually prevent sharp interfaces. In addition, film  contamination by flux melt constituents (if it is not \na self-flux without foreign components) is unavoida ble in most cases. Nevertheless, it was recently \ndemonstrated, that epitaxial films of 100 nm or thi nner are also accessible with this technique \n[50,51].  \nIn this study, we will show that we are able to dep osit nanometer-thin YIG LPE films with low FMR \nlosses and consistently high magnetic quality down to a thickness of 20 nm. There is no thinnest \n\"ultimate\" thickness for iron garnet LPE films, as it is sometimes claimed. \nIt should be pointed out, that, in addition to the damping properties, magnetic anisotropy \ncontributions as a function of the sample stoichiom etry and film/substrate pairing are also of great \nimportance, since they determine the static and dyn amic magnetization of the epitaxial iron garnet \nfilms and thus their possible applications. For exa mple, large negative uniaxial anisotropy fields \nwere usually observed for garnet films under compre ssion, such as for YIG on gadolinium gallium \ngarnet (Gd 3Ga 5O12 ; GGG) or other suitable substrates with smaller latt ice parameters grown by gas \nphase deposition techniques (see e.g. Refs. [35,36, 52-57]), which favors in-plane magnetization. \nLarge perpendicular magnetic anisotropies, on the o ther hand, can be found for films under tensile \nstrain, e.g. on substrates with larger lattice para meter or for rare earth iron garnet films with smal ler \nlattice parameter than GGG (see e.g. Refs. [58-62] ). Between these two extremes are YIG LPE \nfilms, which are usually grown on standard GGG subs trates and exhibit small tensile strain if no \nlattice misfit compensation, e.g. by La ion substit ution [50,63], has been performed. Such films are \ncharacterized  by a small uniaxial magnetic anisotropy and dominan t shape anisotropy when no \nlarger growth–induced anisotropy contributions due to Pb or Bi substitution occurs [64]. \nHowever, only little information about the structur al properties and the thickness-dependent \nmagnetic anisotropy contributions of nanometer-thin  LPE films has been published so far, which is \nwhy we are concentrating on these properties for YI G films with thicknesses down to 10 nm. This \nallowed us to describe the intrinsic damping behavi or over a wide frequency range and to determine \na set of magnetic anisotropy parameters for all inv estigated films. \n \n  3 II. EXPERIMENTAL DETAILS \n \nNanometer-thin YIG films were deposited on 1-inch ( 111) GGG substrates by LPE from PbO-B 2O3-\nbased high-temperature solutions (HTL) at about 865 °C using the isothermal dipping method (see \ne.g. [65]). Nominally pure Y 3Fe 5O12  films with smooth surfaces were obtained within on e minute \ndeposition time on horizontally rotated substrates with rotation rates of 100 rpm. The only variable \ngrowth parameter for all samples in this study was the degree of undercooling ( ∆T = TL-Tepitaxy ) that \nwas restricted to ∆T ≤ 5 K to obtain films with thicknesses between 10 an d 110 nm. Here TL is the \nliquidus temperature of the high-temperature soluti on and Tepitaxy  is the deposition temperature. After \ndeposition, the samples were pulled out of the solu tion followed by a spin-off of most of the liquid \nmelt remnants at 1000 rpm, pulled out of the furnac e and cooled down to room temperature. \nSubsequently, the sample holder had to be stored wi th the sample in a diluted, hot nitric-acetic-acid \nsolution to remove the rest of the solidified solut ion residues. Finally, the reverse side YIG film of  \nthe doubled-sided grown samples was removed by mech anical polishing and samples were cut into \nchips of different sizes by a diamond wire saw. The  film thicknesses were determined by X-ray \nreflectometry (XRR) and by high-resolution X-ray di ffraction (HR-XRD) analysis, and the latter \ndata were used to calculate anisotropy and magnetiz ation values. \nAtomic force microscopy (AFM) using a Park Scientif ic M5 instrument was carried out for each \nsample at three different regions over 400 µm2 ranges to determine the root-mean-square (RMS) \nsurface roughness. \nThe XRR measurements were carried out using a PANan alytical/X-Pert Pro system. For the HR-\nXRD investigations, a Seifert-GE XRD3003HR diffract ometer using a point focus was equipped \nwith a spherical 2D Göbel mirror and a Bartels mono chromator on the source side. Both systems use \nCu Kα1 radiation. Reciprocal space maps (RSMs) were measu red with the help of a position-sensitive \ndetector (Mythen 1k) at the symmetric (444) and (88 8) as well as the asymmetric (088), (624), and \n(880) reflections. To obtain the highest possible a ngular resolution for symmetric θ−2 θ line scans, a \ntriple-axis analyzer in front of a scintillation co unter was installed on the detector. Using a recurs ive \ndynamical algorithm implemented in the commercial p rogram RC_REF_Sim_Win [66], the vertical \nlattice misfits were calculated.  \nRutherford backscattering spectrometry (RBS) was ap plied to investigate the composition of the \ngrown YIG films using 1.8 MeV He ions and a backsca ttering angle of 168°. Backscattering events \nwere registered with a common Si detector. The ener gy calibration of the multichannel analyzer \nrevealed 3.61 keV per channel. A thin carbon layer was deposited on top of the samples to avoid \ncharging during analysis. The samples were tilted b y 5° with respect to the incoming He ion beam \nand rotated around the axis perpendicular to the sa mple surface in order to obtain reliable random \nspectra. The analysis of the measured spectra was p erformed by a home-made software [67] based \non the computer code NDF [68] and then enables the calculation of the RBS spectra. The measured \ndata were fitted by calculated spectra to extract t he film composition. In this way, the Fe-to-Y ratio  \nof the films was determined. Because of the low mas s of oxygen, the O signal of the deposited films \nis too low for quantitative analysis. \nHigh-resolution transmission electron microscopy (H R-TEM) investigations were performed with \nan image C s-corrected Titan 80-300 microscope (FEI) operated a t an accelerating voltage of 300 kV. \nHigh-angle annular dark-field scanning transmission  electron microscopy (HAADF-STEM) imaging \nand spectrum imaging analysis based on energy-dispe rsive X-ray spectroscopy (EDXS) were done \nat 200 kV with a Talos F200X microscope equipped wi th a Super-X EDXS detector system (FEI). \nPrior to TEM analysis, the specimen mounted in a hi gh-visibility low-background holder was placed \nfor 10 s into a Model 1020 Plasma Cleaner (Fischion e) to remove possible contaminations. Classical \ncross-sectional TEM-lamella preparation was done by  sawing, grinding, polishing, dimpling, and  4 final Ar-ion milling. Quantification of the element  maps including Bremsstrahlung background \ncorrection based on the physical TEM model, series fit peak deconvolution, and application of \ntabulated theoretical Cliff-Lorimer factors as well  as absorption correction was done for the \nelements Y (K α line), Fe (K α line), Gd (L α line), Ga (K α line), O (K line), and C (K line) using the \nESPRIT software version 1.9 (Bruker). \nThe ferromagnetic resonance (FMR) absorption spectr a were taken on two different setups. The \nfrequency-swept measurements were recorded on a Roh de & Schwarz ZVA 67 vector network \nanalyzer attached to a broadband stripline. The YIG /GGG sample was mounted face-down on the \nstripline, and the transmission signals S21  and S12  were recorded using a source power of -10 dBm (= \n0.1 mW). The microwave frequency was swept across t he resonance frequency fres , while the in-\nplane magnetic field H remained constant. Each recorded frequency spectru m was fitted by a \nLorentz function and allowed us to define the reson ance frequency fres  and the frequency linewidth \nΔfFWHM  corresponding to the applied field H = Hres . \nIn addition, field-swept measurements were carried out with another setup using an Agilent E8364B \nvector network analyzer and an 80-µm-wide coplanar waveguide. Again, the microwave \ntransmission parameter S21  was recorded as the FMR signal. This time, the mic rowave frequency \nwas kept constant and the external magnetic field w as swept through resonance. This facilitates \ntracking the FMR signals over large frequency range s. The microwave power was set to 0 dBm (= \n1 mW). In addition, this setup allowed for azimutha l and polar angle-dependent measurements to \ndetermine the anisotropy and damping contributions in detail. The FMR spectra were fitted by a \ncomplex Lorentz function to retrieve the resonance field Hres  and field-swept peak-to-peak linewidth \nΔHpp . By fitting the four sets of resonance field data,  i.e. (i) the in-plane and (ii) the perpendicular-\nto-plane frequency dependence as well as (iii) the azimuthal and (iv) polar angular dependences at f \n= 10 GHz, with the resonance equation for the cubic (111) garnet system, a consistent set of \nanisotropy parameters was determined for each sampl e. In addition, the damping parameters and \ncontributions were determined from the frequency- a nd angle-dependent linewidth data. \nThe vibrating sample magnetometer (VSM, MicroSense LLC, EZ-9) was used to measure the \nmagnetic moments of the YIG/GGG samples magnetized along the YIG film surface. The external \nmagnetic field H was controlled within an error of ≤0.01 Oe. To est imate the volume magnetization \nM of the YIG films, the raw VSM signal was corrected  from background contributions (due to the \nsample holder and the GGG substrate) and normalized  to the YIG volume. The Curie temperatures \nTC for the YIG samples were determined by zero-extrap olation of the temperature dependencies M \n(H=const, T) measured in small in-plane magnetic fields. In or der to verify the Curie temperatures \nmeasured by VSM, a differential thermal analysis of  a 0.55 mm thick YIG single crystal slice was \ncarried out and then used as a reference sample for  the VSM temperature calibration. \n \n \nIII. RESULTS AND DISCUSSION \n \nA. Microstructural properties of nanometer-thin YIG  films \n \nThe thickness values reported in this study are der ived from the Laue oscillations observed in the θ-\n2θ patterns of the high-resolution X-ray diffraction (HR-XRD) measurements and are confirmed by \nX-ray reflectivity (XRR) measurements (see Fig. 1).  The differences between both methods for \ndetermining the film thickness are in the range of ±1 nm. The surface roughness of the films, \nmeasured by atomic force microscopy (AFM) reveals R MS values ranging between 0.2 and 0.4 nm, \nindependent of the film thickness. Sometimes, howev er, partial remnants of dendritic overgrowth  5 increase the surface roughness to RMS values above 0.4 nm for inspection areas larger than 400 µm2 \n(see, e.g., the disturbance in the top right corner  of the AFM image inset in Fig. 1). \n-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 10 010 110 210 310 410 510 610 710 810 910 10 \n  \n t = 10.6 nm Intensity (arb.units) \nIncidence angle [deg] 20 \nµm15 \n10 \n5\n010 15 20 5 0µmnm  1.5 \n 1.0 \n 0.5 \n 0.0 \n-0.5 \n-1.0 \n-1.5 \n t = 21.5 nm  t = 30.9 nm  t = 43.2 nm \n \nFIG. 1. XRR plots of LPE-grown YIG films of differe nt thicknesses. Solid lines correspond to the \nexperimental data, while dashed lines represent the  fitted curves. The spectra shifted vertically for \nease of comparison. The inset shows an AFM image of  the surface topography of the 11 nm YIG \nfilm with a RMS roughness of 0.4 nm. \n \n \n1. Epitaxial perfection studied by high-resolution X-ray diffraction  \n \nCombined high-resolution reciprocal space map (HR-R SM) investigations around asymmetric and \nsymmetric Bragg reflections are useful to evaluate the intergrowth relations of epitaxial films on \nsingle-crystalline substrates as well as to disting uish between lattice strain induced by the film \nlattice distortion or compositional changes due to stoichiometric deviations. \n33.10 33.15 33.20 46.0 46.5 47.0 47.5 48.0 48.5 49.0 \nQx (nm -1 )Qz (nm -1 )\n(088) \nYIG (GGG) \nt = 21 nm 5E0 5E1 5E2 5E3 5E4 5E5 5E6 Intensity (a) \n-0.1 0 0.1 34.5 35.0 35.5 36.0 \nQx (nm -1 )Qz (nm -1 ) (444) \n  6 0 20 40 60 80 100 46810 \n118.5 119.0 119.5 120.0 10 110 210 310 410 510 6\n  t  =     9 nm \n  t   =   11 nm \n  t   =   21 nm \n  t   =   30 nm \n  t   =   42 nm \n  t   = 106 nm     (888) \nYIG / GGG \nScattering angle θ− 2θ (deg) Intensity (counts) (b) \nOut-of-plane misfit \n-δd⊥\nfilm  [10 -4]\nFilm thickness (nm) \n \nFIG. 2. (a) Combined high-resolution reciprocal spa ce maps around the asymmetric YIG/GGG \n(088) Bragg reflection of a 21-nm-thin single-cryst alline YIG LPE film. The inset shows the \ncorresponding symmetric YIG/GGG (444) peak: measure ments were carried out using a position-\nsensitive detector. (b) HR-XRD triple-axis θ–2θ scans around the symmetric YIG/GGG (888) peak \nfor various film thicknesses. The inset shows the ( vertical) out-of-plane misfit vs. film thickness (t he \nsolid line is a guide to the eyes).  \n \nFigure 2(a) shows the HR-RSM of the 21 nm YIG film grown on GGG (111) substrate, measured at \nthe asymmetric (088) reflection in steep incidence,  indicating that both the film and the substrate \nBragg peak positions are almost identical. Besides the nearly symmetrical intensity distribution \nalong the [111]  out-of-plane direction (i.e. the Qz axis), there is only very weak diffuse scattering \nclose to the Bragg peak visible, pointing towards a  nearly perfect crystal lattice without significant  \ncompositional strain or geometric mosaicity. In add ition, no shift of in-plane (the Qx axis) film \nBragg peak position with respect to the substrate i s observed. This behavior indicates a fully straine d \npseudomorphic film growth with a perfect coherent i n-plane lattice match with the GGG substrate. \nThe pattern of the diffuse scattering observed alon g the Qx axis of the symmetric (444) reflection \n(inset in Fig. 2(a)) is very similar to the one fou nd for a comparable GGG substrate (not shown), \nindicating that the defect structure of the system is mainly defined by the substrate and/or substrate  \nsurface. Within the experimental error of ∆Q/Q ∼ 5×10 −6 nm -1 of the high-resolution diffractometer, \nthe same performance was found for all investigated  LPE films with thicknesses below 100 nm, \nclearly demonstrating coherent YIG film growth with out signs of film relaxation. \nHigh-resolution triple-axis coupled θ–2θ scans at the (888) and (444) symmetrical reflectio ns \n(angular accuracy better than 1.5\") were carried ou t to define the strain and film thicknesses of the \nYIG films. Figure 2(b) shows the results obtained a t the (888) reflection. Under these conditions, the  \nBragg reflection of the 106 nm thick YIG layer is c learly visible as a shoulder of the (888) GGG \nsubstrate reflection at higher diffraction angles a nd this indicates a smaller out-of-plane value for the \nlattice parameter d888  than for the GGG substrate. This is characteristic  for tensely stressed “pure” \nYIG LPE films [50,63]. For LPE films with a thickne ss significantly less than 100 nm, however, \nonly simulations can provide the structural paramet ers. For this reason, the diffracted signals shown \nin Fig. 2(b) were simulated and fitted. Using the b est fit of both, the (444) and (888) reflections, t he \nout-of-plane lattice misfit values ( )⊥ ⊥ ⊥ ⊥ ⊥ ⊥∆ − = − = QQ d d d d / /substrate substrate film filmδ  were determined (see \nTable 1). Assuming a fully pseudomorphic [111]-orie nted system, the in-plane stress of the YIG \nfilm can be calculated by σ′|| = -2c 44 δd⊥\nfilm  (see the Supplemental Material [69 ] for a detailed \nderivation and references therein [61,70,71]). The in-plane biaxial ε|| and out-of-plane uniaxial ε⊥  7 strains can be calculated as well using the stiffne ss tensor components c11 , c12 , and c44  for which we \nuse averaged values taken from [72,73] (see also Su pplemental Material [69 ]). The resulting \nparameters are listed in Table I. \nThe inset in Fig. 2(b) shows the out-of-plane misfi t as a function of the film thickness. A weak \nmonotonous increase of ⊥\nfilmdδ with decreasing film thickness is observed between 106 nm and \n21 nm. The same behavior was reported by Ortiz et al . [61] for compressively strained EuIG and \nTbIG PLD-grown films with film thicknesses down to 4 nm and 5 nm, respectively. However, for \nour thinnest LPE films with t ∼ 10 nm, the out-of-plane misfit rapidly drops. Such  a significant \nchange of the misfit with respect to the film thick ness was only mentioned for considerably \ncompressively strained YIG PLD films by O. d’Allivy  Kelly et al.  [17]. They assume, that this \neffect indicated a critical film thickness (below 1 5 nm) for strain relaxation, but did not explain it  in \ntheir letter. \nFor semiconductor LPE films, however, it is known, that interdiffusion processes at the \nfilm/substrate interfaces can generate continuous c omposition profiles in the diffusion zone without \nabrupt changes in the lattice parameters, which lea d to modified stress profiles depending on the \nthickness of the epilayers (see e.g. [74]). A possi ble explanation for the observed behavior could \ntherefore be the presence of a smoothly changing la ttice parameter value in the interface region.  \nSuch composition profiles have recently been discus sed for YIG films grown on GGG substrates by \nhigh-temperature and long-time laser MBE deposition  experiments [75] , and transition layer \nthicknesses have been modeled based on polarized ne utron and X-ray reflectometry techniques. The \nprobability of the existence of such a thin continu ous transition layer and its influence on the \nmagneto-static film properties will be discussed be low. \n \nTABLE I. Structural parameters of the YIG LPE films  grown on GGG (111) substrates: film \nthickness t measured by HR-XRD, RMS roughness obtained by AFM,  vertical lattice misfit ⊥\nfilmdδ  \nobtained by HR-XRD, in-plane strain ε|| and out-of-plane strain ε⊥ and the resulting  in-plane stress σ′||. \n \nt \n(nm)  roughness  \n(nm)  δd⊥\nfilm   \n×10 -4 ε|| \n×10 -4 ε⊥ \n×10 -4 σ′|| \n×10 8 Pa  \n9 - -4.3 2.3 -2.0 0.7 \n11 0.4 -6.1 3.3 -2.8 0.9 \n21  0.2 -10.4 5.6 -4.8 1.6 \n30 0.2 -9.4 5.1 -4.3 1.4 \n42  0.3 -9.2 5.0 -4.2 1.4 \n106  0.4 -8.5 4.6 -3.9 1.3 \n±1 ±0.1 ±0.7 ±0.4 ±0.4 ±0.1 \n \n \n2. Chemical composition studied by Rutherford Backs cattering spectrometry \n \nBesides the epitaxial perfection, the chemical comp osition of the films is of interest to estimate \ndeviations from the ideal Y 3Fe 5O12  stoichiometry and to detect impurity elements. The refore, RBS \nmeasurements were performed for selected LPE films.  As an example, Fig. 3 shows the random \nspectrum of a 30 nm thick YIG film on GGG substrate . The inset presents the main part of the \nspectrum. Applying the NDF software, the computed c urve (solid line) matches perfectly the \nexperimental one (symbols). This enables us to dete rmine the Fe:Y ratio. As for all investigated LPE \nfilms, the Fe:Y ratio was determined to be R = 1.67, which corresponds to the ideal iron garnet   8 stoichiometry with Fe:Y = 5:3. At higher magnificat ions of the backscattering yield in Fig. 3, a very \nlow intensity signal can be observed at ion energie s higher than for backscattering on gadolinium \natoms from the GGG substrate. Although the intensit y is rather low, it can be attributed to heavy \nimpurity elements present to a very low amount over  all in the YIG film. We assign this signal to  \nlead and platinum. These elements may come from the  solvent and the crucible during the \ndeposition of the YIG film. After background correc tion a total quantity of (0.08 ± 0.02) at.% for the \nsum of both elements could be determined. This corr esponds to 0.01 < x + y < 0.02 formula units of \nthe nominal film composition (Y 3-x-yPb xPt y)(Fe 5-x-yPb xPt y)O 12 . In a first approximation, for the \ncalculation of the RBS spectra, it was assumed, tha t both elements contribute in equal parts to the \nhigh-energy signal. So, the calculated spectrum tak es into account the existence of 0.04 at.% lead \nand 0.04 at.% platinum within the YIG layer. This y ields a good representation of the separated \nsignal for these two elements. \n1550 1600 1650 1700 1750 1800 0100 200 300 400 \n  \n measurement       simulation \n separated Pb + Pt signal Backscattering yield  (counts) \nIon energy  (keV) YIG/GGG  \nt = 30 nm \nGd signal \nfrom GGG \nsubstrate 1000 1500 02000 4000 6000 8000 \n   \n Gd Ga YFe \n \nFIG. 3. Energy spectrum of 1.8 MeV He ions backscat tered on the YIG/GGG sample with a YIG \nfilm thickness of t = 30 nm. The inset shows the main part of the spec trum with the edges of the \nsubstrate elements Gd and Ga and the Fe and Y peak from the YIG film. \n \n \n3. Crystalline perfection studied by high-resolutio n transmission electron microscopy \n \nTo analyze the film lattice perfection as well as t he heteroepitaxial intergrowth behavior, HR-TEM \ninvestigations were performed. A cross-sectional im age of an 11 nm thin YIG film on a GGG \nsubstrate makes it possible to visualize both, the entire YIG film volume up to the film surface and \nthe interface in a magnified HR-TEM microscope imag e (see Fig. 4(a)). Besides the perfect \nfilm/substrate interface, neither structural lattic e defects nor significant misalignment could be \nobserved in the coherently strained YIG film lattic e up to the film surface. \nTo prove the homogeneity of the bulk composition an d the performance of the film/substrate \ninterface, HAADF-STEM imaging (Fig. 4(b)) together with element mapping, based on EDXS \nanalysis (Figs. 4(c)-(g)), were performed. The corr esponding HAADF-STEM image in Fig. 4(b) \nallows clearly resolving the film/interface region due to the significant difference of the atomic \nnumber contrast. Because of the uniform spatial dis tribution of both, the film (Y, Fe, O) and the \nsubstrate elements (Gd, Ga, O), which are independe ntly represented by different colors in Figs.  9 4(c)-(g), a homogeneous composition over the entire  YIG film can be confirmed. Small brightness \nvariations within the element maps (on the right ha nd side) result from slight thickness variations of  \nthe classically prepared TEM lamella. Neither an in termixing of the substrate nor of the film \nelements at the YIG/GGG interface is observed in th e element maps within the EDXS detection \nlimit, which is estimated to be slightly below 1 at .-% for the measuring conditions used. For that \nreason, tiny Pb and Pt contributions in the YIG fil m, as shown by RBS (see Fig. 3), where not \ndetected here. \nTo evaluate the lateral element distributions acros s the film near the film/substrate interface, \nquantified line scans were performed as presented i n Fig. 4(h). Using the 10%-to-90% edge \nresponse criterion, it shows a  transition width of (1.9 ± 0.4) nm at the interface . This is lower than \nthe observed 4-6 nm non-magnetic dead layer reporte d for YIG films deposited by RF magnetron \nsputtering [76], and the about 4 nm or the 5–7 nm d eep Ga diffusion observed for PLD [77] or laser \nmolecular beam epitaxy (MBE) [75], respectively . However, at some positions of the sample’s \ncross-section we found a reduced YIG film thickness  on a wavy GGG surface (not shown), which \nwe attribute to a possible etch-back of the substra te at the beginning of film growth or an already \nexisting wavy substrate surface. For further growth  experiments, a careful characterization of the \nsubstrate surfaces by AFM should, therefore, be per formed. The TEM investigations show, that the \nLPE technology is suitable for growing nanometer-th in YIG films without lattice defects and \nwithout significant interdiffusion at the film/subs trate interface, which are necessary preconditions \nfor undisturbed spin-wave propagation and low ferro magnetic damping losses. \n \n \nGGG YIG resist (a) \n  10  0 2 4 6 8 10 12 14 16 18 Composition \n(at.-%) \nDistance (nm) \n(c) Y \n(d) Fe \n(e) Gd \n(f) Ga \n(g) O \n(h) \nGd 3Ga 5O12 Y3Fe 5O12 surface \ninterface \n(b) HAADF Line scan \nY\nFe \nGd \nGa \nO\n \n \nFIG. 4. (a) Cross-sectional high-resolution TEM ima ge of the 11-nm-thin YIG/GGG (111) film. The \narrows mark the YIG/GGG interface. (b) HAADF-STEM i mage highlighting the well-separated \nYIG/GGG interface. (c-g) EDXS element maps of the 1 1-nm-thin YIG/GGG (111) film cross-\nsection. (h) Line scan as marked in (b) of the elem ental concentrations across the film thickness.  \n \n \nB. Static and dynamic magnetization characterizatio n of nanometer-thin YIG films \n \nAfter gaining insight into the YIG film microstruct ure, we want to link these properties to the FMR \nperformance to find out, which of them plays an ess ential role in the observed magneto-static and \ndynamic behavior. Therefore, FMR measurements were carried out within a frequency range of 1 to \n40 GHz, with the external magnetic field either par allel to the surface plane of the sample along the \nH || [11-2] film direction or perpendicular to it ( H || [111 ]). In addition, angle-dependent \nmeasurements, i.e., varying the angle θH of the external magnetic field (polar angular depe ndence, \nwhere θH = 0 is the sample’s normal [111 ] direction) or the azimuth angle φH (in-plane angular \ndependence, where φH = 0 is the sample’s horizontal  [1-10] direction), were performed at \nf = 10 GHz. These four measurement ‘geometries’ allo w to determine Landé’s g-factor, effective \nmagnetization 4π Meff , and anisotropy fields from the resonance field de pendence and to disentangle \nthe damping contributions from the linewidth depend ence [78,79].  \nThe FMR resonance equations to fit the angle- and f requency-dependencies (see eqs. (S22), (S23) in \nthe Supplemental Material [69] for in-plane and out -of-plane bias field conditions after Baselgia et \nal.  [80 ])  are derived from the free energy density of a cubi c (111) system [81]: \n \n ( ) [ ]\n( ) ( )\n\n\n\n− + +− − − ++ − ⋅ −=\n⊥\nϕ θ θ θ θϕϕ θ θ πθ θ ϕϕ θ θ\n3sin cos sin32sin41cos31cos sin cos 2cos cos cos sin sin\n3 4 4\n42 2\n|| 22\n22\nKK K MH M F\nu sH H H s\n, (1) \n  11  where K2⊥, K2|| , and K4 are the uniaxial out-of plane, uniaxial in-plane, and cubic anisotropy \nconstants, respectively.  φ and θ are the angles of the magnetization. Angle φu allows for a rotation of \nthe uniaxial anisotropy direction with respect to t he cubic anisotropy direction.  \n \n \n1. Frequency-dependent FMR linewidth analysis \n \nTo investigate the influence of different contribut ions on the overall magnetic damping, we model \nthe field-swept peak-to-peak linewidth Δ Hpp  of our YIG (111) films as a sum of four contributi ons \n[79,82]:  \n TMS 0 mos G pp H H H H H ∆ + ∆ + ∆ + ∆ = ∆ , (2) \n \nwhere Δ HG is the Gilbert damping, Δ Hmos  the mosaicity, Δ H0 the inhomogeneous broadening, and \nΔHTMS  is the two-magnon scattering contribution, respect ively. Note, that all linewidths in this paper \nare peak-to-peak linewidths, even if not explicitly  stated. \nThe intrinsic Gilbert damping is given by \n  \n f H\nΞ= ∆\nγπα\n34\nG , (3) \n \nwhere γ = gµBħ is the gyromagnetic ratio and Ξ is the dragging function. The dragging function is  a \ncorrection factor to the linewidth needed in field- swept FMR measurements if H and M are not \ncollinear (see e.g . [82]). For H || M follows Ξ = 1. \nThe inhomogeneity term ∆Hmos  accounts for a spread (distribution) of the effect ive magnetization \n4π Meff  [82,83] given by the parameter δ4πMeff : \n \n eff\neffres\nmos 4432MMHH πδπ∂∂= ∆ . (4) \n \n∆H0, i.e. the zero-frequency linewidth, is a general b roadening term accounting for other \ninhomogeneities of the sample, such as the microwav e power dependence of the linewidth in YIG \n(see, e.g., [84]) and systematic fit errors: for ex ample, consistently narrower total full-width at ha lf-\nmaximum linewidths ∆HFWHM  of up to 0.5 Oe were determined by additional freq uency-swept \nmeasurements at a microwave power of -10 dBm compar ed to the field-swept measurements at \n0 dBm discussed here. \nAll kinds of inhomogeneous broadening (including ∆Hmos ) are caused by slightly different \nresonance fields in parts of the sample. These indi vidual resonance lines might be still resolvable at  \nlow frequencies, where Gilbert damping is not large  enough yet–especially for YIG. However, at \nhigher frequencies, these lines become broader and eventually coalesce to a single (apparently \nbroadened) line, which even might exhibit small sho ulders or other kinds of asymmetry. Hence, \nwhat might be nicely fit with a single line at high  frequencies might cause difficulties at low \nfrequencies and sub-mT linewidths. The effect on fi tting the anisotropy constants from the \nresonance fields is not so sensitive. If the resona nce lines cannot be disentangled or the line is not  \nentirely Lorentzian-shaped anymore, the fit might o verestimate the true linewidth resulting in a \nsystematic broader line accounted for by ∆H0.  12  The last term in Eq. (2), ∆HTMS , covers the two-magnon scattering contribution, wh ich is an \nextrinsic damping mechanism due to randomly distrib uted defects. For the in-plane frequency-\ndependence it reads [78,79,82,85-87]:  \n \n \n2 22 2sin\n32\n02\n0 202\n0 2\n1\nTMS\nf fff ff\nH\n+\n\n+−\n\n+\n⋅ Γ\nΞ= ∆−, (5) \n \nwhere f0 = γ4π Meff  and Γ is the two-magnon scattering strength. \nEach of the contributions has a characteristic angl e and frequency dependence. Overall, the \nlinewidth vs. frequency dependencies and the linewi dth vs. angle dependencies can be described \nwith one set of parameters. \nAs we will see, the applied model fits very well to  the experimental results and allows for \ndisentangling the contributions that are responsibl e for the frequency dependence of the linewidth. \nAt first, we discuss the different damping contribu tions. Then, we go into details for the individual \nmagneto-static parameters, the relevant anisotropy contributions mentioned above, which provided \nalso the base input for the fit parameters for the frequency-dependent FMR linewidth of our YIG \nfilms. \nIn Figure 5, the obtained frequency-dependent peak- to-peak linewidths ∆Hpp  (symbols) for the four \nthicknesses 11 nm, 21 nm, 30 nm, and 42 nm are pres ented. The red (solid) curves represent fits \nusing Eq. (2). Figure 5(a) shows data and fits for the out-of-plane bias field configuration ( θH = 0°) \nand Fig. 5(b) for field-in-plane ( θH = 90°), respectively. As mentioned above, due to a  quite complex \nshape of the resonance lines below ~15 GHz for θH = 0° (with more absorption lines needed to \nreflect the shape of the spectrum than for higher f requencies) the linewidths could not anymore be \nevaluated unambiguously with the required precision  for films with thicknesses above 11 nm. \nHowever, for the thinnest film, the evaluation was possible and the overall fit exhibits a linear \nbehavior down to 1 GHz. This means, in the field-ou t-of-plane geometry, the main contribution to \nthe damping is the Gilbert damping α, which can be determined from the linear slope according to \nEq. (3). As it is known from two-magnon scattering (TMS) theory [85,86], there is no TMS \ncontribution if M is perpendicular to the sample plane. The only rem aining contribution is the \ninhomogeneous broadening given by the zero-frequenc y offset \n  13  0369\n036\n036\n0 5 10 15 20 25 30 35 40 036θ H= 0 deg \n 11 nm \n 21 nm ∆Hpp  (Oe) \n 30 nm \n \nf (GHz) 42 nm (a) \n \n \nFIG. 5: Frequency dependence of the linewidth with magnetic field (a) perpendicular-to-plane and \n(b) in-plane. The red (solid) lines are fits to the  data. For the 11-nm sample the individual \ncontributions to the total linewidth are shown in t he top-right panel. Note the different y-axis scaling \nfor the 11 nm sample in the top-left panel. \n \nFrom these out-of-plane measurements, the Gilbert d amping coefficients could be determined, \nranging from α = 0.9 × 10 -4 for the 42-nm-thick sample to α = 2.0 × 10 -4 for the 21 nm sample, \nwhich is about twice the value obtained from in-pla ne measurements (as discussed below). For the \nultrathin 11 nm film, a slightly increased Gilbert damping coefficient of α = 2.7 × 10 -4 and a \nsignificantly enlarged zero-frequency linewidth of 2.8 Oe were found. As mentioned above, the \nreason for the larger offset might be an apparent u nresolvable broadening due to inhomogeneity. For \nthe 21 and 30 nm sample, the zero-frequency interce pt is about ∆H0 = 0.5 Oe, in contrast to ∆H0 = \n1.5 Oe for the 42 nm sample. This indicates, that t he 42 nm sample, in contrast to the thinner \nsamples, seems to have additional microstructural d efects, leading to a superposition of lines. This i s \nvery likely, because the inhomogeneous broadening p reviously reported for 100 nm YIG LPE films \nwas also in the range of ∆H0  = 0.5-0.7 Oe [50].   \nIn Fig. 5(b), the results of the corresponding in-p lane field configuration are given. For the 11 nm \nsample, the four individual fit contributions consi dered in the fit according to Eq. (2) are depicted by \nsolid curves. This sample shows a significant curva ture. The 42 nm sample also shows a small \ncurvature, whereas the other two samples only have a weak curvature at lower frequencies. This \ncurvature usually hints to a contribution from two- magnon scattering, but can also be due to a spread \nof the effective magnetization. Note, the frequency -dependence of the mosaicity and TMS term look \nquite similar at higher frequencies, but show a dif ferent curvature at lower frequencies. Hence, the \nshape of the curve and, thus, the fit reveal, that it is due to a spread of the effective magnetizatio n, \nδ4πMeff  as given by Eq. (4), which lies in the range of 0. 4 to 0.9 G. For the 11 nm sample, this value 036\n036\n036\n0 5 10 15 20 25 30 35 40 03611 nm  Exp.  ∆H  ∆HG ∆Hmos  ∆HTMS  ∆H0\nθ H= 90 deg \n21 nm \n30 nm \nf (GHz) 42 nm ∆Hpp  (Oe) (b) \n 036\n036\n036\n0 5 10 15 20 25 30 35 40 03611 nm  Exp.  ∆H  ∆HG ∆Hmos  ∆HTMS  ∆H0\nθ H= 90 deg \n21 nm \n30 nm \nf (GHz) 42 nm ∆Hpp  (Oe) (b) \n  14  is larger, i.e., δ4π Meff  = 3.2 G, and in addition one needs a small TMS dam ping contribution of Γ = \n1.5×10 7 Hz for a proper fit (see Table II). This is again a distinctive sign, that the 11 nm sample has \nsignificantly different structural and/or magnetic properties, leading to the additional linewidth \ncontributions. The Gilbert damping coefficients of all four samples in in-plane configuration are \nα ≤ 1.3 × 10 -4 and correspond to the best values reported earlier  for 100 nm YIG LPE films [50]. \nThese are also lower than for a recently reported 1 8 nm YIG LPE film [51]. Thus, at room \ntemperature, no significant increase in Gilbert dam ping could be observed for LPE films down to \n10 nm with decreasing thickness. This contrasts wit h various references for PLD and RF-sputtered \nYIG films grown on (111) GGG substrates [88-92].  \n \nTABLE II. Magnetic damping parameters of the LPE (1 11) YIG films: film thickness t, in-plane \nGilbert damping parameter α||, inhomogeneous broadening ∆H0|| , spread of effective magnetization  \nδ4πMeff   and two-magnon scattering contribution Γ. \n \nt \n(nm)  α||  \n (×10 -4) ∆H0||  \n (G)  δ4π Meff   \n(Oe)  Γ \n(10 7 Hz)  \n11 1.2  0.4  3.2  1.5  \n21  1.3  0.6  0.4  0 \n30  1.2  0.4  0.7  0 \n42  1.0  0.4  0.9  0 \naccuracy  ±0.2  ±0.2  ±0.3  ±0.3  \n \n \nAll field-in-plane linewidth parameters of the inve stigated samples are summarized in Table II. It is \nobvious, that inhomogeneous contributions, i.e., th ose originating from magnetic mosaicity δ4πMeff , \nare very small for the samples without two-magnon s cattering. This confirms the high \nmicrostructural perfection and homogeneity of the v olume and interfaces of the LPE-grown films \nwith film thicknesses larger than 11 nm. Contributi ons to two-magnon scattering appear to occur \nonly for LPE films with a thickness of less than 21  nm thick. \n \n \n2. Analysis of magnetic anisotropy contributions \n \nIn the following, we will discuss the anisotropy co ntributions, which provided the base input for the \nfit parameters used for the frequency-dependent FMR  linewidth curves shown above. All curves \nwere fitted iteratively with the respective resonan ce equation (see Eqs. (S22) and (S23) in the \nSupplemental Material [69]) to retrieve a coherent set of fit parameters. The fit parameters are liste d \nin table III. Since the saturation magnetization an d the in-plane stress are known from VSM \nmeasurements and HR-XRD investigations, the anisotr opy constants K can be calculated from the \nanisotropy fields determined by FMR. \n \nTABLE III. Magneto-static parameters of the YIG LPE films of t hickness t: Landé’s g-factor, \neffective magnetization 4 πMeff exp , cubic anisotropy field 2 K4/Ms, and uniaxial in-plane anisotropy \nfield 2 K2||/Ms determined from FMR, saturation magnetization 4 πMs determined from VSM, stress-\ninduced anisotropy field 2 Kσ/Ms calculated from X-ray diffraction data, resulting o ut-of-plane \nuniaxial anisotropy field 2 K2⊥/Ms and effective magnetization 4 πMeff cal , cubic anisotropy constant \nK4, stress-induced anisotropy constant Kσ, and out-of-plane uniaxial anisotropy constant K2⊥. \n  15  t \n(nm) g 4πMeff exp \n(G) 2K4/Ms \n(Oe) 2K2||/Ms \n(Oe) 4πMs \n(G) \n11  2.015 1566 -93 2.0 1494 \n21 2.016 1647 -79 0.8 1819 \n30 2.015 1677 -79 0.6 1830 \n42 2.014 1699 -86 1.1 1860 \naccuracy ±0.002 ±13 ±2 ±3 ±41 \n \nt \n(nm) 2Kσ/Ms \n(G) 2K2⊥/Ms \n(G) 4πMeff cal  \n(G) K4 \n(10 3 erg/cm 3) Kσ \n(10 3 erg/cm 3) K2⊥ \n(10 3 erg/cm 3) \n11 65 127 1368 -5.5 3.9   7.5 \n21 91 143 1676 -5.7 6.6 10.4 \n30 82 135 1696 -5.8 6.0   9.8 \n42 79 136 1724 -6.4 5.8 10.1 \naccuracy ±7 ±4 ±40 ±0.3 ±0.4 ±0.5 \n \n \nThe g-factor of the samples was determined from the freq uency dependencies of the resonance field. \nThere was no significant thickness dependence obser ved yielding a value of g = 2.015(1) for all \nsamples. The cubic anisotropy field 2 K4/Ms was found to be nearly constant, and the average v alue \nis -84(2) Oe, which is in good agreement to reporte d values of -85 Oe for a 120 micrometer thick \nLPE film [81] and of about -80 Oe for a 18 nm thin LPE film [51]. Our calculated anisotropy \nconstants K4 are almost always in the range between -5.7×10 3 and -6.4×10 3 erg/cm 3, which \ncorresponds to YIG single crystal bulk values at 29 5 K [93] . Furthermore, a rather weak in-plane \nuniaxial anisotropy field 2 K2|| /Ms of about 0.6–2 Oe was found, which  had already be en determined \nfor 100 nm YIG LPE films [50].   \nThe stress-induced anisotropy constant Kσ and anisotropy field 2 Kσ/Ms are calculated according to \nRef. [94 ] (for details, see Eqs. (S14), (S15), (S18) in the Supplemental Material [69]). 2 Kσ/Ms is \nsmall and in the same order of magnitude as the cub ic anisotropy field 2 K4/Ms, but with opposite \nsign. Due to the observed monotonous increase of th e out-of-plane lattice misfit (see inset in Fig. \n2(b)), 2 Kσ/Ms grows with decreasing film thickness until it decl ines significantly at a film thickness \nbelow 21 nm. However, the observed stress values ar e almost an order of magnitude smaller than, \ne.g., for as-deposited YIG PLD films on GGG (111) u nder compressive strain (see, e.g., Refs. \n[17,23,35,36]). Only by a complex procedure, applyi ng mid-temperature deposition, cooling, and \npost-annealing treatment, authors of Ref. [95] succ eeded in a change from compressively to tensely \nstrained YIG films. These samples then exhibited th e same stress-induced anisotropy constant as it \nwas observed for our YIG LPE films. \nIn the following, we take a closer look to the cont ributions to the out-of-plane uniaxial anisotropy  \nfield H2⊥ = 2 K2⊥/Ms. A general description for magnetic garnets has been  given for example by \nHansen [94]. Applied to thick [43,64]  as well as to thin epitaxial iron garnet films (see , e.g., \n[37,56,59,60,62]) , the out-of-plane uniaxial anisotropy field H2⊥ is mainly determined by the \nmagnetocrystalline and  uniaxial anisotropy contributions. While the former  refers to the direction of \nmagnetization to preferred crystallographic directi ons in the cubic garnet lattice, the latter origina tes \nfrom lattice strain and growth conditions. Due to t he very low supercooling ( ≤5K), growth-induced \ncontributions, usually observed for micrometer YIG films with larger Pb impurity contents, can be \nneglected in the case of our nanometer-thin YIG LPE  films (see e.g. [64]). Thus,  H 2⊥ can be  16  determined quantitatively by summing the cubic magn etocrystalline anisotropy (first term, \ndetermined by FMR) and the stress-induced anisotrop y (second term, determined by XRD), \n \n \ns sMK\nMKHσ2\n344\n2 + − =⊥ , (6) \n \nor expressed for the (111) substrate orientation (s ee also SM [69 ] and Ref. [93,96]) by:  \n \n \nsMKH39 4111|| 4\n2λσ′ − −=⊥ . (7) \n \nUsing the experimentally determined first-order cub ic anisotropy constant K4 and the in-plane stress \ncomponent ||σ′from Tables I and III along with the room-temperatu re magnetostriction coefficient \nλ111  [94 ], the uniaxial anisotropy field H2⊥ can be calculated, if the saturation magnetization  Ms is \nknown. Ms can be obtained with appropriate accuracy for exam ple from VSM or SQUID \nmeasurements, if the sample volume is exactly known .  \nMagnetic hysteresis loops of YIG LPE films recorded  at room-temperature by VSM measurements \nwith in-plane applied magnetic field are shown in F ig. 6. The paramagnetic contribution of the GGG \nsubstrate was subtracted as described in Ref. [50]. Extremely small coercivity fields with Hc values \nof ∼ 0.2 Oe were obtained for all YIG/GGG samples with the exception of the 21 nm film. These \nvalues are comparable with the best gas phase epita xial films [17,39,76], but the measured saturation \nfields with Hs < 2.0 Oe are significantly smaller. All films exhi bit nearly in-plane magnetization due \nto the dominant contribution of form anisotropy. Ap art from the thinnest sample, the saturation \nmoments determined are not thickness-dependent (see  Table III and Fig. 6) and are very close to \nYIG volume values determined for YIG single crystal s at room temperature (4 πMs ∼ 1800 G) \n[93,97]. However, the observed decrease of the satu ration magnetization in such films with a \nthickness of about 10 nm is significant and will be  discussed below. \n-20 -15 -10 -5 0 5 10 15 20 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 \n10 100 1.4 1.5 1.6 1.7 1.8 1.9 \n  \n H (Oe)  4 πM (10 4 G)  106 nm \n 43 nm \n 30 nm \n 21 nm \n 11 nm \n   4 πM (10 4 G) \n d (nm) \n \nFIG.6: Magnetization loops M(H) of YIG films at room temperature as a function of  the in-plane \nmagnetic field. The inset shows the thickness-depen dent saturation magnetization (the solid line is a \nguide to the eyes). \n  17  However, for nanometer-thin films, it is a big chal lenge to determine Ms precisely enough, because \ntoo large errors can arise from the film’s volume c alculation. While the surface area of the sample \ncan be determined with sufficient precision by opti cal microscopy, thickness measurements with X-\nray or ellipsometry methods can lead to thickness e rrors in the range of ±1 nm due to very small \nmacroscopic morphology or roughness fluctuations. T herefore, for films with thicknesses below \n20 nm, for example, uncertainties up to a maximum o f 10 percent must be considered. This could \nsignificantly affect the effective magnetization 4 πMeff , which can be calculated based on the \nmeasured Ms values by \n ⊥ − =2 eff 4 4 H M Msπ π . (8) \nThis fact can explain the large difference between the calculated 4 πMeff cal  and the measured \n4πMeff exp  values for the 11 nm thin film discussed below, wh ile a much better agreement was \nachieved for the thicker films (see Table III). \nAs expected from micrometer-thick YIG LPE films gro wn on GGG (111) substrates [81], the out-\nof-plane uniaxial anisotropy field H2⊥ and the out-of-plane uniaxial anisotropy constants K2⊥ show, \nthat completely pseudomorphically strained, nanomet er-thin LPE films exhibit no pronounced \nmagnetic anisotropy. Small changes of the in-plane stress σ′|| (see Table I) and thus also in the \nstress-induced anisotropy 2 Kσ/Ms (or Kσ) have no significant influence on the out-of-plane  uniaxial \nanisotropy H2⊥ (see Table III). A comparable H2⊥ value is also expected for films thicker than \n42 nm, since the out-of-plane lattice misfit δd⊥\nfilm  tends to a constant value (see inset in Fig. 2 (b) ). \nThis is in contrast to Ref. [51], where the uniaxia l anisotropy field of YIG LPE films becomes \nnegative above a film thickness of about 50 nm. \n \n \n3. Thickness-dependent analysis of the effective ma gnetization field \n \nTo verify the trend of the calculated 4 πMeff cal  values for decreasing film thicknesses, one can \ncompare the effective magnetization with the experi mentally determined one. This was done for 18 \nYIG films with thicknesses ranging from 10 to 120 n m, including the four samples from above. All \nfilms were grown during the same run under nearly i dentical conditions. Only the growth \ntemperature was varied within a range of 5 K.  This time, the FMR was measured with a constant \nexternal magnetic field applied in-plane and sweepi ng the frequency.  \n0 20 40 60 80 100 120 1500 1550 1600 1650 1700 1750 1800 \n frequency field sweep \n magnetic field sweep Effective Magnetization (G) \nFilm thickness (nm) (a) \n  18  10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 535 540 545 550 555 560 \n fitted VSM values \n Outlier values Curie temperature (K) \nYIG thickness (m) (b) \n \nFig. 7. (a) Thickness dependence of the effective m agnetization 4 πMeff . Blue circles denote \nmeasurements taken by field-sweep and red squares d enote frequency-swept measurements, \nrespectively. (b) Thickness dependence of the Curie  temperature Tc. The dashed line is a guide to the \neyes. \n \n \nIn Fig. 7(a), the obtained thickness dependence of the effective magnetization 4 πMeff  (squares) is \npresented and a monotonous decrease of 4 πMeff  with film thickness reduction can be observed. \nBelow 40 nm, the slope of the curve increases and, for the thinnest films, there is a significant drop  \nof about 100 G. This behavior has been confirmed by  in-plane FMR magnetic field-sweep \nmeasurements for selected samples (circles), as dis cussed before. The values are listed in Table III. \nSimilar results have been reported for YIG PLD film s by Kumar et al. [53]. \nIf we compare the experimental values with the calc ulated ones in Table III, then the same trend of a \nsteady reduction of the effective saturation magnet ization with decreasing film thickness can be \nobserved. The deviation between both 4 πMeff  values is approximately 1–2 %, except for the 11 n m \nfilm. Hence, the saturation magnetization used to c alculate the effective saturation (according to \nequation (8)) does not appear to be as error-prone as it could be due to an inaccuracy in the film \nthickness determination. Therefore, we speculate, t hat the significant drop of 4 πMeff  for the 11 nm \nthin film can be explained by the observed reductio n of 4 πMs (see Table III). \nA similar behavior for 4 πMs was reported for thin PLD or magnetron-sputtered Y IG films, and \ndifferent explanations were given [76,56,91]. One r eason for a reduced saturation magnetization \ncould be an intermixing of substrate and film eleme nts at the GGG/YIG interface, whereby a gradual \nchange of the film composition is assumed [75,77]. In particular, gallium ion diffusion into the first  \nYIG atomic layers will lead to magnetically diluted  ferrimagnetic layers at the interface, due to the \nfact, that magnetic Fe ions are replaced by diamagn etic Ga ions in the various magnetic sublattices. \nThis assumption is supported by recent reports of Y IG films on GGG substrates. One reports about a \n5–7 nm deep Ga penetration found in laser-MBE films  [75 ]. Another group found a Ga penetration \nthroughout a 13-nm-thin PLD film [98]. In these cas es, high-temperature film growth above 850°C \nor prolonged post-annealing at temperatures of 850° C could promote such diffusion processes. In \ncontrast, the deposition time during which the LPE samples were exposed to high temperatures \nabove 860°C was only 5 minutes. Though, the assumed  Ga diffusion depth in our YIG films should \nnot exceed more than 2 nm according to the EDXS ele ment maps in Fig. 4(h). In addition, the Gd  19  diffusion in YIG films, as discussed for RF-magnetr on sputtered [76,99]  or  PLD films [98], could \nlead to the incorporation of paramagnetic ions into  the diamagnetic rare earth sublattice sites, which  \nwould also alter the magnetization [98]. However, n o extended interdiffusion layer was observed at \nthe film/substrate interface for our LPE films, so that the presence of a ‘separate, abrupt’ gadoliniu m \niron garnet interface layer, as reported by Ref. [9 8], is not expected. Therefore, due to possible \ninterdiffusion effects at temperatures of about 860 °C, a gradual reduction of Ms at a postulated \ninterface layer could be the reason for the observe d low saturation value for the thinnest LPE film, \nlisted in Tab. III. \nTo further rule out a discrete magnetic dead layer,  Curie temperature ( Tc) measurements were \nperformed by VSM. It is known from literature, that  Tc remains constant up to a film thickness of \napproximately four YIG unit cells [100], i.e. 2.8 n m, since one YIG unit cell length along the [111 ] \ndirection amounts to d111  ∼ 0.7 nm. Accordingly, the Tc of “pure” YIG films with abrupt interfaces \nand a film thickness of ∼10 nm should be equal to that of bulk material. In order to check this, \ntemperature-dependent VSM measurements (see Fig. 7( b)) were carried out for our LPE films as \nwell as for a bulk YIG single crystal slice, which was used as a reference. We found almost constant \nvalues of Tc = (551±2) K for sample thicknesses between 46 nm ( thin film) and 0.55 mm (bulk). \nThis is in good agreement with the literature, in w hich a Tc of ∼550 K has been reported, e.g. for a \n100-nm-thin sputtered YIG film [76],  while 559 K has been reported for YIG single crysta ls [97]. \nHowever, for our about 10-nm-thin YIG films, Tc decreased significantly to ∼534±1 K (Fig 7(b)), \nwhich is consistent with the observed reduction of 4 πMs listed in Table III. \nHence, the most likely explanation for the observed  reduction of 4 πMs is that the YIG layers at the \nsubstrate/film interface exhibit a reduced saturati on magnetization due to a magnetically diluted iron  \nsublattice, resulting from high-temperature diffusi on of gallium ions from the GGG substrate into \nthe YIG film . While nearly zero gallium content at the film surfa ce leads to a bulk-like value of \n4πMs ∼ 1800 G [93], an increased content of gallium at th e film/substrate interface should, therefore, \nresult in significantly reduced 4 πMs values. In this case, the average saturation magne tization for the \nentire film volume should be reduced and that could  explain the observed decrease in 4 πMs to about \n1500 G for the 11 nm thin LPE film. For thicker fil ms, however, the influence of thin gallium-\nenriched interface layers on the entire film magnet ization decreases, which explains the fast \nachievement of a constant Curie temperature, and th us, a constant Ms with increasing thickness of \nthe YIG volume. In order to confirm our assumptions , additional analyses, such as detailed \nsecondary ion mass spectroscopy (SIMS) investigatio ns, are necessary which, however, go beyond \nthe scope of this report. \n \n \nIV. CONCLUSIONS AND OUTLOOK \n \nIn summary, we have demonstrated that LPE can be us ed to fabricate sub-40 nm YIG films with \nhigh microstructural perfection, smooth surfaces an d sharp interfaces as well as excellent microwave \nproperties down to a minimum film thickness of 11 n m. All LPE films with ≥21 nm thickness \nexhibit extremely narrow FMR linewidths of ∆Hpp  <1.5 Oe at 15 GHz and very low magnetic \ndamping coefficients of α ≤1.3 × 10 -4 which are the lowest values reported within an ext ended \nfrequency range of 1 to 40 GHz. We were able to sho w that LPE-grown YIG films down to a \nthickness of 21 nm have the same magnetization dyna mics influenced by small cubic and stress-\ninduced anisotropy fields. The deviating magnetizat ion dynamics of ultrathin LPE films with \nthicknesses of ∼10 nm are probably caused by an increased inhomogen eous damping and by small \ntwo-magnon scattering contributions, and we specula te that possible inhomogeneities of the \ncomposition in the vicinity of the film/substrate i nterface might be the reason for this. Therefore, i n  20  further studies we will address detailed investigat ions of the composition of the film/substrate \ninterface by high-resolution SIMS measurements and advanced STEM analyses to confirm a gradual \nchange of the LPE film composition at the interface . \nThe results presented here encourage us to take the  next step towards nano- and microscaled \nmagnonic structures, such as directional couplers, logic gates, transistors etc. for a next-generation  \nof computing circuits. The development of nanoscopi c YIG waveguides and nanostructures is \nalready underway and the first circuits are current ly being fabricated [10,12,29]. With its scalabilit y \nto large wafer diameters of up to 3 and 4 inches, L PE technology opens up an alternative way for \nefficient circuit manufacturing for a future YIG pl anar technology on a wafer scale. \n \n \nACKNOWLEDGMENTS \n \nWe thank P. Landeros and R. Gallardo for fruitful d iscussions and A. Khudorozhkov for his help \nduring the measurements. C. D. and O. S. thank R. K öcher for AFM measurements, A. Hartmann \nfor the DSC measurements and R. Meyer and B. Wenzel  for technical support. J. G. thanks A. \nScholz for the support during the XRD measurements.  We would like to thank Romy Aniol for the \nTEM specimen preparation. 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Mater. 170 , L243 (1997).  \n[101 ] Strictly speaking this is valid only for [001] ori ented systems for all others, if the cubic \nlattice constant values of afilm  and asubstrate  are almost identical.  27  Supplemental Material: \nLow damping and microstructural perfection of sub-4 0nm-thin yttrium iron garnet films \ngrown by liquid phase epitaxy  \n \nCarsten Dubs, 1 Oleksii Surzhenko, 1 Ronny Thomas, 2 Julia Osten, 2 Tobias Schneider, 2 Kilian Lenz, 2 \nJörg Grenzer, 2 René Hübner,2 Elke Wendler 3 \n \n1 INNOVENT e.V. Technologieentwicklung, Prüssingstr.  27B, 07745 Jena, Germany \n2 Institute of Ion Beam Physics and Materials Resear ch, Helmholtz-Zentrum Dresden-Rossendorf, \nBautzner Landstr. 400, 01328 Dresden, Germany \n3 Institut für Festkörperphysik, Friedrich-Schiller- Universität Jena, Helmholtzweg 3, 07743 Jena, \nGermany \n \n \nI. STRAIN CALCULATIONS  \n \nIn the following we will derive the in-plane (horiz ontal) stress σ||  as a function of the out-of-plane \n(vertical) lattice misfit ⊥\nfilmdδ  for a [111] oriented cubic system. These calculati ons are based on the \nelasticity theory following mainly Hinckley [70 ], Sander [71 ] and Ortiz et al . [61 ]. Assuming a fully \npseudomorphic system there is only one parameter to  be determined: The out-of-plane lattice misfit \n⊥\nfilmdδ . This value can be directly  obtained from the data of the HR-XRD measurements a nd/or from \nthe corresponding simulations of the symmetrical (4 44) and (888) reflections. \nThe vertical and parallel lattice misfits can be ca lculated by \n , 0||\nsubstrate||\nsubstrate||\nfilm ||\nfilm\nsubstratesubstrate film\nfilm =−=−=⊥⊥ ⊥\n⊥\ndd dddd dd δ δ  (S1) \nwhere dk\ni with i = [substrate, (pseudomorph) film or (cubic) relaxe d film] is the (measured) net \nplane distances for the k = ⊥ (vertical) or || (parallel) direction with respect  to the substrate surface. \nIgnoring any dynamical diffraction effects, the out -of-plane lattice misfit can be directly determined  \nfrom the measurement as follows: \n \nB qq\nqq qq dθθδ δtansubstrate substratesubstrate film\nfilm film∆− =∆− =−− = − =⊥⊥\n⊥⊥ ⊥\n⊥ ⊥, (S2) \nwhere q⊥film  and q⊥substrate are the derived peak positions in the Q-space of th e thin film and the \nsubstrate, respectively. ∆θB is the (kinematical) Bragg angular difference “thi n film – substrate” and \nθB is the Bragg Peak position of the substrate, respe ctively. These formulas follow directly from the \nderivative of Bragg’s law. \nThe in- and out-of-plane strains are given as follo ws: \n .||\nfilm  relaxed||\nfilm  relaxed||\nfilm ||\nfilm  relaxedfilm  relaxed film\ndd d\ndd d −=−=⊥⊥ ⊥\n⊥ε ε  (S3) \nThe general relationship between stress and strain is defined as follows: \n ,kl ijkl ij ε σ C=  (S4) \nwhere σij  are the stress, ε kl  the strain and Cijkl  the second order stiffness tensors and the summati on is \ndone over the repeated indices. The subscripts ij and  kl  refer to the axes of the coordinates system of  28  the unit cell (1,2,3 = x,y,z ). The samples under investigations have a [111] ou t-of-plane orientation; \ntherefore, the corresponding rotation matrices have  to be applied: \n ijkll k j i CUUUU Cδ γ β α αβγδ=′ . (S5) \nFor the [111] oriented surfaces U 111  yields to \n \n\n\n\n −\n=\n310\n3231\n21\n6131\n21\n61\n111U . (S6) \n \n \n \nFigure S1 : Representation of the cubic unprimed and the rota ted, primed coordinate system for an \n<111> oriented thin film; where x´, y´ correspond to the in-plane (||) directions and z´ to the out-of-\nplane ( ⊥) direction; after [71 ]. \n \n \nThe in-plane stress ||σ′ can be expressed in terms of the out-of-plane latt ice misfit ⊥\nfilmdδ obtained by \nXRD measurements.  \n \nFor a cubic pseudomorphic system we can write:  \n 1313 1212 1111 11 ε ε ε σ c c c + + =  (S7) \n ( )⊥′+′+′=′ ε ε σ12 || 12 11 || c c c  (S8) \nwith  \n ⊥′− =ε ν ε||  (S9) \nresulting in: \n ||\n44 12 1112 11\n44 ||4 226 ε σc c cc cc+ ++=′ . (S10) \n \n  29   \nTaking the corresponding rotation matrices and rela tionships into account  [101 ]: \n ⊥ ⊥ ⊥\n+− =+=film 111111\n||\nfilm 1111and11d d δννε δνε , (S11) \nwhere \n .4 4 24 2\n44 12 1144 12 11 111\nc c cc c c\n− ++ +=ν  (S12) \n \nThe in-plane stress can be now expressed in terms o f the out-of-plane lattice misfit by: \n ⊥− =′film 44 || 2 dcδ σ . (S13) \nHere, c44  is the component from the stiffness tensor and ⊥\nfilmdδ is the out-of-plane lattice misfit as \ndefined above. \n \n \nII. ANISOTROPY CALCULATIONS FOR (111) ORIENTED EPIT AXIAL GARNET FILMS  \n \nThe stress-induced anisotropy parameter for the cub ic (111) orientation can be calculated according \nto [94] by \n 111||23λ =Kσ σ′ − , (S14) \nwhere σ´||  is the above calculated in-plane stress for {111} oriented thin films, and λ 111  is the \ncorresponding magnetostriction constant. \nThe stress-induced anisotropy parameter is therefor e given by: \n 111 film 443 λdc=Kσ⊥δ . (S15) \nThe perpendicular magnetic anisotropy field can be calculated according to [43]: \n growth cub 2 H+ H+ H= Hstress ⊥ . (S16) \nAssuming negligible growth-induced contributions Hgrowth  and applying the cubic anisotropy field \nfor (111) film orientation obtained by FMR measurem ents \n \nscubMK= H4\n34− , (S17) \nand taking into account the stress-induced anisotro py field \n \ns sσ\nstressMλ=MK= H111||3 2 σ′\n− , (S18) \nthe effective perpendicular anisotropy field result s in \n \nsMλ +KH39 4111|| 4\n2σ′\n− =⊥ . (S19)  30  From the resonance conditions for the perpendicular  (M ||  [111 ]) magnetized epitaxial thin film, the \neffective saturation magnetization can be obtained by [64 ] \n \n+ − − =⊥\ns ssMK\nMKM Hf2 4\neff2\n344πω,  (S20) \nfrom which the effective saturation magnetization c an be calculated by \n ⊥ −2 eff eff 4 4 H πM= πM=Hs . (S21) \n \nIII. FERROMAGNETIC RESONANCE  \n \nFrom the free energy density given by Eq. (1) of th e main text, the resonance equations have been \ncalculated applying the approach of Baselgia et al.  [80 ]. The resonance conditions for the frequency-\ndependences with field out-of-plane ( f⊥) and in-plane ( f|| ) read: \n \n \n\n\n+ − − \n\n− − =⊥MK\nMKM HMKM H fe e|| 2 4\nff4\nff 23443442π ππγ, (S22)  \n ( ) ( ) ( )\n\n\n\n\n\n− − − − + ×\n\n\n− − = ϕ ϕϕ π ϕϕπγ3 cos 2 cos24 2cos2\n222\n4 2 || 2 4\nff|| 2\n||MK\nMK\nMKM HMKH fu e u\n  (S23). \n \nExamples of angle- and frequency-dependent FMR meas urements with out- and in-plane \nconfiguration of the magnetic bias field are shown in Figure S2.  31  -30° 0° 30° 60° 90° 120° 345f = 10 GHz \n11 nm \n21 nm \n30  nm \n42 nm \n Fits Hres (kOe)\nθH\n0 5 10 15 010 20 30 40 f (GHz)\nH (kOe) \n0 5 10 15 010 20 30 40 \nθH=0° f(GHz)\nH (kOe)  11 nm \n 21 nm \n 30 nm \n 42 nm \n Fit (11 nm) θH=90° \n 11 nm \n 21 nm \n 30 nm \n 42 nm (a) \n(b) \n(c) 2.76 2.77 2.78 2 .7 9 2 .8 0 FMR- S igna l  (arb . units)\nH (kOe) 42 nm \nf = 10 GHz \n4.30 4.32 4.34 4.36 4.38 4 .4 0 FMR-Signa l  (arb . units)\nH (kOe) 11 nm \nf = 8 GHz θH\nφHH→\n[110] _ [112] _M→\nY IG(111) φθ\n \nFigure S2. (a) Polar angular dependencies of the FM R measured at f = 10 GHz. The inset shows the \nFMR coordinate system. Solid lines are fits accordi ng to the resonance equation. (b) Frequency \ndependencies of the resonance field measured with f ield in-plane and (c) out-of-plane. The solid \nblack line is a fit to the 11 nm dataset. Other fit  curves have been omitted for visual clarity. Inset s \nshow FMR spectra and the indicated positions includ ing Lorentzian fits. "    },    {        "title": "2209.02914v2.Convergence_analysis_of_an_implicit_finite_difference_method_for_the_inertial_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "CONVERGENCE ANALYSIS OF AN IMPLICIT FINITE\nDIFFERENCE METHOD FOR THE INERTIAL\nLANDAU-LIFSHITZ-GILBERT EQUATION\nJINGRUN CHEN, PANCHI LI, AND CHENG WANG\nAbstract. The Landau-Lifshitz-Gilbert (LLG) equation is a widely used model\nfor fast magnetization dynamics in ferromagnetic materials. Recently, the iner-\ntial LLG equation, which contains an inertial term, has been proposed to cap-\nture the ultra-fast magnetization dynamics at the sub-picosecond timescale.\nMathematically, this generalized model contains the \frst temporal derivative\nand a newly introduced second temporal derivative of magnetization. Conse-\nquently, it produces extra di\u000eculties in numerical analysis due to the mixed\nhyperbolic-parabolic type of this equation with degeneracy. In this work, we\npropose an implicit \fnite di\u000berence scheme based on the central di\u000berence in\nboth time and space. A \fxed point iteration method is applied to solve the im-\nplicit nonlinear system. With the help of a second order accurate constructed\nsolution, we provide a convergence analysis in H1for this numerical scheme, in\nthe`1(0;T;H1\nh) norm. It is shown that the proposed method is second order\naccurate in both time and space, with unconditional stability and a natural\npreservation of the magnetization length. In the hyperbolic regime, signi\fcant\ndamping wave behaviors of magnetization at a shorter timescale are observed\nthrough numerical simulations.\n1.Introduction\nThe Landau-Lifshitz-Gilbert (LLG) equation [15, 19] describes the dissipative\nmagnetization dynamics in ferromagnetic materials, which is highly nonlinear and\nhas a non-convex constraint. Physically, it is widely used to interpret the experi-\nmental observations. However, recent experiments [5, 16, 17] con\frm that its valid-\nity is limited to timescales from picosecond to larger timescales for which the angular\nmomentum reaches equilibrium in a force \feld. At shorter timescales, e.g. \u0018100 fs,\nthe ultra-fast magnetization dynamics has been observed [17]. To account for this,\nthe inertial Landau-Lifshitz-Gilbert (iLLG) equation is proposed [6, 10, 12]. As a\nresult, the magnetization converges to its equilibrium along a locus with damping\nnutation simulated in [21], when the inertial e\u000bect is activated by a non-equilibrium\ninitialization or an external magnetic \feld.\nFor a ferromagnet over \n 2Rd;d= 1;2;3, the observable states are depicted by\nthe distribution of the magnetization in \n. The magnetization denoted by m(x;t) is\na vector \feld taking values in the unit sphere S2ofR3, which indicates that jmj= 1\nin a point-wise sense. In micromagnetics, the evolution of mis governed by the\nLLG equation. In addition to experiment and theory, micromagnetics simulations\nDate : September 13, 2022.\n2010 Mathematics Subject Classi\fcation. Primary 35K61, 65M06, 65M12.\nKey words and phrases. Convergence analysis, inertial Landau-Lifshitz-Gilbert equation, im-\nplicit central di\u000berence scheme, second order accuracy.\n1arXiv:2209.02914v2  [math.NA]  12 Sep 20222 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nhave become increasingly important over the past several decades. Therefore, nu-\nmerous numerical approaches have been proposed for the LLG equation and its\nequivalent form, the Landau-Lifshitz (LL) equation; see [9, 18] for reviews and ref-\nerences therein. In terms of time marching, the simplest explicit methods, such\nas the forward Euler method and Runge-Kutta methods, were favored in the early\ndays, while small time step size must be used due to the stability restriction [22].\nOf course, implicit methods avoid the stability constraint and these methods pro-\nduce the approximate solution in H1(\n) [1, 2]. However, in order to guarantee the\nconvergence of the schemes, a step-size condition k=O(h2) must be satis\fed in\nboth the theoretical analysis and numerical simulations. To obtain the weak solu-\ntion in the \fnite element framework, an intermediate variable vwith the de\fnition\nv=@tmrepresenting the increment rate at current time is introduced, and to solve\nvin the tangent space of mwhere it satis\fes v\u0001m= 0 in a point-wise sense, then\nthe con\fguration at the next time step can be obtained. Directly, the strong solu-\ntion can be obtained through solving the implicit mid-point scheme [4] and the im-\nplicit backward Euler scheme [13] using \fxed-point iteration methods. By contrast,\nthe semi-implicit methods have achieved a desired balance between stability and\ne\u000eciency for the micromagnetics simulations. The Gauss-Seidel projection meth-\nods [11, 20, 27], the linearized backward Euler scheme [8, 14], the Crank-Nicolson\nprojection scheme [3], and the second order semi-implicit backward di\u000berentiation\nformula projection scheme [7, 28] have been developed in recent years. In prac-\ntice, all these semi-implicit methods inherit the unconditional stability of implicit\nschemes, and achieve the considerable improvement in e\u000eciency.\nThe LLG equation is a nonlinear parabolic system which consists of the gyro-\nmagnetic term and the damping term. It is a classical kinetic equation that only\ncontains the velocity; no acceleration is included in the equation. When relaxing\nthe system from a non-equilibrium state or applying a perturbation, it is natural\nthat an acceleration term will be present, resulting in the inertial term in the iLLG\nequation. More speci\fcally, the time evolution of m(x;t) is described by @tmand\nm\u0002@tmwith the addition of an inertial term m\u0002@ttm. Thus, the iLLG equa-\ntion is a nonlinear system of mixed hyperbolic-parabolic type with degeneracy. To\nnumerically study the hyperbolic behaviors of the magnetization, the \frst-order\naccuracy tangent plane scheme (TPS) and the second-order accuracy angular mo-\nmentum method (AMM) are proposed in [23]. The \fxed-point iteration method is\nused for the implicit marching. These two methods aim to \fnd the weak solution.\nFurthermore, a second-order accurate semi-implicit method is presented in [21], and\n@ttmand@tmare approximated by the central di\u000berence.\nIn this work, we provide the convergence analysis of the implicit mid-point\nscheme on three time layers for the iLLG equation. Subject to the condition\nk\u0014Ch2, it produces a unique second-order approximation in H1(\nT). Owing to\nthe application of the mid-point scheme, it naturally preserves the magnetization\nlength. Moreover, we propose a \fxed-point iteration method to solve the nonlinear\nscheme, which converges to a unique solution under the condition of k\u0014Ch2.\nNumerical simulations are reported to con\frm the theoretic analysis and study the\ninertial dynamics at shorter timescales.\nThe rest of this paper is organized as follows. The iLLG equation and the\nnumerical method are introduced in Section 2. The detailed convergence analysis\nis provided in Section 3. In addition, a \fxed-point iteration method for solvingCONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 3\nthe implicit scheme is proposed in Section 4, and the convergence is established\nupon the condition k\u0014Ch2. Numerical tests, including the accuracy test and\nobservation of the inertial e\u000bect, are presented in Section 5. Concluding remarks\nare made in Section 6.\n2.The physical model and the numerical method\nThe intrinsic magnetization of a ferromagnetic body m=m(x;t) : \nT:=\n\n\u0002(0;T)!S2is modeled by the conventional LLG equation:\n@tm=\u0000m\u0002\u0001m+\u000bm\u0002@tm; (x;t)2\nT; (2.1a)\nm(x;0) =m(0); x2\n; (2.1b)\n@\u0017m(x;t) = 0; (x;t)2@\n\u0002[0;T]; (2.1c)\nwhere\u0017represents the unit outward normal vector on @\n, and\u000b\u001c1 is the\ndamping parameter. If the relaxation starts from a non-equilibrium state or a\nsudden perturbation is applied, the acceleration should be considered in the kinetic\nequation, which is the inertial e\u000bect observed in various experiments at the sub-\npicosecond timescale. In turn, its dynamics is described by the iLLG equation\n@tm=\u0000m\u0002(\u0001m+He) +\u000bm\u0002(@tm+\u001c@ttm); (x;t)2\nT; (2.2a)\nm(x;0) =m(0); x2\n; (2.2b)\n@tm(x;0) = 0; x2\n; (2.2c)\n@\u0017m(x;t) = 0; (x;t)2@\n\u0002[0;T]; (2.2d)\nwhere\u001cis the phenomenological inertia parameter, and Heis a perturbation of\nan applied magnetic \feld. To ease the discussion, the external \feld is neglected in\nthe subsequent analysis and is only considered in micromagnetics simulations. An\nadditional initial condition @tm(x;0) = 0 is added, which implies that the velocity\nis 0 att= 0 and it is a necessary condition for the well-posedness. Then the energy\nis de\fned as\n(2.3)E[m] =1\n2Z\n\n\u0010\njrmj2\u00002m\u0001He+\u000b\u001cj@tmj2\u0011\ndx:\nFor constant external magnetic \felds, it satis\fes the energy dissipation law\n(2.4)d\ndtE[m] =\u0000\u000bZ\n\nj@tmj2dx\u00140:\nTherefore, under the condition of (2.2c), for almost all T02(0;T), we have\n(2.5)1\n2Z\n\n\u0010\njrm(x;T0)j2+\u000b\u001cj@tm(x;T0)j2\u0011\ndx\u00141\n2Z\n\n\u0000\njrm(x;0)j2\u0001\ndx:\nBefore the formal algorithm is presented, here the spatial di\u000berence mesh and\nthe temporal discretization have to be stated. The uniform mesh for \n is con-\nstructed with mesh-size hand a time step-size k > 0 is set. Let Lbe the set of\nnodesfxl= (xi;yj;zk)gin 3-D space with the indices i= 0;1;\u0001\u0001\u0001;nx;nx + 1;j=\n0;1;\u0001\u0001\u0001;ny;ny +1 andk= 0;1;\u0001\u0001\u0001;nz;nz +1, and the ghost points on the bound-\nary of \n are denoted by ix= 0;nx+ 1,jy= 0;ny+ 1 andkz= 0;nz+ 1. We\nuse the half grid points with mi;j;k=m((i\u00001\n2)hx;(j\u00001\n2)hy;(k\u00001\n2)hz). Here\nhx= 1=nx,hy= 1=ny,hz= 1=nzandh=hx=hy=hzholds for uniform4 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nspatial meshes. Due to the homogeneous Neumann boundary condition (2.2d), the\nfollowing extrapolation formula is derived:\n(2.6)mix+1;j;k=mix;j;k;mi;jy+1;k=mi;jy;k;mi;j;k z+1=mi;j;k z;\nfor any 1\u0014i\u0014nx;1\u0014j\u0014ny;1\u0014k\u0014nz. Meanwhile, the temporal derivatives\nare discretized by the central di\u000berence, with the details stated in the following\nde\fnition.\nDe\fnition 2.1. For\u001en+1=\u001e(x;tn+1)and n+1= (tn+1), de\fne\nd+\nt\u001en=\u001en+1\u0000\u001en\nk; d\u0000\nt\u001en=\u001en\u0000\u001en\u00001\nk;\nand\nD+\nt n= n+1\u0000 n\nk; D\u0000\nt n= n\u0000 n\u00001\nk:\nConsequently, we denote\ndt\u001en+1=1\n2(d+\nt\u001en+d\u0000\nt\u001en); Dt n+1=1\n2(D+\nt n+D\u0000\nt n):\nIn particular, the second time derivative is approximated by the central di\u000berence\nform\n(2.7) dtt\u001e=\u001en+1\u00002\u001en+\u001en\u00001\nk2:\nThen for the initial condition (2.2c), there holds\n(2.8) m(xl;0) =m(xl;k);8l2L;\nwhereL=f(i;j;k )ji= 1;\u0001\u0001\u0001;nx;j= 1;\u0001\u0001\u0001;ny;k= 1;\u0001\u0001\u0001;nz:g. Denotemn\nh(n\u0015\n0) as the numerical solution. Given grid functions fh;gh2`2(\nh;R3), we list\nde\fnitions of the discrete inner product and norms used in this paper.\nDe\fnition 2.2. The discrete inner product h\u0001;\u0001iin`2(\nh;R3)is de\fned by\n(2.9) hfh;ghi=hdX\nl2Lfh(xl)\u0001gh(xl):\nThe discrete `2norm andH1\nhnorm ofmhare\n(2.10) kfhk2\n2=hdX\nl2Lfh(xl)\u0001fh(xl);\nand\n(2.11) kfhk2\nH1\nh=kfhk2\n2+krhfhk2\n2\nwithrhrepresenting the central di\u000berence stencil of the gradient operator.\nBesides, the norm k\u0001k1in`1(\nh;R3) is de\fned by\n(2.12) kfhk1= max\nl2Lkfh(xl)k1:\nTherefore, the approximation scheme of the iLLG equation is presented below.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 5\nAlgorithm 2.1. Givenm0\nh;m1\nh2W1;2(\nh;S2). Letmn\u00001\nh;mn\nh2W1;2(\nh;S2),\nwe computemn+1\nhby\n(2.13) dtmn+1\nh\u0000\u000b\u0016mn\nh\u0002\u0000\ndtmn+1\nh+\u001cdttmn\nh\u0001\n=\u0000\u0016mn\nh\u0002\u0001h\u0016mn\nh;\nwhere \u0016mn\nh=1\n2(mn+1\nh+mn\u00001\nh), and \u0001hrepresents the standard seven-point stencil\nof the Laplacian operator.\nThe corresponding fully discrete version of the above (2.13) reads as\nmn+1\nh\u0000mn\u00001\nh\n2k\u0000\u000bmn+1\nh+mn\u00001\nh\n2\u0002\u0012mn+1\nh\u0000mn\u00001\nh\n2k+\u001cmn+1\nh\u00002mn\nh+mn\u00001\nh\nk2\u0013\n=\u0000mn+1\nh+mn\u00001\nh\n2\u0002\u0001h\u0012mn+1\nh+mn\u00001\nh\n2\u0013\n: (2.14)\nWithin three time steps, there have not been many direct discretization methods\nto get the second-order temporal accuracy. Due to the mid-point approximation\nfeature, this implicit scheme is excellent in maintaining certain properties of the\noriginal system.\nLemma 2.1. Given\f\fm0\nh(xl)\f\f= 1, then the sequence fmn\nh(xl)gn\u00150produced by\n(2.13) satis\fes\n(i)jmn\nh(xl)j= 1;8l2L;\n(ii)1\n2Dtkrhmn+1\nhk2\n2+\u000bkdtmn+1\nhk2\n2+1\n2\u000b\u001cD\u0000\ntkd+\ntmn\nhk2\n2= 0.\nProof. On account of the initial condition (2.2c), we see that m0(xl) =m1(xl)\nholding for all l2L. Taking the vector inner product with (2.13) by ( mn+1\nh(xl) +\nmn\u00001\nh(xl)), it obvious that we can get\njmn+1\nhj=jmn\nhj=\u0001\u0001\u0001=jm1\nhj=jm0\nhj= 1;\nin the point-wise sense. This con\frms (i). In order to verify (ii), we take inner\nproduct with (2.13) by \u0000\u0001h\u0016mn\nhand get\n1\n2Dtkrhmn+1\nhk2\n2\u0000\u000bh\u0016mn\nh\u0002dtmn+1\nh;\u0000\u0001h\u0016mn\nhi\u0000\u000b\u001chmn\nh\u0002dttmn\nh;\u0000\u0001h\u0016mn\nhi= 0:\nSubsequently, taking inner products with dtmn+1\nhanddttmn+1\nhseparately leads to\nthe following equalities:\nkdtmn+1\nhk2\n2\u0000\u000b\u001chmn\nh\u0002dttmn\nh;dtmn+1\nhi=\u0000h\u0016mn\nh\u0002dtmn+1\nh;\u0000\u0001h\u0016mn\nhi;\nand\n1\n2D\u0000\ntkd+\ntmn\nhk2\n2+\u000bhmn\nh\u0002dttmn\nh;dtmn+1\nhi=\u0000hmn\nh\u0002dttmn\nh;\u0000\u0001h\u0016mn\nhi:\nA combination of the above three identities yields (ii). \u0003\nIn lemma 2.1, taking k!0 gives\n(2.15)d\ndt\u00121\n2krhmn+1\nhk2\n2+\u000b\u001c\n2k@tmn\nhk2\n2\u0013\n=\u0000\u000bk@tmn+1\nhk2\n2;\nwhich is consistent with the continuous energy law (2.4). Accordingly, in the ab-\nsence of the external magnetic \feld, the discretized version energy dissipation law\nwould be maintained with a modi\fcation\n(2.16)E(mn+1\nh;mn\nh) =\u000b\u001c\n2\r\r\rmn+1\nh\u0000mn\nh\nk\r\r\r2\n2+1\n4(krhmn+1\nhk2\n2+krhmn\nhk2\n2):6 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nTheorem 2.1. Givenmn\u00001\nh;mn\nh;mn+1\nh2W1;2(\nh;S2), we have a discrete energy\ndissipation law, for the modi\fed energy (2.16) :\n(2.17) E(mn+1\nh;mn\nh)\u0014E(mn\nh;mn\u00001\nh):\nProof. Denote a discrete function\n\u0016n:=\u000b\u0010mn+1\nh\u0000mn\u00001\nh\n2k+\u001cmn+1\nh\u00002mn\nh+mn\u00001\nh\nk2\u0011\n\u00001\n2\u0001h(mn+1\nh+mn\u00001\nh):\nTaking a discrete inner product with (2.13) by \u0016ngives\n\u000b\n4k2hmn+1\nh\u0000mn\u00001\nh;mn+1\nh\u0000mn\u00001\nhi+\u000b\u001c\n2k3hmn+1\nh\u0000mn\u00001\nh;mn+1\nh\u00002mn\nh+mn\u00001\nhi(2.18)\n+\u000b\n4kD\nmn+1\nh\u0000mn\u00001\nh;\u0000\u0001h(mn+1\nh+mn\u00001\nh)E\n=D\n\u0000mn+1\nh+mn\u00001\nh\n2\u0002\u0016n;\u0016nE\n= 0:\nMeanwhile, the following estimates are available:\nhmn+1\nh\u0000mn\u00001\nh;mn+1\nh\u0000mn\u00001\nhi=kmn+1\nh\u0000mn\u00001\nhk2\n2\u00150; (2.19)\nhmn+1\nh\u0000mn\u00001\nh;mn+1\nh\u00002mn\nh+mn\u00001\nhi\n=D\n(mn+1\nh\u0000mn\nh) + (mn\nh\u0000mn\u00001\nh);(mn+1\nh\u0000mn\nh)\u0000(mn\nh\u0000mn\u00001\nh)E\n;\n=kmn+1\nh\u0000mn\nhk2\n2\u0000kmn\nh\u0000mn\u00001\nhk2\n2;(2.20)\nD\nmn+1\nh\u0000mn\u00001\nh;\u0000\u0001h(mn+1\nh+mn\u00001\nh)E\n=D\nrh(mn+1\nh\u0000mn\u00001\nh);rh(mn+1\nh+mn\u00001\nh)E\n=krhmn+1\nhk2\n2\u0000krhmn\u00001\nhk2\n2\n=(krhmn+1\nhk2\n2+krhmn\nhk2\n2)\u0000(krhmn\nhk2\n2+krhmn\u00001\nhk2\n2):(2.21)\nGoing back to (2.18), we arrive at\n\u000b\u001c\n2k\u0010\r\r\rmn+1\nh\u0000mn\nh\nk\r\r\r2\n2\u0000\r\r\rmn\nh\u0000mn\u00001\nh\nk\r\r\r2\n2\u0011\n(2.22)\n+1\n4k\u0010\n(krhmn+1\nhk2\n2+krhmn\nhk2\n2)\u0000(krhmn\nhk2\n2+krhmn\u00001\nhk2\n2)\u0011\n=\u0000\u000b\r\r\rmn+1\nh\u0000mn\u00001\nh\n2k\r\r\r2\n2\u00140;\nwhich is exactly the energy dissipation estimate (2.17). This \fnishes the proof of\nTheorem 2.1. \u0003\nMeanwhile, it is noticed that, given the initial pro\fle of matt= 0, namelym0,\nan accurate approximation to m1andm2has to be made. In more details, an\nO(k2+h2) accuracy is required for both m1,m2andm1\u0000m0\nk,m2\u0000m1\nk, which is\nneeded in the convergence analysis.\nThe initial pro\fle m0could be taken as m0=m(\u0001;0). This in turn gives a\ntrivial zero initial error for m0. Form1andm2, a careful Taylor expansion revealsCONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 7\nthat\nm1=m0+k@tm0+k2\n2@ttm0+O(k3)\n=m0+k2\n2@ttm0+O(k3); (2.23)\nm2=m0+ 2k@tm0+ 2k2@ttm0+O(k3)\n=m0+ 2k2@ttm0+O(k3); (2.24)\nin which the initial data (2.2c), @tm(\u0001;0)\u00110, has been applied in the derivation.\nTherefore, an accurate approximation to m1andm2relies on a precise value of\n@ttmatt= 0. An evaluation of the original PDE (2.2a) implies that\nm0\u0002(@ttm0) =1\n\u000b\u001cm0\u0002(\u0001m0+H0\ne); (2.25)\nin which the trivial initial data (2.2c) has been applied again. Meanwhile, motivated\nby the point-wise temporal di\u000berentiation identity\n(2.26) m\u0001@ttm=\u0000(@tm)2+1\n2@tt(jmj2) =\u0000(@tm)2;\nand the fact that jmj\u00111, we see that its evaluation at t= 0 yields\n(2.27) m0\u0001@ttm0=\u0000(@tm0)2= 0:\nSubsequently, a combination of (2.26) and (2.27) uniquely determines @ttm0:\n(2.28) @ttm0=\u00001\n\u000b\u001cm0\u0002(m0\u0002(\u0001m0+H0\ne));\nand a substitution of this value into (2.23), (2.24) leads to an O(k3) approximation\ntom1andm2.\nMoreover, with spatial approximation introduced, an O(k2+h2) accuracy is\nobtained for both m1,m2andm1\u0000m0\nk,m2\u0000m1\nk. This \fnishes the initialization\nprocess.\n3.Convergence analysis\nThe theoretical result concerning the convergence analysis is stated below.\nTheorem 3.1. Assume that the exact solution of (2.2) has the regularity me2\nC3([0;T]; [C0(\u0016\n)]3)\\C2([0;T]; [C2(\u0016\n)]3)\\L1([0;T]; [C4(\u0016\n)]3). Denote a nodal\ninterpolation operator Phsuch thatPhmh2C1(\n), and the numerical solution mn\nh\n(n\u00150) obtained from (2.13) with the initial error satisfying kep\nhk2+krhep\nhk2=\nO(k2+h2), whereep\nh=Phme(\u0001;tp)\u0000mp\nh,p= 0;1;2, andkeq+1\nh\u0000eq\nh\nkk2=O(k2+h2),\nq= 0;1. Then the following convergence result holds for 2\u0014n\u0014\u0004T\nk\u0005\nash;k!0+:\nkPhme(\u0001;tn)\u0000mn\nhk2+krh(Phme(\u0001;tn)\u0000mn\nh)k2\u0014C(k2+h2); (3.1)\nin which the constant C>0is independent of kandh.\nBefore the rigorous proof is given, the following estimates are declared, which\nwill be utilized in the convergence analysis. In the sequel, for simplicity of notation,\nwe will use a uniform constant Cto denote all the controllable constants throughout\nthis part.8 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nLemma 3.1 (Discrete gradient acting on cross product) .[7]For grid functions fh\nandghover the uniform numerical grid, we have\nkrh(fh\u0002gh)k2\u0014C\u0010\nkfhk2\u0001krhghk1+kghk1\u0001krhfhk2\u0011\n: (3.2)\nLemma 3.2 (Point-wise product involved with second order temporal stencil) .For\ngrid functions fhandghover the time domain, we have\nfn+1\nh\u00002fn\nh+fn\u00001\nh\nk2\u0001gn\nh=\u0000fn\nh\u0000fn\u00001\nh\nk\u0001gn\nh\u0000gn\u00001\nh\nk\n+1\nk\u0010fn+1\nh\u0000fn\nh\nk\u0001gn\nh\u0000fn\nh\u0000fn\u00001\nh\nk\u0001gn\u00001\nh\u0011\n: (3.3)\nNow we proceed into the convergence estimate. First, we construct an approxi-\nmate solution m:\n(3.4) m=me+h2m(1);\nin which the auxiliary \feld m(1)satis\fes the following Poisson equation\n\u0001m(1)=^Cwith ^C=1\nj\njZ\n@\n@3\n\u0017meds; (3.5)\n@zm(1)jz=0=\u00001\n24@3\nzmejz=0; @zm(1)jz=1=1\n24@3\nzmejz=1;\nwith boundary conditions along xandydirections de\fned in a similar way.\nThe purpose of such a construction will be illustrated later. Then we extend the\napproximate pro\fle mto the numerical \\ghost\" points, according to the extrapo-\nlation formula:\n(3.6) mi;j;0=mi;j;1;mi;j;nz +1=mi;j;nz;\nand the extrapolation for other boundaries can be formulated in the same man-\nner. Subsequently, we prove that such an extrapolation yields a higher order\nO(h5) approximation, instead of the standard O(h3) accuracy. Also see the re-\nlated works [24, 25, 26] in the existing literature.\nPerforming a careful Taylor expansion for the exact solution around the boundary\nsectionz= 0, combined with the mesh point values: z0=\u00001\n2h,z1=1\n2h, we get\nme(xi;yj;z0) =me(xi;yj;z1)\u0000h@zme(xi;yj;0)\u0000h3\n24@3\nzme(xi;yj;0) +O(h5)\n=me(xi;yj;z1)\u0000h3\n24@3\nzme(xi;yj;0) +O(h5); (3.7)\nin which the homogenous boundary condition has been applied in the second step.\nA similar Taylor expansion for the constructed pro\fle m(1)reveals that\nm(1)(xi;yj;z0) =m(1)(xi;yj;z1)\u0000h@zm(1)(xi;yj;0) +O(h3)\n=m(1)(xi;yj;z1) +h\n24@3\nzme(xi;yj;0) +O(h3); (3.8)\nwith the boundary condition in (3.5) applied. In turn, a substitution of (3.7)-(3.8)\ninto (3.4) indicates that\n(3.9) m(xi;yj;z0) =m(xi;yj;z1) +O(h5):\nIn other words, the extrapolation formula (3.6) is indeed O(h5) accurate.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 9\nAs a result of the boundary extrapolation estimate (3.9), we see that the discrete\nLaplacian of myields the second-order accuracy at all the mesh points (including\nboundary points):\n(3.10)\n\u0001hmi;j;k= \u0001me(xi;yj;zk)+O(h2);80\u0014i\u0014nx+1;0\u0014j\u0014ny+1;0\u0014k\u0014nz+1:\nMoreover, a detailed calculation of Taylor expansion, in both time and space, leads\nto the following truncation error estimate:\nmn+1\nh\u0000mn\u00001\nh\n2k=mn+1\nh+mn\u00001\nh\n2\u0002\u0010\n\u000bmn+1\nh\u0000mn\u00001\nh\n2k+\u000b\u001cmn+1\nh\u00002mn\nh+mn\u00001\nh\nk2\n\u0000\u0001h\u0010mn+1\nh+mn\u00001\nh\n2\u0011\u0011\n+\u001cn; (3.11)\nwherek\u001cnk2\u0014C(k2+h2). In addition, a higher order Taylor expansion in space\nand time reveals the following estimate for the discrete gradient of the truncation\nerror, in both time and space:\n(3.12) krh\u001cnk2;k\u001cn\u0000\u001cn\u00001\nkk2\u0014C(k2+h2):\nIn fact, such a discrete k\u0001kH1\nhbound for the truncation comes from the regularity as-\nsumption for the exact solution, me2C3([0;T]; [C0(\u0016\n)]3)\\C2([0;T]; [C2(\u0016\n)]3)\\\nL1([0;T]; [C4(\u0016\n)]3), as stated in Theorem 3.1, as well as the fact that m(1)2\nC1([0;T]; [C1(\u0016\n)]3)\\L1([0;T]; [C2(\u0016\n)]3), as indicated by the Poisson equation (3.5).\nWe introduce the numerical error function en\nh=mn\nh\u0000mn\nh, instead of a di-\nrect comparison between the numerical solution and the exact solution. The error\nfunction between the numerical solution and the constructed solution mhwill be\nanalyzed, due to its higher order consistency estimate (3.9) around the boundary.\nTherefore, a subtraction of (2.14) from the consistency estimate (3.11) leads to the\nerror function evolution system:\nen+1\nh\u0000en\u00001\nh\n2k=mn+1\nh+mn\u00001\nh\n2\u0002~\u0016n\nh+en+1\nh+en\u00001\nh\n2\u0002\u0016n\nh+\u001cn;(3.13)\n\u0016n\nh:=\u000b\u0010mn+1\nh\u0000mn\u00001\nh\n2k+\u001cmn+1\nh\u00002mn\nh+mn\u00001\nh\nk2\u0011\n\u0000\u0001h\u0010mn+1\nh+mn\u00001\nh\n2\u0011\n;(3.14)\n~\u0016n\nh:=\u000b\u0010en+1\nh\u0000en\u00001\nh\n2k+\u001cen+1\nh\u00002en\nh+en\u00001\nh\nk2\u0011\n\u0000\u0001h\u0010en+1\nh+en\u00001\nh\n2\u0011\n:(3.15)\nBefore proceeding into the formal estimate, we establish a W1\nhbound for\u0016n\nh,\nwhich is based on the constructed approximate solution m(by (3.14)). Because of\nthe regularity for me, the following bound is available:\nk\u0016`\nhk1;krh\u0016`\nhk1;k\u0016n\nh\u0000\u0016n\u00001\nh\nkk1\u0014C; ` =n;n\u00001: (3.16)\nIn addition, the following preliminary estimate will be useful in the convergence\nanalysis.10 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nLemma 3.3 (A preliminary error estimate) .We have\nke`\nhk2\n2\u00142ke0\nhk2\n2+ 2Tk`\u00001X\nj=0kej+1\nh\u0000ej\nh\nkk2\n2;8`\u0001k\u0014T: (3.17)\nProof. We begin with the expansion:\ne`\nh=e0\nh+k`\u00001X\nj=0ej+1\nh\u0000ej\nh\nk;8`\u0001k\u0014T: (3.18)\nIn turn, a careful application of the Cauchy inequality reveals that\nke`\nhk2\n2\u00142\u0010\nke0\nhk2\n2+k2k`\u00001X\nj=0ej+1\nh\u0000ej\nh\nkk2\n2\u0011\n; (3.19)\nk2k`\u00001X\nj=0ej+1\nh\u0000ej\nh\nkk2\n2\u0014k2\u0001`\u0001`\u00001X\nj=0kej+1\nh\u0000ej\nh\nkk2\n2\u0014Tk`\u00001X\nj=0kej+1\nh\u0000ej\nh\nkk2\n2; (3.20)\nin which the fact that `\u0001k\u0014Thas been applied. Therefore, a combination of\n(3.19) and (3.20) yields the desired estimate (3.17). This completes the proof of\nLemma 3.3. \u0003\nTaking a discrete inner product with the numerical error equation (3.13) by ~\u0016n\nh\ngives\n1\n2khen+1\nh\u0000en\u00001\nh;~\u0016n\nhi=hmn+1\nh+mn\u00001\nh\n2\u0002~\u0016n\nh;~\u0016n\nhi\n+hen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;~\u0016n\nhi+h\u001cn;~\u0016n\nhi: (3.21)\nThe analysis on the left hand side of (3.21) is similar to the ones in (2.19)-(2.21):\n1\n2khen+1\nh\u0000en\u00001\nh;~\u0016n\nhi=\u000b\u001c\n2k3hen+1\nh\u0000en\u00001\nh;en+1\nh\u00002en\nh+en\u00001\nhi\n+\u000b\n4k2hen+1\nh\u0000en\u00001\nh;en+1\nh\u0000en\u00001\nhi\n+1\n4kD\nrh(en+1\nh\u0000en\u00001\nh);rh(en+1\nh+en\u00001\nh)E\n; (3.22)\nhen+1\nh\u0000en\u00001\nh;en+1\nh\u0000en\u00001\nhi=ken+1\u0000en\u00001\nhk2\n2\u00150; (3.23)\nhen+1\nh\u0000en\u00001\nh;en+1\nh\u00002en\nh+en\u00001\nhi\n=ken+1\nh\u0000en\nhk2\n2\u0000ken\nh\u0000en\u00001\nhk2\n2; (3.24)\nD\nen+1\nh\u0000en\u00001\nh;\u0000\u0001h(en+1\nh+en\u00001\nh)E\n=D\nrh(en+1\nh\u0000en\u00001\nh);rh(en+1\nh+en\u00001\nh)E\n=krhen+1\nhk2\n2\u0000krhen\u00001\nhk2\n2\n=(krhen+1\nhk2\n2+krhen\nhk2\n2)\u0000(krhen\nhk2\n2+krhen\u00001\nhk2\n2): (3.25)\nThis in turn leads to the following identity:\n1\n2khen+1\nh\u0000en\u00001\nh;~\u0016n\nhi=1\nk(En+1e;h\u0000Ene;h) +\u000b\n4k2ken+1\nh\u0000en\u00001\nhk2\n2; (3.26)\nEn+1e;h=\u000b\u001c\n2ken+1\nh\u0000en\nh\nkk2\n2+1\n4(krhen+1\nhk2\n2+krhen\nhk2\n2): (3.27)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 11\nThe \frst term on the right hand side of (3.21) vanishes, due to the fact that\nmn+1\nh+mn\u00001\nh\n2\u0002~\u0016n\nhis orthogonal to ~\u0016n\nh, at a point-wise level:\n(3.28) hmn+1\nh+mn\u00001\nh\n2\u0002~\u0016n\nh;~\u0016n\nhi= 0:\nThe second term on the right hand side of (3.21) contains three parts:\nhen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;~\u0016n\nhi=I1+I2+I3; (3.29)\nI1=\u000bhen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;en+1\nh\u0000en\u00001\nh\n2ki; (3.30)\nI2=\u000b\u001chen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;en+1\nh\u00002en\nh+en\u00001\nh\nk2i; (3.31)\nI3=hen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;\u0000\u0001h\u0010en+1\nh+en\u00001\nh\n2\u0011\ni: (3.32)\nThe \frst inner product, I1, could be bounded in a straightforward way, with the\nhelp of discrete H older inequality:\nI1=\u000bhen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;en+1\nh\u0000en\u00001\nh\n2ki\n\u0014\u000b\n4ken+1\nh+en\u00001\nhk2\u0001k\u0016n\nhk1\u0001ken+1\nh\u0000en\u00001\nh\nkk2\n\u0014Cken+1\nh+en\u00001\nhk2\u0001ken+1\nh\u0000en\u00001\nh\nkk2\n\u0014C(ken+1\nhk2\n2+ken\u00001\nhk2\n2+ken+1\nh\u0000en\u00001\nh\nkk2\n2): (3.33)\nFor the second inner product, I2, we denotegn\nh:=en+1\nh+en\u00001\nh\n2\u0002\u0016n\nh. An application\nof point-wise identity (3.3) (in lemma 3.2) reveals that\nI2=\u000b\u001chgn\nh;en+1\nh\u00002en\nh+en\u00001\nh\nk2i\n=\u0000\u000b\u001chen\nh\u0000en\u00001\nh\nk;gn\nh\u0000gn\u00001\nh\nki\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;gn\nhi\u0000hen\nh\u0000en\u00001\nh\nk;gn\u00001\nhi\u0011\n: (3.34)\nMeanwhile, the following expansion is observed:\ngn\nh\u0000gn\u00001\nh\nk=1\n4(en+1\nh\u0000en\nh\nk+en\u00001\nh\u0000en\u00002\nh\nk)\u0002(\u0016n\nh+\u0016n\u00001\nh)\n+en+1\nh+en\nh+en\u00001\nh+en\u00002\nh\n4\u0002\u0016n\nh\u0000\u0016n\u00001\nh\nk: (3.35)12 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nThis in turn indicates the associated estimate:\nkgn\nh\u0000gn\u00001\nh\nkk2\u00141\n4(ken+1\nh\u0000en\nh\nkk2+ken\u00001\nh\u0000en\u00002\nh\nkk2)\u0001(k\u0016n\nhk1+k\u0016n\u00001\nhk1)\n+ken+1\nhk2+ken\nhk2+ken\u00001\nhk2+ken\u00002\nhk2\n4\u0001k\u0016n\nh\u0000\u0016n\u00001\nh\nkk1\n\u0014C\u0010\nken+1\nh\u0000en\nh\nkk2+ken\u00001\nh\u0000en\u00002\nh\nkk2\n+ken+1\nhk2+ken\nhk2+ken\u00001\nhk2+ken\u00002\nhk2\u0011\n; (3.36)\nin which the bound (3.16) has been applied. Going back to (3.34), we see that\n\u0000\u000b\u001chen\nh\u0000en\u00001\nh\nk;gn\nh\u0000gn\u00001\nh\nki\u0014\u000b\u001cken\nh\u0000en\u00001\nh\nkk2\u0001kgn\nh\u0000gn\u00001\nh\nkk2\n\u0014C\u0010\nken+1\nh\u0000en\nh\nkk2+ken\u00001\nh\u0000en\u00002\nh\nkk2+ken+1\nhk2\n+ken\nhk2+ken\u00001\nhk2+ken\u00002\nhk2\u0011\nken\nh\u0000en\u00001\nh\nkk2\n\u0014C\u0010\nken+1\nh\u0000en\nh\nkk2\n2+ken\u00001\nh\u0000en\u00002\nh\nkk2\n2+ken+1\nhk2\n2\n+ken\nhk2\n2+ken\u00001\nhk2\n2+ken\u00002\nhk2\n2+ken\nh\u0000en\u00001\nh\nkk2\n2\u0011\n; (3.37)\nI2\u0014C\u0010\nken+1\nh\u0000en\nh\nkk2\n2+ken\u00001\nh\u0000en\u00002\nh\nkk2\n2+ken+1\nhk2\n2\n+ken\nhk2\n2+ken\u00001\nhk2\n2+ken\u00002\nhk2\n2+ken\nh\u0000en\u00001\nh\nkk2\n2\u0011\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;gn\nhi\u0000hen\nh\u0000en\u00001\nh\nk;gn\u00001\nhi\u0011\n: (3.38)\nFor the third inner product part, I3, an application of summation by parts formula\ngives\nI3=hen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;\u0000\u0001h\u0010en+1\nh+en\u00001\nh\n2\u0011\ni\n=hrh\u0010en+1\nh+en\u00001\nh\n2\u0002\u0016n\nh\u0011\n;rh\u0010en+1\nh+en\u00001\nh\n2\u0011\ni: (3.39)\nMeanwhile, we make use of the preliminary inequality (3.2) (in lemma 3.1) and get\nkrh\u0010en+1\nh+en\u00001\nh\n2\u0002\u0016n\nh\u0011\nk2\n\u0014C\u0010\nken+1\nh+en\u00001\nh\n2k2\u0001krh\u0016n\nhk1+k\u0016n\nhk1\u0001krh(en+1\nh+en\u00001\nh\n2)k2\u0011\n\u0014C\u0010\nken+1\nh+en\u00001\nh\n2k2+krh(en+1\nh+en\u00001\nh\n2)k2\u0011\n\u0014C\u0010\nken+1\nhk2+ken\u00001\nhk2+krhen+1\nhk2+krhen\u00001\nhk2\u0011\n: (3.40)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 13\nAgain, the bound (3.16) has been applied in the derivation. Therefore, the following\nestimate is available for I3:\nI3\u0014krh\u0010en+1\nh+en\u00001\nh\n2\u0002\u0016n\nh\u0011\nk2\u0001krh\u0010en+1\nh+en\u00001\nh\n2\u0011\nk2\n\u0014C\u0010\nken+1\nhk2+ken\u00001\nhk2+krhen+1\nhk2+krhen\u00001\nhk2\u0011\n\u0001\u0010\nkrhen+1\nhk2+krhen\u00001\nhk2\u0011\n\u0014C\u0010\nken+1\nhk2\n2+ken\u00001\nhk2\n2+krhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n: (3.41)\nThe estimate of I3can also be obtained by a direct application of discrete H older\ninequality:\nI3=h\u0010en+1\nh+en\u00001\nh\n2\u0002rh\u0016n\nh\u0011\n;rh\u0010en+1\nh+en\u00001\nh\n2\u0011\ni\n\u00141\n4ken+1\nh+en\u00001\nhk2\u0001krh\u0016n\nhk1\u0001krh(en+1\nh+en\u00001\nh)k2\n\u0014C\u0010\nken+1\nhk2\n2+ken\u00001\nhk2\n2+krhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n: (3.42)\nA substitution of (3.33), (3.38) and (3.42) into (3.29) yields the following bound:\nhen+1\nh+en\u00001\nh\n2\u0002\u0016n\nh;~\u0016n\nhi=I1+I2+I3\n\u0014C\u0010\nken+1\nh\u0000en\nh\nkk2\n2+ken\nh\u0000en\u00001\nh\nkk2\n2+ken\u00001\nh\u0000en\u00002\nh\nkk2\n2\n+ken+1\nhk2\n2+ken\nhk2\n2+ken\u00001\nhk2\n2+ken\u00002\nhk2\n2+krhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;gn\nhi\u0000hen\nh\u0000en\u00001\nh\nk;gn\u00001\nhi\u0011\n: (3.43)\nThe third term on the right hand side of (3.21) could be analyzed in a similar\nfashion:\nh\u001cn;~\u0016n\nhi=I4+I5+I6; (3.44)\nI4=\u000bh\u001cn;en+1\nh\u0000en\u00001\nh\n2ki; I 5=\u000b\u001ch\u001cn;en+1\nh\u00002en\nh+en\u00001\nh\nk2i; (3.45)\nI6=h\u001cn;\u0000\u0001h\u0010en+1\nh+en\u00001\nh\n2\u0011\ni; (3.46)\nI4=\u000bh\u001cn;en+1\nh\u0000en\u00001\nh\n2ki\u0014\u000b\n2k\u001cnk2\u0001ken+1\nh\u0000en\u00001\nh\nkk2\n\u0014\u000b\n4(k\u001cnk2\n2+ken+1\nh\u0000en\u00001\nh\nkk2\n2); (3.47)\nI5=\u000b\u001ch\u001cn;en+1\nh\u00002en\nh+en\u00001\nh\nk2i\n=\u0000\u000b\u001chen\nh\u0000en\u00001\nh\nk;\u001cn\u0000\u001cn\u00001\nki\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;\u001cni\u0000hen\nh\u0000en\u00001\nh\nk;\u001cn\u00001i\u0011\n; (3.48)14 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\n\u0000hen\nh\u0000en\u00001\nh\nk;\u001cn\u0000\u001cn\u00001\nki\u0014ken\nh\u0000en\u00001\nh\nkk2\u0001k\u001cn\u0000\u001cn\u00001\nkk2\n\u0014C(k2+h2)ken\nh\u0000en\u00001\nh\nkk2\u0014C(k4+h4) +1\n2ken\nh\u0000en\u00001\nh\nkk2\n2; (3.49)\nI5\u0014C(k4+h4) +\u000b\u001c\n2ken\nh\u0000en\u00001\nh\nkk2\n2\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;\u001cni\u0000hen\nh\u0000en\u00001\nh\nk;\u001cn\u00001i\u0011\n; (3.50)\nI6=h\u001cn;\u0000\u0001h\u0010en+1\nh+en\u00001\nh\n2\u0011\ni=hrh\u001cn;rh\u0010en+1\nh+en\u00001\nh\n2\u0011\ni\n\u0014krh\u001cnk2\u0001krh\u0010en+1\nh+en\u00001\nh\n2\u0011\nk2\u0014C(k2+h2)krh\u0010en+1\nh+en\u00001\nh\n2\u0011\nk2\n\u0014C(k4+h4) +1\n2\u0010\nkrhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n: (3.51)\nNotice that the truncation error estimate (3.12) has been repeatedly applied in the\nderivation. Going back to (3.44), we obtain\nh\u001cn;~\u0016n\nhi=I4+I5+I6\n\u0014C(k4+h4) +\u000b\n2ken+1\nh\u0000en\nh\nkk2\n2+\u000b(\u001c+ 1)\n2ken\nh\u0000en\u00001\nh\nkk2\n2\n+1\n2\u0010\nkrhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;\u001cni\u0000hen\nh\u0000en\u00001\nh\nk;\u001cn\u00001i\u0011\n: (3.52)\nFinally, a substitution of (3.26)-(3.27), (3.28), (3.43) and (3.52) into (3.21) leads\nto the following inequality:\n1\nk(En+1e;h\u0000Ene;h) +\u000b\n4k2ken+1\nh\u0000en\u00001\nhk2\n2\n\u0014C(k4+h4) +C\u0010\nken+1\nh\u0000en\nh\nkk2\n2+ken\nh\u0000en\u00001\nh\nkk2\n2+ken\u00001\nh\u0000en\u00002\nh\nkk2\n2\n+ken+1\nhk2\n2+ken\nhk2\n2+ken\u00001\nhk2\n2+ken\u00002\nhk2\n2+krhen+1\nhk2\n2+krhen\u00001\nhk2\n2\u0011\n+\u000b\u001c\nk\u0010\nhen+1\nh\u0000en\nh\nk;gn\nh+\u001cni\u0000hen\nh\u0000en\u00001\nh\nk;gn\u00001\nh+\u001cn\u00001i\u0011\n: (3.53)\nSubsequently, a summation in time yields\nEn+1e;h\u0014E2e;h+CT(k4+h4) +Ck\u0010nX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2+n+1X\nj=0(kej\nhk2\n2+krhej\nhk2\n2)\u0011\n+\u000b\u001c\u0010\nhen+1\nh\u0000en\nh\nk;gn\nh+\u001cni\u0000he2\nh\u0000e1\nh\nk;g1\nh+\u001c1i\u0011\n: (3.54)CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 15\nFor the term \u000b\u001chen+1\nh\u0000en\nh\nk;gn\nh+\u001cni, the following estimate could be derived\n\u000b\u001chen+1\nh\u0000en\nh\nk;gn\nh+\u001cni\u0014\u000b\u001c\n4ken+1\nh\u0000en\nh\nkk2\n2+ 2\u000b\u001c(kgn\nhk2\n2+k\u001cnk2\n2); (3.55)\nkgn\nhk2=ken+1\nh+en\u00001\nh\n2\u0002\u0016n\nhk2\u0014ken+1\nh+en\u00001\nh\n2k2\u0001k\u0016n\nhk1\n\u0014Cken+1\nh+en\u00001\nh\n2k2\u0014C(ken+1\nhk2+ken\u00001\nhk2); (3.56)\nin which the bound (3.16) has been used again. Then we get\n\u000b\u001chen+1\nh\u0000en\nh\nk;gn\nh+\u001cni\u0014\u000b\u001c\n4ken+1\nh\u0000en\nh\nkk2\n2+ 2\u000b\u001ck\u001cnk2\n2\n+C(ken+1\nhk2\n2+ken\u00001\nhk2\n2)\n\u00141\n2En+1e;h+ 2\u000b\u001ck\u001cnk2\n2+C(ken+1\nhk2\n2+ken\u00001\nhk2\n2); (3.57)\nin which the expansion identity, En+1e;h=\u000b\u001c\n2ken+1\nh\u0000en\nh\nkk2\n2+1\n4(krhen+1\nhk2\n2+krhen\nhk2\n2)\n(given by (3.27)), has been applied. Its substitution into (3.54) gives\nEn+1e;h\u00142E2e;h+CT(k4+h4) +Ck\u0010nX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2+n+1X\nj=0(kej\nhk2\n2+krhej\nhk2\n2)\u0011\n+C(ken+1\nhk2\n2+ken\u00001\nhk2\n2) + 4\u000b\u001ck\u001cnk2\n2\u00002\u000b\u001che2\nh\u0000e1\nh\nk;g1\nh+\u001c1i: (3.58)\nMoreover, an application of the preliminary error estimate (3.17) (in Lemma 3.3)\nleads to\nEn+1e;h\u00142E2e;h+CT(k4+h4) +C(T2+ 1)knX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2+CTke0\nhk2\n2\n+Ckn+1X\nj=0krhej\nhk2\n2+ 4\u000b\u001ck\u001cnk2\n2\u00002\u000b\u001che2\nh\u0000e1\nh\nk;g1\nh+\u001c1i; (3.59)\nin which we have made use of the following fact:\nkn+1X\nj=0kej\nhk2\n2\u0014k\u0001(n+ 1)\u0010\n2ke0\nhk2\n2+ 2TknX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2\u0011\n\u00142Tke0\nhk2\n2+ 2T2knX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2: (3.60)16 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nIn addition, for the initial error quantities, the following estimates are available:\nE2e;h=\u000b\u001c\n2ke2\nh\u0000e1\nh\nkk2\n2+1\n4(krhe2\nhk2\n2+krhe1\nhk2\n2)\u0014C(k4+h4); (3.61)\nke0\nhk2\n2\u0014C(k4+h4); (3.62)\n4\u000b\u001ck\u001cnk2\n2\u0014C(k4+h4); (3.63)\nkg1\nhk2=ke2\nh+e0\nh\n2\u0002\u00161\nhk2\u0014ke2\nh+e0\nh\n2k2\u0001k\u00161\nhk1\u0014C(k2+h2); (3.64)\n\u00002\u000b\u001che2\nh\u0000e1\nh\nk;g1\nh+\u001c1i\u00142\u000b\u001cke2\nh\u0000e1\nh\nkk2\u0001(kg1\nhk2+k\u001c1k2)\n\u0014C(k4+h4); (3.65)\nwhich comes from the assumption in Theorem 3.1. Then we arrive at\nEn+1e;h\u0014C(T2+ 1)knX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2+Ckn+1X\nj=0krhej\nhk2\n2+C(T+ 1)(k4+h4)\n\u0014C(T+ 1)(k4+h4) +C(T2+ 1)knX\nj=0Ej+1e;h; (3.66)\nin which the fact that Ej+1e;h=\u000b\u001c\n2kej+1\nh\u0000ej\nh\nkk2\n2+1\n4(krhej+1\nhk2\n2+krhej\nhk2\n2), has been\nused. In turn, an application of discrete Gronwall inequality results in the desired\nconvergence estimate:\nEn+1e;h\u0014CTeCT(k4+h4);for all (n+ 1) :n+ 1\u0014\u0016T\nk\u0017\n; (3.67)\nken+1\nh\u0000en\nh\nkk2+krhen+1\nhk2\u0014C(k2+h2): (3.68)\nAgain, an application of the preliminary error estimate (3.17) (in Lemma 3.3) im-\nplies that\nken+1\nhk2\n2\u00142ke0\nhk2\n2+ 2TknX\nj=0kej+1\nh\u0000ej\nh\nkk2\n2\u0014C(k4+h4);\nso thatken+1\nhk2\u0014C(k2+h2): (3.69)\nA combination of (3.68) and (3.69) \fnishes the proof of Theorem 3.1.\n4.A numerical solver for the nonlinear system\nIt is clear that Algorithm 2.1 is a nonlinear scheme. The following \fxed-point\niteration is employed to solve it.\nAlgorithm 4.1. Setmn+1;0\nh= 2mn\nh\u0000mn\u00001\nhandp= 0.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 17\n(i)Computemn+1;p+1\nhsuch that\n(4.1)mn+1;p+1\nh\u0000mn\u00001\nh\n2k=\u0000mn+1;p+1\nh+mn\u00001\nh\n2\u0002\u0001h \nmn+1;p\nh+mn\u00001\nh\n2!\n+\u000bmn+1;p+1\nh+mn\u00001\nh\n2\u0002 \nmn+1;p+1\nh\u0000mn\u00001\nh\n2k+\u001cmn+1;p+1\nh\u00002mn\nh+mn\u00001\nh\nk2!\n:\n(ii)Ifkmn+1;p+1\nh\u0000mn+1;p\nhk2\u0014\u000f, then stop and set mn+1\nh=mn+1;p+1\nh.\n(iii) Setp p+ 1and go to (i).\nDenote the operator\n(4.2)Lp=I\u0000\u000bmn\u00001\nh\u0002\u00002\u000b\u001c\nkmn\nh\u0002\u0000k\n2\u0001h(mn+1;p\nh+mn\u00001\nh)\u0002;\nand make the \fxed-point iteration solve the following equation\n(4.3)Lpmn+1;p+1\nh=mn\u00001\nh+2\u000b\u001c\nkmn\nh\u0002mn\u00001\nh\u0000k\n2mn\u00001\nh\u0002\u0001h(mn+1;p\nh+mn\u00001\nh);\nin its inner iteration. Under the condition k\u0014Ch2withCa constant, the following\nlemma con\frms the convergence of Algorithm 4.1. For any l2Land owing to the\nproperty ofjmh(xl)j= 1, it is clear that 0 <kmhk1\u00141. For the discretized `2\nnorm of ^m2C3([0;T]; [C0(\u0016\n)]3)\\C2([0;T]; [C2(\u0016\n)]3)\\L1([0;T]; [C4(\u0016\n)]3), we\nhave\n(4.4) krh^mk2\u00142h\u00001k^mk2:\nThen\n(4.5)k\u0001h^mk2\n2=\u0000hrh^m;rh\u0001h^mi\n\u0014krh^mk2krh\u0001h^mk2\u00142h\u00001krh^mk2k\u0001h^mk2;\nwhich in turn implies the following inverse inequality:\n(4.6) k\u0001h^mk2\u00144h\u00002k^mk2:\nLemma 4.1. Letjmn\u00001\nhj=jmn\nhj= 1, there exists a constant c0such thatkmn\u00001\nhk1,\nkmn\nhk1\u0014c0. The solution mn+1;p\nhcalculated by (4.1) satis\fesjmn+1;p\nhj=jmn\u00001\nhj\nforp= 1;2;\u0001\u0001\u0001, which means that we can still \fnd the constant c0\u00141satisfying\nkmn+1;p\nhk1\u0014c0. Then, for all p\u00151, there exists a unique solution mn+1;p\nhin\n(4.1) and the following inequality is valid:\n(4.7)kmn+1;p+1\nh\u0000mn+1;p\nhk2\u00144c0kh\u00002kmn+1;p\u0000mn+1;p\u00001k2:\nProof. For anymh2S2, the following identity is clear:\nhmh;Lpmhi= 1;\nfor allp\u00151. Thus the operator Lpis positive de\fnite for all p\u00151, which\nprovides the unique solvability of (4.1). Taking the discrete inner product with\n(4.1) bymn+1;p+1\nh+mn\u00001\nh, we havejmn+1;p+1\nhj= 1 in a point-wise sense, which\nmeans that the length of the magnetization is preserved at each step in the inner18 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\niteration. Thus, we can \fnd a constant c0\u00141 to control the `1norm ofmn\u00001\nh,\nmn\nhandmn+1;p\nhforp= 1;2;\u0001\u0001\u0001simultaneously.\nSubtraction of two subsequent equations in the \fxed-point iteration yields\n1\n2k(mn+1;p+1\nh\u0000mn+1;p\nh) =\u00001\n4(mn+1;p+1\nh\u0000mn+1;p\nh)\u0002\u0001hmn+1;p\nh\n\u00001\n4mn+1;p\nh\u0002\u0001h(mn+1;p\nh\u0000mn+1;p\u00001\nh)\n\u00001\n4mn\nh\u0002\u0001h(mn+1;p\u0000mn+1;p\u00001)\n\u00001\n4(mn+1;p+1\nh\u0000mn+1;p)\u0002\u0001hmn\u00001\nh\n+\u000b\n2kmn\u00001\nh\u0002(mn+1;p+1\nh\u0000mn+1;p\nh) +\u000b\u001c\n2k2mn\nh\u0002(mn+1;p+1\nh\u0000mn+1;p\nh):\nTaking the inner product with ( mn+1;p+1\nh\u0000mn+1;p\nh) by the above equation produces\nkmn+1;p+1\nh\u0000mn+1;p\nhk2\u0014k\n2kmn+1;p\nhk1k\u0001h(mn+1;p\nh\u0000mn+1;p\u00001\nh)k2\n+k\n2kmn\nhk1k\u0001h(mn+1;p\nh\u0000mn+1;p\u00001\nh)k2\n\u0014c0kk\u0001h(mn+1;p\u0000mn+1;p\u00001)k2:\nIn turn, the convergence result becomes\n(4.8)kmn+1;p+1\nh\u0000mn+1;p\nhk2\u00144c0kh\u00002kmn+1;p\nh\u0000mn+1;p\u00001\nhk2;\nwhich completes the proof of Lemma 4.1. \u0003\n5.Numerical experiments\n5.1.Accuracy tests. Consider the 1-D iLLG equation\n@tm=\u0000m\u0002@xxm+\u000bm\u0002(@tm+\u001c@ttm) +f:\nThe exact solution is chosen to be me= (cos(\u0016x) sin(t2);sin(\u0016x) sin(t2);cos(t2))T\nwith \u0016x=x2(1\u0000x)2, and the forcing term is given by f=@tme+me\u0002@xxme\u0000\n\u000bme\u0002(@tme+\u001c@ttme). Fixing the tolerance \u000f= 1.0e-07 for the \fxed-point\niteration, we record the discrete `2and`1errors between the exact solution and\nnumerical solution with a sequence of temporal step-size and spatial mesh-size. The\nparameters in the above 1-D equation are set as: \u000b= 0:1,\u001c= 10:0, and the \fnal\ntimeT= 0:01. The temporal step-sizes and spatial mesh-sizes are listed in the\nTable 1 and Table 2.\nTable 1. The discrete `2and`1errors in terms of the temporal\nstep-size. The spatial mesh-size is \fxed as h= 0:001 over \n =\n(0;1) and the \fnal time is T= 0:01.\nkkmh\u0000mek2kmh\u0000mek1\nT/40 1.2500e-11 1.2584e-11\nT/60 5.6887e-12 5.6024e-12\nT/80 3.2008e-12 3.1525e-12\nT/100 2.0487e-12 2.0174e-12\norder 1.97 2.00CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 19\nTable 2. The discrete `2and`1error in terms of the spatial\nmesh-size. The parameters are set as: the temporal step-size k=\n2.0e-06, \n = (0 ;1) and the \fnal time T= 0:5.\nhkmh\u0000mek2kmh\u0000mek1\n1/20 1.9742e-05 2.5320e-05\n1/40 4.9846e-06 6.3459e-06\n1/60 2.2340e-06 2.8201e-06\n1/80 1.2720e-06 1.5853e-06\norder 1.98 2.00\nIn addition, the 3-D iLLG equation is also considered:\n@tm=\u0000m\u0002\u0001m+\u000bm\u0002(@tm+\u001c@ttm) +f:\nThe exact solution is chosen to be me= (cos(\u0016x\u0016y\u0016z) sin(t2);sin(\u0016x\u0016y\u0016z) sin(t2));cos(t2))T\nwith \u0016y=y2(1\u0000y)2and \u0016z=z2(1\u0000z)2, and the forcing term f=@tme+me\u0002\n\u0001me\u0000\u000bme\u0002(@tme+\u001c@ttme). Similarly, we record the discrete `2and`1er-\nrors between exact and numerical solutions with a sequence of temporal step-sizes\nand spatial mesh-sizes. The corresponding parameters are set as: \u000b= 0:01 and\n\u001c= 1000:0. Besides, the \fnal time of this simulation is given by T= 0:01, with\nthe temporal step-size and spatial mesh-size listed in Table 3 and Table 4.\nTable 3. The discrete `2and`1errors in terms of the temporal\nstep-size. The spatial mesh-size is \fxed as h= 0:001 and \fnal time\nisT= 0:01.\nkkmh\u0000mek2kmh\u0000mek1\nT/100 1.2678e-05 1.2765e-05\nT/120 8.8067e-06 8.8830e-06\nT/140 6.4725e-06 6.5419e-06\nT/160 4.9576e-06 5.0224e-06\norder 2.00 1.98\nTable 4. The discrete `2and`1errors in terms of spatial mesh-\nsize. The temporal step-size is \fxed as k= 2.0e-06.\nhkmh\u0000mek2kmh\u0000mek1\n1/8 1.4392e-07 3.4940e-07\n1/10 9.6832e-08 2.2864e-07\n1/12 6.9825e-08 1.6079e-07\n1/14 5.2828e-08 1.1895e-07\norder 1.79 1.92\n5.2.Micromagnetics tests. The inertial e\u000bect can be observed during the relax-\nation of a system with a non-equilibrium initialization. To visualize this, we conduct\nmicromagnetics simulations for both the LLG equation and the iLLG equation.20 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nIn the following simulations, a 3-D domain \n = [0 ;1]\u0002[0;1]\u0002[0;0:4] is uniformly\ndiscretized into 10 \u000210\u00024 cells, with uniform initialization m0= (p\n2=2;p\n2=2;0)T.\nFor comparison, the LLG equation is discretized by the mid-point scheme proposed\nin [4] with the \fxed-point iteration solver proposed in this work. The damping pa-\nrameter is\u000b= 0:5 and the \feld is \fxed as He= (10;0;0)T, which indicates that the\nsystem shall converge to m= (1;0;0)T. Here the relaxation of the magnetization\nbehavior controlled by the LLG equation is visualized in Figure 1.\nTime0 1 2 3 4 5/angbracketleftm/angbracketright\n-0.500.51\n/angbracketleftm1/angbracketright\n/angbracketleftm2/angbracketright\n/angbracketleftm3/angbracketright\nFigure 1. The relaxation of the spatially averaged magnetization\ncontrolled by the LLG equation. The \fnal time is T= 5:0 with\nk= 0:001, and the damping parameter is \u000b= 0:5.\nAs for the counterpart of the LLG equation, wtih a given reference \feld He, the\ndiscrete energy of the iLLG equation becomes\n(5.1) E[mn+1;mn] =1\n4(krhmn+1\nhk2\n2+krhmn\nhk2\n2)+\n\u000b\u001c\n2\r\r\rmn+1\nh\u0000mn\nh\nk\r\r\r2\n2\u00001\n2hmn+1\nh+mn\nh;Hei:\nSetting the inertial parameter \u001c= 1:0, the spatially averaged magnetization is\nrecorded to depict the inertial e\u000bect in Figure 2(a). Meanwhile, the energy decay is\nalso numerically veri\fed by Figure 2(b). The inertial e\u000bect is observed at shorter\ntimescales for magnetization dynamics during the relaxation of the system with a\nnon-equilibrium initialization.\nFurthermore, the inertial e\u000bect can also be activated by an external perturbation\napplied to an equilibrium state. Here we set the damping parameter \u000b= 0:02 and\n\u001c= 0:5, then the time step-size must be reduced to 0.001 with T= 3:0. For the\nequilibrium state m0= (1;0;0)T, the perturbation 4 :0\u0002sin(2\u0019ft) is applied along\nydirection over the time interval [0 ;0:05], withf= 20. The relaxation of the\niLLG equation, revealed by the evolution of the spatially averaged magnetization,\nis visualized in Figure 3.CONVERGENCE ANALYSIS OF AN IMPLICIT FDM FOR ILLG EQUATION 21\nTime0 5 10 15 20/angbracketleftm/angbracketright\n-0.500.51\n/angbracketleftm1/angbracketright\n/angbracketleftm2/angbracketright\n/angbracketleftm3/angbracketright\n(a)Spatially averaged magnetization\nTime0 2 4 6 8 10 12 14 16 18 20E[mn+1,mn]\n-4-3.8-3.6-3.4-3.2-3-2.8\n(b)Energy decay\nFigure 2. The spatially averaged magnetization (A) and the en-\nergy evolution (B) in the iLLG equation. Parameters setting:\nT= 20,k= 0:02,\u001c= 1:0 and\u000b= 0:5.\n6.Conclusion\nIn this work, an implicit mid-point scheme, with three time steps, is proposed to\nsolve the inertial Landau-Lifshitz-Gilbert equation. This algorithm preserves the\nproperties of magnetization dynamics, such as the energy decay and the constant\nlength of magnetization and is proved to be second-order accurate in both space\nand time. In the convergence analysis, we \frst construct a second-order approxi-\nmationmof the exact solution. It is found that mproducesO(h5) accuracy at\nthe mesh points around the boundary sections, which simpli\fes the estimation at\nboundary points. Then, by analyzing the error function between the numerical22 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nTime0 0.5 1 1.5 2 2.5 3/angbracketleftm/angbracketright\n-0.200.20.40.60.8/angbracketleftm1/angbracketright\n/angbracketleftm2/angbracketright\n/angbracketleftm3/angbracketright\nFigure 3. The response of the spatially averaged magnetization\nfor the magnetic perturbation in the presence of the inertial e\u000bect.\nFor the equilibrium initialization m0= (1;0;0)T, a perturbation\n4:0\u0002sin(2\u0019ft) is applied along ydirection during time interval\n0\u00180:05, withf= 20. The basic simulation parameters are:\n\u000b= 0:02,\u001c= 0:5,T= 3:0 andk= 0:001.\nsolution and the constructed solution mh, we derive the convergence result in the\nH1(\nT) norm. Furthermore, a \fxed-point iteration method is proposed to solve\nthis implicit nonlinear scheme under the time-step restriction k\u0014Ch2. Numerical\nresults con\frm the theoretical analysis and clearly show the unique inertial e\u000bect\nin micromagnetics simulations.\nAcknowledgments\nThis work is supported in part by the grant NSFC 11971021 (J. Chen), the\nprogram of China Scholarships Council No. 202106920036 (P. Li), the grant NSF\nDMS-2012269 (C. Wang).\nReferences\n1. F. Alouges and P. 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Phys. 404(2020), 109104.24 JINGRUN CHEN, PANCHI LI, AND CHENG WANG\nSchool of Mathematical Sciences and Suzhou Institute for Advanced Research, Uni-\nversity of Science and Technology of China, China\nSuzhou Institute for Advanced Research, University of Science and Technology of\nChina, Suzhou, Jiangsu 215123, China\nEmail address :jingrunchen@ustc.edu.cn\nSchool of Mathematical Sciences, Soochow University, Suzhou, 215006, China\nEmail address :lipanchi1994@163.com\nMathematics Department, University of Massachusetts, North Dartmouth, MA 02747,\nUSA\nEmail address :cwang1@umassd.edu"    },    {        "title": "1508.00629v2.A_Critical_Analysis_of_the_Feasibility_of_Pure_Strain_Actuated_Giant_Magnetostrictive_Nanoscale_Memories.pdf",        "content": " 1 A Critical Analysis  of the Feasibility of Pure Strain -Actuated Giant Magnetostrictive \nNano scale Memories  \nP.G. Gowtham1, G.E. Rowlands1, and R.A. Buhrman1 \n1Cornell University, Ithaca, New York, 14853, USA \n \nAbstract  \n  Concepts for memories  based on the  manipulation  of giant magnetostrictive \nnano magnets  by stress pulses  have garnered recent attention  due to their potential for ultra -low \nenergy operation in the high storage density limit . Here we discuss the feasibility of making such \nmemories in light of the fact that the Gilbert damping of such materials is typically quite  high. \nWe report the results of numerical simulations for  several classes of  toggle precessional and non -\ntoggle dissipative  magnetoelastic  switching modes. M aterial candidates  for each o f the several \nclasses are analyzed  and f orms for the anisotropy energy density and range s of material \nparameters appropriate for each material class are employed.   Our study indicates that the Gilbert \ndamping  as well as the anisotropy and demagnetization energies are all crucial for  determining \nthe feasibility of  magnetoelastic toggle -mode precessional switching schemes.  The role s of \nthermal stability  and thermal fluctuations for stress -pulse switching of gia nt magnetostrictive \nnanomagnets are also discussed in detail and are shown to be important in the viability, design, \nand footprint of magnetostrictive switching schemes.  \n \n \n \n  2 I. Introduction  \n \nIn recent years pure electric -field based control of magnetization has become a subject of \nvery active research. It has been demonstrated in a variety of systems ranging from multiferroic \nsingle phase materials, gated dilute ferromagnetic semiconductors 1–3, ultra -thin metallic \nferromagnet/oxide interfaces 4–10 and piezoelectric /magnetoelastic composites 11–15. Beyond the \ngoal of establishing an understanding of the physics involved in each of these systems, this work \nhas been strongly motivated by the fact that  electrical -field based manipulation of magnetization \ncould form the basis for a new generation of ultra -low power, non -volatile memories.  Electric -\nfield based magnetic devices are not necessarily limited by Ohmic losses during the write cycle \n(as can be the case in current based memories such as spin -torque magnetic random access \nmemory (ST -MRAM) ) but rather by the capacitive charging/decharging energies incurred per \nwrite cycle. As the capacitance of these devices scale with area the write energies have the \npotential to be as low as 1 aJ per write cycle or less.  \nOne general approach to the electrical control of magnetism utilizes a magnetostrictive \nmagnet/piezoelectric transducer hybrid as the active component of a nanoscale memory element. \nIn this appro ach a mechanical strain is generated by an electric field within the piezoelectric \nsubstrate or film and is then transferred to a thin, nanoscale magnetostrictive magnet that is \nformed on top of the piezoelectric. The physical interaction driving the write  cycle of these \ndevices is the magnetoelastic interaction that describes the coupling between strain in a magnetic \nbody and the magnetic anisotropy energy. The strain imposed upon the magnet creates an \ninternal effective magnetic field via the magnetoelast ic interaction that can exert a direct torque \non the magnetization. If successfully implemented this torque can switch the magnet from one \nstable configuration to another, but whether imposed stresses and strains can be used to switch a  3 magnetic element be tween two bi -stable states depend s on the strength of the magnetoelastic \ncoupling (or the magnetostriction). Typical values of the magnetostriction ( = 0.5 -60 ppm) in \nmost ferromagnets yield strain and stress scales that make the process of strain -induced \nswitching inefficient or impossible. However, considerable advances have been made in \nsynthesizing materials both in bulk and in thin film form that have magnetostrictions that are one \nto two orders of magnitude larger than standard transition metal ferrom agnets.  These giant \nmagnetostrictive materials allow the efficient conversion of strains into torque on the \nmagnetization.  However it is important to note that a large magnetostrictive (or magnetoelastic) \neffect tends to also translate into very high magnetic damping by virtue of the strong coupling \nbetween magnons and the phonon thermal bath, which has important implicati ons, both positive \nand negative, for piezoelectric based magnetic devices.   \nIn this paper we provide an analysis of the switching modes  of several different \nimplementations of piezoelectric/magnetostrictive devices.  We discuss how the high damping \nthat is generally associated with giant magnetoelasticity affects the feasibility of different \napproaches, and we also take other key material  properties into consideration, including the \nsaturation magnetization of the magnetostrictive element, and the form and magnitude of its \nmagnetic anisotropy. Th e scope of th is work excludes device concepts and physics \ncircumscribed by magneto -elastic mani pulation of domain walls in magnetic films, wires, and \nnanoparticle arrays 11,12,16. Instead we focus  here on analyzing various magnetoelastic reversal \nmodes, principally within the single domain approximation, but we do extend this work to \nmicromagnetic modeling in cases where it is not clear that the macrospin approximation prov ides \na fully successful description of the essential physics. We enumerate potential material \ns 4 candidates for each of the modes evaluated and discuss the various challenges inherent in \nconstructing reliable memory cells based on each of the reversal modes t hat we consider.  \nII. Toggle -Mode Precessional Switching  \n \nStress pulsing of a magnetoelastic element can be used to construct a toggle mode \nmemory. The toggling mechanism between two stable states relies on transient dynamics of the \nmagnetization that are initi ated by an abrupt change in the anisotropy energy that is of fixed and \nshort duration. This change in the anisotropy is created by the stress pulse and under the right \nconditions can generate precessional dynamics about a new effective field. This effectiv e field \ncan take the magnetization on a path such that when the pulse is turned off the magnetization \nwill relax to the other stable state. This type of switching mode is referred to as toggle switching \nbecause the same sign of the stress pulse will take t he magnetization from one state to the other \nirrespective of the initial state. We can divide the consideration of the toggle switching modes \ninto two cases; one that utilizes a high \nsM  in-plane magnetized element, and the other that \nemploys perpendicular magnetic anisotro py (PMA) materials with a lower  \nsM. We make this \ndistinction largely because of differences in the structure of the torques and stress fields required \nto induce a switch in these two class es of systems.  The switching of in -plane giant \nmagnetostrictive nanomagnets with sizeable out -of-plane demagnetization fields relies on the use \nof in-plane uniaxial stress -induced effective fields that overcome the in -plane anisotropy (~O( 102 \nOe)). The mo ment will experience a torque canting the moment out of plane and causing \nprecession about the large demagnetization field. Thus the precessional time scales for toggling \nbetween stable in -plane states will be largely determined by the d emagnetization fiel d (and thus \nsM\n). The dynamics of this mode bears striking resemblance to the dynamics in hard -axis field  5 pulse switching of nanomagnets 17. On the other hand, the dominant energy scale in PMA giant \nmagnetostrictive materials is the perpendicular anisotropy energy. This energy scale can vary \nsubstantially (anywhere from \nuK  ~ 105-107 ergs/cm3) depending on the materials utilized and the \ndetails of their growth. The anisotropy energy scale in these materials can be tuned into a region \nwhere stress -induced anisotropy energies can be comparable to it. A biaxial stress -induced \nanisotropy energy, i n this geometry, can induce switching by cancelling and/or overcoming the \nperpendicular anisotropy energy. As we shall see, this fact and the low\nsM  of these systems \nimply dynamical time scales that are substantially different from the case where in -plane \nmagnetized materials are employed.  \nA. In-Plane Magnetized Magnetostrictive Materials  \n \nWe first treat the macrospin switching dynamics of an in -plane magnetized \nmagnetostrictive nanomagnet with uniaxial anisotropy under a simple rectangular uniaxial stress \npulse. Giant magnetostriction in in -plane magnetized systems have been demonstrated for \nsputtered polycrystalline Tb 0.3Dy0.7Fe2 (Terfenol -D) 18, and more recently in quenched Co xFe1-x  \nthin film systems 19. We assume that the uniaxial anisotropy is defined completely by the shape \nanisotropy of the elliptical element and that any magneto -crystalline anisotropy in the film is \nconsiderably weaker. This is a reasonabl e assumption for the materials considered here in the \nlimit where the grain size is considerably smaller than the nanomagnet’s dimensions. The stress \nfield is applied by voltage pulsing an anisotropic piezoelectric film that is in contact with the \nnanomagn et. The proper choice of the film orientation of a piezoelectric material such as <110> \nlead magnesium niobate -lead titanate (PMN -PT) can ensure that an effective uniaxial in -plane \nstrain develops along a particular crystalline axis after poling the piezo in the z -direction. We  6 assume that the nanomagnet major axis lies along such a crystalline direction (the <110> -\ndirection of PMN -PT) so that the shape anisotropy is coincident with the strain axis  (see Figure 1 \nfor the relevant geometry) . For the analysis below we use material values appropriate to \nsputtered, nanocrystalline Tb0.3Dy0.7Fe2 18 (\nsM= 600 emu/cm3, \ns = 670 ppm is the saturation \nmagnetostriction). Nanocrystalline Tb0.3Dy0.7Fe2  films, with a mean crystalline grain diameter \ngraind\n < 10 nm, can  have an extremely high magnetostriction while being relatively magnetica lly \nsoft with coercive fields,  \ncH  ~ 50-100 Oe, results which can be achieved by thermal processing \nduring sputter growth at T  ~ 375 ºC 20. The nanomagnet dimensions were as sumed to be 80 nm \n(minor axis) × 135 nm (major axis) × 5 nm (thickness) yielding a shape anisotropy field \n4 ( )k y x sH N N M\n = 323 Oe and \n4 ( )demag z y sH N N M = 5.97 kOe. We use \ndemagnetization factors that are correct for an elliptical cylinder 21. \nThe value of the Gilbert damping parameter \n  for the magnetostrictive element is quite \nimportant in determining its dynamical behavior during in -plane stress -induced toggle switching. \nPrevious simulation results 22–24 used a value (\n0.1  for Terfenol -D) that, at least arguably, is \nconsid erably lower than is reasonable since that value was extracted from spin pumping in a Ni \n(2 nm) /Dy(5 nm) bilayer 25. However, that bilayer material is not a good surrogate for a rare -\nearth transition -metal alloy (especially for \n0L  rare earth ions). In the latter case the loss \ncontribution from direct magnon to short w avelength phonon conversion is important, as has \nbeen directly confirmed by studies of \n0L  rare earth ion doping into transition metals 26,27. For \nexample in -plane magnetized nanocrystalline 10% Tb -doped Py shows \n~ 0.8  when magnetron \nsputtered at 5 mtorr Ar pressure, even though the magnetostriction is small within this region of \nTb doping 27. We contend that a substantial increase in the magnetoelastic interaction in alloys  7 with higher Tb content is likely to make \n even larger. Magnetization rotation in a highly \nmagnetostrictive magnet will efficiently generate longer wavelength acoustic  phonons as well \nand heat loss will be generated when these phonons thermalize. Unfortunately, measurements of \nthe magnetic damping parameter in polycrystalline Tb0.3Dy0.7Fe2 do not appear to be available in \nthe literature. However, some results on the amo rphous Tb x[FeCo] 1-x system, achieved by using \nrecent ultra -fast demagnetization techniques, have extracted  \n~ 0.5 for compositions (x  ~ 0.3) \nthat have high magnetostriction 28. We can also estimate the scale for the Gilbert damping by \nusing a formalism that takes into account direct magnon to long wavelength phonon conversion  \nvia the magnetoelastic interaction  and subsequent phonon relaxation to the thermal phonon \nbath29. The damping can be estimated by the following formula:  \n \n2\n2236 1 1\n22s\nsT s L s\neff ex eff exMc M c M\nAA\n\n\n\n \n   \n            \n(1) \n \nUsing \nsM  = 600 emu/cm3, the exchange stiffness \nexA = 0.7x10-6 erg/cm, a mass density ρ \n= 8.5 g/cm3, Young’s modulus of 65 GPa 30, Poisson ratio \n0.3 , and an acoustic damping time \n\n= 0.18 ps 29 the result is an estimate of  \n~1 . Given the uncertainties in the various  parameter s \ndetermining the Gilbert damping , we examine the magnetization dynamics for values of  \n\nranging from 0.3 to 1.0.  \n We simulate the switching dynamics of the magnetic moment of a Terfenol -D \nnanomagnet at T=300 K using the Landau -Lifshitz -Gilbert form of the equation  describing the \nprecession of a magnetic moment  \nm:  8  \n( ) ( )eff eff eff Langevinddttdt dt       mmm H m H m   \n(2) \n \nwhere\neff  is the gyromagnetic ratio. As Tb0.3Dy0.7Fe2 is a rare earth – transition metal (RE-TM) \nferrimagnet (or more accurately a speromagnet), the gyromagnetic ratio cannot simply be \nassumed to be the free electron value. Instead we use the value\neff  = 1.78 107 Hz/Oe as \nextracted from a spin wave resonance study in the TbFe 2 system 31 which appears appropriate \nsince Dy and Tb are similar in magnetic moment/atom (10\nB  and 9\nB  respectively) and g factor \n( ~4/3 and ~3/2 respectively).  \nThe first term in Equation (2) represents the torque on the magnetization from any \napplied fields, the effective stress field, and any anisotropy and demagnetization fields that might \nbe present. The third term in the LLG represents the damping torque that acts to relax the \nmagnetization towards the direction of the effective field and hence damp out precessional \ndynamics. The second term is the Gaussian -distributed Langevin field that takes into account the \neffect thermal fluctuations on the magnetization dynamics. From the fluctuation -dissipation \ntheorem, \n2RMS B\nLangevin\neff skTHM V t\n  where \nt  is the simulation time -step 32. Thermal fluc tuations \nare also accounted for in our modeling by assuming that the equilibrium azimuthal and polar \nstarting angles (\n0  and \n0 /2  respectively) have a random mean fluctuation given by \nequipartition as  \n00 2\n2RMS BkT\nEV\n\n\n\n and \n0 24 ( )RMS B\nz y skT\nN N M V  . A \nbiasH   of 100 Oe was \n 9 used for our simulations which creates two stable energy minima at  \n0arcsin ~ 18bias\nkH\nH\n\n and \n1162\n symmetric about \n/2 . This non -zero starting angle ensures that \n00RMS  . \nThis field bias is essential as the initial torque from a stress pulse depends on the initial starting \nangle. This angular dependence generates much larger thermally -induced fluctu ations in the \ninitial torque than a hard -axis field pulse. The hard axis bias field also reduces the energy barrier \nbetween the two stable states. For Hbias = 100 Oe the energy barrier between the two states is Eb \n= 1.2 eV yielding a room temperature \n/bBE k T = 49. This ensures the long term thermal \nstability required for a magnetic memory.  \nTo incorporate the effect of a stress pulse in Equation (2) we employ a free energy form \nfor the effective field,  \n( ) /efftE  Hm that expresses the effect of a stress pulse along the x -\ndirection of our in -plane nanomagnet with a uniaxial shape anisotropy in the x -direction. The \nstress enters the energy as an effective in -plane anisotropy term that adds to the shape anisotropy \nof the  magnet (first term in Equation (3) below). The sign convention here is such that \n0\nimplies a tensile stress on the x -axis while \n0 implies a compressive strain. We also include \nthe possibility of a bias field applied along the hard axis in the final term in Equation (3).     \n \n22\n223( , , ) [2 ( ) ( )]2\n2 ( )x y z y x s s x\nz y s z bias s yE m m m N N M t m\nN N M m H M m \n  \n     \n(3) \n \n The geometry that we have assumed allows only for fast compressive -stress pulse based \ntoggle mode switching. The application of a DC compressive stress along the x -axis only reduces \nthe magnitude of the anisotropy and changes the position of the equilibriu m magnetic angles \n0 10 and \n10180\n  while keeping the potential wells associated with these states symmetric as \nwell. Adiabatically increasing the value of the compressive stress moves the angles toward \n/2\n until \n3()2sutK  but obviously can never induce a magnetic switch.  \n Thus the magnetoelastic memory in this geometry must make use of the transient \nbehavior of the magnetization under a stress pulse as opposed to re lying on quasistatic changes \nto the energy landscape. A compressive stress pulse where \n3()2sutK  creates a sudden \nchange in the effective field. The resultant effective field\n32ˆsu\neff y bias\nsKmHM  \nHy  \npoints in the y -direction and causes a  torque that brings the magnetization out of plane. At this \npoint the magnetization rotates rapidly about the very large perpendicular demagnetization field\nˆ 4demag s z Mm Hz\n and if the pulse is turned off at the right time will relax down to the \nopposite state at \n1  = 163. Such a switching trajectory for our simulated nanomagnet is shown in \nthe red curve in  Figure 2. This mode of switching is set by a minimum characteristic time scale\n1~ 7.54sw\nspsM\n, but the precession time will in general be longer than \nsw  for moderate \nstress pulse amplitudes,  \n( ) 2 / 3us tK , as the magnetization then cants out of plane enough to \nsee only a fraction of the maximum possible  \ndemagH . Larger stress pulse amplitudes result in \nshorter pulse duratio ns being required as the magnetization has a larger initial excursion out of \nplane. For pulse durations that are longer than required for a rotation (blue and green curves \nin Figure 2) \nm will exhibit damped elliptical precession about  \n/2 . If the stress is released \nduring the correct portion of any of these subsequent precessional cycles the magnetization \n180\n 11 should relax down to the \n1  state [blue curve in  Figure 2], but otherwise it will relax down to the \noriginal state [green curve in  Figure 2].  \nThe prospect of a practical device working reliably in the long pulse regime appears to be \nrather poor. The high damping of giant magnetostrictive magnets and the large field scale of the \ndemagnetization field yield very stringent pulse timing requirements  and fast damping times for \nequilibration to \n/2 . The natural time scale for magnetization damping in the in -plane \nmagnetized thin film case is  \n1\n2d\nsM  , which ranges from 50 ps down to 15 ps for\n0.3 1\n with \nsM = 600 emu/cm3. This high damping also results in the influence of thermal \nnoise on the magnetization dynamics being quite strong since  \nLangevinH  . Thus large stress \nlevels with extremely short pulse durations are required in order to rotate the magnetization \naround the \n/2 minimum within the damping time, and to keep the precession amplitude \nlarge enough that the magnetization  will deterministically relax to the reversed state. Our \nsimulation results for polycrystalline Tb0.3Dy0.7Fe2 show  that a high stress pulse amplitude of\n85 MPa\nwith a pulse duration ~ 65  ps is required if  \n0.5  (Figure 3a). However, the \npulse duration window for which the magnetization will deterministically switch is extremely \nsmall in this case (<5 ps). This is due to the fact that the precession amplitude about the \n/2  \nminimum at this damping gets small enough that thermal fluctuations allow only a very small \nwindow for which switching is reliable. For the lowest damping that we consider reasonable to \nassume,  \n0.3 , reliable switching is possible between \npulse ~ 30-60 ps at \n85 MPa . At a \nlarger damping \n0.75  we find that the switching is non -deterministic for all pulse widths as \nthe magnetization damps too quickly; instead very high stresses , \n200 MPa are required to \n1 12 generate deterministic switching of the magnetization with a pulse duration w indow \npulse ~ 25-\n45 ps ( Figure 3b).  \nGiven the high value of the expected damping we have also simulated the magnetization \ndynamics in the Landau Lifshitz (LL) form:  \n \n2(1 ) ( ( ) ( ))LL eff Langevinddttdt dt       mmm H H m   \n(4) \n \nThe LL form and the LLG form are equivalent in low damping limit (\n1 ) but they \npredict different dynamics at higher damping values.  Which of these norm -preserving forms for \nthe dynamics has the right damping form is still a subject of debate 33–37. As one increases α in \nthe LL form the precessional speed is kept the same while the damping is assumed to affect only \nthe rate of decay of the precession amplitude. The damping in the LLG dynamics, on the other \nhand, is a viscosity term and retards the pre cessional speed. The effect of this retardation can be \nseen in the LLG dynamics as the precessional cycles move to longer times as a function of \nincreasing damping. Our simulations show that the LL form (for fixed \n ) predicts highe r \nprecessional speeds than the LLG and hence an even shorter pulse duration window for which \nswitching is deterministic than the LLG, ~12 ps for LL as opposed to ~ 30 ps for LLG ( Figure \n3c). \nThe damping clearly plays a crucial role in the stress amplitude scale and pulse duration \nwindows for which deterministic switching is possible, regardless of the form used to describe \nthe dynamics. Even though the magnetostriction of Tb 0.3Dy0.7Fe2 is high and the stress required \nto entirely overcome the anisotropy energy is only 9.6 MPa, the fast damping time scale and \nincreased thermal noise (set by the large damping and the out -of-plane demagnetization) means  13 that the stress -amplitude that is required to achieve deterministic toggle switching is 10 -20 times \nlarger. In addition, the pulse duration for in -plane toggling must be extremely short, with typical \npulse durations of 10 -50 ps with tight time windows of 20 -30 ps within which the acoustic pulse \nmust be turned off. Given ferroelectric switching rise times on the order of ~50 ps extracted from \nexperiment38 and considering the acoustical resonant response of the entire piezoelectric / \nmagnetostrictive nanostructure and acoustic ringing and inertial terms in the lattice dynamics, \ngeneration of such large stresses with the strict pulse time requirem ents needed for switching in \nthis mode is likely unfeasible.  In addition, the stress scales  required to successfully toggle switch \nthe giant magnetostrictive nanomagnet in this geometry are nearly as high or even higher than \nthat for transition metal ferromagnets such as Ni (\n~ 38 ppms with \n0.045 ). For example, \nwith a 70 nm × 130 nm elliptical Ni nanomagnet with a thickness of 6 nm and a hard axis bias \nfield of 120 Oe we should obtain  switching at stress values \n = +95 MPa and \npulse  = 0.75 ns. \nTherefore  the use of giant magnetostrictive nanomagnets with high damping in this toggle mode \nscheme confers no clear advantage over the use of a more conventional transition metal \nferromagnet, and in neither case does this approach appear particularly viable for t echnological \nimplementation.  \nB. Magneto -Elastic Materials with PMA: Toggle Mode Switching  \n \nCertain amorphous sputtered RE/TM alloy films with perpendicular magnetic anisotropy \nsuch as a -TbFe 2 39–42 and a - Tb0.3Dy0.7Fe2 43 have  properties that may make these materials \nfeasible for use in  stress -pulse toggle switching. In certain composition ranges they exhibit large \nmagnetostriction (\ns  > 270 ppm for a -TbFe 2, and both  \ns  and the effective out of plane  14 anisotropy can be tuned over fairly wide ranges by varying the process gas pressure during \nsputter deposition, the target atom -substrate incidence angle, and the substrate temperature.  \nWe consider the energy of such an out -of-plane magnetostrictive material under the \ninfluence of a magnetic field  \nbiasH  applied in the \nˆx  direction and a pulsed biaxial stress:  \n \n223( , , ) [ 2 ( )]2u\nx y z s s biaxial z s bias xE m m m K M t m M H m        \n(5) \n \nSuch a biaxial stress could be applied to the magnet if it is part of a patterned [001] -poled PZT \nthin film/ferromagnet bilayer. A schematic of this device geometry is depicted in Figure 4.When\n0biasH\n, it is straightforward to see the stress pulse will not result in reliable switching since, \nwhen the tensile biaxial stress is large enough, the out of plane anisotropy becomes an easy -plane \nanisotropy and the equator presents a zero -torque condition on t he magnetization, resulting in a \n50%, or random, probability of reversal when the pulse is removed. However, reliable switching \nis possible for \n0biasH  since that results in a finite canting of \nm towards the x -axis. This \ncanting is required for the same reasons a hard -axis bias field was needed for the toggle \nswitching of an in -plane magnetized element as discussed previously.  A pulsed biaxial stress \nfield can then in principle lead to deterministic precessional toggle switching between  the +z and \n–z energy minima . This mode of pulsed switching is analogous to voltage pulse switching in the \nultra-thin CoFeB|MgO using the voltage -controlled magnetic anisotropy effect.5,8 Previous \nsimulation results have also di scussed  this class of  macrospin  magnetoelast ic switc hing in the \ncontext  of a Ni|Barium -Titatate  multilayer44 and a zero -field, biaxial stress -pulse induced  toggle \nswitching  scheme  taking advantage of micromagnetic  inhomogeneities has recently appeared in \nthe literature45. Here we discuss biaxial stress -pulse  switching for a broad class of  giant  15 magnetostrictive  PMA  magnets where we argue  that the monodomain limit strictly applies \nthroughout the switching process and extend past previous macrospin modeling by \nsystematically think ing about  how pulse -timing requirements  and critical write stress amplitudes \nare determined  by the damping, the PMA strength, and \nsM  for values  reasonable for these \nmaterials.  \nFor our simulation study of stress -pulse toggle switching of a PMA magnet, we \nconsidered a Tb 33Fe67 nanomagnet with an \nsM  = 300 emu/cm3, \neffK  = 4.0×105 ergs/cm3 and \ns  \n= 270 ppm.  To estimate the appropriate value for the damping parameter we noted that ultrafast \ndemagnetization measurements on Tb 18Fe82 have yielded  \n0.27 . This 18 -82 composition lies \nin a region where the magnetostriction is moderate (\ns ~50 ppm) 43 so we assumed that the \ndamping will be on the same order or higher for a -TbFe 2 due to its high magnetostriction. \nTherefore we ran simulations for the range of  \n= 0.3 -1. For the gyromagnetic ratio  we used\neff  \n= 1.78×107 s-1G-1 which  is appropriate for a -TbFe 2 31. We assumed an effective exchange \nconstant \n611 10effA erg cm   46 implying an exchange length  \nexeff no stress\neffAlK\n = 15.8 nm (in \nthe absences of an applied str ess) and \n22exeff pulse\nsAlM = 13.3 nm (assuming that the stress pulse \namplitude is just enough to cancel the out of plane anisotropy). A monodomain crossover \ncriterion of \ncd ~  ~ 56 nm  (with the pulse off) and \ncd ~\n22ex\nsA\nM ~ 47 nm  (with the pulse \non) can be calculated by considering the minimum length -scale associated with supporting \nthermal λ/2 confined spin wave modes 47. The important point here is that the low \nsM  of these \nsystems ensures that the exchange length is still fairly long even during the switching process, \n4ex\nuA\nK 16 which suggests that the macrospin approximation should be valid for describing the switching \ndynamics of this system for reasonably sized nanomagnets.  \nWe simulated a circular element with a diameter of 60 nm and a thickness of 10 nm, \nunder an x -axis bias field, \nbiasH = 500 Oe which creates an initial canting angle of 11 degrees \nfrom the  vertical  (z-axis).  This starting angle is sufficient to enable deterministic toggle \nprecessional switching between the +z and –z minima via biaxial stress pulsing.  The assumed \ndevice geometry, anisotropy energy density and bias field corresponded to an energy barrier \nbE   \n= 4.6  eV for thermally activated reversal, and hence a room temperature thermal stability factor  \n\n = 185.  \nWe show selected results of the macrospin simulations of stress -pulse toggle switching of \nthis modeled TbFe 2 PMA nanomagnet. Typical switching trajectories are shown in Figure 5a. The \nswitching transition can be divided into two stages (see Figure 5b): the precessional stage that \noccurs when the stress field is applied, during which the dynamics of the magnetization are \ndominated by precession about the effective field that arises from the sum of the bias field and \nthe easy -plane anisotropy field \n3 ( ) 2eff\ns\nz\nstKmM , and the dissipative stage that begins when the \npulse is turned off and where the large \neffK and the large \n  result in a comparatively quick \nrelaxation to the other energy minimum. Thus most of the switching process is spent in the \nprecessional phase and the entire switching process is not much longer than the actual stress \npulse duration.  For pulse amplitudes a t or not too far above the critical stress for reversal,\n2 / 3eff\ns K\n the two relevant timescales for the dynamics are set approximately by the \nprecessional period\n1/ 100 pssw bias H of the nanomagnet and the damping time 17 \n~ 2 /d bias H  . Both of these timescales are much longer than the timescales set by precession \nand damping about the demagnetization field in the in -plane magnetized toggle switching case. \nThe result is that even with quite high damping one can have reliable s witching over much \nbroader pulse width windows, 200 -450 ps . (Figure 6a,b). The relatively large pulse duration \nwindows within which reliable switching is possible (as compared to the in -plane toggle mode) \nhold for both the LL and LLG damping. However, the diffe rence between the two forms is \nevident in the PMA case ( Figure 6c). At fixed \n , the LLG damping predicts a larger pulse \nduration window than the LL damping. Also the effective viscosity implicit within the LLG \nequation ensures that the switching time scales are slower than in the LL case as can also be seen \nin Figure 6c. \nAn additional and important point concerns the factors that determine the critical \nswitching amplitude. In the in -plane toggle mode switching of the previous section, it was found \nthat the in-plane anisotropy field was not the dominant factor in determining the stress scale \nrequired to transduce a deterministic toggle switch. Instead, we found that the stress scale was \nalmost exclusively dependent on the need to generate a high enough preces sion \namplitude/precession speed during the switching trajectory so as to not be damped out to the \ntemporary equilibrium at \n/2  (at least within the damping range considered). This means \nthat the critical stress scale to transduce a deterministic switch is essentially determined by the \ndamping. We find that the situation is fundamentally different for the PMA based toggle \nmemories.  The critical amplitude \nc  is nearly independent of the damping from a range of \n0.3 0.75\n up until \n~1 where the damping is sufficiently high (i.e. damping times equaling \nand/or exceeding the p recessional time scale) that at \n85  MPa the magnetization traverses \ntoo close to the minimum at \n/2 ,\n0 . The main reason for this difference between the  18 PMA toggle based memories and the in-plane toggle based memory lies in the role that the \napplication of stress plays in the dynamics. First, in the in -plane case, the initial elliptical \namplitude and the initial out of plane excursion of the magnetization is set by the stress pulse \nmagnitu de. Therefore the stress has to be high to generate a large enough amplitude such that the \ndamping does not take the trajectory too close to the minimum at which point Langevin \nfluctuations become an appreciable part of the total effective field. This is n ot true in the PMA \ncase where the initial precession amplitude about the bias field is large and the effective stress \nscale for initiating this precession about the bias field is the full cancellation of the perpendicular \nanisotropy.  \nSince the minimum stre ss-pulse amplitude required to initiate a magnetic reversal in out -\nof-plane toggle switching scales with \neffK in the range of damping values considered, lowering \nthe PMA of the nanomagnet is a straightforward way to reduce the stress and write energy \nrequirements for this type of memory cell. Such reductions can be achieved by strain engineering \nthrough the choice of substrate, base electrode and transducer layers, by the choice of deposition \nparameters, and/or by post -growth annealing  protocols. For example growing a TbFe 2 film with a \nstrong tensile biaxial strain can substantially lower \neffK . If the P MA of such a nanomagnet can \nbe reliably r educed to \neffK = 2105 ergs/cm3 our simulations indicate that this would result in \nreliable pulse toggle switching at \n ~ -50 MPa (corresponding to a strain amplitude on the TbFe 2 \nfilm of less than 0.1%) with \npulse  ≈ 400 ps, for 0.3 ≤ \n  ≤  0.75 and \nbiasH ~ 250  Oe . Electrical \nactuation of this level of stress/strain in the sub -ns regime, while challenging, may  be possible to \nachieve.48 If we again assume \nsM  =300 emu/cm3, a diameter of 60 nm and a thickness of  10 nm, \nthis low PMA nanomagnet would still have a high thermal stability with  \n92 . The challenge, \n 19 of course, is to consistently and uniformly control the residual strain  in the magnetostrictive \nlayer. It is important to note that no  such tailoring (short of systematically lowering the damping) \ncan exist in the in -plane toggle mode case.  \nIII. Two -State Non -Toggle Switching  \n \nSo far we have discussed toggle mode switching where the same polarity strain pulse is \napplied to reverse the magnetization between two bi -stable states. In this case the strain pulse \nacts to create a temporary field around which the magnetization precesse s and the pulse is timed \nso that the energy landscape and magnetization relax the magnetization to the new state with the \ntermination of the pulse.  Non -toggle mode magneto -elastic switching differs fundamentally \nfrom the precessional dynamics of toggle -mode switching, being an example of dissipative \nmagnetization dynamics where a strain pulse of one sign destabilizes the original state (A) and \ncreates a global energy minimum for the other state (B). The energy landscape and the damping \ntorque completely de termine the trajectory of the magnetization and the magnetization \neffectively “rolls” down to its new global energy minimum. Reversing the sign of the strain pulse \ndestabilizes state B and makes state A the global energy minimum – thus ensuring a switch ba ck \nto state A. There are some major advantages to this class of switching for magneto -elastic \nmemories over toggle mode memories. Precise acoustic pulse timing is no longer an issue. The \nswitching time scales, for reasonable stress values, can range  from q uasi-static to nanoseconds. \nIn addition, the large damping typical of magnetoelastic materials does not present a challenge \nfor achieving robust switching trajectories in deterministic switching as it does in toggle -mode \nmemories. Below we will discuss det erministic switching for magneto -elastic materials that have \ntwo different types of magnetic anisotropy.   20 C. The Case of Cubic Anisotropy  \n \nWe first consider magneto -elastic materials with cubic anisotropy under the influence of a \nuniaxial stress field pulse. T here are many epitaxial Fe -based magnetostrictive materials that \nexhibit a dominant cubic anisotropy when magnetron -sputter grown on oriented C u underlayers \non Si or on MgO,  GaAs , or PMN -PT substrates. For example, Fe 81Ga19 grown on MgO [100] or \non GaAs ex hibit a cubic anisotropy 49–51. Given the low cost of these Fe -based materials \ncompared to rare -earth alloys, it is worth investigating whether such films can be used to \nconstruct a two state memory. Fe 81Ga19 on MgO exhibits easy axes along <100>. In ad dition, \nepitaxial Fe 81Ga19 films have been found to have a reasonably high magnetostriction λ100=180 \nppm making them suitable for stress induced switching. If we assume that the  cubic \nmagnetoelastic thin-film nanomagnet has circular cross section, that the  stress field is applied by \na transducer along the [100] direction , and that a bias field  is applied  at \n4  degrees, the  \nmagnetic free energy is :  \n \n2 2 2 2 2\n11\n2( , ) (1 ) 2 ( )\n3( ) ( )2 2x y x y z z z s z\ns bias\nx y s xE m m K m m K m m N N M m\nMHm m t m\n    \n  \n   \n(6) \n \nEquation (6) shows that, in the absence of a bias field, the anisotropy energy is 4 -fold \nsymmetric in the film -plane. It is rather easy to see that it is im possible to make a two -state non -\ntoggle switching with a simple cubic anisotropy energy and uniaxial stress field along [100].  \nFigure 7a shows the free energy landscape described by Equation (6) without stress applied. To \ncreate a two -state deterministic magnetostrictive device ,  \nbiasH  needs to be strong enough to \neradicate the energy minima at \n and \n3 / 2  which strictly requires that  \n1 0.5 /bias sH K M .  21 Finite temperature considerations can lower this minimum bias field requirement considerably. \nThis is due to the fact that the bias field can make the lifetime to escape the energy minima in th e \nthird quadrant and fourth qua drant small and the energy bar rier to return them from the energy \nminima in the first quadrant extremely large.  We arbitrarily set this requirement for the bias \nfield to correspond to a lifetime of 75 μs. The typical energy barriers to hop from back to the \nmetastable minima in the thi rd and fourth quadrant for device volumes we will consider are on \nthe order of several eV.  \nThe requirement for thermal stability of the two minima in the first quadrant , given a \ndiameter\nd  and a thickness \nfilmt  for the nanomagnet,  sets an upper  bound on  \nbiasH  as we require \n/ 40bbE k T \n at room temp erature between the two states (see Figure 7c). It is desirable that \nthis upper bound is high enough that there is some degree of tolerance to the value of the bias \nfield at device dimensions that are employed. This sets requirement s on the minimum volume of \nthe cylindical  nanomagnet that are dependent on\n1K . \nFor a circular element with \nd  = 100 nm, \nfilmt  = 12.5 nm and  \n1K= 1.5 105 ergs/cm3, two -\nstate non -toggle switching with the required thermal stability can only occur for \nbiasH  between \n50 - 56 Oe. This is too small a range of acceptable bias fields. However , by increasing \nfilmt  to 15 \nnm the bias field range grows to \nbiasH = 50 - 90 Oe wh ich is an acceptable range. For\n1K = \n2.0×105 erg/cm3 with \nd= 100 nm and \nfilmt = 12.5 nm , there is an appreciable region of bias field \n(~65-120 Oe) for which \n/barrier BE k T  > 42. For\n1K  = 2.5 105 ergs/cm3, the bias range goes from \n90 – 190 Oe for the same volume.  The main po int here is that, given the scale  for the cubic \nanisotropy in Fe 81Ga19, careful attention must be paid to the  actual  values of the anisotropy \n\n 22 constants, device lateral dimensions, film thickness, and the exchange bias strength in order  to \nensure device stability  in the sub -100 nm diameter  regime .  \n We now discuss the dynamics for a simulated case where  \nd = 100 nm,  \nfilmt= 12.5 nm,  \n1K\n= 2.0×105 ergs/cm3, \nbiasH  = 85 Oe, and \nsM  = 1300 emu/cm3.  Two stable minima exist at \n\n=10o and \n = 80o. Figure 7b shows the effect of the stress pulse on the energy landscape.  When \na compressive stress \nc  is applied, the potential minimum at \n =10o is rendered unstable \nand the magnetization follows the free energy gradient to \n = 80o (green curve). Since the stress \nfield is applied along [100] the magnetization first switches to a minima very close to but greater \nthan \n = 80o and when the stress is released it gently relaxes down to the zero stress minimum at \n\n= 80o. In order to switch from \n = 80o  to \n = 10o we need to reverse the sign of the applied \nstress field to tensile (red curve). A memory constructed on these principles is thus non -toggle.  \nThe magnetization -switching trajectory is simple and follows the dissipative dynamics \ndictated by the free energy landscape (see Figure 8a). We have assumed a damping of \n0.1  \nfor the Fe 81Ga19 system,  based on previous measurements52 and as confirmed by our own. Higher \ndamping only ends up speeding up the sw itching and ri ng-down process. Figure 8b shows the \nsimulated stress amplitude and pulse switching probability phas e diagram at room temperature.  \nUltimately, we must take the macrospin estimates for device parameters as  only a roug h \nguide. The macrospin dynamics approximate the true micromagnetics less and less well as the \ndevice diameter gets larger. The mai n reason for this is the large\nsM  of Fe 81Ga19 and the \ntendency of the magnetization to curl at the sample edges. Accordingly we have  performed T  = 0 \nºK micromagnetic simulations in  OOMMF.53 An exchange bias field \nbiasH  = 85 Oe was applied  23 at \n  = 45º and we assume \n1K  = 2.0×105 ergs/cm3,  \nsM  = 1300 emu/cm3, and \nexA = 1.9 × 10-6 \nerg/cm. Micromagnetics show that the macrospin picture quantitatively captures the switching \ndynamics, the angular positions of the stables states (\n0~ 10 and\n1~ 80 ) and the critical \nstress amplitude at (\n ~ 30 MPa) when the device diameter \nd < 75 nm. The switching is \nessentially a rigid in -plane rotation of the magnetization from  \n0 to \n1 . However, we cho se to \nshow the switching for  an element with \nd  = 100 nm because it allowed for thermal stability of \nthe devices in a region of thicknes s (\nfilmt  = 12-15 nm) where \nbiasH ~ 50-100 Oe at room \ntemperature could be reasonably expected. The initial average magnetization angle is larger (\n0~ 19\nand \n1~ 71 ) than would b e predicted by macrospin for a \nd = 100 nm element. \nThis is due to the magnetization c urling at the devices edges at\nd = 100 nm (see Figure 8c). \nDespite the fact that magnetization profile differs from the macrospin picture we find that there \nis no appreciable difference between the stress scales required for switching , or the basi c \nswitching mechanism.  \nThe stress amplitude scale for writing the simulated Fe 81Ga19 element at ~ 30 MPa is not \nexcessively high and  there are essentially no demands on the acoustic pulse width requirements.  \nThese memories can thus be written at pulse amplitudes of ~ 30 MPa with acoustical pulse \nwidths of ~ 10 ns. These numbers do not represent a major challenge from the acoustical \ntransduction point of view.  The drawback s to this scheme are  the necessity of growing high \nquality single crystal thin film s of Fe 81Ga19 on a piezoelectric substrate that can generate large \nenough strain to switch the magnet (e.g. PMN -PT) and difficulties  associated with  tailoring the \nmagnetocrystalline anisotropy \n1K  and ensuring thermal stability at low lateral device \ndimensions.   24 D. The Case of Uniaxial Anisotropy   \n \nLastly we discuss deterministic (non -toggle) switching of an in -plane giant \nmagnetostrictive magnet with uniaxial anisotropy. In -plane magnetized polycrystalline TbDyFe \npatterned into ellipti cal nanomagnets could serve as a potential candidate material in  such a \nmemory scheme. To implement deterministic switching in this geometry a bias field  \nbiasH  is \napplied along the hard axis of the nanomagnet. This generates two stable minima at \n0 and \n0 180\n symmetric about the hard axis. The axis of the stress pulse then needs to be non -\ncollinear with respect to the e asy axis in order to break the symmetry of the potential wells and \ndrive the transition to the selected equilibrium position. Figure 9 below shows a schematic of the \nsituation. When a stress pulse is applied in the direction that makes an angle\n with respect to the \neasy axis of the nanom agnet,  \noo0 90 , the free energy within the macrospin approximation \nbecomes:  \n \n2 2 2 2\n2( , , ) [2 ( ) 2 ( )\n3( ) (cos( ) sin( ) )2x y z y x s x z y s z bias s y\ns y x\nsE m m m N N M m N N M m H M m\nt m mM\n      \n     \n(7) \n \nFrom Equation (7) it can be seen that a sufficiently strong compressive stress pulse can switch \nthe magnetization between \n0  and \no\n0 180 , but only if \n0 is between\n and . To see why \nthis condition is necessary, we look at the magnetization dynamics in the high stress limit when \n0 0\n.  During such a strong pulse the magnetization will s ee a hard axis appear at\n  \nand hence will rotate towards the new easy axis at \n90 , but when the stress pulse is \no90 25 turned off the magnetization will equilibrate back to \n0 . This situation is represented by the \ngreen trajectory shown in  Figure 11a.   \nBut when \no\n090 , a sufficiently strong compressive stress pulse defines a new easy \naxis close to \no90  and when the pulse is turned off the magnetization will relax to\n0 180\n (blue trajectory in Figure 11a). Similarly the possibility of switching from \no180  \nto \nwith a tensile  strain depends on whether \no o o90 180 90     . Thus\no45 is the \noptimal situation as then the energy landscape becomes mirror symmetric about the hard axis and \nthe amplitude of the required switching stress (voltage) are equal. This scheme is quite similar to \nthe case of deterministic switching in biaxial anisotropy systems (with the coordinate system \nrotated by ). We note that a set of papers54–56 have previously proposed this particular case as \na candidate for non -toggle magnetoelectric memory  and have experimentally demonstrated \noperation of such a memory in the large feature -size (i.e. extended film ) limit .55  \nWe argue here that in-plane giant magnetostrictive magnets operated in the non -toggle \nmode could be a good candidate for  construct ing memories with  low write stress amplitude, and \nnanosecond -scale write time operation. However , as we will discuss , the prospects of this type of \nswitching mode  being suitable  for implementation in ultrahigh density memory appear to be  \nrather poor. The m ain reason for this lies in the hard axis bias field requirements for maintaining \nlow write error rates and the effect that such a hard axis bias field will have on  the long term \nthermal stability  of the element . At T = 0 ºK the requirement on  \nbiasH  is only that it be strong \nenough that  \n0 > 45º. However, this is no longer sufficient  at finite temperature where thermal \nfluctuations impl y a thermal, Gaussian distribution of the initial orientation of the magnetization \no45 26 direction \n0 about  \n0. If a significant componen t of this angular distribution falls below 45 \ndegrees there will be a high write error rate. Thus we must ensure that \nbiasH  is high enough that \nthe probability of \n  < 45º is extremely low. We have selected the re quirement that \n  < 45º  is a \n8\n event  where \n  is the standard deviation of \n about  \n0 and is  given by the relation\n. However,  \nbiasH  must be low enough to be technologically feasible,  but also \nmust not exceed a value that compromises the energy barrier between the two potential minima – \nthus rendering the nanomagnet thermally unstable . These minimum and maximum  requirement s \non \nbiasH  puts significant constraints on the minimum size of the nanomagnet that can be used in \nthis device approach. It also sets some rather tight requirements on the hard axis bias field, as we \nshall see.  \nWe first disc uss the effects of these requirements  in the case of a relatively large \nmagnetostrictive  device.  We assume the use  of a polycrystalline Tb 0.3Dy0.7Fe2 element having \nsM\n = 600 emu/cm3 and an elliptical cross section of 400×900 nm2 and a thickness \nfilmt  = 12.5 \nnm. This results in a  shape anisotropy field \nkH  ≈ 260 Oe. We find that for an applied hard axis \nbias field \nbiasH  ~ 200 Oe, a field strength that can be reasonably engineered on -chip, the \nequilibrium angle of the  element is \n0 ≈ 51º and its root mean square (RMS)  angular fluctuation  \namplitude is \nRMS ≈ 0.75º. Thus element ’s anisotropy field  and the assumed hard axis biasing \ncondition s just satisfy  the assumed requirement that  \n08RMS   > 45º (see Figure 10b). The \nmagnetic energy barrier to thermal energy ratio for the element at \nbiasH  = 200 Oe is \n/bBE k T\n02\n2BkT\nEV\n\n\n\n 27 ≈ 350, which  easily  satisf ies the long-term thermal  stability  requirement  (see Figure 10a), and \nwhich also provides some latitude for the use of a slightly higher\nbiasH  if desired to further reduce \nthe write error rate .  \nIt is straightforward to see from these numbers that if the area of the magnetostrictive \nelement is substantially reduced below 400 ×900 nm2 there must be a corresponding increase in \nkH\n  and hence in\nbiasH if the write error rate for the device is to remain acceptable.  Of course an \nincrease in the thickness of the element can partially reduce the increase in fluctuation amplitude \ndue to the decrease in the magnetic a rea, but the feasible range of thickness variation cannot \nmatch the effect of, for example, reducing the cross -sectional area by a factor of 10 to 100, with \nthe latter, arguably, being the minimum required for high density memory applications.    While \nperhaps a strong shape anisotropy and an increased \nfilmt  can yield the required \nkH  ≥ 1 kOe, the \nfact that in this deterministic mode of magnetostrictive switching we must also have \nbiasH  ~ \nkH  \nresults in a bias field requirement that is not technologically feasible.  We could of course allow \nthe write error rate to be much larger than indicated by an 8\nfluctuation probability, but this \nwould only relax the requirement on \nbiasH  marginally, which always must be such that \n0  > \n45o.Thus the deterministic  magneto strictive  device  is not a viable  candidate for ultra -high density \nmemory.  Instead this approach is only feasible for device s with lateral area  ≥ 105 nm2 . \nWhile the requir ement of a large footprint is a limitation of the deterministic  \nmagneto strictive  memory element , this device does have the significant advantage that the stress \nscale required to switch the memory is quite low. We have simulated T = 300 ºK macrospin \nswitching dynamics for a 400×900 nm2 ellipse with thickness \nfilmt  = 12.5 nm with \nbiasH = 200 Oe \nsuch that \n0 ~ 51º.  The Gilbert damping parameter was set to  \n0.5 and magnetostriction  \ns =  28 670 ppm. The magnetization switches by simple rotation from \n0  = 51º to \n1129\n that is \ndriven by the stress pulse induced change in the energy landscape (see Figure 11a). Phase \ndiagram results are provided in Figure 11b where the switching from  \n0 = 51º to \n1 = 129 º \nshows a 100% switching probability for stresses as low as \n  = - 5 MPa for pulse widths as short \nas 1 ns.  \nSince the dimensions of the ellipse are large enough that t he macrospin picture is  not strictly \nvalid, we have also conducted T = 0 K micromagnetic simulations of the stress -pulse induced \nreversal in this geometry. We find that the trajectories are essentially well described by a quasi -\ncoherent rotation with  non-uniformities in the magnetization being more pronounced at the \nellipse edges (see Figure 11c). The minimum stress pulse amplitude for swi tching is even lower \nthan that predicted by macrospin at \n  = - 3 MPa. This stress scale for switching is substantially \nlower than any of the switching mode schemes discussed before. Despite the fact that this \nscheme is not scalable down into the 100 -200 nm size regime, it can be  appropriate for larger \nfootprint memori es that can be written at very low write stress pulse amplitudes.  \nIV. CONCLUSION  \n   \nThe physical properties of giant magnetostrictive magnets (particularly of the rare -earth \nbased TbFe 2 and Tb 0.3Dy0.7Fe2 alloys) place severe restrictions on the viability of such materials \nfor use in fast, ultra -high density , low energy consumption  data storage. We have enumerated the \nvarious potential problems that might arise from the characteristically high damping of giant \nmagnetostrictive nanoma gnets in toggle -mode switch ing. We have also discussed  the rol e that \nthermal fluctuation s have on the various switching modes and the challenges involved in  29 maintaining long -time device thermal stability that arise mainly from the necessity of employing \nhard axis bias fields .  \nIt is clear that the task of constructing a reliable memory using pure stress induced \nreversal of g iant magnetostrictive magnets will be , when pos sible, a question of trade -offs and \ncareful engineering . PMA based giant magnetostrictive nanomagnets can be made extremely \nsmall (\nd  < 50 nm) while still maintaining thermal stability.  The small diameter and low cross -\nsectional area of these PMA giant magnetostrictive devices could , in principle,  lead to very low \ncapacitive write energies.   The counterpoint is that  the stress fields required to switch the device \nare not  necessarily  small and the acoustical pulse timing  requirements  are demanding.  However, \nit might  be possible t o tune the magnetostriction  \ns ,\nK , and \nsM  (either by adjustment of the \ngrowth conditions of the magnetostrictive magnet or by engineering the RE-TM multilayers \nappropriately) in order to significantly reduce the pulse amplitudes required f or switching (down \ninto the 20-50 MPa  range) and reduce th e required in -plane bias field – without compromising \nthermal stability of the bit . Such tuning must be carried out carefully. As we have discussed , the \nGilbert dampi ng \n, \ns ,\nK , and \nsM  can all affect the pure stress -driven switching process  and \ndevice thermal stability  in ways that are certainly interlinked and not necessarily complementary.   \nTwo state  non-toggle memories  such as we described in Section  III D  could have  extremely low \nstress write amplitudes and non-restrictive pulse requirements . However, the trade -off arises \nfrom thermal stability considerations and such a switching scheme is not scala ble down into the \n100-200 nm size regime . Despite this limitation there  may well be a place for  durable  memories \nwith very low write stress pulse amplitudes  and low write energies that operate reliably in the \nnanosecond regime .  30 ACKNOWLEDGEMENTS  \nWe thank  R.B. van Dover,  W.E. Bailey, C. Vittoria,  J.T. Heron, T. Gosavi, and S. Bhave \nfor fruitful discussions.  We also thank D.C. Ralph and T. Moriyama for comments and \nsuggestions on the manuscript. This work was supported by the Office of Naval Research and the \nArmy Research Office.   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  31 REFERENCES  \n \n1 H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, Nature \n408, 944 (2000).  \n2 D. Chiba, M. Sawicki, Y. Nishitani, Y. Nakatani, F. Matsukura, and H. Ohno, Nature 455, 515 \n(2008).  \n3 D. Chiba, M. 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Appl. \nPhys. 113, 17C719 (2013).  \n56 S. Giordano, Y. Dusch, N. Tiercelin, P. Pernod, and V. Preobrazhensky, J. Phys. D. Appl. \nPhys. 46, 325002 (2013).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  35  \n \n \nFigure 1. Magnetoelastic elliptical memory element schematic with associated coordinate system for in -\nplane stress -pulse induced toggle switching.  Here  \nM is the magnetization  vector  with \n and \n being \npolar and azimuthal angles . For the in -plane t oggle switching case, the initial normalized magnetization  \n0 0 0ˆˆ cos sinm x y\n and is in the film plane with \n0arcsin[ / ]bias kHH   and \nˆbias bias H Hy . \n \n \n \nFigure 2. Toggle switching trajectory for an in -plane magnetized polycrystalline Tb 0.3Dy 0.7Fe2 element \nwith \nLLG  = 0.3, \n  = -120 MPa, and \npulse  = 50 ps (red) and 125 ps (blue) and 160 ps (green).  \n 36  \nFigure 3. a) Effect of the Gilbert damping on pulse switching probability statistics for\n  = -85 MPa. b) \nEffect of increasing stress pulse amplitude for high damping  \nLLG  = 0.75. Very high stress pulses ( >200 \nMPa) are required to allow precession to be fast enough to cause a switch before dynamics are damped \nout. c) Comparison of switching statistics for the LL and LLG dynamics at \n  = -200 MPa, \n  = 0.75. \nThe LL dynamics exhibits faster precession than the LLG for a given torque implying shorter windows of \nreliability and requirements for faster pulses.  \n \n \nFigure 4.  Schematic of TbFe 2 magnetic element under biaxial stress generated by a PZT layer.  \nHere the initial normalized magnetization  \n0 0 0ˆˆ cos sinm z x   is predominantly out of the \nfilm plane with a cant  \n0arcsin[ / ]bias kHH   in the x -direction provided by  \nˆbias bias H Hx . \n  \n 37  \n \nFigure 5. a) Switching trajectories for a TbFe 2 nanomagnet under a pulsed biaxial stress \n  = -85 MPa, \npulse\n = 400 ps ( green ) and \n  = -120 MPa and\npulse  = 300 ps (blue ) b) Switching trajectory time \ntrace for {m x,my,mz} for \n  = -85 MPa . The pulse is initiated at t  = 500 ps. The blue region \ndenotes when precession about \nbiasH  dominates (i.e. while the pulse is on) and the red when the \ndissipative dynamics rapidly damp the system down to the other equilibrium point.  \n \n 38 Figure 6. a) Dependence of the simulated pulse switching probability on \n  for \n  = -85 MPa . b) \nDependence of pulse switching probability on stress amplitude. Stress -induced switching is possible even \nfor \n  = 1.0. c) Comparison of pulse switching probability  for LL and LLG dynamics for \n = -85 MPa \nand \n  = 0.75. Here the difference between the LL and LLG dynamics has a significant effect on the \nwidth of the pulse window where reliable switching is predicted by the simulations (\nLL  = 200 ps and \nLLG\n=320 ps.)  \n \n \nFigure 7. a) Energy (normalized to \n1K ) landscape as a function of angle for various values of exchange \nbias energy. b)  \n= 80º (\n= 10 º) is the only stab le equilibrium for compressive ( tensi le) stress. \nDissipative dynamics and the free energy landscape then dictate the non -toggle switching dynamics. c) \nShows the energy barrier dependence  on the [110] bias field for a \nd   = 100 nm, \nfilmt = 12.5 nm circular \nelement with (curve 1) \n1K = 2.5x105 ergs/cm3, (curve 2) \n1K = 2.0×105 ergs/cm3, and ( curve 4) \n1K\n=1.5×105 ergs/cm3. Curve 3 shows the energy barrier dependence for  \n1K=1.5x105 ergs/cm3 and \nd  = 100 \nnm & \nfilmt  = 15 nm . \n \n \n \n \n 39  \nFigure 8. a) Magnetoelastic switching trajectory for Fe 81Ga19 with \n  = -45 MPa and \npulse = 3 ns. The \nmain part of the switching occurs within 200 ps. The magnetization relaxes to the equilibrium defined \nwhen the pulse is on and then relaxes to the final equilibrium when the pulse is turned off. b) Switchin g \nprobability phase diagram for Fe 81Ga19 with biaxial anisotropy at T  = 300 ºK. c)  T = 0 ºK OOMMF \nsimulations showing the equilibrium m icromagnetic configuration for \n1K  = 2×105 ergs/cm3 and \nsM  = \n1300 emu/cm3. Subsequent shots show the rotational switching mode for a 45 MPa uniaxial compressive \nstress along [100].  Color scale is blue -white -red indicating the local projection \n1xm  (blue), \n0xm\n(white), \n1xm (red).  \n  \n 40  \n \n \nFigure 9. Schematic of magnetostrictive device geometry that utilizes uniaxial anisotropy to achieve \ndeterministic switching. Polycrystalline Tb 0.3Dy 0.7Fe2 on PMN -PT with 1 axis oriented at angle \n  with \nrespect to the easy axis.  In this geometry, \nM  lies in the x -y plane (film -plane) with the normalized  \nˆˆ cos sinm x y\n.  \n  \n 41  \n \nFigure 10. a) In-plane shape anisotropy field (\nkH ) and hard axis bias field (\nbiasH ) for a 400×900 nm2 \nellipse as a function of film thickness required to ensure \n0  = 51º . Thermal stability parameter\n  plotted \nversus film thickness with\nkH , \nbiasH  such that  \n0 = 51º .  b) Eight  times the RMS angle fluctuation \nabout three different  average \n0 > 45º versus film thickness for a 400×900 nm2 ellipse at T = 300 ºK. \n \n \n 42 Figure 11. a) Magnetization trajectories for\n = 45º, \n= -5 MPa ,\npulse = 3 ns, with ~ 200 Oe \nyielding \n0  = 51º  ( red) and\n = 45º,\n = -20 MPa with \nbiasH  = 120 Oe yielding  \n0 = 28º ( green). b) T = \n300 ºK stress pulse (compressive) switching prob ability phase diagram for a 400×90 0 nm2 ellipse with \nfilmt\n = 12.5 nm , \n= 45º, \n0 = 51º c) Micromagneti c switching trajectory of a 400×90 0 nm2 ellipse under \na DC compressive stress of -3 MPa transduced along 45 degrees.  Color scale is blue -white -red indicating \nthe local projection \n1xm  (blue), \n0xm (white), \n1xm (red).  \n \n \n \nbiasH"    },    {        "title": "1612.02360v1.Gilbert_damping_of_magnetostatic_modes_in_a_yttrium_iron_garnet_sphere.pdf",        "content": "Gilbert damping of magnetostatic modes in a yttrium iron garnet sphere\nS. Klingler,1, 2,a)H. Maier-Flaig,1, 2C. Dubs,3O. Surzhenko,3R. Gross,1, 2, 4H. Huebl,1, 2, 4\nS.T.B. Goennenwein,1, 5, 6and M. Weiler1, 2\n1)Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching,\nGermany\n2)Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3)INNOVENT e.V. Technologieentwicklung, 07745 Jena, Germany\n4)Nanosystems Initiative Munich, 80799 Munich, Germany\n5)Institut f ur Festk orperphysik, Technische Universit at Dresden, 01062 Dresden,\nGermany\n6)Center for Transport and Devices of Emergent Materials, Technische Universit at Dresden, 01062 Dresden,\nGermany\n(Dated: 8 December 2016)\nThe magnetostatic mode (MSM) spectrum of a 300 \u0016m diameter single crystalline sphere of yttrium iron\ngarnet is investigated using broadband ferromagnetic resonance (FMR). The individual MSMs are identi\fed\nvia their characteristic dispersion relations and the corresponding mode number tuples ( nmr) are assigned.\nTaking FMR data over a broad frequency and magnetic \feld range allows to analyze both the Gilbert\ndamping parameter \u000band the inhomogeneous line broadening contribution to the total linewidth of the\nMSMs separately. The linewidth analysis shows that all MSMs share the same Gilbert damping parameter\n\u000b= 2:7(5)\u000210\u00005irrespective of their mode index. In contrast, the inhomogeneous line broadening shows a\npronounced mode dependence. This observation is modeled in terms of two-magnon scattering processes of\nthe MSMs into the spin-wave manifold, mediated by surface and volume defects.\nThe ferrimagnetic insulator yttrium iron garnet (YIG)\nhas numerous applications in technology and funda-\nmental research due to its low intrinsic Gilbert damp-\ning and large spin-wave propagation length.1It is used\nas prototypical material in various experiments in spin\nelectronics2{4and spin caloritronics5,6and is indispens-\nable for microwave technology.\nRecently, YIG spheres attracted attention in the\n\feld of quantum information technology.7{15For exam-\nple, strong coupling between magnons and photons in\nYIG/cavity hybrid systems can be employed for the up-\nand down-conversion of quantum signals between mi-\ncrowave and optical frequencies, enabling a long-range\ntransmission of quantum information between microwave\nquantum circuits.14{16Here, the damping of the mag-\nnetic excitation plays a crucial role, since it limits the\ntime-scale in which energy and information is exchanged\nand stored in the magnon-photon hybrid system.\nOne type of magnetic excitations in YIG spheres17{19\nare magnetostatic modes (MSMs) which resemble stand-\ning spin-wave patterns within the sphere. Although the\nlinewidth of MSMs in YIG spheres has been studied at\n\fxed frequencies in the past,20{22the respective contri-\nbutions of intrinsic Gilbert damping and inhomogeneous\nline broadening23to the total linewidth have not yet been\ninvestigated. In particular, it is not evident from the\nliterature, whether di\u000berent MSMs feature the same or\ndi\u000berent Gilbert damping.24,25\nHere, we report on the study of dynamic properties of\nmultiple MSMs for a 300 \u0016m diameter YIG sphere us-\ning broadband ferromagnetic resonance. The frequency\na)Electronic mail: stefan.klingler@wmi.badw.deand magnetic \feld resolved FMR data allows to separate\nGilbert damping and inhomogeneous line broadening of\nthe MSMs. One and the same Gilbert damping parame-\nter\u000b= 2:7(5)\u000210\u00005is found for all MSMs, independent\nof their particular mode index. However, the inhomoge-\nneous line broadening markedly di\u000bers between the ob-\nserved MSMs. This \fnding is attributed to two-magnon\nscattering processes of the MSMs into the spin-wave man-\nifold, mediated by surface and volume defects.\nThe MSM pro\fles and eigenfrequencies of a magnetic\nsphere can be calculated in the magnetostatic approx-\nimationr\u0002H= 0,17{19using the Landau-Lifshitz-\nGilbert equation (LLG).26,27The resonance frequencies\n\n of the MSMs are obtained by solving the characteristic\nequation:17{19\nn+ 1 +\u00180dPm\nn(\u00180)=d\u00180\nPmn(\u00180)\u0006m\u0017= 0; (1)\nwhere\u00182\n0= 1 + 1=\u0014,\u0014= \n H=\u0000\n\n2\nH\u0000\n2\u0001\n,\u0017=\n\n=\u0000\n\n2\nH\u0000\n2\u0001\n, \nH=\u00160Hi=\u00160Msand \n =!=\r\u0016 0Ms.\nHere,\r=gJ\u0016B=~is the gyromagnetic ratio, gJis the\nLand\u0013 eg-factor,\u0016Bis the Bohr magneton, ~is the reduced\nPlanck constant, \u00160is the vacuum permeability and Ms\nis the saturation magnetization. The angular frequency\nof the applied microwave \feld is denoted as != 2\u0019f.\nThe internal \feld is given by Hi=H0+Hani+Hdemag ,\nwhereH0is the applied static magnet \feld, Haniis the\nanisotropy \feld, and Hdemag =\u0000Ms=3 is the demagneti-\nzation \feld of a sphere.\nThe mode pro\fles of the MSMs have the form of asso-\nciated Legendre polynomials Pm\nn, where the localization\nof the MSMs at the surface is related to the mode index\nn2N.21The indexjmj\u0014ncorresponds to an angular-\nmomentum quantum number of the MSM,28where thearXiv:1612.02360v1  [cond-mat.mtrl-sci]  7 Dec 20162\nbar above the mode index mis used for indices m < 0.\nThe index r\u00150 enumerates the solutions of the char-\nacteristic equation (1) for given nandmfor increasing\nfrequencies.18,29In total, each MSM is uniquely identi-\n\fed by the index tuple ( nmr). For more information and\nplots of the MSM mode patterns, the review of Ref. 19\nis recommended.\nThe Gilbert damping parameter phenomenologically\naccounts for the viscous (linearly frequency-dependent)\nrelaxation of magnetic excitations. Assuming a domi-\nnant Gilbert-type damping for all MSM modes, the full\nlinewidth at half maximum (FWHM) \u0001 f(nmr)of a MSM\nresonance line at frequency f(nmr)\nres is given by:30\n\u0001f(nmr)= 2\u000bf(nmr)\nres + \u0001f(nmr)\n0: (2)\nHere, \u0001f0denotes the inhomogeneous line broadening\ncontributions to the total linewidth. For a two-magnon\nscattering process mediated by volume and surface de-\nfects the latter can be written as:21\n\u0001f(nmr)\n0 = \u0001fm-mF(nmr)+ \u0001f0\n0: (3)\nHere, \u0001fm-maccounts for the two-magnon scattering pro-\ncess of the MSMs into the spin-wave manifold.21,22,31The\nfactorF(nmr)represents the ratio of the linewidth of a\nparticular MSM with respect to the uniform precessing\n(110)-mode.21,22,32,33It therefore accounts for the surface\nsensitivity of the speci\fc mode compared to the (110)-\nmode. The two-magnon scattering processes can be sup-\npressed if a perfectly polished YIG sphere is used, due to\nthe vanishing ability of the system to transfer linear and\nangular momentum from and to the lattice.21The term\n\u0001f0\n0represents a constant contribution to the linewidth\nin which all other frequency-independent broadening ef-\nfects are absorbed. The complete scattering theory used\nin this letter is presented in Ref. 21.\nFig. 1 (a) shows a sketch of the measurement setup.\nThe YIG sphere with a diameter of d= 300\u0016m is placed\nin a disk shaped Vespel sample holder (diameter 6 mm,\nnot shown), which has a centered hole with a diameter\nof 350\u0016m. The sphere in the sample holder is exposed\nto a static magnetic \feld in order to align the easy [111]-\ndirection of the YIG crystal parallel to the \feld direc-\ntion. The orientation of the sphere is subsequently \fxed\nusing photoresist and the alignment is con\frmed by Laue\ndi\u000braction.\nThe oriented YIG sphere is placed on a 50 \n impedance\nmatched coplanar waveguide (CPW) structure. The\nsphere is placed in the middle of the w= 300\u0016m wide\ncenter conductor, with the YIG [110]-axis aligned par-\nallel to the long axis of the center conductor of the\nCPW. Additionally, a pressed crumb of Diphenylpicryl-\nhydrazyl (DPPH) is glued on the center conductor, where\nthe distance between the YIG sphere and the DPPH is\nl\u00191 cm. DPPH is a spin marker with a g-factor34of\ngDPPH = 2:0036(3). The measurement of its resonance\nfrequency\nfDPPH =gDPPH\u0016B\n2\u0019~\u00160HDPPH\n0 (4)\nP1 P2 z, [111] y, [110] \nx, h xVNAelectro magnet top view (a)  side view \nYIG \nDPPH CPW \nH0\nIm ∆S 21   ,Re ∆S 21 (a.u.)  \n-10 0 10\nf-f res  (MHz)(b) (530)-mode  \nH0w\nlaa/2 P1 \nP2 hrf FIG. 1. (a) The CPW with the YIG sphere and the DPPH\nis positioned in the homogeneous \feld of an electromagnet.\nThe CPW is connected to port 1 (P1) and port 2 (P2) of a\nvector network analyzer (VNA). The YIG sphere is placed\non top of the center conductor of the CPW with its [111]-\naxis parallel to the applied magnetic \feld H0inz-direction.\n(b) Typical normalized transmission spectrum of the (530)-\nmode at\u00160H0= 0:8 T (symbols) including a \ft to Eq. (5)\n(lines).\nprovides an independent magnetic \feld reference at the\nsample position, in addition to Hall probe measurements.\nThe static magnetic \feld calculated from the DPPH\nresonance frequency is denoted as HDPPH\n0 . The stray\n\feld originating from the YIG sphere at the location of\nthe DPPH creates a systematic measurement error of\n\u000e\u00160Hstray\u001440\u0016T, as estimated using a dipole approxi-\nmation.\nFor the broadband FMR experiments, the CPW is po-\nsitioned between the pole shoes of an electromagnet with\na maximum \feld strength of j\u00160H0j\u00142:25 T. The pole\nshoe diameter is a= 6 cm, while the pole shoe sepa-\nration isa=2, to ensure a su\u000ecient homogeneity of the\napplied magnetic \felds. The measured radial \feld gra-\ndient creates a systematic \feld measurement error of\n\u000e\u00160Hdisp= 0:3 mT forl= 1 cm displacement from the\ncenter axis.\nThe CPW is connected to port 1 (P1) and port 2 (P2)\nof a vector network analyzer (VNA) and the complex\nscattering parameter S21is recorded as a function of H0\nandf\u001426:5 GHz. The applied microwave power is -\n20 dBm to avoid non-linear e\u000bects causing additional line\nbroadening. The microwave current \rowing along the\ncenter conductor generates a microwave magnetic \feld\npredominately in the x-direction at the location of the\nYIG sphere. This results in an oscillating torque on\nthe magnetization, which is aligned in parallel to the z-\ndirection by the external static \feld H0. Forf=f(nmr)\nres ,\nthe excited resonant precession of the magnetization re-\nsults in an absorption of microwave power.\nIn order to eliminate the e\u000bect of the frequency depen-\ndent background transmission of the CPW, the following\nmeasurement protocol is applied: First, S21is measured\nfor \fxedH0in a frequency range fDPPH\u00061 GHz. Second,\nS21is measured for the same frequency range at a slightly3\nFIG. 2. (a) Normalized transmission magnitude j\u0001S21j\nplotted versus applied magnetic \feld \u00160H0and microwave\nfrequencyfrelative to the DPPH resonance fDPPH . The\ncontrast between the dashed lines is stretched for better vis-\nibility. (b) Calculated and measured dispersions of various\nMSMs (lines and open circles, respectively).\nlarger magnetic \feld H0+ \u0001H0, with\u00160\u0001H0= 100 mT.\nSince for this \feld no YIG and DPPH resonances are\npresent in the observed frequency range, the latter mea-\nsurement contains the pure background transmission.\nThird, the normalized transmission spectra is obtained\nas \u0001S21=S21(H0)=S21(H0+ \u0001H0), which corrects the\nmagnitude and the phase of the signal. This procedure is\nrepeated for all applied magnetic \felds. The transmitted\nmagnitude around the resonance can be expressed as:30\n\u0001S21(f) =A+Bf+Z\n\u0010\nf(nmr)\nres\u00112\n\u0000if2\u0000if\u0001f(nmr):(5)\nHere,Ais a complex o\u000bset parameter, Bis a complex lin-\near background and Zis a complex scaling parameter.35\nFig. 1 (b) exemplary shows the real and imaginary part of\n\u0001S21for the (530)-mode at \u00160H0= 0:8 T. In addition, a\n\ft of Eq. 5 to the data is shown, which adequately models\nthe shape of the resonances.\nFig. 2 (a) shows the normalized transmitted magnitude\nj\u0001S21jas a function of H0andf\u0000fDPPH on a linear\ncolor-coded scale. The frequency axis is chosen relative\nto the DPPH resonance frequency, so that all modes with\na linear dispersion f(nmr)\nres/H0appear as straight lines,whereas modes with a non-linear dispersion are curved.\nNote, that the \feld values displayed on the y-axis repre-\nsent the magnetic \feld strength measured with the Hall\nprobe.\nThe di\u000berent modes appearing in the color plot in\nFig. 2 (a) can be identi\fed in a straightforward manner.\nAt \frst, all visible resonances are \ftted using Eq. (5)\nin order to extract f(nmr)\nres and \u0001f(nmr). Furthermore,\nthe DPPH resonance line is identi\fed as straight line at\nf\u0000fDPPH = 0 MHz and the resonance \felds HDPPH\n0 are\ncalculated using Eq. (4).\nSecond, the straight lines at about f\u0000fDPPH\u0019\n\u000060 MHz and f\u0000fDPPH\u0019\u0000740 MHz are identi\fed as\nthe (110)- and (210)-mode, respectively. A simultaneous\n\ft of the dispersion relations18\nf(110)\nres =gYIG\u0016B\n2\u0019~\u00160(H0+Hani) (6)\nand\nf(210)\nres =gYIG\u0016B\n2\u0019~\u00160\u0012\nH0+Hani\u00002\n15Ms\u0013\n(7)\nto the measured values of f(110)\nres,f(210)\nres and\u00160HDPPH\n0\nyieldsgYIG = 2:0054(3),\u00160Ms= 176:0(4) mT and\n\u00160Hani=\u00002:5(4) mT. The error of gYIGis given by the\nsystematic error introduced by the \feld normalization\nusinggDPPH . The errors in \u00160Haniand\u00160Msare given\nby\u000e\u00160Hdisp+\u000e\u00160Hstray. All values are in good agree-\nment with previously reported material parameters36{40\nfor YIG (gYIG = 2:005(2),\u00160Hani=\u00005:7 mT and\n\u00160Ms= 180 mT) and, hence, justify the (110)- and (210)-\nmode assignments.\nThird, the complete MSM manifold is computed using\nthe extracted material parameters. The mode numbers\nof the remaining modes are determined from the charac-\nteristic dispersions. Fig. 2 (b) shows the dispersions of\nthe identi\fed modes as function of f(nmr)\nres\u0000fDPPH and\nHDPPH\n0 , with very good agreement of theory (lines) and\nexperiment (circles). Slight deviations between model\npredictions and data might be attributed to a non-perfect\nspherical shape of the sample, which would change the\nboundary conditions for the magnetization dynamic in\nthe YIG spheroid, and thus the dispersion relations.\nIn Fig. 3 (a) the linewidth \u0001 f(nmr)of each MSM is\nplotted versus its resonance frequency f(nmr)\nres . The o\u000bset\n\u0001f(nmr)\n0 is magni\fed by a factor of 5 to emphasize the\ndi\u000berences in the inhomogeneous line broadening. Indi-\nvidual \fts of all \u0001 f(nmr)to Eq. (2) yield identical slopes\nfor all modes within a small scatter, which is also evident\nfrom the linewidth data in Fig. 3 (a). Hence, the Gilbert\ndamping parameter and inhomogeneous line broadening\nare obtained from a simultaneous \ft of Eq. (2) to the\nextracted data points. Here, \u000bis a shared \ft parameter\nfor all MSMs, but the inhomogeneous line broadening\n\u0001f(nmr)\n0 is \ftted separately for each mode. To avoid\n\ftting errors, the linewidths data are disregarded when\na mode anti-crossing is observed, since this results in a4\n5 10 15 20 25 ∆f (nmr) (MHz) \nfres       (GHz) (a) \n0(110)\n(440)\n(531)\n(530)\n(511)\n(631)\n(502)\n(nmr) 246810 Offset x5 \n(b) \n0.00.51.0 1.5 2.0 ∆f0(nmr) (MHz) \n-500 0 500 -250 250 ∆f00=0.3 MHz (110) \n(440) (531) (530) (511) \n(631) \n(502) Measurement \nTheory \n fres      - fDPPH  (MHz) (nmr) \nFIG. 3. (a) Linewidth vs. resonance frequency of the\nmeasured MSMs. The Gilbert damping of all MSMs is\n\u000b= 2:7(5)\u000210\u00005as evident from the same slope of all\ncurves. The inhomogeneous line broadening is di\u000berent for\neach MSM. Note that the data points are plotted with an o\u000b-\nset proportional to the inhomogeneous line broadening. (b)\nInhomogeneous line broadening as a function of f\u0000fDPPH .\npronounced increase in linewidth.41As evident from the\nsolid \ft curves in Fig. 3 (a) the evolution of the linewidth\nwith resonance frequency of all measured MSMs can be\nwell described with a shared Gilbert damping parameter\nof\u000b= 2:7(5)\u000210\u00005, independent of the mode num-\nber and the mode intensity. The latter strongly sug-\ngests a negligible e\u000bect of radiative damping on the mea-\nsured linewidths.42The error in \u000bis given by the scat-\nter of\u000bfrom the independent \fts. Other groups report\nGilbert damping parameters for YIG \flms43{49larger\nthan\u000b= 6:15\u000210\u00005, whereas for bulk YIG37,49,50values\nof\u000b= 4\u000210\u00005are found. Hence, the Gilbert damp-\ning parameter obtained here is the smallest experimen-\ntal value reported so far. The results are in agreement\nwith the notion, that the Gilbert damping parameter is a\nbulk property which only depends on intrinsic damping\ne\u000bects. However, the inhomogeneous line broadening is\nindeed di\u000berent for the various MSMs.\nFig. 3 (b) shows the extracted values for the inhomo-\ngeneous line broadening (\flled dots) as a function of\nf(nmr)\nres\u0000fDPPH . The error bars indicate the variation of\nthe inhomogeneous line broadening between global andindividual \fts. In order to show the approximate posi-\ntion of the modes in comparison to Fig. 2, the x-scale is\ncalculated for a magnetic \feld strength of \u00160H= 0:5 T.\nAdditionally, the linewidths \u0001 f(nmr)\n0 for all modes are\ncalculated using the two-magnon scattering theory, given\nin Eq. (4) of Ref. 21 (open circles). For the calculations of\nthe linewidths, a pit radius R= 350 nm and a constant\nlinewidth contribution of \u0001 f0\n0= 30 kHz was assumed.\nSince the calculated \u0001 fm-mare slightly frequency depen-\ndent, the average linewidth values for the measured \feld\nand frequency range are used and the standard deviation\nis indicated by the error bars of the open symbols. For\nmost MSMs the variation is smaller than 10 kHz. Never-\ntheless, the (440)-mode should show a prominent peak in\nthe linewidth measurement at about f(440)\nres = 10 GHz in\nFig. 3 (a),21which is however not observed in the experi-\nmental data. Additionally, the (110)-MSM shows a much\nlarger linewidth than expected from the calculations. In\na perfect sphere the (110)-mode is degenerate with the\n(430)-mode,18but in a real sphere this degeneracy might\nbe lifted. If the di\u000berence of the (110)- and (430)-mode\nfrequencies is smaller than the linewidth of the measured\nresonance, an additional inhomogeneous line broadening\nis expected. Indeed, a careful analysis of the (110)-MSM\nline shape reveals a second resonance line in very close\nvicinity to the (110)-mode, yielding an arti\fcial inhomo-\ngeneous line broadening of this mode. Besides these two\nMSMs, an excellent quantitative agreement between the\ntwo-magnon scattering model and experiment is found.\nIn conclusion, broadband ferromagnetic resonance ex-\nperiments on magnetostatic modes in a YIG sphere are\npresented and various magnetostatic modes are identi-\n\fed. The linewidth analysis of the data allows to distin-\nguish between the Gilbert damping and inhomogeneous\nline broadening. A very small Gilbert damping parame-\nter of\u000b= 2:7(5)\u000210\u00005is found for all MSMs, indepen-\ndent of their mode indices. 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Islama,<\naPhysics Discipline, Khulna University, Khulna 9208, Bangladesh\nARTICLE INFO\nKeywords :\nMagnetization reversal\nShape anisotropy\nEnergy barrier\nStimulatedenergyabsorption _emissionABSTRACT\nWe investigate the influence of shape anisotropy on the magnetization reversal of a single-domain\nmagnetic nanoparticle driven by a circularly polarized linear down-chirp microwave field pulse\n(DCMP). Based on the Landau-Lifshitz-Gilbert equation, numerical results show that the three con-\ntrollingparametersofDCMP,namely,microwaveamplitude,initialfrequencyandchirprate,decrease\nwith the increase of shape anisotropy. For certain shape anisotropy, the reversal time significantly\nreduces. Thesefindingsarerelatedtothecompetitionofshapeanisotropyanduniaxialmagnetocrys-\ntalline anisotropy and thus to the height of energy barrier which separates the two stable states. The\nresultofdampingdependenceofmagnetizationreversalindicatesthatforacertainsampleshape,there\nexistsanoptimaldampingsituationatwhichmagnetizationisfastest. Moreover,itisalsoshownthat\ntherequiredmicrowavefieldamplitudecanbeloweredbyapplyingthespin-polarizedcurrentsimulta-\nneously. Theusageofanoptimumcombinationofbothmicrowavefieldpulseandcurrentissuggested\ntoachievecostefficiencyandfasterswitching. Sothesefindingsmayprovidetheknowledgetofabri-\ncate the shape of a single domain nanoparticle for the fast and power-efficient magnetic data storage\ndevice.\n1. Introduction\nObtaining fast and energy-efficient magnetization rever-\nsalofsingle-domainoftheperpendicularlymagnetizednanopar-\nticle is an interesting issue owing to its potential applica-\ntion in high-density data storage devices [1, 2, 3] and rapid\ndata access [4]. For high density, high thermal stability and\nlow error rate in device application, high anisotropy mate-\nrials [5] are needed. But it is a challenge to find out a way\nwith low energy to achieve the fastest magnetization rever-\nsalforhigh-anisotropymagneticnanoparticle. Overthepast\ntwo decades, several controlling parameters _driving forces\narediscoveredtoachievefastestmagnetizationreversalwith\nlow cost. Namely, magnetization reversal using a constant\nmagnetic field [6, 7], reversal by microwave field of con-\nstant frequency or time dependent frequency, either with or\nwithout a polarized electric current [8, 9, 10, 11, 12, 13,\n14, 15, 16, 17, 18, 19] and by spin transfer torque (STT)\nor spin orbit torque (SOT) [20, 21, 22, 23, 24, 25, 26, 27,\n28, 29, 30, 31, 32, 33, 34]. All methods mentioned above\naresufferingfromspecificdrawbacks[6,35,36,37,38,39,\n40, 41]. For example, magnetic field or microwave field\ndriven magnetization is not energy efficient and fast as ex-\npected. Incaseofcurrent(bySTT)orSOTdriven,thehigher\nthreshold current requirement is a bottleneck. Later on, re-\nsearchers are digressed to employ the microwave chirped\npulses (microwave with time-dependent frequency) which\ninduce fast and energy-efficient magnetization reversal [13,\n14, 15, 16, 17, 19] but still the required field amplitude, ini-\ntial frequency and chirp rate are not small as desired prac-\n<Corresponding authors:\ntorikul@phy.ku.ac.bd (M.T. Islam)\nORCID(s): 0000-0002-4168-8695 (Z.K. Juthy); 0000-0002-7500-1572\n(M.A.J. Pikul); 0000-0002-6742-2158 (M.A.S. Akanda);\n0000-0002-3846-4009 (M.T. Islam)tically. Recent studies [42, 43] have demonstrated that the\nfastmagnetizationreversalofacubicnanoparticle(i.e.,with\nzerodemagnetization _shapeanisotropy)obtainedbythelin-\neardownchirpmicrowavepulse(DCMP).Thephysicalmech-\nanismofthismodelisthattheDCMP(whosefrequencylin-\nearlydecreasesfrominitial +f0tofinal*f0)stimulatesmi-\ncrowave energy absorption (emission) by the magnetic mo-\nment before (after) crossing the energy barrier. The effi-\nciencyofthemicrowaveenergyabsorption _emissiondepends\nonhowcloselythemicrowavefrequencyandmagnetization\nintrinsic frequency match during reversal. This frequency-\nmatching is related to the energy barrier [44, 45]. However,\ninsmall-scalenanoparticle,demagnetizationfield _shapeanisotropy\n(whichwasexcludedpreviously)hasagreatimportance[46,\n47,44]because,itinteracts _opposesthedominateduniaxial\nmagnetocrystalline anisotropy and thus reduces the energy\nbarrier between two stable states. So, the magnetization re-\nversal of a nanoparticle with finite shape anisotropy might\nbefasterandenergyefficient. Inaddition,fabricatingperfect\ncubicnanoparticle(i.e.,withzerodemagnetizationfield)isa\nchallengingissuefrompracticalpointofview. Forthemost\ndevice applications, the desired microwave pulse (DCMP)\nshould be withsmaller amplitude, initial frequency and low\nchirp rate.\nTherefore,thisstudyfocusesonhowtheshapeanisotropy\nHshapeaffects the energy consumption i.e., the amplitude\nHmw, initial frequency foand pulse chirp rate \u0011of DCMP\nand the magnetization switching time ts. Interestingly, we\nfoundthattheparametersofDCMPi.e., Hmw,fo,and\u0011de-\ncrease with the increase of shape anisotropy Hshape. In ad-\ndition,thereisacritical Hshapeorcross-sectionalarea A,for\nwhichorlarger, tssignificantlyreducesto0.346nswhichis\nclose to the theoretical limit [40]. For instance, in case of\na sample of cross-sectional area 3232nm2, orHshape=\n0:598T, we estimated ts= 0:346ns,Hmw= 0:0187T,\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 1 of 7arXiv:2108.10965v2  [cond-mat.mes-hall]  2 Oct 2021Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\nfo=5:2GHz,andoptimal \u0011=24:96ns*2whicharesignif-\nicantly smaller than ts=0:56ns,Hmw=0:045T,fo=21\nGHz,andoptimal \u0011=67:2ns*2forthecubicshapedorzero\nHshape. Thephysicalreasonofthesefindingsisrelatedtothe\ncompetitionbetweenshapeanisotropyandtheuniaxialmag-\nnetocrystalline anisotropy. The findings of damping depen-\ndence of magnetization reversal indicate that, for a specific\nAorHshape,thereexistsaoptimaldampingsituation. More-\nover,thisstudyshowsthattherequiredmicrowavefieldam-\nplitude of DCMP can be lowered by applying direct current\nsimultaneously along with DCMP. Therefore, an optimum\ncombination of microwave field pulse and current could be\napplied to achieve faster and power efficient magnetization\nreversal. So these findings might be significant to design\ntheshapeofsingledomainnanoparticletoproducethelow-\nenergy cost magnetic storage device with high speed data\nprocessing.\n2. Analytical Model and Method\nWe consider a spin valve system that consists of a free\nand fixed ferromagnetic layers by sandwiching a nonmag-\nneticspacer,asshownschematicallyinFigure1(a). Thefree\nlayerisasquarenanoparticleofarea Aandthickness d. The\nmagnetization direction of the fixed and free layer is along\nzdirection, i.e., both are perpendicularly magnetized. The\nmagnetizationofthefreelayercanbetreatedasamacrospin\nrepresented by the unit moment mwith total magnetization\nAdMs, whereMsrepresents the saturation magnetization.\nThe demagnetization field, which generated by the magne-\ntization of nanoparticle, can be approximated by an easy-\nplane shape anisotropy. The demagnetization field _shape\nanisotropy field is expressed as Hshape=Hshapemzzsuch\nthatHshape=*\u00160.Nz*Nx/Ms,istheshapeanisotropyco-\nefficient,where NzandNxaredemagnetizationfactors[45]\nand\u00160=4\u001910*7N_A2is the vacuum magnetic perme-\nability. Because of the strong uniaxial magnetocrystalline\nanisotropy Hani=Hanimzz, the magnetization of the free\nlayer prefers to stay in two ground states, i.e., mparallel to\nzand*z.\nUnderthecircularlypolarizedDCMPandspinpolarized\ncurrent, the magnetization dynamics mis governed by the\nLandau-Lifshitz-Gilbert (LLG) equation [7, 9, 37, 48]\ndm\ndt=*\rmHeff+\u000bmdm\ndt*\rhsm.pm/(1)\nwhere\u000bisthedimensionlessGilbertdampingconstantand,\n\risgyromagneticratio. Heffisthetotaleffectivefieldwhich\ncontains the microwave magnetic field Hmw, the exchange\nfield2A\nMs(2mwhereAistheexchangestiffnessconstant,and\ntheeffectiveanisotropyfield Hkalongzdirection,i.e., Hk=\nHkmzz.hsis the strength of spin transfer torque(STT),\nhs=`PJ\n2e\u0016oMsd(2)\nwhereJ,e,`,P,\u00160andddenote the current density, elec-\ntron charge, the Planck’s constant, spin polarization of cur-\nmf0\n0-f(b) (a)\nτ\n-\n+\npFigure 1: (a) Schematic diagram of the system in where mand\npdenote the unit vectors of the magnetization of free layer\nand fixed layer, respectively. A linear down-chirp microwave\nfield and an electric current are applied on a single domain\nnanoparticle. (b) The frequency profile illustrates (sweeping\nfrom+f0to*f0) the linear down-chirp microwave field.\nrent,thevacuumpermeability,andthicknessofthefreelayer,\nrespectively.\nTheHkhas two components: an uniaxial magnetocrys-\ntalline anisotropy Haniand a shape anisotropy Hshapefield\nwhich can be expressed as Hk=Hani+Hshape= [Hani*\n\u00160.Nz*Nx/Ms]mzz. Thus, the resonant frequency of the\nnanoparticlecanbefoundfromtheKittelformula f0=\r\n2\u0019[Hani*\n\u00160.Nz*Nx/Ms]:\nWefirstapplyonlythemicrowavefield-drivenmagneti-\nzationreversal,i.e.,withoutelectriccurrent(SST)term,the\nrate of energy change is expressed as\n \u000f=*\u000b\róómHeffóó2*m\fHmw (3)\nSince damping is positive, so, the first term is always nega-\ntive. For time-dependent external microwave field, depend-\ning on the angle between the instantaneous magnetization\nmandHmw, the second term can be either positive or neg-\native, in other words, the microwave field pulse can trigger\nstimulated energy absorption or emission.\nIt has been shown that the fast magnetization reversal is\nachieved by DCMP [42] with the physical picture: the fre-\nquency of DCMP matches the frequency change of magne-\ntizationprecession. Hence,ittriggersstimulatedmicrowave\nabsorption (emission) by (from) the magnetization before\n(after) it crosses over the energy barrier.\nThemicrowavefieldthatwehaveappliedtakestheform,\nHmw=Hmw\u0004cos\u001e.t/x+sin\u001e.t/y\u0005(4)\nwhereHmwistheamplitudeofthemicrowavefieldand \u001e.t/\nis the phase. In this paper, we use a DCMP whose instanta-\nneous frequency is defined as f.t/ =1\n2\u0019d\u001e\ndt, which linearly\ndecreases with time at constant rate \u0011(in units of ns*2) as\nshown in Figure 1(b). The phase \u001e.t/and the instantaneous\nfrequencyf.t/are given by,\n\u001e.t/=2\u0019.f0t*\u0011\n2t2/;f.t/=f0*\u0011t (5)\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 2 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\n/s40/s98/s41/s40/s99/s41\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s49/s50/s116\n/s115/s32/s40/s110/s115/s41\n/s104 /s32/s40/s110/s115/s45/s50\n/s41/s32/s32 /s65\n/s48/s32\n/s32/s32 /s65\n/s49/s32/s32/s32\n/s32/s32 /s65\n/s50/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s49/s53/s51/s48/s52/s53/s54/s48/s104 /s32/s40/s110/s115/s45/s50\n/s41\n/s72\n/s115/s104/s97/s112/s101/s40/s84/s41/s48/s46/s48 /s48/s46/s51 /s48/s46/s54 /s48/s46/s57/s45/s49/s48/s49/s109\n/s122\n/s116/s32 /s40/s110/s115/s41/s32/s32 /s65\n/s48\n/s32/s32 /s65\n/s49/s32/s32\n/s32/s32 /s65\n/s50/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52\n/s32/s32 /s65\n/s53\n/s32/s32 /s65\n/s54\n/s32/s32 /s65\n/s55\n/s32/s32 /s65\n/s56/s40/s97/s41\nFigure 2: Model parameters of the free layer, Ms=106A/m,Aex=1310*12J_m,Hk=0:75T,\r=1:761011rad/(T \fs),\nand\u000b= 0:01. (a) Temporal evolution of mzinduced by DCMP with Hmw=0.045 T for the samples of different A. (b) The\ndependence of switching time tson the chirp rate \u0011for different cross-sectional areas while keeping the microwave field amplitudes\nHmwconstant. The vertical dashed lines indicate the lower and upper limits of \u0011for magnetization switching. (c) The dependence\nof chirp rate \u0011on shape anisotropy Hshapefor switching time within 1 ns.\n/s40/s99/s41/s40/s98/s41\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120\n/s45/s57/s48 /s48 /s57/s48/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51\n/s115/s115/s115/s115/s115\n/s105/s105/s105/s105/s105/s105/s101 /s47/s77\n/s115/s72\n/s107\n/s113 /s32/s40/s100/s101/s103/s41/s32/s32 /s65\n/s48\n/s32/s32 /s65\n/s50/s32/s32\n/s32/s32 /s65\n/s52/s32\n/s32/s32 /s65\n/s53/s32\n/s32/s32 /s65\n/s54\n/s32/s32 /s65\n/s55\n/s32/s32 /s65\n/s56/s105/s115/s40/s97/s41\nFigure 3: (a) The potential \"along the line \u001e= 0 for the\nsamples of different A. Angles\u0012and\u001eare defined as cos*1mz\nandtan*1.mx_my/. The symbols i and s represent the initial\nstateandthesaddlepoint,respectively. Magnetizationreversal\ntrajectories of (b) A0=88nm2and (c)A8=3232 nm2.\nwherefoistheinitialfrequencyat t=0. Thedurationofthe\nmicrowavepulseis \u001c=2fo\n\u0011whichmeansthefinalfrequency\nis*fo. AccordingtotheappliedDCMP,thesecondtermof\nEq. (3) can be expressed as\n \"=*Hmwsin\u0012.t/sin\b.t/d\u001e\ndt(6)\nwhere\b.t/istheanglebetween mt(thein-planecomponent\nofmandHmw). Therefore, the microwave field pulse can\ntrigger the stimulated energy absorption (for *\b.t/) which\nis occurred before crossing the energy barrier and emission\n(for\b.t/) after crossing the energy barrier.\nFor this study, the parameters of Permalloy are chosen\nfrom typical experiments on microwave-driven magnetiza-\ntion reversal as Ms=106A_m,Hani=0:75T,\r=1:76\n1011rad_.T\fs/,Aex= 1310*12J_m and\u000b= 0:01. Al-\nthough, the demonstrated strategy and findings would still\nbe valid for other materials. The cell size used throughout\nthis study is .222/ nm3. We solved the LLG equa-tionnumericallyusingtheMUMAX3package[49]fortime-\ndependent circularly polarized microwave field. We con-\nsider the switching time window of 1ns and least magne-\ntization reversal mz=*0:9.\n3. Numerical Results\nIn a small-scale nanoparticle, demagnetization field or\nshapeanisotropyfield Hshapeisanimportantpropertytotake\ninto account in the investigation of the magnetization rever-\nsal. In order to approximate demagnetization field _shape\nanisotropyHshape,wechoosecuboidshapedsample _nanoparticle\nofvolumeV=Ad,whereAisthecross-sectionalareaand d\nistheheight. Toincreasethedemagnetizationfieldstrength\norHshape,Aisgraduallyincreasedwhilethethickness d=8\nnm is kept fixed. Specifically, we intend to investigate the\nmagnetization reversal of the samples, A1=1010 ,A2=\n1212,A3=1414 ,A4=1616 ,A5=2020 ,A6=\n2424,A7=2828 ,A8=3232 nm2. Foreachsample,\nby finding the demagnetization factors [45] NzandNx, the\nshape anisotropy coefficient Hshape=\u00160.Nz*Nx/Msare\ncalculated analytically. Hshapeactually acts along *zi.e.,\nit opposes the intrinsic easy axis anisotropy field Haniand\nhence reduces the effective anisotropy. Therefore, for each\nA,theresonance _theoreticalfrequencyisestimatedbywell-\nknownKittel-formula f0=\r\n2\u0019\u0004Hani*\u00160.Nz*Nx/Ms\u0005as\nshown in the Table 1.\nToinvestigatetheDCMP-drivenmagnetizationreversals\nof nanoparticles of different A, by keeping Hmw=0:045T\nandf0(close to resonant frequency corresponding to Aor\nHshape) fixed, we estimate the optimal \u0011for each sample or\nHshape. Figure2(b)showsthe \u0011dependenceof tsforseveral\nsamples orHshape. It is noted that there exists a window of\n\u0011denoted by vertical dashed line. For a small range of \u0011in\nthe right edge of window, the tsis minimal. It means that\nthereisaflexiblespacetochoose \u0011. Inaddition,theoptimal\n\u0011.Hshape/decreaseswith HshapeasshownintheFigure2(c).\nThese findings are useful in device application since it was\nreported that generating DCMP with higher chirp rate is a\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 3 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\nTable 1\nSimulated values of the controlling parameters for different samples of cross-sectional area A\nCross*sectional area Shape anisotropy field Natural frequency Simulated minimal frequency\nA(nm2) hshape(T) f0(GHz) f0(GHz)\nA1 0.09606 18.3 18.3\nA2 0.17718 16 17\nA3 0.2465 14.1 14.\nA4 0.3064 12.4 12.4\nA5 0.4049 9.6 10.2\nA6 0.4827 7.49 9\nA7 0.5457 5.72 7.5\nA8 0.5980 4.26 5.2\nchallenge [42]. Afterward, DCMP (with Hmw= 0:045,f0\ncorrespondingto Hshapeandoptimal\u0011(Hshape))drivenmag-\nnetization reversals are investigate and thus the time evolu-\ntions ofmzfor different HshapeorAare shown in the Fig-\nure 2(a). It is found that with the increase of Hshape, the\nmagnetization switching time tsremains almost same until\nA7= 2828 nm2but forA8g3232nm2, the mag-\nnetizationreversesverysmoothlyi.e., tsdropsto=0.34 ns\nwhichisclosetothetheoreticallimit(0.39 ns)[40]. So,itis\nmentionedthatthereisacritical Aandforlargervaluesthan\nit,theswitchingtimebecomessignificantlysmallerwhichis\nan ultimate demand in device application.\nTo understand the reason why fast reversal is obtained,\none can recall that the energy barrier is proportional to the\neffectiveanisotropyfield[39,40]. Sincethe Hshapeopposes\nthe anisotropy field Haniand thus reduces the height of the\nenergy barrier (energy difference between the initial state\nand saddle point). This reduction of energy barrier for dif-\nferentAispresentedintheFigure3(a). Therefore,themag-\nnetizationreversalbecomeseasierandfasterbecausethemi-\ncrowave frequency and magnetization precession frequency\nclosely matches as expected. To be more explicit, one may\nlookatthetrajectoriesofmagnetizationreversalfortwocases\nofAorHshapein Figure 3(b) and 3(c). For A0= 88\nnm2orHshape= 0, the magnetization takes a longer time\nbecause of some additional processions while crossing the\nhigher energy barrier. However, for A8= 3232 nm2\norHshape= 0:598T, the magnetization reversal is fairly\nsmoothi.e.,withoutanyadditionalprocessionssincetheen-\nergy barrier is reduced by Hshape. For more justification,\nFig4(a)showsthechangetheangle \b(blue)andmz(black\nline)withtimefor A8=3232 nm2. Notethattheangle \b\nremains almost 90oduring magnetization reversal which is\ndesiredfortriggeringanefficientmicrowaveenergyabsorp-\ntion (emission) by magnetization. Thus the corresponding\nenergychangingrate  \u000f(red),inFig4(c)isobtainedwithab-\nsorption and emission peaks (no distortion) which lead fast\nreversal. However, Fig 4(b), for A0= 88nm2i.e., for\nzeroHshape, shows the  \u000f(red line) absorption and emission\npeaks (distorted) indicating inefficient i.e., slower magneti-\nzation reversal (black line).\nThen,fortheDCMPwithfixed Hmw=0:045Tandop-\ntimal chirp rate \u0011.Hshape/, we obtain the fast magnetization\n/s40/s99/s41/s40/s98/s41/s40/s97/s41\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s45/s49/s48/s49\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s50/s55/s48/s45/s49/s56/s48/s45/s57/s48/s48/s57/s48/s49/s56/s48/s50/s55/s48/s70 /s32/s40/s100/s101/s103/s41\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52/s45/s49/s48/s49\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s49/s48/s49/s101 /s32/s40/s49/s48/s49/s53\n/s74/s47/s115/s32/s109/s51\n/s41/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s45/s49/s48/s49\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51/s101 /s32/s40/s49/s48/s49/s53\n/s74/s47/s115/s32/s109/s51\n/s41/s40/s97/s41Figure 4: (a) ForA8= 3232 nm2, the plot of the relative\nangle\bagainst time (blue line) and the time dependence of\nmz(black line). Plots of the energy changing rate  \u000fof mag-\nnetization against time (red line) and the time dependence of\nmz(black line) for the cross-sectional area (b) A0=88nm2\nand (c)A8=3232 nm2\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s53/s49/s48/s49/s53/s50/s48/s32/s102\n/s111/s32/s40/s84/s104/s101/s111/s114/s101/s116/s105/s99/s97/s108/s41\n/s32/s102\n/s111/s32/s40/s83/s105/s109/s117/s108/s97/s116/s101/s100/s41\n/s32/s72\n/s109/s119\n/s72\n/s115/s104/s97/s112/s101/s32/s40/s84/s41/s102\n/s111/s32/s40/s71/s72/s122/s41\n/s48/s46/s48/s50/s48/s48/s46/s48/s50/s52/s48/s46/s48/s50/s56/s48/s46/s48/s51/s50/s72\n/s109/s119/s32/s40/s84/s41\nFigure 5: The dependence of initial frequency foand min-\nimum required field amplitude Hmwas a function of shape\nanisotropyHshapefor switching time within 1 ns.\nreversal of different Afor a certain range of f0around each\nresonance_naturalfrequency. Thereexistaminimal f0,i.e.,\nminimalf0correspondingtoeachsample,forwhichswitch-\ning time is lowest. Figure 5 shows the simulated minimal\nfrequencyf0(bluecircles)asafunctionof Hshapeandfound\nthatthesimulated f0(bluecircles)ofDCMPdecreaseswith\nincreasingHshape. These can be explained as the Hshapere-\nduces the effective anisotropy Hk. For further justification,\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 4 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\n/s48/s46/s48/s48 /s48/s46/s48/s51 /s48/s46/s48/s54/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s116\n/s115/s40/s110/s115/s41\n/s97/s32/s32 /s65\n/s48/s32\n/s32/s32 /s65\n/s49/s32/s32\n/s32/s32 /s65\n/s50/s32/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52/s32\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120/s40/s99/s41/s40/s98/s41\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120/s40/s97/s41\nFigure 6: (a) Minimal switching time for the samples of dif-\nferentAas a function of Gilbert damping \u000b. Magnetization\nreversal trajectories of A3=1414 nm2for damping (b) \u000b=\n0.010 and (c) \u000b=0:017.\nthetheoreticallycalculatedfrequencies(blackline)areplot-\nted in same Figure 5 and noted that the decreasing trends\nare consistent as expected. Later on, by keeping the min-\nimalf0.Hshape/and optimal \u0011.Hshape/(i.e., corresponding\ntodifferentAorHshape)fixed,weinvestigatetheminimally\nrequiredamplitude Hmw(redsquare)asafunctionof Hshape.\nFigure5showsthat Hmwalsodecreaseswiththeincreaseof\nHshape. This is explained by the same reason of reducing\nthe height of the energy barrier with Hshapeas shown in the\nFigure 2(b). So even the smaller amplitude Hmwcan drive\nfaster magnetization reversal.\nTheGilbertdampingcoefficient \u000bisanimportantmate-\nrial parameter which has a significant effect on the magne-\ntization reversal process as well as switching time. In gen-\neral, damping has two influences. First, it hinders energy\nabsorption (cost more energy) during climbing the energy\nbarrier for magnetization reversal [50]. Second, after cross-\ning the energy barrier, it dissipates the energy of magne-\ntization promptly and thus it accelerates the magnetization\nto reach the ground _stable state [51, 52]. However, in the\nmagnetizationswitchingprocess,themagnetizationrequires\nto climb an energy barrier (originated from the anisotropy\nfield) and then goes to a stable state. Ideally, for the fast\nandenergy-efficientmagnetizationreversal,smaller(larger)\ndamping (\u000b) before (after) cross over the energy barrier is\nbetter. Therefore, this study emphasizes finding the optimal\n\u000bofthesampleofdifferent Aatwhichthefastestreversalis\nvalid. Accordingly, DCMP-driven magnetization as a func-\ntion of\u000bfor different Ais studied. Figure 6(a) shows the\nGilbert damping dependence of switching time for different\nA. It is found that for A0= 88nm2(uniaxial sample),\nthe range of \u000bis small but for finite shape anisotropy (biax-\nial sample), the range of \u000bis large which is useful in device\napplication. Moreover, for the sample of 1414nm2, the\nswitchingtimeislowest( ts=0:482ns)at\u000b=0:017. Tobe\nmore explicit, Figure 6(b) and 6(c) show the trajectories of\nmagnetization reversal for \u000b= 0:01and\u000b= 0:017respec-\ntively and observed that for \u000b= 0:017, the reversal path is\nshorter. This is happened because, after cross over the en-\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s72\n/s109/s119/s32/s40/s84/s41\n/s74 /s32/s40/s49/s48/s49/s50\n/s32/s65/s47/s109/s50\n/s41/s32/s32 /s65\n/s48/s32\n/s32/s32 /s65\n/s49/s32/s32\n/s32/s32 /s65\n/s50/s32/s32\n/s32/s32 /s65\n/s51/s32\n/s32/s32 /s65\n/s52/s32Figure 7: Phase diagram, for magnetization switching within\n1 ns, in terms of DCMP field amplitudes Hmwand current\ndensityJfor the samples of different A.\nergy barrier, the larger \u000breduces the magnetization preces-\nsion by dissipating the magnetization energy promptly and\nthus,itleadstofastreversal. Itisindicatedthatthereisaspe-\ncific sample that has an optimal damping situation at which\nthe magnetization reversal might be fastest.\nFormostdeviceapplication,thedesiredmicrowavepulse\n(DCMP) should be of smaller amplitude Hmw, smaller ini-\ntialfrequency foandlowchirprate \u0011,butwehaveobtained,\nforasampleof A=1616 nm2,minimalHmw=0:0278T,\noptimumfo=12:4GHz, and\u0011=40:672ns*2. In order to\nfurther reduce Hmw, we simultaneously apply a direct cur-\nrent such that it assists the magnetization reversal induced\nbyDCMP.ForDCMPcontribution,wetune Hmwwhilethe\nminimalfoandoptimal \u0011correspondingtodifferent Akept\nfixed. Figure 7 shows the Hmw*Jphase diagram for the\nsample of different Aunder the condition of magnetization\nreversal time 1 ns. As an example, for A= 1616 nm2,\nwithout DCMP, the required current to obtain fast switch-\ning gets very high i.e., J= 2:31012A_m2while without\ncurrenti.e.,J=0,theHmw=0:0278Tislargefrompracti-\ncal point of view. Therefore, with simultaneous applying of\nDCMPandcurrent,therequiredvaluesof HmwandJcould\nbe smaller than the above mentioned values (for individual\ncase)whichgivesalargescopeofpracticalrealizationofthis\nDCMP-driven magnetization reversal strategy.\n4. Discussion and Conclusions\nThisstudyinvestigatestheinfluenceofshapeanisotropy\nontheDCMP-drivenmagnetizationreversaltime tsandthe\nthreecontrollingparameters( Hmw,foand\u0011)ofDCMP,and\ndamping dependence of magnetization reversal. The shape\nanisotropy interestingly assists magnetization reversal since\nthe parameters of DCMP Hmw,foand\u0011decrease with the\nthe increase of shape anisotropy Hshape. For the sample of\nA8g3232nm2,tssignificantly reduces to 0.346 ns.\nThe minimal parameters of DCMP are estimated as Hmw=\n0:0187T,fo=5:2GHz,andoptimal \u0011=24:96ns*2which\nZ. K. Juthy et al.: Preprint submitted to Elsevier Page 5 of 7Shape anisotropy effect on magnetization reversal induced by linear down chirp pulse\nare useful in device application. These findings can be at-\ntributed to the shape anisotropy Hshapewhich reduces the\neffective anisotropy and thus reduces the height of the en-\nergybarrier. TheDCMPstimulatesefficientenergyabsorp-\ntion and emission during magnetization reversal. A recent\nstudy [43] reported that the temperature effect also assists\nthe magnetization reversal, i.e., it reduces the parameters of\nmicrowave field pulse. Therefore, it is desired that, at finite\ntemperature,theestimatedvaluesof Hmw,foand\u0011wouldbe\ndecreased further (which is beyond the scope of this study).\nThe result of damping dependence of magnetization rever-\nsal indicates that for a specific AorHshape, there exists an\noptimal damping situation. Moreover, it is also shown that\nthe required microwave field amplitude can be lowered by\napplying the direct current simultaneously. The usage of an\noptimumcombinationofbothmicrowavefieldpulseandcur-\nrent is suggested to achieve cost efficient and faster switch-\ning. So these findings may provide the knowledge to fabri-\ncate the shape of a single domain nanoparticle to generate\nthe magnetic data storage device.\n5. Acknowledgements\nThisworkwassupportedbytheMinistryofScienceand\nTechnology (Grant No. 440 EAS) and the Ministry of Edu-\ncation (BANBEIS, Grant No. SD2019972).\nReferences\n[1] S. Sun, C. B. Murray, D. Weller, L. Folks, A. Moser, Monodisperse\nFePt nanoparticles and ferromagnetic FePt nanocrystal superlattices,\nScience 287 (5460) (2000) 1989–1992.\n[2] S. Woods, J. Kirtley, S. Sun, R. Koch, Direct investigation of su-\nperparamagnetism in Co nanoparticle films, Physical Review Letters\n87 (13) (2001) 137205.\n[3] D. Zitoun, M. Respaud, M.-C. Fromen, M. J. 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Thonig2\n1Argonne National Laboratory, Lemont, IL 60439, United States\n2Department of Physics and Astronomy, Materials Theory, University Uppsala, SE-75120 Uppsala, Sweden\n3Peter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich & JARA, D-52425 J ulich, Germany\n4School of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden\n(Dated: February 21, 2018)\nBy using a multiscale approach based on \frst-principles density functional theory combined with\natomistic spin dynamics, we investigate the electronic structure and magnetization dynamics of an\ninverse Heusler and a Heusler compound and their alloys, i. e. Mn 2\u0000xZxCoAl and Mn 2\u0000xZxVAl,\nwhere Z= Mo, W, Os and Ru, respectively. A signature of the ferrimagnetic ordering of Mn 2CoAl\nand Mn 2VAl Heusler alloys is re\rected in the calculated Heisenberg exchange constants. They decay\nvery rapidly with the interatomic distance and have short range, which is a consequence of the\nexistence of the \fnite gap in the minority spin band. The calculated Gilbert damping parameter\nof both Mn 2CoAl and Mn 2VAl is high compared to other half-metals, but interestingly in the\nparticular case of the inverse Mn 2CoAl alloys and due to the spin-gapless semiconducting property,\nthe damping parameters decrease with the doping concentration in clear contradiction to the general\ntrend. Atomistic spin dynamics simulations predict ultrafast magnetisation switching in Mn 2CoAl\nand Mn 2VAl under the in\ruence of an external magnetic \feld, starting from a threshold \feld of\n2 T. Our overall \fnding extends with Heusler and inverse Heusler alloys, the class of materials that\nexhibits laser induced magnetic switching.\nI. INTRODUCTION\nThe \feld of the ultrafast magnetization dynamics has\nbecome one of the most important topics in magnetism,\nstarting from the pioneering experiment on ferromag-\nnetic nickel from Beaurepaire et al.1in 1996. Since\nthen, numerous experiments were carried out on 3 d\n(Fe2, Co3, Ni4,5), 4f(Tb and Gd6) ferromagnets, as\nwell as on several alloys (GdFeCo7{13, TbCo14, CoPt15)\nand half metallic systems (CrO 216, Co 2Cr0:6Fe0:4Al17,\nCo2FeSi, Co 2MnGe, Co 2FeAl18, and Co 2FexMn1\u0000xSi19\nand Co 2MnSi19,20) aiming to \fnd faster ways of manip-\nulating spins in a controllable way, opening a new \feld\nin the advanced information/data storage and data pro-\ncessing technologies.\nExperimental observations revealed that the charac-\nteristic demagnetization times of 3 delements are within\nthe 100 fs time scale, much faster than that of the 4 f-\nferromagnets, which show more complex behavior in-\nvolving a two-step demagnetization process in 10 ps time\nscale. Surprisingly, recent pump-probe experiments on\nhalf-metallic Heusler alloys measured distinguished and\ntypically larger all-optical switching times when com-\npared to 3d-ferromagnets8,19. In these materials, one of\nthe spin channels is completely or partially unoccupied\naround the Fermi energy, consecutively the magneto op-\ntical excitations from one channel to another channel are\nforbidden.\nAttempts to understand the momentum transfer be-\ntween the electrons, spins and phonons after a short\nlaser pulse have opened a new debate in the \feld. Sev-\neral quantitative models had been proposed to describe\nthe mechanism of the ultrafast demagnetization suchas the microscopic three-temperature model21, stochas-\ntic atomistic descriptions22, models using the stochas-\ntic Landau-Lifshitz-Bloch equation23,24and models sug-\ngesting the presence of di\u000busive or superdi\u000busive spin\ncurrents6,25{28. The \frst three models relate the spin-\nscattering to the Gilbert damping parameter, \u000b, that de-\nscribes the energy dissipation in a magnetic system via\nelementary spin-\rip processes29,30. Here, we combine the\nab initio description of the magnetic exchange interaction\nand Gilbert damping31{33parameter with the Landau-\nLifshitz-Gilbert equation to investigate the demagneti-\nzation process in half-metallic ferrimagnetic Heusler and\ninverse Heusler alloys.\na0\nAl\nV\nMn\na0\nAl\nCo\nMna) b)\nFIG. 1. (Color online) Schematic crystal structures of a)\nthe Heusler alloy Mn 2VAl and b) the inverse Heusler alloy\nMn2CoAl. Di\u000berent atom types are represented by di\u000berent\ncolours. Solid and dashed lines indicate the bond between the\natoms and are added to guide the eye. The lattice constant\na0is also indicated.arXiv:1802.07195v1  [cond-mat.mtrl-sci]  20 Feb 20182\nHeusler and inverse Heusler alloys are de\fned as\nternary intermetallic compounds with a composition of\nX2YT(cf. Fig. 1). Heusler alloys crystallize in the L2 1\nstructure (space group Fm \u00163m, 225), with the 4 a(0, 0, 0),\n4b(1\n2,1\n2,1\n2) and 8c(1\n4,1\n4,1\n4) Wycko\u000b positions. XandY\nare transition metals occupying the 8 cand 4apositions,\nrespectively, and Tis a main group III, IV or V element\nsitting in the 4 bposition. Inverse Heusler alloys adopt the\nHg2CuTi prototype structure (space group F \u001643m, 216),\nwith 4a(0,0,0), 4b(1\n2,1\n2,1\n2), 4c(1\n4,1\n4,1\n4) and 4d(3\n4,3\n4,\n3\n4) positions. In this case, XandYare transition metals,\nXoccupying the 4 aand 4dpositions while Yis the 4cpo-\nsition. The main element T sits in the 4 bposition. Both\nstructures may be regarded as a cubic unit cell, which\nconsists of four interpenetrating fcc sublattices. There\nare four atoms in the diagonal of the cube following the\nX-Y-X-Tsequence for Heusler alloys and X-X-Y-Tfor\nthe inverse Heuslers.\nHere, we study the demagnetization dynamics of a\nHeusler and an inverse Heusler compound and their al-\nloys, i.e. Mn 2\u0000xZxVAl and Mn 2\u0000xZxCoAl, where Z=\nMo, W, Os and Ru. Mn 2VAl is a well known half-metallic\nferrimagnetic Heusler compound34{39where the minority\nspin channel is the conducting one40. Mn 2CoAl adopts\nthe inverse Heusler structure41and it is predicted41and\ncon\frmed42to be a spin gapless magnetic semiconductor.\nThese peculiarities of the band structure are re\rected in\nthe Gilbert damping parameter and a\u000bect the magneti-\nsation dynamics under the in\ruence of a laser pulse, as\nwill be described below.\nThe paper is divided as follows: In Section II we intro-\nduce our numerical technique to study materials proper-\nties and magnetization dynamics in Heusler alloys. Elec-\ntronic and magnetic properties of the parent Heusler al-\nloys Mn 2CoAl and Mn 2VAl as well as doping of these\nmaterials with Os, Ru, W, and Mo is discussed in Sec-\ntion III A. Demagnetisation studies of these alloys caused\nby a femtosecond laser are described in Section III E. Fi-\nnally, the article concludes in Section IV with an outlook.\nII. METHODS\nA. Electronic structure calculation\nThe electronic and magnetic properties of the studied\nmaterials are obtained from \frst principle calculations\nby applying full-relativistic multiple scattering theory\nas formulated in the Korringa-Kohn-Rostocker (KKR)\napproach43. This method is implemented in the SPR-\nKKR package44,45. Solving the Dirac equation, relativis-\ntic e\u000bects are fully accounted for, especially the spin-orbit\ninteraction which is essential for heavy elements such as\nthe here considered dopants Os, W, Ru, and Mo. The\npotentials are treated by the atomic sphere approxima-\ntion (ASA) and obtained by self-consistently solving the\nKohn-Sham density functional theory (DFT) equation\nwithin the local density (LDA) or generalized gradientapproximation (PBE) as devised by Perdew, Burke and\nErnzerhof46,47. Note that we applied the PBE functional\nif not further speci\fed. The irreducible Brillouin zone is\nsampled by\u0019500 k-points. To describe substitutional\ndisorder in the sub-lattices of the alloys we make use\nof the coherent potential approximation (CPA)48. The\nspin-polarized scalar relativistic full-potential (SR-FP)\nmode49of the KKR approach is used to calculate the to-\ntal energies as a function of volume [ E(V)], which gives\nan estimate of the lattice constant a0.\nB. Calculation of Heisenberg exchange and Gilbert\ndamping\nThe angular momentum transfer in terms of Heisen-\nberg exchange interactions Jijand energy dissipation\nrelated to the Gilbert damping parameter \u000bis deter-\nmined by an ab-initio method with the aim to ad-\ndress the magnetic ground state and also the dynami-\ncal properties by using the Landau-Lifshitz-Gilbert equa-\ntion. The interatomic exchange interactions, Jij, were\ncalculated via the Liechtenstein-Katsnelson-Antropov-\nGubanov (LKAG) formalism50\nJij=1\n\u0019Z\"F\n\u00001Im Tr\u0010\n\u0001i\u001c\"\nij\u0001j\u001c#\nji\u0011\nd\": (1)\nwhere \u0001i=t\u00001\ni;\"\u0000t\u00001\ni;#is the spin-resolved di\u000berence of the\nsingle-site scattering matrix tiat siteiand\u001cijis the scat-\ntering path operator, describing the propagation of the\nelectrons between two sites iandj. The Fermi energy\nis denoted by \"F. Note that in CPA, the multiple scat-\ntering matrix is replaced by the scattering properties of\nthe e\u000bective medium ^ \u001ci\u0016;j\u0017 =Xi\u0016\u001cCPA\nijXj\u0017constructed\nfrom a defect of type \u0016;\u0017at sitei;j, respectively. The\ndefects are taken into account by the defect matrix Xi\u0016.\nFrom the calculated exchange interactions, it is possible\nto obtain the spin wave sti\u000bness, D, which is expressed\nas:51\nD= lim\n\u0011!02\n3X\nje\u0000\u0011jr0jj\na0J0jjrijj2(2)\nby using super cell calculation with random con\fgura-\ntions of the dopants in 12 ensembles and starting from\na reference site i= 0. The distance between site iand\njis given by rijand the parameter \u0011is introduced to\nguarantee convergence within a Pade interpolation ap-\nproximation.\nThe Gilbert damping parameter is identi\fed on the\nbasis of the linear response theory33by means of the\nmultiple scattering formalism52. The diagonal elements\n\u0016=x;y;Zof the Gilbert damping tensor can be written\nas33:\n\u000b\u0016\u0016=g\n\u0019mtotX\njTr\nT\u0016\n0~\u001c0jT\u0016\nj~\u001cj0\u000b\nc; (3)3\nwhere the e\u000bective g-factor g= 2(1 +morb=mspin) and\ntotal magnetic moment mtot=mspin+morbare given by\nthe spin and orbital moments, mspinandmorb, respec-\ntively, ascribed to a unit cell. Equation (3) gives \u000b\u0016\u0016for\nthe atomic cell at lattice site 0 and implies a summation\nover contributions from all sites indexed by j, including\nj= 0. Moreover, ~ \u001cijis related to the imaginary part of\nthe multiple scattering operator that is evaluated only at\nthe Fermi energy \"F. Finally,T\u0016\nirepresents the matrix\nelements of the torque operator ^T\u0016=\f\u001b\u0016Bxc(r). The\nnotationh:::icrepresents the con\fgurational average, in-\ncluding vertex corrections33derived by Butler53and ac-\ncounting for \fnite temperature using the alloy analogy\nmodel within CPA54.\nC. Atomistic spin dynamics\nThe evolution of atomistic spins in a thermal bath is\ndescribed by the Landau-Lifshitz-Gilbert (LLG) equa-\ntion55,56, where the dynamics of a magnetic moment is\nexpressed in terms of precession and damping:\ndmi(t)\ndt=\u0000\r\n(1 +\u000b2)\u0012\nmi(t)\u0002Bi(t)\n+\u000b\nmimi(t)\u0002(mi(t)\u0002Bi(t))\u0013\n:(4)\nHere\ris the gyromagnetic ratio, \u000brepresents the di-\nmensionless Gilbert damping constant, and mi=miei\nis an individual atomic moment on site i. The e\u000bective\nmagnetic \feld is given by Bi=\u0000@H\n@mi+bi, whereH=\n\u0000P\ni6=jJijei\u0001ejandbiis an stochastic \feld. The latter\ndescribes white noise ( hbi(t)\u0001bj(t0)i= 2D\u000eij\u000e(t\u0000t0)),\nwhere the \ructuation width is D=\u000bkBTs=\rm. Thus, the\nspin temperature Tsdirectly passes into LLG equation\nvia the stochastic magnetic \feld biand is obtained from\nsolving the two-temperature (2T) model57. The analyti-\ncal expression of this two temperature model reads,\nTs=T0+ (5)\n(TP\u0000T0)\u0002(1\u0000exp(\u0000t=\u001cinitial ))\u0002exp(\u0000t=\u001cfinal)+\n(TF\u0000T0)\u0002(1\u0000exp(\u0000t=\u001cfinal))\nwhereT0is the initial temperature of the system, TP\nis the peak temperature after the laser pulse is applied\nandTFis the \fnal temperature. Both the initial and\n\fnal temperature are set to 300 K, where the peak tem-\nperature is a parameter in the simulations. The time-\ndependent parameters \u001cinitial and\u001c\fnalare exponential\nparameters, \fxed by \u001cinitial = 10 fs and \u001c\fnal= 20 ps from\nRef. [58]. Note that both relaxation times are materials\nspeci\fc and kBis the Boltzmann constant.III. RESULTS AND DISCUSSION\nThis current section is divided in \fve parts. In the \frst\nand second part we discuss the electronic structure and\nthe magnetic moments, respectively, of pure and doped\nHeusler and inverse Heusler materials based on DFT-\noptimized lattice constants. The third part deals with\nthe Heisenberg interaction, spin wave sti\u000bness, as well\nas the ordering temperature. The Gilbert damping is\ndiscussed in part four. The last part focuses on the de-\nmagnetisation and reliable switching in Heusler materials\nbased on the LLG equation.\nA. Electronic structure calculations\nLattice parameters are estimated from total energy cal-\nculations, compared to Refs. [38] and [59], and listed in\nTable I. For undoped Mn 2CoAl and Mn 2VAl, we im-\nproved the theoretically predicted values used in Ref. [59]\nby 10% and they are closer to the experimentally mea-\nsured lattice constant. The improvement comes from tak-\ning into account the full-potential, which is known to im-\nprove lattice constants60. By doping Mn with 4d and\n5d metals Mo, Ru, W, and Os, we observe an expected\nincrease of the lattice constant with the concentration\nof the dopants, since the atomic radius of the dopant is\nlarger than the one of Mn. For Mn 2VAl, the increase\nof the lattice constant is substantially bigger ( \u00191% for\nx= 1% doping) than for Mn 2CoAl (\u00190:1% forx= 1%\ndoping).\nThermal switching within our classical atomistic model\nis completely determined by the Heisenberg exchange\nand the Gilbert damping of the system61, which are in\nturn identi\fed by the scattering-path matrices and the\nsingle-site scattering matrices of the Kohn-Sham prob-\nlem in Eqs. (1) and (3). Hence, we \frst have to ad-\ndress the electronic structure by means of the density of\nstates (DOS; Fig. 2). The here studied inverse Heusler\nMn2CoAl is known to be a spin gapless semiconductor,\nwhere an almost zero-width energy gap at the Fermi level\nexists in the majority-spin channel (the majority states\nare plotted with positive values and the minority spin\nstates with negative values) but a regular energy gap oc-\ncurs in the minority spin-channel (see inset in the bottom\npanel of Fig. 2). This was already reported, for example,\nin Ref. [59]. The density of states and, consequently,\nthe gap are sensitive to the applied exchange correlation\nfunctional. Using local density approximation (LDA),\nstates are shifted up in energy (not shown here) com-\npared to the PBE by about 10 meV and, consequently,\nno gap at the Fermi energy is observed. Note that the\no\u000bset of the energy from the real axis in Fig. 2 (the\nspectral width of the electron bands) is small and about\n1 meV, which causes sharp features in the DOS. A \fnite\nspectral width also gives rise to an overlap of the states\naround the Fermi level and `hide' the zero-width energy\ngap; a \fnite density of states at \"Fis observed. Bands4\n-120-80-4004080120Density of states (1/Ryd)\n-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1\nE−EF(Ryd)-120-80-4004080120\n-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1-120-80-4004080120\n-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1-120-80-4004080120\n-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1-120-80-4004080120\n-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1Mn 2CoAlDOS\n-0.02 0.0 0.02\nE−EF(Ryd)-0.02 0.0 0.02 -0.02 0.0 0.02 -0.02 0.0 0.02 -0.02 0.0 0.02\n-80-4004080Density of states (1/Ryd)\n-80-4004080\n-80-4004080\n-80-4004080\n-80-4004080W\nRu\nMo\nOs\nMn 2VAlDOS\nE−EF(Ryd)\nFIG. 2. (Color online) Density of states (DOS) for Mn 2CoAl\n(lower panel) and Mn 2VAl (upper panel) without doping(gray\nbackground) and with doping of W (red lines), Ru (blue lines),\nMo (green lines), and Os (orange lines). Positive (negative)\nDOS values correspond to the majority-(minority-) spin elec-\ntrons and are indicated by bold up-(down-) arrays. The inset\nis a magni\fcation of the DOS around the Fermi level.\nthat cross the Fermi level, are mainly allocated to Mn1\nand Co (band structure is not shown here, but it can be\nfound elsewhere41). Note that the superscripts 1 and 2\nbetween the two Mn atoms. In contrast to Ref. [59], the\nFermi energy is not located at the centre of the minority\nband gap, which will a\u000bect the coupling between the col-\nlective and single-electron excitations, i.e. the exchange\ninteractions.\nThe chemical compound Mn 2VAl, however, is half-\nmetallic (cf. Fig. 2) with a gap in the majority spin-\nchannel. The width of the majority band gap (0 :7 eV) is\nbigger than the minority spin gap in Mn 2CoAl (0:4 eV),\nwhich signi\fcantly a\u000bects the magnetic properties. In\nthe minority spin channel and at the Fermi energy Mn\nprojected states cause a strong peak in the DOS that hy-bridize with V atoms. States above the Fermi energy are\ndominated by the d-states of V atoms.\nThe spin-gapless semiconducting or half-metallic be-\nhaviour in Mn 2\u0000xZxCoAl and Mn 2\u0000xZxVAl is destroyed\nby replacing some of the Mn atoms with heavy metals, Z\n= Mo, W, Os, Ru of a given concentration x= 0:05 and\n0:1. Comparing total energies (not shown here) allows us\nto conclude that for the inverse Heusler Mn 2\u0000xZxCoAl\ndoping at both Mn-sites (Mn1-Mn2) has the lowest en-\nergy. We obtained a maximal energy di\u000berence of \u0001 E\u0019\n40 meV when doping at Mn1-Y, Mn2-Y, orYwith 1%\nof the dopants W, Ru, Mo, Os. There is no major vari-\nation found in \u0001 Ebetween the di\u000berent dopands. Note\nthat we used here the same lattice constant as shown in\nTable I, but in principle it will vary when doping at Mn1-\nMn2, Mn1-Y, Mn2-Y, orY. However, Mn 2\u0000xZxVAl has\nthe lowest energy when doping only the V atom, but to\ntreat both material on the same footing, we consider also\nMn2\u0000xZxVAl to be doped at the Mn1-Mn2atoms.\nIn the case of Mn 2CoAl, W and Mo generate states\nat the spin-gap majority states at the Fermi level, where\non the other hand the gap in the minority spin chan-\nnel survives. In terms of a rigid band model, W- and\nMo-doping decreases the Fermi energy, which relocates\nthe DOS to higher energies. The dopants Os and Ru\nhave one electron more than Mn in the valence band\nand, consequently, a\u000bect the density of states in the op-\nposite way: Minority states are added and become occu-\npied. The Fermi energy increases, which shifts the den-\nsity of states to smaller energies. For Mn 2VAl, doping\nwith Ru and Os preserves the half-metallic behaviour; it\nadd states below the Fermi energy and typically at the\nenergy\"=\u00000:025 Ryd. Doping with Mo and W reduces\nthe width of the band-gap and shifts it above the Fermi\nenergy. Related to the alloying, the density of states\nsmears out in the whole energy range.\nB. Magnetic moments\nThe exchange splitting in the DOS and, consequently,\nthe total magnetic moment is a\u000bected by doping (see\nFig. 3). Both Heusler materials are ferrimagnetic.\nAn antiferromagnetic coupling between the Mn atoms\nwas observed for the inverse Heusler alloy Mn 2CoAl\n(cf. Table I), caused by the inequivalence of the two\nMn atoms. These results are in good agreement with\nexperiments41,42and existing theoretical predictions59,62.\nAccording to the Bethe-Slater curve63, transition-metal\natoms such as Mn tend to have an antiferromagnetic\nspin moment when they are close to each other. In\nMn2\u0000xZxVAl, the Mn atoms are equivalent and, thus,\nhave the same magnetic moment that couple ferromag-\nnetically. The V atom, however, is antiferromagnetic\nwith respect to the Mn atoms and has a strong induced\nmagnetic moment of 0 :91\u0016B. Opposite to the total mag-\nnetic moment, the size of the element resolved magnetic\nmoments is sensitive to the lattice constant of the system5\n1.71.81.92.02.12.2m(µB)\n0.0 0.05 0.1\nxW\nRu\nMo\nOsMn 2−2xZ2xVAl\nMn 2−2xZ2xCoAl\nFIG. 3. (Color online) Total magnetic moments of\nMn2\u0000xZxCoAl (triangles) and Mn 2\u0000xZxVAl (circles) as a\nfunction of dopant concentration x. The symbol Z represents\nMo (green lines and symboles), Os (orange lines and symbols),\nRu (blue lines and symbols), and W (red lines and symbols).\nand moments can vary up to 13 %, which was also found\nin Ref. 59.\nThe size but not the sign of the elemental magnetic\nmoments changes by doping the Heusler materials with\n4d and 5d heavy metals, and, thus, also the total mag-\nnetic moment. Typically, the induced magnetic moments\nof dopants are parallel to the magnetic moment of Mn\natoms and they become larger if the magnetic moment\nof the Mn atom is smaller. In the case of Mn 2CoAl, the\ndopants W, Ru, Mo, and Os cause a decay of the to-\ntal magnetic moment of about 0 :1\u00000:2\u0016Bforx= 1%,\nwhile in the case of Mn 2VAl, only the dopants Ru and\nMo decrease the magnetic moment. This is caused by a\nsigni\fcant change of the Mn magnetic moments of about\n\u0001m\u00190:1\u00000:2\u0016B, but also for Co atoms the moment\nvariation is about \u0001 m\u00190:2\u0016B.\nC. Heisenberg exchange parameter and Curie\ntemperatures\nBased on our electronic structure analysis in the Sec-\ntion III B, we calculated the Heisenberg exchange param-\neterJij(see Fig. 4). The already revealed ferrimag-\nnetic behaviour is re\rected also in the exchange con-\nstantsJ. The magnetic exchange parameters decay very\nrapidly with the interatomic distance, rij, which is as-\ncribed to the existence of the \fnite spin gap in the\nminority-channel51,64. Our results for Mn 2CoAl are sim-\nilar to the ones already reported in Refs. [62,59]. Note\nthe factor of 2 in Ref. [59] may be caused by a di\u000berent\ndouble-counting convention of the Heisenberg Hamilto-\nnian. For the compound Mn 2CoAl, the antiferromagnetic\ninteraction between Mn1and Mn2dominates the ferri-\nmagnetism, whereas the Mn2-Co interatomic exchange\ninteraction is ferromagnetic. In Mn 2VAl, the situation is\nthe opposite: the Mn to V interaction is dominating and\nantiferromagnetic, where only the Mn1-Mn2contributes\n-2.5-2.0-1.5-1.0-0.50.00.51.01.5Jij(mRyd)\n0.4 0.8 1.2 1.6 2.0\nrij·a−1\n0Mn1-Mn1\nMn1-Mn2\nMn1-Co\nMn2-Mn2\nMn2-Co\n-0.8-0.6-0.4-0.20.00.20.4Jij(mRyd)\n0.4 0.8 1.2 1.6 2.0\nrij·a−1\n0Mn 1-Mn 1\nMn 1-Mn 2\nMn 1-V\nMn 2-Mn 2\nMn 2-VFIG. 4. (Color online) Intersublattice Heisenberg exchange\nparameter as a function of renormalized interatomic distance\nfor a) Mn 2VAl and b) Mn 2CoAl. Di\u000berent colours represents\nthe coupling between Mn1-Mn1(red dotes), Mn1-Mn2(blue\ndotes), Mn1-Co or Mn1-V (green dotes), Mn2-Mn2(orange\ndotes) and Mn2-Co or Mn2-V (cyan dotes).\nwith a ferromagnetic coupling but with half the strength\nof the Mn-V interaction. The coupling between equiv-\nalent Mn atoms in Mn 2VAl (Mn1-Mn1and Mn2-Mn2)\nis small and negligible. The calculated interactions de-\npend to some extent on the details of the calculations.\nIn particular, the JMn-CoandJMn-Vinteractions depend\nstrongly on the applied exchange-correlation functional,\nbut also on the lattice constant of the system. Notice that\nforJMn-CoandJMn-Vin LDA we obtain twice the size of\ntheJ's from PBE (not shown here). The other couplings\n(e.g.JMn-Al,JCo-Al,JV-Al) turned out to be negligible,\nprimarily caused by a vanishing magnetic moment on the\nAl atom.\nAs shown in Fig. 5, doping with 4d and 5d elements\nreduces nearest-neighbour interactions and the correla-\ntion length between magnetic moments, which is a direct\nconsequence of the disorder and the coherent potential\napproximation65. Nearest neighbour interactions are af-\nfected mostly by the doping. In general, the exchange6\nCompound a0(\u0017A)m[Mn1]m[Z1]m[Mn2]m[Z2]m[Y]\nMn2CoAl 5 :73 [ 59] \u00001:64 2 :77 0 :93\nMn1:8W0:2CoAl \u00001:52\u00000:52 2:75 0:26 0:78\nMn1:8Ru0:2CoAl \u00001:62\u00000:10 2:76 0:06 0:91\nMn1:8Mo0:2CoAl \u00001:53\u00000:56 2:75 0:33 0:78\nMn1:8Os0:2CoAl \u00001:56\u00000:12 2:76 0:18 0:92\nMn2VAl 5 :69 [ 38] 1 :32 1 :32 \u00000:66\nMn1:8W0:2VAl 1 :32 0:19 1:32 0:19\u00000:57\nMn1:8Ru0:2VAl 1 :31 0:08 1:31 0:08\u00000:61\nMn1:8Mo0:2VAl 1 :36 0:25 1:36 0:25\u00000:58\nMn1:8Os0:2VAl 1 :31 0:09 1:31 0:09\u00000:60\nMn2CoAl 5:79 [5:84 exp] \u00001:81 2 :91 0 :96\nMn1:8W0:2CoAl 5 :79 \u00001:37\u00000:46 2:62 0:22 0:77\nMn1:8Ru0:2CoAl 5 :79 \u00001:81\u00000:10 2:90 0:07 0:96\nMn1:8Mo0:2CoAl 5 :79 \u00001:71\u00000:60 2:89 0:39 0:82\nMn1:8Os0:2CoAl 5 :80 \u00001:80\u00000:12 2:92 0:19 0:98\nMn2VAl 5:84 [5:88 exp] 1:47 1 :47 \u00000:91\nMn1:8W0:2VAl 5 :92 1:73 0:37 1:73 0:37\u00000:99\nMn1:8Ru0:2VAl 5 :86 1:50 0:05 1:50 0:05\u00000:89\nMn1:8Mo0:2VAl 5 :91 1:72 0:45 1:72 0:45\u00000:98\nMn1:8Os0:2VAl 5 :92 1:52 0:06 1:52 0:06\u00000:91\nTABLE I. Lattice constant and atom resolved magnetic moments (in \u0016B) of the host Mn 2CoAl and Mn 2VAl . The upper panel\nshows results for a \fxed lattice constant obtained from literature, where the lower panel is for lattice constants calculated from\ntotal energy minimization. The superscripts 1 and 2 distinguish between the two Mn atoms. The symbol Yrepresents either\nCo or V. The magnetic moment of Al is negligibly small.\ncouplings diminish with doping concentration xup to\n0:6 mRyd for W and x= 0:1. For Os and Ru doping,\nthere is a slight increase of the exchange coupling (about\n0:03 mRyd).\nWith knowledge about the trends in the exchange cou-\nplingsfJg, one can estimate the spin-wave sti\u000bness D\nand the phase transition temperature from both mean\n\feld theory via kBTMF\nC =3=2P\njJ0jor from Monte\nCarlo simulations. The results are shown in Fig. 6. The\nspin-wave sti\u000bness (upper panel in Fig. 6) for Mn 2\u0000xCoAl\nis in good agreement with Ref. [59], while for Mn 2\u0000xVAl\nwe reproduce the spin wave sti\u000bness constant Dalready\nreported in Ref. [66] ( D= 324 meV \u0017A2), but not the ex-\nperimentally measured sti\u000bness67(D= 534 meV \u0017A2).\nFor the Co based Heusler compounds we obtain a hard-\nening of the spin-waves after an initial softening, where\nfor the V based Heusler compound, only hardening of\nthe spin-waves with doping is observed. The phase tran-\nsition temperature TC, which turns out to be inversely\nproportional to D, decreases with doping concentration\nxfor two reasons, namely: i)reduction of the magnetic\nmoment due to doping and, consequently, stronger \ruc-tuations at a given temperature as well as ii)reduction of\ncorrelation. The critical temperature TCis obtained from\nMonte Carlo simulations on the Metropolis algorithm68,\nfrom Binder's fourth cumulant68for di\u000berent simulated\nsystem sizes but also from the spin susceptibility \u001f. Note\nthat the \frst method could fail for antiferro- and ferri-\nmagnets. Thus, we obtain a systematic error of about\n\u00065 K.\nOur simulations of ordering temperature (680 K for\nMn2CoAl and 475 K for Mn 2VAl) underestimate the\ntransition temperature observed from experiment (720 K\nfor Mn 2CoAl42and 768 K for Mn 2VAl). This discrep-\nancy that is most notable for Mn 2\u0000xVAl was reported\nearlier66and could have multiple reasons. First, magnetic\nproperties in Heusler alloys are sensitive to the intersti-\ntial region spanned by the mu\u000en tin potential. Thus,\nfull-potential simulations are required as it was shown in\nRefs. [41, 42, and 69]. Also the results depend crucially on\nthe choice of the exchange-correlation functional and on\nelectron correlations e.g. addressed by including a Hub-\nbardU66. Second, the Heisenberg exchange is calculated\nfor a collinear ferrimagnetic state but when the magnetic7\n-2.0-1.5-1.0-0.50.00.51.0Jij(mRyd)\n0.4 0.8 1.2 1.6\nrij·a−1\n0W\nMn1-Mn1\nMn1-Mn2\nMn1-Co\nMn2-Mn2\nMn2-Co\n0.4 0.8 1.2 1.6 2.0\nrij·a−1\n0Ru-2.0-1.5-1.0-0.50.00.51.0Jij(mRyd)Mo Os\n-1.0-0.8-0.6-0.4-0.20.00.2Jij(mRyd)\n0.4 0.8 1.2 1.6\nrij·a−1\n0W\nMn 1-Mn 1,Mn 2-Mn 2\nMn 1-Mn 2\nMn 1-V,Mn 2-V\nMn 1-X1\nMn 1-X2\n0.4 0.8 1.2 1.6 2.0\nrij·a−1\n0Ru-1.0-0.8-0.6-0.4-0.20.00.20.4Jij(mRyd)Mo Os\nFIG. 5. (Color online) Intersublattice Heisenberg exchange\nparameter as a function of renormalized interatomic distance\nfor a) Mn 2\u0000xZxVAl and b) Mn 2\u0000xZxCoAl, where the di\u000ber-\nent subpanels show the dopants W (bottom left), Ru (bottom\nright), Mo (top left), and Os (top right). Di\u000berent colours\nrepresents the coupling between Mn1-Mn1(red dotes), Mn1-\nMn2(blue dotes), Mn1-Co or Mn1-V (green dotes), Mn2-Mn2\n(orange dotes) and Mn2-Co or Mn2-V (cyan dotes).\ndisorder is taken into account in the electronic structure,\nusually the exchange interaction is biased65. Based on\nthe alloy analogy model54, we modelled also the temper-\nature stability of the magnetic properties (magnetic mo-\nments and magnetic exchange) coming from electronic\nstructure by the partial disordered local moment (DLM)\n600700800900100011001200TMF\nC(K)\n0.0 0.05 0.1\nxW\nRu\nMo\nOsMn 2−2xX2xVAl\nMn 2−2xX2xCoAl450500550600650TC(K)250300350400450D(meV ˚A2)FIG. 6. (Color online) Spin wave sti\u000bness D, critical tem-\nperaturesTC, and mean \feld critical temperatures TMF\nCof\nMn2\u0000xZxCoAl (triangles) and Mn 2\u0000xZxVAl (circles) as a\nfunction of dopand concentration x. Dopands are Mn (black\ncircles), Os (red squares), Ru (green diamonds), and W (or-\nange triangles).\napproximation within the Ising model65. DLM approach\nis believed to accurately describe `spin temperature' in\nthe electronic structure70. However, it turned out that\nthe disordered local moment theory can not be applied\nto both Heusler and inverse Heusler for similar reasons as\nfor Ni71: the magnetic moments in Al and Co/V disap-\npear. For Mn 2CoAl, our simulations show furthermore\nthat the magnetic moment of the Mn2atom is zero in\nthe paramagnetic phase and, consequently, the magnetic\nexchange and the phase transition temperature are zero.\nThis result is independent of the doping with 4d and 5d\nelements. These results indicate the inconsistency of the\nDLM model for Heusler materials. It is still an open\nquestion, if results get improved by applying relativis-\ntic DLM theory72. Third, we consider only a simpli\fed\napproach for electron correlation in the LDA and GGA\ndensity functional. However, it is known66that improved\nmodels for electron correlation have the trend to increase\nslightly the phase transition temperature.\nD. Gilbert damping\nPrevious studies61have shown that Gilbert damping is\na crucial parameter in the ultrafast switching procedure\nand, thus, call for ab-initio footing. Figure 7 shows the8\nGilbert damping \u000bas a function xatT= 300 K. Note\nthat for these calculations both lattice and magnetic \ruc-\ntuations terms are considered, where the magnetic \ructu-\nations are assumed from a linear correlation between the\nmagnetization and the temperature. This could result in\nerrors, in particular at high temperatures.\n012345α·10−3\n0.00.40.81.21.6\nn(EF) (eV)\n0.0 0.05 0.1\nxMn 2−2xX2xCoAl2345678910α·10−3\n2.83.23.64.04.4\nn(EF) (eV)Mn 2−2xX2xVAl\nFIG. 7. (Color online) Gilbert damping parameters \u000b\n(solid lines) and density of states at the Fermi level n(EF)\n(dotted lines) of Mn 2\u0000xZxCoAl (triangular symboles) and\nMn2\u0000xZxVAl (circle symboles) vs dopand concentration x.\nDopands are W (red color), Ru (blue color), Mo (green color),\nand Os (orange color).\nThe Gilbert damping of both undoped Heusler materi-\nals (Mn 2CoAl:\u000b= 0:0030, Mn 2VAl:\u000b= 0:0029) is high\ncompared to other half-metals reported, e.g., in Ref. [66]\nor low-damping alloys like Fe 0:75Co0:2573. The trends of\nthe Gilbert damping parameters with dopant concentra-\ntion are di\u000berent for Heusler and inverse Heusler materi-\nals. In Mn 2\u0000xZxVAl, doping leads to an increase of the\ndamping with x, except for the case of Ru. The slope\nof\u000bversus concentration xfollows the general increase\nof the total density of states at the Fermi level as it is\nproposed in Refs. [33, 73, and 74], but not linear to it.\nThis non-linearity was already observed for Heusler ma-\nterials in Ref. [66] or doped permalloy with the heavy\n4d and 5d elements used here75. The observed damping\n\u000bis di\u000berent from zero, however, small. This is in line\nwith the theory proposed in Ref. [74], in which damping\nis proportional to the product of the spin-polarised DOS\nand, consequently \u000b\u00190. The increase of damping can be\nalso understood in terms of the Kambersk\u0013 y model76,77:\nAlloying broadens the electron bands and more spin-\rip\ntransitions between the electron states occur. This is\ntrue only, if interband transitions are already dominat-\ning. In the inverse Heusler material Mn 2CoAl we even\n\fnd a decrease with x. This is due to the spin-gaplesssemiconducting behaviour (cf. Fig. 2): Only a low num-\nber of states exist at the Fermi energy, making interband\ntransitions unlikely. The damping is dominated by intra-\nband transitions, that tend to decrease with very small\nx. With increasing x, however, states appear within the\ngap and interband transition are preferred. Thus, a small\nincrease with even higher concentration is expected and\nobserved. However, not only the number of states at the\nFermi energy and the spectral width of the states con-\ntribute to the damping, but also the spin-orbit coupling\n(SOC), the Land\u0013 e factor, and the saturation magnetiza-\ntion a\u000bect the damping parameter. Since we dope with\nrather heavy elements W, Mo, Ru, and Os, spin orbit\ncoupling strongly contributes to the variation of damp-\ning with concentration x: the higher the `mass' of the\ndopant atom (W and Os compared to Ru and Mo) is,\nthe higher is the damping parameter.\nAfter we addressed all relevant parameters for the sim-\nulation based on the Landau-Lifshitz-Gilbert equation,\nwe are able to perform ultrafast switching calculations.\nE. Ultrafast switching\n-1.0-0.50.00.51.0M/M s\n0 5 10 15 20\nt (ps)V\nMn1\nMn2-1.0-0.50.00.51.0M/M sCo\nMn1\nMn2\nFIG. 8. (Color online) Ultrafast switching behaviour of\nMn2CoAl (upper panel) and Mn 2VAl (lower panel). The de-\nmagnetization is shown element resolved (blue and green lines\n- Mn atoms, red line - Co/ V atom). The peak temperature is\n600 K for Mn 2VAl and 900 K for Mn 2CoAl. The external mag-\nnetic \feld is B= 2:5 T and damping parameter is \u000b= 0:009.\nThe arrow indicates the crossing point at where the switching\ntakes place.9\n7007508008509009501000TP(K)\n0 2 4 6 8 10\nB(T)α= 0.006\n0 2 4 6 8 10\nB(T)α= 0.009\n0.02.04.06.08.0\nt(ps)\n450500550600650700750TP(K)\n0 2 4 6 8 10\nB(T)α= 0.006\n0 2 4 6 8 10\nB(T)α= 0.009\n0.02.04.06.08.0\nt(ps)b)a)\nFIG. 9. (Color online) Thermal switching phase diagram for\ndi\u000berent damping parameter (0.006 and 0.009) in a) Mn 2CoAl\nand b) Mn 2VAl. The peak temperature is represented versus\nthe strength of the external magnetic \feld. The colour scale\n(fast switching - blue colour, slow switching - red colour) rep-\nresents the time in units of ps where the switching (indicated\nby an arrow in Fig. (8)) takes place. No switching is repre-\nsented by the black background.\nIn order to study the ultrafast switching process in\nHeusler alloys we combined the two temperature model\nwith an atomistic spin dynamics code78. Here, we consid-\nered a very long thermal pulse of 20 ps with di\u000berent peak\ntemperatures TP. Typical timescales of the ultrafast de-\nmagnetization and remagnetization process for Mn 2VAl\nand Mn 2CoAl are in the orders of picoseconds (1 \u00005 ps)\n(see Fig. 8). The time scales are mainly dictated by the\nGilbert damping \u000b, which is varied in our studies between\n0:003, 0:006, and 0:009, but can depend on the Heisen-\nberg exchange12. As demonstrated above, these damping\nvalues are achievable by doping the `pure' Heusler mate-\nrials. There is only a slight shift observable in the demag-\nnetization time of each individual element in Mn 2CoAl,\nwhere for Mn 2VAl, it is not. After demagnetization, the\nHeusler material undergoes reliable switching only when\nan external magnetic \feld induced by the pump-pulse is\npresent. Thus, three parameters | damping, peak tem-\nperature and pulse induced external magnetic \feld |\nspan a phase space for observing reliable switching, as\nshown in Fig. 9.\nWe did not observe any magnetic switching for both\nHeusler materials with \u000b= 0:003 (data not shown here).\nTypically for certain threshold peak temperatures TP\nabove the magnetic phase transition temperature ( TC=\n700 K for Mn 2CoAl andTC= 475 K for Mn 2VAl) switch-\ning occurs. The peak temperature can be tuned by the\nlaser intensity and the pulse duration. The presence of\nan e\u000bective magnetic \feld during pumping is discussedin literature79,80. It was argued that the electric \feld of\nthe pump pulse induces a strong material speci\fc mag-\nnetic \feld of 10\u0000100 T. Even below but above certain\nminimum magnetic \feld of 1 \u00002 T, we observed reliable\nswitching. This threshold magnetic \feld as well as the\nswitching time (indicated by reduced contrast in Fig. 9)\ndecreases with increasing damping. The time when the\nswitching occurs (crossing point in Fig. 8 and colour scale\nin Fig. 9) typically passes a maximum at certain and de-\ncreases for larger peak temperatures. However, there is\nalso a minimum switching time of around 2 \u00003 ps, con-\ntrolled by the demagnetization rate. Note that due to the\ndi\u000berent spin polarization and resulting di\u000berent atomic\nmagnetic moments and magnetic states, an asymmetry\nin the phase diagram between Mn 2CoAl and Mn 2VAl oc-\ncurs.\nNevertheless, our approach has certain limitations. For\ninstance, we explicitly neglect the electronic motion and\ne\u000bects like super di\u000busion or spin-\rip scattering, as dis-\ncussed in Ref.25. We also assume the damping to be `spin-\nand phonon-temperature' independent. This is a rough\napproximation, in particular, due to the important role of\nphonons in the demagnetization process (e.g. Ref. [81])\nand for energy dissipation in magnetic systems33. Fur-\nthermore, we neglect the change of the magnetic ex-\nchange interaction with temperature, although magnetic\nmoments of Co and V atoms vanish in the DLM approx-\nimation. This behaviour in the disordered local moment\ntheory is well studied71and occurs also for Ni atoms. But\nwe have shown elsewhere58but also others82{84, that our\nmethodology is applicable for demagnetization in bulk\nbcc Fe and hcp Co compounds and, likely, for the Heusler\nmaterials studied here. We also neglect possible struc-\ntural phase transition to A2 or B2 disorder during de-\nmagnetization.\nIV. CONCLUSION\nWe have demonstrated thermal switching in Heusler\nand inverse Heusler materials making use of magnetic\n\feld pulse induced by the pump-pulse. We found a sensi-\ntive dependence of the possible switching and the switch-\ning time on the magnetic \feld pulse strength, the peak\ntemperature in the e\u000bective two-temperature model as\nwell as intrinsic materials properties, say the Heisenberg\nexchange and the Gilbert damping parameter. We have\nshown that the latter can be tuned by doping heavy ele-\nments, say W, Mo, Ru, Os, to both, higher and lower\ndamping values, especially in the case of spin-gapless\nsemiconductor. This calls for further investigations on\nother spin-gapless semiconductor59, aiming for tuning the\nGilbert damping to very low values, which may enable in-\nteresting spintronic and magnonic applications85. Within\nour methodology, we could reproduce exchange parame-\nter and, consequently, phase transition temperatures re-\nported in literature59. Our overall \fnding extends with\nHeusler and inverse Heusler alloys the class of materials10\nthat exhibits laser induced magnetic switching and calls\nfor future theoretical and experimental studies.V. ACKNOWLEDGEMENT\nWe acknowledge \fnancial support from the Swedish\nResearch Council. O.E. and and E.K.D.-Cz. ac-\nknowledged support from KAW (projects 2013.0020\nand 2012.0031) as well as acknowledges eSSENCE and\nSTandUP. The calculations were performed at NSC\n(Link oping University, Sweden) under a SNAC project.\n1E. Beaurepaire, J. C. Merle, A. Daunois, and J. Y. Bigot,\nPhys. Rev. 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China \n3IBM Almaden Research Center, San Jose, California 95120, USA \n4Max Planck Institute for Microstructu re Physics, 06120 Halle (Saale), Germany \n5Department of Physics and Astronomy, Univers ity of California, Riverside, California 92521, \nUSA \n†These authors contributed equally to the work \n*Correspondence to be addressed to: jing.shi @ucr.edu (J.S.) and weihan@pku.edu.cn (W.H.) \n \n \nAbstract \nThe Gilbert damping of ferromagnetic materials is arguably the most important but least \nunderstood phenomenological parameter that dictates real-time magnetization dynamics. \nUnderstanding the physical origin of  the Gilbert damping is highly relevant to developing future \nfast switching spintronics devices such as magnetic sensors and magnetic random access memory. Here, we report an experimental stud y of temperature-dependent Gilbert damping in \npermalloy (Py) thin films of varying thicknesses  by ferromagnetic resonance. From the thickness \ndependence, two independent cont ributions to the Gilbert damping are identified, namely bulk \ndamping and surface damping. Of particular inte rest, bulk damping decreases monotonically as \nthe temperature decreases, while surface da mping shows an enhancement peak at the 2 temperature of ~50 K. These results provide an important insight to the physical origin of the \nGilbert damping in ultr athin magnetic films.  \n \nIntroduction \nIt is well known that the magnetization dynamics  is described by the Landau-Lifshitz-Gilbert \nequation with a phenomenological parameter called the Gilbert damping ( α),1,2:   \n eff\nSdM dMMH Mdt M dtαγ=− × + × \n   (1) \nwhere M\nis the magnetization vector,  γis the gyromagne tic ratio, and SM M=\n is the saturation \nmagnetization. Despite intense theore tical and experimental efforts3-15, the microscopic origin of \nthe damping in ferromagnetic (FM) metallic ma terials is still not well understood. Using FM \nmetals as an example, vanadium doping decreases the Gilbert damping of Fe3 while many other \nrare-earth metals doping increase s the damping of permalloy (Py)4-6,16. Theoretically, several \nmodels have been developed to explain some  key characteristics. For example, spin-orbit \ncoupling is proposed to be the intrinsic or igin for homogenous time-varying magnetization9. The \ns-d exchange scattering model assumes that damp ing results from scattering of the conducting \nspin polarized electrons with the magnetization10. Besides, there is the Fermi surface breathing \nmodel taking account of the spin scattering with  the lattice defects ba sed on the Fermi golden \nrule11,12. Furthermore, other damping mechanisms in clude electron-electron scattering, electron-\nimpurity scattering13 and spin pumping into the adjacent nonmagnetic layers14, as well as the two \nmagnon scattering model, which refers to that pa irs of magnon are scatte red by defects, and the \nferromagnetic resonance (FMR) mode  moves into short wavelength spin waves, leading to a 3 dephasing contribution to the linewidth15. In magnetic nanostructu res, the magnetization \ndynamics is dictated by the Gilbert damping of the FM materials which can be simulated by \nmicromagnetics given the boundaries and dimens ions of the nanostructures. Therefore, \nunderstanding the Gilbert damping in FM materials is particularly important for characterizing \nand controlling ultrafast responses in magnetic  nanostructures that ar e highly relevant to \nspintronic applications such as  magne tic sensors and magnetic random access memory17.  \nIn this letter, we report an expe rimental investigation of the G ilbert damping in Py thin films \nvia variable temperature FMR in a modified  multi-functional insert  of physical property \nmeasurement system with a coplanar waveguide (see methods for details). We choose Py thin \nfilms since it is an interesting FM metallic material for spintronics due to its high permeability, nearly zero magnetostriction, low coercivity, a nd very large anisotropi c magnetoresistance. In \nour study, Py thin films are gr own on top of ~25 nm SiO\n2/Si substrates with a thickness ( d) range \nof 3-50 nm by magnetron sputtering (see methods  for details). A capping layer of TaN or Al 2O3 \nis used to prevent oxidation of the Py during m easurement. Interestingly, we observe that the \nGilbert damping of the thin Py films ( d <= 10 nm) shows an enhanced peak at ~ 50 K, while \nthicker films ( d >= 20 nm) decreases monotonically as the temperature decreases. The distinct \nlow-temperature behavior in the Gilbert dampi ng in different thickness regimes indicates a \npronounced surface contribution in the thin limit. In  fact, from the linear  relationship of the \nGilbert damping as a function of the 1/ d, we identify two contribu tions, namely bulk damping \nand surface damping. Interestingl y, these two contributions show  very different temperature \ndependent behaviors, in whic h the bulk damping decreases m onotonically as the temperature \ndecreases, while the surface damping indicates an  enhancement peak at ~ 50 K. We also notice \nthat the effective magnetization sh ows an increase at the same temperature of ~50 K for 3 and 5 4 nm Py films.  These observations could be all related to the magnetization reorientation on the \nPy surface at a certain temperatur e. Our results are important for theoretical investigation of the \nphysical origins of Gilbert damping and also us eful for the purpose of designing fast switching \nspintronics devices.  \nResults and Discussion \nFigure 1a shows five representative curves  of the forward amplitude of the complex \ntransmission coefficients (S 21) vs. in plane magnetic field meas ured on the 30 nm Py film with \nTaN capping at the frequencies of 4, 6, 8, 10 an d 12 GHz and at 300 K after renormalization by \nsubtracting a constant background. These experiment al results could be fitted using the Lorentz \nequation18: \n 2\n21 0 22()\n() ( )resHSSHH HΔ∝Δ+ −  (2) \nwhere S0 is the constant describing the coefficient for the transmitted microwave power, H is the \nexternal magnetic field, Hres is the magnetic field under the resonance condition, and ΔH is the \nhalf linewidth. The extracted ΔH vs. the excitation frequency ( f) is summarized in Figures 1b and \n1c for the temperature of 300 K and 5 K respect ively. The Gilbert damp ing could be obtained \nfrom the linearly fitted curves (red lin es), based on the following equation:  \n 02()H fHπαγΔ= + Δ   (3) \nin which γ is the geomagnetic ratio and ΔH0 is related to the inhom ogeneous properties of the \nPy films. The Gilbert damping at 300 K and 5 K is calculated to be 0.0064 ± 0.0001 and 0.0055 \n± 0.0001 respectively.  5 The temperature dependence of the Gilbert damp ing for 3-50 nm Py films with TaN capping \nlayer is summarized in Figure 2a. As d decreases, the Gilbert damping increases, indicative of \nthe increasing importance of the film surfaces. Interestingly, fo r thicker Py films (e.g. 30 nm), \nthe damping decreases monotonically as the temper ature decreases, which is expected for bulk \nmaterials due to suppressed sca ttering at low temperature. As d decreases down to 10 nm, an \nenhanced peak of the damping is obser ved at the temperature of ~ 50 K. As d decreases further, \nthe peak of the damping becomes more pronounce d. For the 3 nm Py film, the damping shows a \nslight decrease first from 0.0126 ± 0.0001 at 3 00 K to 0.0121 ± 0.0001 at 175 K, and a giant \nenhancement up to 0.0142 ± 0.0001 at 50 K, and then a sharp decrease back down to 0.0114 ± \n0.0003 at 5 K.  \nThe Gilbert damping as a function of the Py th icknesses at each temperature is also studied. \nFigure 2b shows the thickness dependence of the Py damping at 300 K. As d increases, the \nGilbert damping decreases, which indicates a surface/interface enhanced damping for thin Py \nfilms19. To separate the damping due to the bul k and the surface/interface contribution, the \ndamping is plotted as a function of 1/ d, as shown in Figure 2c, and it follows this equation as \nsuggested by theories19-21. \n 1()BSdαα α=+   (4) \nin which the Bα and Sα represent the bulk and surface da mping, respectively. From these \nlinearly fitted curves, we are able to separate  the bulk damping term and the surface damping \nterm out. In Figure 2b, the best fitted parameters for Bα and Sα are 0.0055 ± 0.0003 and 0.020 ± \n0.002 nm. To be noted, there are two insulating mate rials adjacent to the Py films in our studies. 6 This is very different from previous studies on Py/Pt bilayer systems, where the spin pumping \ninto Pt leads to an enhanced magnetic dampi ng in Py. Hence, the enhanced damping in our \nstudies is very unlikely resulti ng from spin pumping into SiO 2 or TaN. To our knowledge, this \nsurface damping could be related to  interfacial spin f lip scattering at the interface between Py \nand the insulating layers, which ha s been included in a generalized  spin-pumping theory reported \nrecently21. \nThe temperature dependence of the bulk damp ing and the surface damping are summarized \nin Figures 3a and 3b. The bulk damping of Py is  ~0.0055 at 300 K. As the temperature decreases, \nit shows a monotonic decrea se and is down to ~0.0049 at 5 K. Th ese values are consistent with \ntheoretical first principle calculations21-23 and the experimental valu es (0.004-0.008) reported for \nPy films with d ≥ 30 nm24-27. The temperature dependence of the bulk damping could be \nattributed to the magnetization rela xation due to the spin-lattice scattering in the Py films, which \ndecreases as the temperature decreases. \nOf particular interest, the surface damping sh ows a completely different characteristic, \nindicating a totally different mechanism from th e bulk damping. A strong enhancement peak is \nobserved at ~ 50 K for the surface damping. Could this enhancement of this surface/interface \ndamping be due to the strong spin-orbit coupli ng in atomic Ta of Ta N capping layer? To \ninvestigate this, we measure the damping of the 5 nm and 30 nm Py films with Al 2O3 capping \nlayer, which is expected to exhibit much lo wer spin-orbit coupling compared to TaN. The \ntemperature dependence of the Py damping is su mmarized in Figures 4a and 4b. Interestingly, \nthe similar enhancement of the damping at ~ 50 K is observed for 5 nm Py film with either Al 2O3 \ncapping layer or TaN layer, whic h excludes that the origin of the feature of the enhanced 7 damping at ~50 K results from th e strong spin-orbit coupling in TaN layer. These results also \nindicate that the mechanism of this feature is most likely related to the common properties of Py \nwith TaN and Al 2O3 capping layers, such as the crysta lline grain boundary and roughness of the \nPy films, etc. \nOne possible mechanism for the observed peak of  the damping at ~50 K could be related to a \nthermally induced spin reorientation transition  on the Py surface at that temperature. For \nexample, it has been show n that the spin reorientation of Py  in magnetic tunnel junction structure \nhappens due to the competition of different magne tic anisotropies, which c ould give rise to the \npeak of the FMR linewidth around the temperature of ~60 K28. Furthermore, we measure the \neffective magnetization ( Meff) as a function of temperature. Meff is obtained from the resonance \nfrequencies ( fres) vs. the external magnetic field via the Kittel formula29:  \n 12() [ ( 4 ) ]2res res res efffH H Mγππ=+   (4) \nin which Hres is the magnetic field at the resonance condition, and Meff is the effective \nmagnetization which contains the saturation ma gnetization and other anisotropy contributions. \nAs shown in Figures 5a and 5b, the 4π*M eff for 30 nm Py films w ith TaN capping layer are \nobtained to be ~10.4 and ~10.9 kG at 300 K and 5 K respectively. The temperature dependences \nof the 4π*M eff for 3nm, 5 nm, and 30 nm Py films are s hown in Figures 6a-6c. Around ~50 K, an \nanomaly in the effective magnetization for thin Py films (3 and 5 nm) is observed. Since we do \nnot expect any steep change in Py’s saturation magnetization at this temperature, the anomaly in \n4π*M eff should be caused by an anisot ropy change which coul d be related to a sp in reorientation. \nHowever, to fully understand the underlying mechan isms of the peak of the surface damping at ~ \n50 K, further theoretical and e xperimental studies are needed. 8 Conclusion  \nIn summary, the thickness and temperature dependences of the Gilbert damping in Py thin \nfilms are investigated, from which the contributio n due to the bulk damping and surface damping \nare clearly identified. Of particular interest, the bulk damping decreases monotonically as the \ntemperature decreases, while the surface damping develops an enhancement peak at ~ 50 K, \nwhich could be related to a thermally induced spin reorientation for the surface magnetization of \nthe Py thin films. This model is also consistent  with the observation of an enhancement of the \neffective magnetization below ~50 K. Our expe rimental results will contribute to the \nunderstanding of the intrinsic and ex trinsic mechanisms of the Gilber t damping in FM thin films.  \n \nMethods   \nMaterials growth. The Py thin films are deposited on ~25 nm SiO 2/Si substrates at room \ntemperature in 3×10- 3 Torr argon in a magnetron sputtering sy stem with a base pressure of ~ \n1×10-8 Torr. The growth rate of the Py is ~ 1  Å/s. To prevent ex situ  oxidation of the Py film \nduring the measurement,  a ~ 20 Å TaN or Al 2O3 capping layer is grown in situ environment. The \nTaN layer is grown by reactive sputtering of a Ta target in an argon-nitrogen gas mixture (ratio: \n90/10). For Al 2O3 capping layer, a thin Al (3 Å) layer is deposited first, and the Al 2O3 is \ndeposited by reactive spu ttering of an Al target in an ar gon-oxygen gas mixture (ratio: 93/7).  \nFMR measurement.  The FMR is measured using the vector network analyzer (VNA, Agilent \nE5071C) connected with a coplanar wave guide30 in the variable temperature insert of a \nQuantum Design Physical Properties Measuremen t System (PPMS) in the temperature range \nfrom 300 to 2 K. The Py sample is cut to be 1 × 0.4 cm and attached to the coplanar wave guide 9 with insulating silicon paste. For each temper ature from 300 K to 2 K, the forward complex \ntransmission coefficients (S 21) for the frequencies between 1 - 15 GHz are recorded as a function \nof the magnetic field sweeping from ~2500 Oe to 0 Oe.  \n \nContributions  \nJ.S. and W.H. proposed and supervised the studies. Y.Z. and Q.S. performed the FMR \nmeasurement and analyzed the data. T.S. and W.Y.  helped the measurement. S.H.Y. and S.S.P.P. \ngrew the films. Y.Z., J.S. and W.H. wrote the manuscript. All authors commented on the \nmanuscript and contributed to its final version. \n \nAcknowledgements   \nWe acknowledge the fruitful discussions with  Ryuichi Shindou, Ke Xia, Ziqiang Qiu, Qian \nNiu, Xincheng Xie and Ji Feng and the support of National Basic Research Programs of China \n(973 Grants 2013CB921903, 2014CB920902 and 2015 CB921104). Wei Han also acknowledges \nthe support by the 1000 Talents Program for Young Scientists of China. \n \nCompeting financial interests \nThe authors declare no compe ting financial interests. \n \n \nReferences: \n \n1 Landau, L. & Lifshitz, E. On the theory of the dispersion of magnetic permeability in \nferromagnetic bodies. Phys. Z. Sowjetunion  8, 153 (1935). \n2 Gilbert, T. L. A phenomenological theory  of damping in ferromagnetic materials. \nMagnetics, IEEE Transactions on  40, 3443-3449, doi:10.1109/TMAG.2004.836740 \n(2004). 10 3 Scheck, C., Cheng, L., Barsukov, I., Frait, Z.  & Bailey, W. E. 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B  78, 020404 (2008). \n14 Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Scattering Theory of Gilbert Damping. \nPhys. Rev. Lett.  101, 037207 (2008). \n15 Arias, R. & Mills, D. L. Extrinsic contri butions to the ferromagne tic resonance response \nof ultrathin films. Phys. Rev. B  60, 7395-7409 (1999). \n16 Walowski, J., Müller, G., Djordjevic, M., M ünzenberg, M., Kläui, M., Vaz, C. A. F. & \nBland, J. A. C. Energy Equilibration Pro cesses of Electrons, Magnons, and Phonons at \nthe Femtosecond Time Scale. Phys. Rev. Lett.  101, 237401 (2008). \n17 Stiles, M. D. & Miltat, J. in Spin Dynamics in Confined Magnetic Structures III  Vol. 101 \nTopics in Applied Physics  (eds Burkard Hillebrands & André Thiaville) Ch. 7, 225-308 \n(Springer Berlin Heidelberg, 2006). \n18 Celinski, Z., Urquhart, K. B. & Heinrich, B. Using ferromagnetic resonance to measure \nthe magnetic moments of ultrathin films. J. Magn. Magn. Mater.  166, 6-26 (1997). \n19 Barati, E., Cinal, M., Edwards, D. M. & Umerski, A. Gilbert damping in magnetic \nlayered systems. Phys. Rev. B  90, 014420 (2014). \n20 Tserkovnyak, Y., Brataas, A., Bauer, G. E. W.  & Halperin, B. I. Nonlocal magnetization \ndynamics in ferromagnetic heterostructures. Rev. Mod. Phys.  77, 1375-1421 (2005). \n21 Liu, Y., Yuan, Z., Wesselink, R. J. H., St arikov, A. A. & Kelly, P. J. Interface \nEnhancement of Gilbert Damping from First Principles. Phys. Rev. Lett.  113, 207202 \n(2014). 11 22 Starikov, A. A., Kelly, P. J ., Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Unified \nFirst-Principles Study of Gilbert Dampi ng, Spin-Flip Diffusion, and Resistivity in \nTransition Metal Alloys. Phys. Rev. Lett.  105, 236601 (2010). \n23 Mankovsky, S., Ködderitzsch, D., Woltersdo rf, G. & Ebert, H. First-principles \ncalculation of the Gilbert damping paramete r via the linear response formalism with \napplication to magnetic transition metals and alloys. Physical Review B  87, 014430 \n(2013). \n24 Bailey, W., Kabos, P., Mancoff, F. & Russe k, S. Control of magnetization dynamics in \nNi81Fe19 thin films through the us e of rare-earth dopants. Magnetics, IEEE Transactions \non 37, 1749-1754 (2001). \n25 Rantschler, J. O., Maranville, B. B., Malle tt, J. J., Chen, P., McMichael, R. D. & \nEgelhoff, W. F. Damping at no rmal metal/permalloy interfaces. Magnetics, IEEE \nTransactions on  41, 3523-3525 (2005). \n26 Luo, C., Feng, Z., Fu, Y., Zha ng, W., Wong, P. K. J., Kou, Z. X., Zhai, Y., Ding, H. F., \nFarle, M., Du, J. & Zhai, H. R. Enhancem ent of magnetization damping coefficient of \npermalloy thin films with dilute Nd dopants. Phys. Rev. B  89, 184412 (2014). \n27 Ghosh, A., Sierra, J. F., Auffret, S., Ebels,  U. & Bailey, W. E. Dependence of nonlocal \nGilbert damping on the ferromagnetic layer t ype in ferromagnet/Cu/Pt heterostructures. \nAppl. Phys. Lett.  98, 052508 (2011). \n28 Sierra, J. F., Pryadun, V. V., Russek, S. E ., García-Hernández, M., Mompean, F., Rozada, \nR., Chubykalo-Fesenko, O., Snoeck, E., Miao, G. X., Moodera, J. S. & Aliev, F. G. \nInterface and Temperature Dependent Magnetic  Properties in Permalloy Thin Films and \nTunnel Junction Structures. Journal of Nanoscience and Nanotechnology  11, 7653-7664 \n(2011). \n29 Kittel, C. On the Theory of Ferromagnetic Resonance Absorption. Phys. Rev.  73, 155 \n(1948). \n30 Kalarickal, S. S., Krivosik, P., Wu, M., Patt on, C. E., Schneider, M. L., Kabos, P., Silva, \nT. J. & Nibarger, J. P. Ferromagnetic reso nance linewidth in metallic thin films: \nComparison of measurement methods. J. Appl. Phys.  99, 093909 (2006). \n 12  \nFigure Captions  \n \nFigure 1.  Measurement of Gilbert damping in Py thin films via ferromagnetic resonance \n(Py thickness = 30 nm).   a, Ferromagnetic resonance spectra of the absorption for 30 nm Py thin \nfilms with TaN capping layer at gigahertz frequencies of 4, 6, 8, 10 and 12 GHz at 300 K after \nnormalization by background subtraction. b, c, The half linewidths as a function of the resonance \nfrequencies at 300 K and 5 K respectively. The red solid lines indicate the fitted lines based on \nequation (3), where the Gilbert damp ing constants could be obtained.  \n \nFigure 2. Temperature dependence of the Gilber t damping of Py thin films with TaN \ncapping.  a, The temperature dependence of the Gilbert damping fo r 3, 5, 10, 15, 20, 30, and 50 \nnm Py films. b, The Gilbert damping as a function of the Py thickness, d, measured at 300 K. c, \nThe Gilbert damping as a function of 1/ d measured at 300 K. The linear fitting corresponds to \nequation (4), in which the slope and the intercep t are related to the surf ace contribution and bulk \ncontribution to the total Gilber t damping. Error bars correspond to one standard deviation. \n Figure 3. Bulk and surface damping of Py  thin films with TaN capping layer.   a, b,  The \ntemperature dependence of the bulk damping an d surface damping, respectively. The inset table \nsummarizes the experimental values reported in early studies. Error bars correspond to one \nstandard deviation.  \nFigure 4. Comparison of the Gilbert damping of Py films with different capping layers.  a, \nb, Temperature dependence of the Gilbert dampi ng of Py thin films with TaN capping layer 13 (blue) and Al 2O3 capping layer (green) for 5 nm Py a nd 30 nm Py, respectively. Error bars \ncorrespond to one standard deviation.  \nFigure 5. Measurement of effective magnetizat ion in Py thin films via ferromagnetic \nresonance (Py thickness = 30 nm).   a, b, The resonance frequencies vs. the resonance magnetic \nfield at 300 K and 5 K, respectively. The fitted li nes (red curves) are obtained using the Kittel \nformula.   \nFigure 6. Effective magnetization of Py fi lms as a function of the temperature.  a, b, c, \nTemperature dependence of the effective magnetizati on of Py thin films of a thickness of 3 nm,  \n5 nm and 30 nm Py respectively. In b, c, the blue/green symbols correspond to the Py with \nTaN/Al\n2O3 capping layer.  \n \n 0\n500\n1000\n1500\n2000\n-0.3\n-0.2\n-0.1\n0.0\n0.1\n   4 \n   6 \n   8\n 10 \n 12 \n \nS\n21\n (dB) \n \nH (Oe)\nT=300 K\nf\n (GHz)\n0\n4\n8\n12\n16\n0\n10\n20\n30\n \n\nH (Oe)\n \nf (GHz)\nT=300 K\n0\n4\n8\n12\n16\n0\n10\n20\n30\n \n\nH (Oe)\n \nf (GHz)\nT=5 K\nb\nc\na\nFigure 10\n50\n100\n150\n200\n250\n300\n0.006\n0.008\n0.010\n0.012\n0.014\nd\n (nm)\n 3      \n 15        \n 5      \n 20\n 10    \n 30    \n               \n 50 \n \na\n \nTemperature (K)\n0.0\n0.1\n0.2\n0.3\n0.004\n0.006\n0.008\n0.010\n0.012\n0.014\n \na\n \n \n1/\nd\n (nm\n-1\n)\n0\n10\n20\n30\n0.006\n0.008\n0.010\n0.012\n0.014\n \n \nd\n (nm)\n \na\na\nb\nc\nFigure \n20\n50\n100\n150\n200\n250\n300\n0.0040\n0.0045\n0.0050\n0.0055\n0.0060\nTheory\n Ref. 21, 22\n Ref. 23\n Temperature (K)\n \na\nB\n \na\na\nExp.\n0.006\nRef. 24\n0.004\n-\n0.008\nRef.\n25\n0.007\nRef.\n26\n0.0067\nRef. 27\n0\n50\n100\n150\n200\n250\n300\n0.016\n0.018\n0.020\n0.022\n0.024\n0.026\n0.028\n0.030\n Temperature (K)\na\nS\n (nm)\n \nb\nFigure \n30\n50\n100\n150\n200\n250\n300\n0.004\n0.006\n0.008\n0.010\n 5 nm Py/TaN\n 5 nm Py/Al\n2\nO\n3\n Temperature (K)\na\n \n \n0\n50\n100\n150\n200\n250\n300\n0.004\n0.005\n0.006\n0.007\n 30 nm Py/TaN\n 30 nm Py/Al\n2\nO\n3\n Temperature (K)\na\n \n \na\nb\nFigure \n4a\nb\n0\n500\n1000\n1500\n2000\n0\n4\n8\n12\n16\n \nf\n (GHz)\n \nH\n (Oe)\nT=300 K\n0\n500\n1000\n1500\n2000\n0\n4\n8\n12\n16\n \nf\n (GHz)\n \nH (Oe)\nT=5 K\nFigure \n58.6\n8.8\n9.0\n9.2\n9.4\n9.6\n4\n\nM\neff\n (kG) \n 5 nm Py/TaN\n 5 nm Py/Al\n2\nO\n3\n \n6.2\n6.3\n6.4\n6.5\n6.6\n6.7\n6.8\n6.9\n4\n\nM\neff\n (kG) \n 3 nm Py/TaN\n \n0\n50\n100\n150\n10.6\n10.7\n10.8\n10.9\n11.0\n 30 nm Py/TaN\n 30 nm Py/Al\n2\nO\n3\n4\n\nM\neff\n (kG) \n \na\nb\nc\nTemperature (K) \nFigure \n6"    },    {        "title": "1907.11853v1.Two_improved_Gauss_Seidel_projection_methods_for_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "Two improved Gauss-Seidel projection methods for\nLandau-Lifshitz-Gilbert equation\nPanchi Lia, Changjian Xiea, Rui Dua,b,\u0003, Jingrun Chena,b,\u0003, Xiao-Ping Wangc,\u0003\naSchool of Mathematical Sciences, Soochow University, Suzhou, 215006, China.\nbMathematical Center for Interdisciplinary Research, Soochow University, Suzhou, 215006, China.\ncDepartment of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay,\nKowloon, Hong Kong, China\nA B S T R A C T\nMicromagnetic simulation is an important tool to study various dynamic behaviors of\nmagnetic order in ferromagnetic materials. The underlying model is the Landau-Lifshitz-\nGilbert equation, where the magnetization dynamics is driven by the gyromagnetic torque\nterm and the Gilbert damping term. Numerically, considerable progress has been made in\nthe past decades. One of the most popular methods is the Gauss-Seidel projection method\ndeveloped by Xiao-Ping Wang, Carlos Garc\u0013 \u0010a-Cervera, and Weinan E in 2001. It \frst solves\na set of heat equations with constant coe\u000ecients and updates the gyromagnetic term in the\nGauss-Seidel manner, and then solves another set of heat equations with constant coe\u000ecients\nfor the damping term. Afterwards, a projection step is applied to preserve the length con-\nstraint in the pointwise sense. This method has been veri\fed to be unconditionally stable\nnumerically and successfully applied to study magnetization dynamics under various controls.\nIn this paper, we present two improved Gauss-Seidel projection methods with uncondi-\ntional stability. The \frst method updates the gyromagnetic term and the damping term\nsimultaneously and follows by a projection step. The second method introduces two sets of\napproximate solutions, where we update the gyromagnetic term and the damping term simul-\ntaneously for one set of approximate solutions and apply the projection step to the other set\nof approximate solutions in an alternating manner. Compared to the original Gauss-Seidel\nprojection method which has to solve heat equations 7 times at each time step, the improved\nmethods solve heat equations 5 times and 3 times, respectively. First-order accuracy in time\nand second-order accuracy in space are veri\fed by examples in both 1D and 3D. In addi-\ntion, unconditional stability with respect to both the grid size and the damping parameter is\ncon\frmed numerically. Application of both methods to a realistic material is also presented\nwith hysteresis loops and magnetization pro\fles. Compared with the original method, the\nrecorded running times suggest that savings of both methods are about 2 =7 and 4=7 for the\nsame accuracy requirement, respectively.\nKeywords: Landau-Lifshitz-Gilbert equation, Gauss-Seidel projection method, unconditional\nstability, micromagnetic simulation\n2000 MSC: 35Q99, 65Z05, 65M06\n1. Introduction\nIn ferromagnetic materials, the intrinsic magnetic order, known as magnetization M=\n(M1;M2;M3)T, is modeled by the following Landau-Lifshitz-Gilbert (LLG) equation [1, 2, 3]\n@M\n@t=\u0000\rM\u0002H\u0000\r\u000b\nMsM\u0002(M\u0002H) (1)\n\u0003Corresponding authors\ne-mail: LiPanchi1994@163.com (Panchi Li), 20184007005@stu.suda.edu.cn (Changjian Xie),\ndurui@suda.edu.cn (Rui Du), jingrunchen@suda.edu.cn (Jingrun Chen), mawang@ust.hk (Xiao-Ping Wang)\n1arXiv:1907.11853v1  [math.NA]  27 Jul 2019with\rthe gyromagnetic ratio and jMj=Msthe saturation magnetization. On the right-\nhand side of (1), the \frst term is the gyromagnetic term and the second term is the Gilbert\ndamping term with \u000bthe dimensionless damping coe\u000ecient [2]. Note that the gyromagnetic\nterm is a conservative term, whereas the damping term is a dissipative term. The local \feld\nH=\u0000\u000eF\n\u000eMis computed from the Landau-Lifshitz energy functional\nF[M] =1\n2Z\n\n\u001aA\nM2sjrMj2+ \b\u0012M\nMs\u0013\n\u00002\u00160He\u0001M\u001b\ndx+\u00160\n2Z\nR3jrUj2dx; (2)\nwhereAis the exchange constant,A\nM2sjrMjis the exchange interaction energy; \b\u0010\nM\nMs\u0011\nis the anisotropy energy, and for simplicity the material is assumed to be uniaxial with\n\b\u0010\nM\nMs\u0011\n=Ku\nM2s(M2\n2+M2\n3) withKuthe anisotropy constant; \u00002\u00160He\u0001Mis the Zeeman\nenergy due to the external \feld with \u00160the permeability of vacuum. \n is the volume occupied\nby the material. The last term in (2) is the energy resulting from the \feld induced by the\nmagnetization distribution inside the material. This stray \feld Hs=\u0000rUwhereU(x)\nsatis\fes\nU(x) =Z\n\nrN(x\u0000y)\u0001M(y)dy; (3)\nwhereN(x\u0000y) =\u00001\n4\u00191\njx\u0000yjis the Newtonian potential.\nFor convenience, we rescale the original LLG equation (1) by changes of variables t!\n(\u00160\rMs)\u00001tandx!LxwithLthe diameter of \n. De\fne m=M=Msandh=MsH. The\ndimensionless LLG equation reads as\n@m\n@t=\u0000m\u0002h\u0000\u000bm\u0002(m\u0002h); (4)\nwhere\nh=\u0000Q(m2e2+m3e3) +\u000f\u0001m+he+hs (5)\nwith dimensionless parameters Q=Ku=(\u00160M2\ns) and\u000f=A=(\u00160M2\nsL2). Here e2= (0;1;0),\ne3= (0;0;1). Neumann boundary condition is used\n@m\n@\u0017j@\n= 0; (6)\nwhere\u0017is the outward unit normal vector on @\n.\nThe LLG equation is a weakly nonlinear equation. In the absence of Gilbert damping,\n\u000b= 0, equation (4) is a degenerate equation of parabolic type and is related to the sympletic\n\row of harmonic maps [4]. In the large damping limit, \u000b!1 , equation (4) is related to\nthe heat \row for harmonic maps [5]. It is easy to check that jmj= 1 in the pointwise sense\nin the evolution. All these properties possesses interesting challenges for designing numerical\nmethods to solve the LLG equation. Meanwhile, micromagnetic simulation is an important\ntool to study magnetization dynamics of magnetic materials [3, 6]. Over the past decades,\nthere has been increasing progress on numerical methods for the LLG equation; see [7, 8, 9]\n2for reviews and references therein. Finite di\u000berence method and \fnite element method have\nbeen used for the spatial discretization.\nFor the temporal discretization, there are explicit schemes such as Runge-Kutta methods\n[10, 11]. Their stepsizes are subject to strong stability constraint. Another issue is that the\nlength of magnetization cannot be preserved and thus a projection step is needed. Implicit\nschemes [12, 13, 14] are unconditionally stable and usually can preserve the length of magne-\ntization automatically. The di\u000eculty of implicit schemes is how to solve a nonlinear system\nof equations at each step. Therefore, semi-implicit methods [15, 16, 17, 18, 19] provide a com-\npromise between stability and the di\u000ecult for solving the equation at each step. A projection\nstep is also needed to preserve the length of magnetization.\nAmong the semi-implicit schemes, the most popular one is the Gauss-Seidel projection\nmethod (GSPM) proposed by Wang, Garc\u0013 \u0010a-Cervera, and E [15, 18]. GSPM \frst solves a\nset of heat equations with constant coe\u000ecients and updates the gyromagnetic term in the\nGauss-Seidel manner, and then solves another set of heat equations with constant coe\u000ecients\nfor the damping term. Afterwards, a projection step is applied to preserve the length of mag-\nnetization. GSPM is \frst-order accurate in time and has been veri\fed to be unconditionally\nstable numerically.\nIn this paper, we present two improved Gauss-Seidel projection methods with uncondi-\ntional stability. The \frst method updates the gyromagnetic term and the damping term\nsimultaneously and follows by a projection step. The second method introduces two sets of\napproximate solutions, where we update the gyromagnetic term and the damping term simul-\ntaneously for one set of approximate solutions and apply the projection step to the other set\nof approximate solutions in an alternating manner. Compared to the original Gauss-Seidel\nprojection method, which solves heat equations 7 times at each time step, the improved\nmethods solve heat equations 5 times and 3 times, respectively. First-order accuracy in time\nand second-order accuracy in space are veri\fed by examples in both 1D and 3D. In addi-\ntion, unconditional stability with respect to both the grid size and the damping parameter is\ncon\frmed numerically. Application of both methods to a realistic material is also presented\nwith hysteresis loops and magnetization pro\fles. Compared with the original method, the\nrecorded running times suggest that savings of both methods are about 2 =7 and 4=7 for the\nsame accuracy requirement, respectively.\nThe rest of the paper is organized as follows. For completeness and comparison, we \frst\nintroduce GSPM in Section 2. Two improved GSPMs are presented in Section 3. Detailed\nnumerical tests are given in Section 4, including accuracy check and e\u000eciency check in both\n1D and 3D, unconditional stability with respect to both the grid size and the damping\nparameter, hysteresis loops, and magnetization pro\fles. Conclusions are drawn in Section 5.\n32. Gauss-Seidel projection method for Landau-Lifshitz-Gilbert equation\nBefore the introduction of the GSPM [15, 18], we \frst use the \fnite di\u000berence method\nfor spatial discretization. Figure 1 shows a schematic picture of spatial grids in 1D. Let\ni= 0;1;\u0001\u0001\u0001;M;M + 1,j= 0;1;\u0001\u0001\u0001;N;N + 1, andk= 0;1;\u0001\u0001\u0001;K;K + 1 be the indices of\ngrid points in 3D.\n0 1𝑥−1\n2𝑥1\n2𝑥𝑁−1\n2𝑥𝑁+1\n2𝑥3\n2𝑥𝑁−3\n2\nFig. 1. Spatial grids in 1D. Nodes x\u00001\n2andxN+1\n2are ghost points.\nSecond-order centered di\u000berence for \u0001 mreads as\n\u0001hmi;j;k=mi+1;j;k\u00002mi;j;k+mi\u00001;j;k\n\u0001x2\n+mi;j+1;k\u00002mi;j;k+mi;j\u00001;k\n\u0001y2\n+mi;j;k+1\u00002mi;j;k+mi;j;k\u00001\n\u0001z2; (7)\nwhere mi;j;k=m((i\u00001\n2)\u0001x;(j\u00001\n2)\u0001y;(k\u00001\n2)\u0001z). For the Neumann boundary condition,\na second-order approximation yields\nm0;j;k=m1;j;k;mM;j;k =mM+1;j;k; j = 1;\u0001\u0001\u0001;N;k = 1;\u0001\u0001\u0001;K;\nmi;0;k=mi;1;k;mi;N;k =mi;N+1;k; i= 1;\u0001\u0001\u0001;M;k = 1;\u0001\u0001\u0001;K;\nmi;j;0=mi;j;1;mi;j;K =mi;j;K +1; i= 1;\u0001\u0001\u0001;M;j = 1;\u0001\u0001\u0001;N:\nTo illustrate the main ideas, we \frst consider the following simpli\fed equation\nmt=\u0000m\u0002\u0001m\u0000\u000bm\u0002(m\u0002\u0001m);\nwhich can be rewritten as\nmt=\u0000m\u0002\u0001m\u0000\u000bm(m\u0001\u0001m) +\u000b\u0001m: (8)\nWe split (8) into two equations\nmt=\u0000m\u0002\u0001m; (9)\nmt=\u000b\u0001m: (10)\nHowever, (9) is still nonlinear. Therefore, we consider a fractional step scheme to solve\n(9)\nm\u0003\u0000mn\n\u0001t= \u0001hm\u0003\nmn+1=mn\u0000mn\u0002m\u0003\n4or\nmn+1=mn\u0000mn\u0002(I\u0000\u0001t\u0001h)\u00001mn;\nwhereIis the identity matrix. This scheme is subject to strong stability constraint, and thus\nthe implicit Gauss-Seidel scheme is introduced to overcome this issue. Let\ngn\ni= (I\u0000\u0001t\u0001h)\u00001mn\ni; i= 1;2;3: (11)\nWe then have 0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=0\n@mn\n1+ (gn\n2mn\n3\u0000gn\n3mn\n2)\nmn\n2+ (gn\n3mn+1\n1\u0000gn+1\n1mn\n3)\nmn\n3+ (gn+1\n1mn+1\n2\u0000gn+1\n2mn+1\n1)1\nA: (12)\nThis scheme solve (9) with unconditional stability. (10) is linear heat equation which can be\nsolved easily. However, the splitting scheme (9) - (10) cannot preserve jmj= 1, and thus a\nprojection step needs to be added.\nFor the full LLG equation (4), the GSPM works as follows. De\fne\nh=\u000f\u0001m+^f; (13)\nwhere ^f=\u0000Q(m2e2+m3e3) +he+hs.\nThe original GSPM [15] solves the equation (4) in three steps:\n\u000fImplicit Gauss-Seidel\ngn\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(mn\ni+ \u0001t^fn\ni); i= 2;3;\ng\u0003\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^fn\ni); i= 1;2; (14)\n0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA=0\n@mn\n1+ (gn\n2mn\n3\u0000gn\n3mn\n2)\nmn\n2+ (gn\n3m\u0003\n1\u0000g\u0003\n1mn\n3)\nmn\n3+ (g\u0003\n1m\u0003\n2\u0000g\u0003\n2m\u0003\n1)1\nA: (15)\n\u000fHeat \row without constraints\n^f\u0003=\u0000Q(m\u0003\n2e2+m\u0003\n3e3) +he+hn\ns; (16)\n0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA=0\n@m\u0003\n1+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n1+^f\u0003\n1)\nm\u0003\n2+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n2+^f\u0003\n2)\nm\u0003\n3+\u000b\u0001t(\u000f\u0001hm\u0003\u0003\n3+^f\u0003\n3)1\nA: (17)\n\u000fProjection onto S2\n0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003\u0003j0\n@m\u0003\u0003\n1\nm\u0003\u0003\n2\nm\u0003\u0003\n31\nA: (18)\nHere the numerical stability of the original GSPM [15] was founded to be independent of\ngridsizes but depend on the damping parameter \u000b. This issue was solved in [18] by replacing\n(14) and (16) with\ng\u0003\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^f\u0003\ni); i= 1;2;\n5and\n^f\u0003=\u0000Q(m\u0003\n2e2+m\u0003\n3e3) +he+h\u0003\ns;\nrespectively. Update of the stray \feld is done using fast Fourier transform [15]. It is easy\nto see that the GSPM solves 7 linear systems of equations with constant coe\u000ecients and\nupdates the stray \feld using FFT 6 times at each step.\n3. Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert\nequation\nBased on the description of the original GSPM in Section 2, we introduce two improved\nGSPMs for LLG equation. The \frst improvement updates both the gyromagnetic term and\nthe damping term simultaneously, termed as Scheme A. The second improvement introduces\ntwo sets of approximate solution with one set for implicit Gauss-Seidel step and the other set\nfor projection in an alternating manner, termed as Scheme B. Details are given in below.\n3.1. Scheme A\nThe main improvement of Scheme A over the original GSPM is the combination of (13)\n- (17), or (9) - (10).\n\u000fImplicit-Gauss-Seidel\ngn\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(mn\ni+ \u0001t^fn\ni); i= 1;2;3;\ng\u0003\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^f\u0003\ni); i= 1;2; (19)\n0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA=0\n@mn\n1\u0000(mn\n2gn\n3\u0000mn\n3gn\n2)\u0000\u000b(mn\n1gn\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n1+\u000bgn\n1\nmn\n2\u0000(mn\n3g\u0003\n1\u0000m\u0003\n1gn\n3)\u0000\u000b(m\u0003\n1g\u0003\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n2+\u000bgn\n2\nmn\n3\u0000(m\u0003\n1g\u0003\n2\u0000m\u0003\n2g\u0003\n1)\u0000\u000b(m\u0003\n1g\u0003\n1+m\u0003\n2g\u0003\n2+mn\n3gn\n3)mn\n3+\u000bgn\n31\nA:(20)\n\u000fProjection onto S2\n0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003j0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA: (21)\nIt is easy to see that Scheme A solves 5 linear systems of equations with constant coe\u000ecients\nand uses FFT 5 times at each step.\n3.2. Scheme B\nThe main improvement of Scheme B over Scheme A is the introduction of two sets of\napproximate solutions, one for (19) - (20) and the other for (21) and the update of these two\nsets of solutions in an alternating manner.\nGiven the initialized g0\ng0\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m0\ni+ \u0001t^f0\ni); i= 1;2;3; (22)\nScheme B works as follows\n6\u000fImplicit Gauss-Seidel\ngn+1\ni= (I\u0000\u0001t\u000f\u0001h)\u00001(m\u0003\ni+ \u0001t^f\u0003\ni); i= 1;2;3 (23)\nm\u0003\n1=mn\n1\u0000(mn\n2gn\n3\u0000mn\n3gn\n2)\u0000\u000b(mn\n1gn\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n1+\n\u000b((mn\n1)2+ (mn\n2)2+ (mn\n3)2)gn\n1\nm\u0003\n2=mn\n2\u0000(mn\n3gn+1\n1\u0000m\u0003\n1gn\n3)\u0000\u000b(m\u0003\n1gn+1\n1+mn\n2gn\n2+mn\n3gn\n3)mn\n2+\n\u000b((m\u0003\n1)2+ (mn\n2)2+ (mn\n3)2)gn\n2\nm\u0003\n3=mn\n3\u0000(m\u0003\n1gn+1\n2\u0000m\u0003\n2gn+1\n1)\u0000\u000b(m\u0003\n1gn+1\n1+m\u0003\n2gn+1\n2+mn\n3gn\n3)mn\n3+\n\u000b((m\u0003\n1)2+ (m\u0003\n2)2+ (mn\n3)2)gn\n3 (24)\n\u000fProjection onto S2\n0\n@mn+1\n1\nmn+1\n2\nmn+1\n31\nA=1\njm\u0003j0\n@m\u0003\n1\nm\u0003\n2\nm\u0003\n31\nA: (25)\nHere one set of approximate solution fm\u0003gis updated in the implicit Gauss-Seidel step and\nthe other set of approximate solution fmn+1gis updated in the projection step. Note that\n(23) is de\fned only for fm\u0003gwhich can be used in two successive temporal steps, and thus\nonly 3 linear systems of equations with constant coe\u000ecients are solved at each step and 3\nFFT executions are used for the stray \feld. The length of magnetization can be preserved\nin the time evolution.\nThe computational cost of GSPM and its improvements comes from solving the linear\nsystems of equations with constant coe\u000ecients. To summarize, we list the number of linear\nsystems of equations to be solved and the number of FFT executions to be used at each step\nfor the original GSPM [18], Scheme A, and Scheme B in Table 1. The savings represent the\nratio between costs of two improved schemes over that of the original GSPM.\nGSPM Scheme Number of linear systems Saving Execution of FFT Saving\nOriginal 7 0 4 0\nScheme A 5 2=7 3 1=4\nScheme B 3 4=7 3 1=4\nTable 1. The number of linear systems of equations to be solved and the number of FFT\nexecutions to be used at each step for the original GSPM [18], Scheme A, and Scheme B. The\nsavings represent the ratio between costs of two improved schemes over that of the original\nGSPM.\n4. Numerical Experiments\nIn this section, we compare the original GSPM [15, 18], Scheme A, and Scheme B via a\nseries of examples in both 1D and 3D, including accuracy check and e\u000eciency check, uncon-\nditional stability with respect to both the grid size and the damping parameter, hysteresis\n7loops, and magnetization pro\fles. For convenience, we de\fne\nratio\u0000i=Time(GSPM)\u0000Time(Scheme i)\nTime(GSPM);\nfori= A and B, which quanti\fes the improved e\u000eciency of Scheme A and Scheme B over\nthe original GSPM [15, 18].\n4.1. Accuracy Test\nExample 4.1 (1D case). In 1D, we choose the exact solution over the unit interval \n =\n[0;1]\nme= (cos(\u0016x) sin(t);sin(\u0016x) sin(t);cos(t));\nwhich satis\fes\nmt=\u0000m\u0002mxx\u0000\u000bm\u0002(m\u0002mxx) +f\nwith \u0016x=x2(1\u0000x)2, and f=met+me\u0002mexx+\u000bme\u0002(me\u0002mexx). Parameters are\n\u000b= 0:00001 andT= 5:0e\u00002.\nWe \frst show the error kme\u0000mhk1withmhbeing the numerical solution with respect\nto the temporal stepsize \u0001tand the spatial stepsize \u0001x. As shown in Figure 2(a) and Fig-\nure 2(c), suggested by the least squares \ftting, both \frst-order accuracy in time and second-\norder accuracy in space are observed. Meanwhile, we record the CPU time as a function\nof accuracy (error) by varying the temporal stepsize and the spatial stepsize in Figure 2(b)\nand Figure 2(d), Table 2 and Table 3, respectively. In addition, from Table 2 and Table 3,\nthe saving of Scheme A over GSPM is about 2=7, which equals 1\u00005=7, and the saving of\nScheme B over GSPM is about 4=7, respectively. This observation is in good agreement with\nthe number of linear systems being solved at each step for these three methods, as shown in\nTable 1.\nXXXXXXXXXXCPU time\u0001tT/1250 T/2500 T/5000 T/10000 Reference\nGSPM 7.7882e-01 1.5445e+00 3.1041e+00 6.2196e+00 -\nScheme A 4.8340e-01 9.9000e-01 2.0527e+00 4.4917e+00 -\nScheme B 3.3010e-01 6.3969e-01 1.2281e+00 2.5510e+00 -\nratio-A 0.38 0.36 0.34 0.28 0.29(2/7)\nratio-B 0.58 0.59 0.60 0.59 0.57(4/7)\nTable 2. Recorded CPU time in 1D with respect to the approximation error when only \u0001tis\nvaried and \u0001x= 1=100.\nExample 4.2 (3D case). In 3D, we choose the exact solution over \n = [0;2]\u0002[0;1]\u0002[0;0:2]\nme= (cos(\u0016x\u0016y\u0016z) sin(t);sin(\u0016x\u0016y\u0016z) sin(t);cos(t));\nwhich satis\fes\nmt=\u0000m\u0002\u0001m\u0000\u000bm\u0002(m\u0002\u0001m) +f\n8log(∆t)-12.5 -12 -11.5 -11 -10.5 -10log(error)\n-12.5-12-11.5-11-10.5-10\nGSPM\nScheme A\nScheme B(a) Temporal accuracy\nlog(error)-12.5 -12 -11.5 -11 -10.5 -10log(time)\n-1.5-1-0.500.511.52\nGSPM\nScheme A\nScheme B (b) CPU time versus approximation error (\u0001 t)\nlog(∆x)-5.1 -5 -4.9 -4.8 -4.7 -4.6log(error)\n-16-15.9-15.8-15.7-15.6-15.5-15.4-15.3-15.2-15.1-15\nGSPM\nScheme A\nScheme B\n(c) Spatial accuracy\nlog(error)-16 -15.8 -15.6 -15.4 -15.2 -15log(time)\n77.588.599.5\nGSPM\nScheme A\nScheme B (d) CPU time versus approximation error (\u0001 x)\nFig. 2. Approximation error and CPU time in 1D. (a) Approximation error as a function of the\ntemporal step size; (b) CPU time as a function of the approximation error when \u0001tis varied\nand \u0001xis \fxed; (c) Approximation error as a function of the spatial step size; (d) CPU time\nas a function of the approximation error when \u0001xis varied and \u0001tis \fxed.\nwith \u0016x=x2(1\u0000x)2,\u0016y=y2(1\u0000y)2,\u0016z=z2(1\u0000z)2andf=met+me\u0002\u0001me+\u000bme\u0002(me\u0002\n\u0001me). Parameters are T= 1:0e\u000005and\u000b= 0:01.\nLike in the 1D case, we \frst show the error kme\u0000mhk1withmhbeing the numerical\nsolution with respect to the temporal stepsize \u0001tand the spatial stepsize \u0001x. As shown in\nFigure 3(a) and Figure 3(c), suggested by the least squares \ftting, both \frst-order accuracy in\ntime and second-order accuracy in space are observed. Meanwhile, we record the CPU time\nas a function of accuracy (error) by varying the temporal stepsize and the spatial stepsize in\nFigure 3(b) and Figure 3(d), Table 4 and Table 5, respectively. In addition, from Table 4 and\nTable 5, the saving of Scheme A over GSPM is about 2=7, and the saving of Scheme B over\nGSPM is about 4=7, respectively. This observation is in good agreement with the number of\nlinear systems being solved at each step for these three methods, as shown in Table 1.\nIt worths mentioning that all these three methods are tested to be unconditionally stable\nwith respect to the spatial gridsize and the temporal stepsize.\n9XXXXXXXXXXCPU time\u0001x1/100 1/120 1/140 1/160 Reference\nGSPM 3.3752e+03 5.2340e+03 9.0334e+03 1.0495e+04 -\nScheme A 2.4391e+03 3.7175e+03 6.5149e+03 8.0429e+03 -\nScheme B 1.4740e+03 2.2448e+03 3.9152e+03 4.8873e+03 -\nratio-A 0.28 0.29 0.28 0.23 0.29(2/7)\nratio-B 0.56 0.57 0.57 0.53 0.57(4/7)\nTable 3. Recorded CPU time in 1D with respect to the approximation error when only \u0001xis\nvaried and \u0001t= 1:0e\u00008.\nXXXXXXXXXXCPU time\u0001tT/10 T/20 T/40 T/80 Reference\nGSPM 3.5188e+01 6.8711e+01 1.4146e+02 2.9769e+02 -\nScheme A 2.3015e+01 4.3920e+01 8.6831e+01 1.7359e+02 -\nScheme B 1.3984e+01 2.6313e+01 5.1928e+01 1.0415e+02 -\nratio-A 0.35 0.36 0.39 0.42 0.29(2/7)\nratio-B 0.60 0.62 0.63 0.65 0.57(4/7)\nTable 4. Recorded CPU time in 3D with respect to the approximation error when only \u0001tis\nvaried and the spatial mesh is 128\u000264\u000210.\n4.2. Micromagnetic Simulations\nTo compare the performance of Scheme A and Scheme B with GSPM, we have carried out\nmicromagnetic simulations of the full LLG equation with realistic material parameters. In\nall our following simulations, we consider a thin \flm ferromagnet of size \n = 1 \u0016m\u00021\u0016m\u0002\n0:02\u0016m with the spatial gridsize 4 nm \u00024 nm\u00024 nm and the temporal stepsize \u0001 t= 1\npicosecond. The demagnetization \feld (stray \feld) is calculated via FFT [15, 18].\n4.2.1. Comparison of hysteresis loops\nThe hysteresis loop is calculated in the following way. First, a positive external \feld\nH0=\u00160His applied and the system is allowed to reach a stable state. Afterwards, the\nexternal \feld is reduced by a certain amount and the system is relaxed to a stable state\nagain. The process continues until the external \feld attains a negative \feld of strength H0.\nThen the external \feld starts to increase and the system relaxes until the initial applied\nexternal \feld H0is approached. In the hysteresis loop, we can monitor the magnetization\ndynamics and plot the average magnetization at the stable state as a function of the strength\nof the external \feld. The stopping criterion for a steady state is that the relative change of\nthe total energy is less than 10\u00007. The applied \feld is parallel to the xaxis. The initial state\nwe take is the uniform state and the damping parameter \u000b= 0:1.\nIn Figure 4, we compare the average magnetization in the hysteresis loop simulated by\nGSPM, Scheme A and Scheme B. Pro\fles of the average magnetization of these three methods\nare in quantitative agreements with approximately the same switch \feld 9 ( \u00060:4) mT.\n4.2.2. Comparison of magnetization pro\fles\nIt is tested that GSPM in [15] was unstable with a very small damping parameter \u000band\nwas resolved in [18]. This section is devoted to the unconditional stability of Scheme A and\n10log(∆t)-14 -13.5 -13 -12.5 -12 -11.5log(error)\n-14-13.5-13-12.5-12-11.5\nGSPM\nScheme A\nScheme B(a) Temporal accuracy\nlog(error)-14 -13.5 -13 -12.5 -12 -11.5log(time)\n2.533.544.555.56\nGSPM\nScheme A\nScheme B (b) CPU time versus approximation error (\u0001 t)\nThe spatial step size log( ∆x)-2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7log(error)\n-32.5-32-31.5-31\nGSPM\nScheme A\nScheme B\n(c) Spatial accuracy\nlog(error)-32.5 -32 -31.5 -31log(time)\n22.533.544.555.56\nGSPM\nScheme A\nScheme B (d) CPU time versus approximation error (\u0001 x)\nFig. 3. Approximation error and CPU time in 3D. (a) Approximation error as a function of the\ntemporal step size; (b) CPU time as a function of the approximation error when \u0001tis varied\nand \u0001x= \u0001y= \u0001zis \fxed; (c) Approximation error as a function of the spatial step size; (d)\nCPU time as a function of the approximation error when space is varied uniformly and \u0001tis\n\fxed.\nScheme B with respect to \u000b. We consider a thin \flm ferromagnet of size 1 \u0016m\u00021\u0016m\u00020:02\u0016m\nwith the spatial gridsize 4 nm \u00024 nm\u00024 nm and the temporal stepsize is 1 picosecond.\nFollowing [18], we consider the full LLG equation with \u000b= 0:1 and\u000b= 0:01 and without\nthe external \feld. The initial state is m0= (0;1;0) ifx2[0;Lx=5][[4Lx=5;Lx] and\nm0= (1;0;0) otherwise. The \fnal time is 10 ns. In Figures 5 to 7, we present a color plot\nof the angle between the in-plane magnetization and the xaxis, and an arrow plot of the\nin-plane magnetization for the original GSPM [15], Scheme A, and Scheme B, respectively.\nIn these \fgures, \u000b= 0:1 is presented in the top row and \u000b= 0:01 is presented in the bottom\nrow; a color plot of the angle between the in-palne magnetization and the xaxis is presented\nin the left column and an arrow plot of the in-plane magnetization is presented in the right\ncolumn.\n5. Conclusion\nIn this paper, based on the original Gauss-Seidel projection methods, we present two\nimproved Gauss-Seidel projection methods with the \frst-order accuracy in time and the\nsecond-order accuracy in space. The \frst method updates the gyromagnetic term and the\n11XXXXXXXXXXCPU time\u0001x1/6 1/8 1/10 1/12 Reference\nGSPM 2.1066e+01 9.2615e+01 1.9879e+02 3.7820e+02 -\nScheme A 1.5278e+01 6.5953e+01 1.4215e+02 2.6725e+02 -\nScheme B 8.9698e+00 3.8684e+01 8.4291e+01 1.5977e+02 -\nratio-A 0.27 0.29 0.28 0.29 0.29(2/7)\nratio-B 0.57 0.58 0.58 0.58 0.57(4/7)\nTable 5. Recorded CPU time in 3D with respect to the approximation error when only the\nspatial gridsize is varied with \u0001x= \u0001y= \u0001zand \u0001t= 1:0e\u000009.\n-50 -40 -30 -20 -10 0 10 20 30 40 50\n0 H (mT)-1-0.8-0.6-0.4-0.200.20.40.60.81M/Ms\n GSPM\n Scheme A\n Scheme B\nFig. 4. Comparison of hysteresis loops for GSPM, Scheme A and Scheme B. Pro\fles of the av-\nerage magnetization of these three methods are in quantitative agreements with approximately\nthe same switch \feld 9 (\u00060:4) mT . The applied \feld is parallel to the xaxis and the initial state\nis the uniform state.\ndamping term simultaneously and follows by a projection step, which requires to solve heat\nequations 5 times at each time step. The second method introduces two sets of approximate\nsolutions, where we update the gyromagnetic term and the damping term simultaneously for\none set of approximate solutions and apply the projection step to the other set of approximate\nsolutions in an alternating manner. Therefore, only 3 heat equations are needed to be solved\nat each step. Compared to the original Gauss-Seidel projection method, which solves heat\nequations 7 times at each step, savings of these two improved methods are about 2 =7 and\n4=7, which is veri\fed by both 1D and 3D examples for the same accuracy requirement. In\naddition, unconditional stability with respect to both the grid size and the damping parameter\nis con\frmed numerically. Application of both methods to a realistic material is also presented\nwith hysteresis loops and magnetization pro\fles.\nAcknowledgments\nThis work is supported in part by the grants NSFC 21602149 (J. Chen), NSFC 11501399\n(R. Du), the Hong Kong Research Grants Council (GRF grants 16302715, 16324416, 16303318\n12(a) Angle pro\fle ( \u000b= 0:1)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (b) Magnetization pro\fle ( \u000b= 0:1)\n(c) Angle pro\fle ( \u000b= 0:01)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (d) Magnetization pro\fle ( \u000b= 0:01)\nFig. 5. Simulation of the full Landau-Lifshitz-Gilbert equation using GSPM without any exter-\nnal \feld. The magnetization on the centered slice of the material in the xyplane is used. Top\nrow:\u000b= 0:1; Bottom row: \u000b= 0:01. Left column: a color plot of the angle between the in-plane\nmagnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization.\n13(a) Angle pro\fle ( \u000b= 0:1)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (b) Magnetization pro\fle ( \u000b= 0:1)\n(c) Angle pro\fle ( \u000b= 0:01)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (d) Magnetization pro\fle ( \u000b= 0:01)\nFig. 6. Simulation of the full Landau-Lifshitz-Gilbert equation using Scheme A without any\nexternal \feld. The magnetization on the centered slice of the material in the xyplane is used.\nTop row: \u000b= 0:1; Bottom row: \u000b= 0:01. Left column: a color plot of the angle between the in-\nplane magnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization.\n14(a) Angle pro\fle ( \u000b= 0:1)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (b) Magnetization pro\fle ( \u000b= 0:1)\n(c) Angle pro\fle ( \u000b= 0:01)\n0 0.2 0.4 0.6 0.8 1\nx (m)00.20.40.60.81y (m) (d) Magnetization pro\fle ( \u000b= 0:01)\nFig. 7. Simulation of the full Landau-Lifshitz-Gilbert equation using Scheme B without any\nexternal \feld. The magnetization on the centered slice of the material in the xyplane is used.\nTop row: \u000b= 0:1; Bottom row: \u000b= 0:01. Left column: a color plot of the angle between the in-\nplane magnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization.\n15and NSFC-RGC joint research grant N-HKUST620/15) (X.-P. Wang), and the Innovation\nProgram for postgraduates in Jiangsu province via grant KYCX19 1947 (C. Xie).\nReferences\n[1] L. Landau, E. Lifshitz, On the theory of the dispersion of magetic permeability in ferromagnetic bodies,\nPhys. Z. Sowjetunion 8 (1935) 153{169.\n[2] T. Gilbert, A lagrangian formulation of gyromagnetic equation of the magnetization \feld, Phys. Rev.\n100 (1955) 1243{1255.\n[3] W. F. B. Jr., Micromagnetics, Interscience Tracts on Physics and Astronomy, 1963.\n[4] P. Sulem, C. Sulem, C. Bardos, On the continuous limit limit for a system of classical spins, Comm.\nMath. Phys. 107 (1986) 431{454.\n[5] M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Di\u000berential Geom. 28 (1988)\n485{502.\n[6] I. \u0014Zuti\u0013 c, J. Fabian, S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys. 76\n(2004) 323{410.\n[7] M. Kruzik, A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism,\nSIAM Rev. 48 (2006) 439{483.\n[8] I. Cimr\u0013 ak, A survey on the numerics and computations for the Landau-Lifshitz equation of micromag-\nnetism, Arch. Comput. Methods Eng. 15 (2008) 277{309.\n[9] C. J. Garc\u0013 \u0010a-Cervera, Numerical micromagnetics: a review, Bol. Soc. Esp. Mat. Apl. 39 (2007) 103{135.\n[10] A. Fran\u0018 cois, J. Pascal, Convergence of a \fnite element discretization for the Landau-Lifshitz equations\nin micromagnetism, Math. Models Methods Appl. Sci. 16 (2006) 299{316.\n[11] A. Romeo, G. Finocchio, M. Carpentieri, L. Torres, G. Consolo, B. Azzerboni, A numerical solution of\nthe magnetization reversal modeling in a permalloy thin \flm using \ffth order runge-kutta method with\nadaptive step size control, Physica B. 403 (2008) 1163{1194.\n[12] Y. H, H. N, Implicit solution of the Landau-Lifshitz-Gilbert equation by the Crank-Nicolson method,\nJ. Magn. Soc. Japan 28 (2004) 924{931.\n[13] S. Bartels, P. Andreas, Convergence of an implicit \fnite element method for the Landau-Lifshitz-Gilbert\nequation, SIAM J. Numer. Anal. 44 (2006) 1405{1419.\n[14] A. Fuwa, T. Ishiwata, M. Tsutsumi, Finite di\u000berence scheme for the Landau-Lifshitz equation, Japan\nJ. Indust. Appl. Math. 29 (2012) 83{110.\n[15] X. Wang, C. J. Garc\u0013 \u0010a-Cervera, W. E, A gauss-seidel projection method for micromagnetics simulations,\nJ. Comput. Phys. 171 (2001) 357{372.\n[16] W. E, X. Wang, Numerical methods for the Landau-Lisfshitz equation, SIAM J. Numer. Anal. 38 (2000)\n1647{1665.\n[17] J. Chen, C. Wang, C. Xie, Convergence analysis of a second-order semi-implicit projection method for\nLandau-Lifshitz equation, arXiv 1902.09740 (2019).\n[18] C. J. Garc\u0013 \u0010a-Cervera, W. E, Improved gauss-seidel projection method for micromagnetics simulations,\nIEEE Trans. Magn. 39 (2003) 1766{1770.\n[19] I. Cimr\u0013 ak, Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation\nwith an exchange \feld, IMA J. Numer. Anal. (2005) 611{634.\n16"    },    {        "title": "1909.02738v2.The_interplay_of_large_two_magnon_ferromagnetic_resonance_linewidths_and_low_Gilbert_damping_in_Heusler_thin_films.pdf",        "content": "The interplay of large two-magnon ferromagnetic resonance linewidths and low\nGilbert damping in Heusler thin \flms\nW. K. Peria,1T. A. Peterson,1A. P. McFadden,2T. Qu,3C. Liu,1C. J. Palmstr\u001cm,2;4and P. A. Crowell1\n1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455\n2Department of Electrical & Computer Engineering,\nUniversity of California, Santa Barbara, California 93106\n3Department of Electrical and Computer Engineering,\nUniversity of Minnesota, Minneapolis, Minnesota 55455\n4Department of Materials, University of California, Santa Barbara, California 93106\nWe report on broadband ferromagnetic resonance linewidth measurements performed on epitaxial\nHeusler thin \flms. A large and anisotropic two-magnon scattering linewidth broadening is observed\nfor measurements with the magnetization lying in the \flm plane, while linewidth measurements with\nthe magnetization saturated perpendicular to the sample plane reveal low Gilbert damping constants\nof (1:5\u00060:1)\u000210\u00003, (1:8\u00060:2)\u000210\u00003, and<8\u000210\u00004for Co 2MnSi/MgO, Co 2MnAl/MgO, and\nCo2FeAl/MgO, respectively. The in-plane measurements are \ft to a model combining Gilbert and\ntwo-magnon scattering contributions to the linewidth, revealing a characteristic disorder lengthscale\nof 10-100 nm.\nI. INTRODUCTION\nThe theoretical understanding of the damping mech-\nanism believed to govern longitudinal magnetization re-\nlaxation in metallic ferromagnets, originally due to Kam-\nbersk\u0013 y [1, 2], has in recent years resulted in quantita-\ntive damping estimates for realistic transition metal band\nstructures [3{5]. Although of great interest where engi-\nneering of damping is desired [6], these calculations re-\nmain largely uncompared to experimental data. Kam-\nbersk\u0013 y damping may be characterized by the so-called\nGilbert damping constant \u000bin the Landau-Lifshitz-\nGilbert macrospin torque equation of motion, and for-\nmally describes how the spin-orbit interaction in itinerant\nelectron systems results in damping of magnetization dy-\nnamics [2]. Schoen et al. [7] have reported that \u000bis mini-\nmized for Co-Fe alloy compositions at which the density-\nof-states at the Fermi level is minimized, in reasonable\nagreement with Kambersk\u0013 y model predictions [8]. Fur-\nthermore, half-metallic, or near half-metallic ferromag-\nnets such as full-Heusler compounds have been predicted\nto demonstrate an ultralow Kambersk\u0013 y \u000b(\u001410\u00003) due\nto their spin-resolved band structure near the Fermi level\n[9]. Finally, anisotropy of the Kambersk\u0013 y damping in sin-\ngle crystals has been predicted, which is more robust for\nFermi surfaces with single-band character [5, 10].\nThe Gilbert damping constant is often reported\nthrough measurements of the ferromagnetic resonance\n(FMR) linewidth \u0001 H, which may be expressed as a sum\nof individual contributions\n\u0001H=2\u000bf\n\r+ \u0001H0+ \u0001HTMS; (1)\nwhere the \frst term is the Gilbert damping linewidth\n(fis the FMR frequency, \ris the gyromagnetic ratio),\n\u0001H0is a frequency-independent inhomogeneous broad-\nening, and \u0001 HTMS represents an extrinsic two-magnon\nscattering (TMS) linewidth contribution [11, 12] that is,\nin general, a nonlinear function of frequency. In recentyears it has been realized that TMS linewidths are per-\nvasive for the conventional in-plane geometry of thin \flm\nFMR measurements, requiring either the perpendicular-\nto-plane FMR geometry [13] (for which TMS processes\nare suppressed) or su\u000eciently broadband measurements\n[14] to extract the bare Gilbert \u000b. For instance, recent\nFMR linewidth studies on Heusler compounds have re-\nported distinct TMS linewidths [15, 16], which challenged\nsimple inference of the Gilbert \u000b.\nIn this article, we present FMR linewidth measure-\nments for epitaxial Heusler thin \flms for all principal ori-\nentations of the magnetization with respect to the sym-\nmetry axes. For the in-plane con\fguration, large and\nanisotropic TMS-dominated linewidths are observed. In\nthe perpendicular-to-plane con\fguration, for which the\nTMS process is inactive [11], the Gilbert \u000band inhomo-\ngeneous broadening are measured. We \fnd evidence of a\nlow (\u001810\u00003) Gilbert\u000bin these Heusler thin \flms, accom-\npanied by a large and anisotropic TMS contribution to\nthe linewdith for in-plane magnetization. We conclude\nby discussing the interplay of low Gilbert \u000band large\nTMS, and we emphasize the nature by which the TMS\nmay conceal the presence of anisotropic Kambersk\u0013 y \u000b.\nII. SAMPLES\nThe Heusler alloy \flms used for these measurements\nwere grown by molecular beam epitaxy (MBE) by co-\nevaporation of elemental sources in ultrahigh vacuum\n(UHV). The MgO(001) substrates were annealed at\n700\u000eC in UHV followed by growth of a 20 nm thick MgO\nbu\u000ber layer by e-beam evaporation at a substrate temper-\nature of 630\u000eC. The 10 nm thick Co 2MnAl and Co 2MnSi\n\flms were grown on the MgO bu\u000ber layers at room tem-\nperature and then annealed at 600\u000eC for 15 minutes\nin situ in order to improve crystalline order and surface\nmorphology. The 24 nm thick Co 2FeAl sample was grown\nusing the same MgO substrate and bu\u000ber layer prepa-arXiv:1909.02738v2  [cond-mat.mtrl-sci]  9 Apr 20202\nration, but at a substrate temperature of 250\u000eC with\nno post-growth anneal. Re\rection high energy electron\ndi\u000braction (RHEED) was monitored during and after\ngrowth of all samples and con\frmed the expected epitax-\nial relationship of MgO(001) h110ijjHeusler(001)h100i.\nX-ray di\u000braction (XRD) demonstrated the existence of\na single phase of (001)-oriented Heusler, along with the\npresence of the (002) re\rection, con\frming at least B2\nordering in all cases. In addition, for the Co 2MnSi \flm\nonly, the (111) re\rection was observed, indicating L2 1\nordering [see Fig. 1(a)]. All of the \flms were capped\nwith several nm of e-beam evaporated AlOx for pas-\nsivation prior to atmospheric exposure. The e\u000bective\nmagnetization for the 24 nm thick Co 2FeAl \flm was\ndetermined from anomalous Hall e\u000bect saturation \feld\nto be 1200 emu/cm3, which is consistent with measure-\nments of Ref. [17] for L2 1or B2-ordered \flms, along\nwith 990 emu/cm3and 930 emu/cm3for the Co 2MnSi\nand Co 2MnAl \flms, respectively. Hereafter, we will refer\nto the Co 2MnSi(10 nm)/MgO as the \\CMS\" \flm, the\nCo2MnAl(10 nm)/MgO \flm as the \\CMA\" \flm, and the\nCo2FeAl(24 nm)/MgO \flm as the \\CFA\" \flm.\nIII. EXPERIMENT\nBroadband FMR linewidth measurements were per-\nformed at room temperature with a coplanar waveguide\n(CPW) transmission setup, similar to that discussed in\ndetail in Refs. [18, 19], placed between the pole faces\nof an electromagnet. A cleaved piece of the sample\n(\u00182 mm\u00021 mm) was placed face-down over the center-\nline of the CPW. A rectifying diode was used to detect\nthe transmitted microwave power, and a \u0018100 Hz mag-\nnetic \feld modulation was used for lock-in detection of\nthe transmitted power, resulting in a signal /d\u001f=dH\n(where\u001fis the \flm dynamic magnetic susceptibility).\nThe excitation frequency could be varied from 0-50 GHz,\nand a microwave power near 0 dBm was typically used. It\nwas veri\fed that all measurements discussed in this arti-\ncle were in the small precession cone angle, linear regime.\nThe orientation of the applied magnetic \feld could be\nrotated to arbitrary angle in the \flm plane (IP), or ap-\nplied perpendicular to the \flm plane (PP). We empha-\nsize again that TMS contributions are suppressed in the\nPP con\fguration [12]. The resonance \felds were \ft as a\nfunction of applied frequency in order to extract various\nmagnetic properties of the \flms.\nThe magnetic free energy per unit volume used to gen-\nerate the resonance conditions for these samples is given\nby\nFM=\u0000M\u0001H+K1sin2\u001ecos2\u001e+ 2\u0019M2\neffcos2\u0012;(2)\nwhere His the applied \feld, \u001eand\u0012are the azimuthal\nand polar angles of the magnetization, respectively, K1\nis a \frst order in-plane cubic anisotropy constant, and\n4\u0019Meffis the PP saturation \feld, which includes the\nusual demagnetization energy and a \frst order uniaxial\n-150-75075150-6-30361\n0 GHz20 GHz30 GHzdχ/dH (arb. u.)H\n - HFMR (Oe)40 GHzCMS\n090180270360101102103104 <111> \n<202>Intensity (arb. u.)φ\n (°)CMS\n-5000 5 00-101F\nield (Oe)M/MSH\n || 〈110〉H\n || 〈100〉H\n || 〈110〉CFA\n(d)(b)(\nc)(a)C\nFAFIG. 1. (a) Wide-angle x-ray di\u000braction \u001e-scans ofh202i\n(blue) andh111i(red) peaks for the CMS \flm. (b) Typical\nderivative susceptibility lineshapes for these samples at dif-\nferent microwave excitation frequencies. The \fts are shown\nas solid lines. (c) In-plane hysteresis loops for CFA obtained\nwith a vibrating-sample magnetometer (VSM). (d) Atomic\nforce microscopy (AFM) image of surface topography for\nCFA. RMS roughness is 0.2 nm.\nanisotropy due to interfacial e\u000bects. The parameters ob-\ntained by \ftting to Eq. 2 are shown in Table I. The uncer-\ntainty in these parameters was estimated by measuring a\nrange of di\u000berent sample pieces, and using the standard\ndeviation of the values as the error bar. The long-range\ninhomogeneity characteristic of epitaxial samples makes\nthis a more accurate estimate of the uncertainty than\nthe \ftting error. The magnetic-\feld-swept FMR line-\nshapes were \ft to the derivative of Lorentzian functions\n[19] in order to extract the full-width at half-maximum\nlinewidths \u0001 H[magnetic \feld units, Fig. 1(b)], which\nare the focus of this article. The maximum resonant fre-\nquency was determined by the maximum magnetic \feld\nthat could be applied for both IP and PP electromagnet\ncon\fgurations, which was 10.6 kOe and 29 kOe, respec-\ntively. For the IP measurement, the angle of the applied\n\feld in the plane of the \flm was varied to determine the\nin-plane magnetocrystalline anisotropy of our samples,\nwhich was fourfold-symmetric for the three \flms char-\nacterized in this article. The anisotropy was con\frmed\nusing vibrating-sample magnetometry (VSM) measure-\nments, an example of which is shown in Fig. 1(c), which\nshows IP easy and hard axis hysteresis loops for the\nCFA \flm. For the PP measurement, alignment was ver-\ni\fed to within\u00180.1\u000eto ensure magnetization saturation\njust above the PP anisotropy \feld, thus minimizing \feld-3\nTABLE I. Summary of the magnetic properties extracted from the dependence of the resonance \feld on applied frequency\nfor both \feld in-plane ( jj) and \feld perpendicular-to-plane ( ?) con\fgurations, along with the Gilbert \u000band inhomogeneous\nbroadening from the perpendicular-to-plane con\fguration. 2 K1=Msand 4\u0019Meffare the in-plane and perpendicular-to-plane\nanisotropy \felds, respectively (see Eq. 2), and gis the Land\u0013 e g-factor.\nSample 2 K1=Ms(Oe) 4\u0019Mjj\neff(kOe) 4 \u0019M?\neff(kOe) gjjg?\u000b001(\u000210\u00003) \u0001H0(Oe)\nCMS 280 12.3 13.3 2.04 2.04 1 :5\u00060:1 9\u00061\nCMA 35 11.3 11.7 2.06 2.08 1 :8\u00060:2 12\u00063\nCFA 230 15.1 15.5 2.06 2.07 <0.8 100\u00066\nCFA 500\u000eC anneal N/A N/A 15.1 N/A 2.07 1 :1\u00060:1 45\u00061\n01 02 03 04 05 00306090120α\n001 = 1.1×10-3CFA 500 °C annealCFAC\nMAα001 < 8×10-4α\n001 = 1.8×10-3ΔH (Oe)F\nrequency (GHz)α001 = 1.5×10-3CMS\nFIG. 2. Linewidths as a function of frequency with the \feld\napplied perpendicular to plane, for which two-magnon scat-\ntering is inactive. The black squares are data for the CMS\n\flm, the red circles are for the CMA \flm, and the blue trian-\ngles are for the CFA \flm. In addition, linewidths are shown\nfor a CFA \flm that was annealed at 500\u000eCex situ (magenta\ndiamonds). Corresponding linear \fts are shown along with\nthe extracted Gilbert damping factor \u000b. The blue dashed\nlines indicate an upper bound of \u000b001= 8\u000210\u00004and a lower\nbound of\u000b001= 0 for CFA.\ndragging contributions to the linewidth.\nIV. RESULTS AND ANALYSIS\nA. Perpendicular-to-plane linewidths\nFirst we discuss the results of the PP measurement. As\nstated in Sec. III, the TMS extrinsic broadening mecha-\nnism is suppressed when the magnetization is normal to\nthe plane of the \flm. We can thus \ft our data to Eq. 1\nwith \u0001HTMS = 0, greatly simplifying the extraction of\nthe Gilbert damping constant \u000band the inhomogeneous\nbroadening \u0001 H0. Prior knowledge of \u0001 H0is particu-\nlarly important for constraining the analysis of the IP\nmeasurements, as we shall discuss.\n3035404550C\nMS〈100〉C\nMS〈110〉ΔH (Oe)CMS2\n0 GHz2\n800300032003400H\nFMR(Oe)-\n450 4 5200400600C\nFA〈100〉ΔH (Oe)A\nngle (°)CFA2\n0 GHzCFA〈110〉2\n600280030003200H\nFMR(Oe)50100150200(a)(\nc)C\nMA〈100〉C\nMA〈110〉ΔH (Oe)CMA1\n5 GHz(b)2\n00020502100H\nFMR(Oe)FIG. 3. Azimuthal angular dependence of the linewidths (left\nordinate, blue circles) and resonance \felds (right ordinate,\nblack squares) for (a) CMS, (b) CMA, and (c) CFA. The\nexcitation frequency was 20 GHz for CMS, 15 GHz for CMA,\nand 20 GHz for CFA. The solid lines are sinusoidal \fts.\nThe dependence of \u0001 Hon frequency for the CMS,\nCMA, and CFA \flms in the PP con\fguration is\nsummarized in Fig. 2, in which \fts to Eq. 1 are\nshown with \u0001 HTMS set to zero. For the CMS\n\flm,\u000b001= (1:5\u00060:1)\u000210\u00003and \u0001H0= 9 Oe,\nwhile for the CMA \flm \u000b001= (1:8\u00060:2)\u000210\u000034\nand \u0001H0= 12 Oe. Co 2MnSi 2=3Al1=3/MgO and\nCo2MnSi 1=3Al2=3/MgO \flms (both 10 nm thick) were\nalso measured, with Gilbert damping values of \u000b001=\n(1:8\u00060:2)\u000210\u00003and\u000b001= (1:5\u00060:1)\u000210\u00003, re-\nspectively (not shown). For CFA, we obtained a damp-\ning value of \u000b001= 3\u000210\u00004with an upper bound of\n\u000b001<8\u000210\u00004and \u0001H0= 100 Oe. These \ft param-\neters are also contained in Table I. The source of the\nlarge inhomogeneous broadening for the CFA \flm is un-\nclear: AFM measurements [Fig. 1(d)] along with XRD in-\ndicate that the \flm is both crystalline and smooth. Note\nthat the range of frequencies shown in Fig. 2 are largely\ngoverned by considerations involving the Kittel equation\n[20]: measurements below 10 GHz were not used due to\nthe increasing in\ruence of slight misalignment on \u0001 H\n(through \feld-dragging) for resonant \felds just above\nthe saturation value. A piece of the CFA sample was\nannealed at 500\u000eCex situ , which reduced the inhomoge-\nnoeus broadening to \u001845 Oe (still a relatively large value)\nand increased the Gilbert damping to \u000b001= 1:1\u000210\u00003\n(similar behavior in CFA was seen in Ref. [21]). The\nconstraint of \u000b001<8\u000210\u00004is among the lowest of re-\nported Gilbert damping constants for metallic ferromag-\nnets, but the \u000b\u001810\u00004range is not unexpected based\non Kambersk\u0013 y model calculations performed for similar\nfull-Heusler compounds [9] or other recent experimental\nreports [22, 23]. It should be noted that Schoen et al. [7]\nhave recently reported \u000b= 5\u000210\u00004for Co 25Fe75thin\n\flms, where spin pumping and radiative damping con-\ntributions were subtracted from the raw measurement.\nSpin pumping contributions to the intrinsic damping are\nnot signi\fcant in our \flms, as no heavy-metal seed layers\nhave been used and the \flms have thicknesses of 10 nm\nor greater. For the radiative damping contribution [13]\nin the geometry of our CPW and sample, we calculate\ncontributions \u000brad<\u00181\u000210\u00004, which is below the uncer-\ntainty in our damping \ft parameter.\nB. In-plane linewidths\nWith the intrinsic damping and inhomogeneous broad-\nening characterized by the PP measurement, we turn our\nattention to the IP linewidth measurements, for which\nTMS contributions are present. For hard-axis measure-\nments, frequencies <\u00185 GHz were not used due to the\nin\ruence of slight magnetic \feld misalignment on the\nlinewidths. For easy-axis measurements, the lower limit\nis determined by the zero-\feld FMR frequency. Fig-\nure 3 shows the dependences of the resonance \felds and\nlinewidths on the angle of the in-plane \feld. An im-\nportant observation seen in Fig. 3 is that the linewidth\nextrema are commensurate with those of the resonance\n\felds and therefore the magnetocrystalline anisotropy en-\nergy. This rules out \feld-dragging and mosaicity contri-\nbutions to the linewidth, which can occur when the reso-\nnance \feld depends strongly on angle [24]. We note that\nsimilar IP angular dependence of the FMR linewidth,\n01 02 03 04 05 002004006008000204060801000\n60120180240300C\nFA(c)ΔH (Oe)F\nrequency (GHz)〈100〉ΔH (Oe)(a)C\nMSC\nMA[\n001]〈110〉(b)ΔH (Oe)FIG. 4. Linewidths along all three principal directions for\nCMS (a), CMA (b), and CFA (c). Heusler crystalline axes\nare labeled byh100i(black),h110i(red), and [001] (blue). In\nall three cases,h110iis the in-plane easy axis and h100iis the\nin-plane hard axis. The corresponding \fts are shown as the\nsolid curves, where the in-plane linewidths are \ft using Eq. 3\nand the out-of-plane linewidths are \ft to the Gilbert damping\nmodel. The \ft parameters are given in Table II.\nwhich was attributed to an anisotropic TMS mechanism\ncaused by a rectangular array of mis\ft dislocations, has\nbeen reported by Kurebayashi et al. [25] and Woltersdorf\nand Heinrich [14] for epitaxial Fe/GaAs(001) ultrathin\n\flms.\nTo further study the anisotropy of the IP \u0001 Hin our\n\flms, we have measured \u0001 Hat the angles correspond-\ning to the extrema of HFMR (and \u0001H) in Fig. 3 over\na range of frequencies. These data are shown in Fig. 4,\nalong with the PP ([001]) measurements for each sample.\nA distinguishing feature of the data shown in Fig. 4 is the\nsigni\fcant deviation between IP and PP linewidths in all\nbut one case (CMS h100i). Large and nonlinear frequency\ndependence of the IP linewidths is strongly suggestive of\nan active TMS linewidth broadening mechanism. In the\npresence of TMS, careful analysis is required to separate5\n0°90°1\n80°2\n70°01x10501 02 03 04 00.00.51.01.52\nx1045x1041\n1.411.82\n4 GHz32 GHz1\n6 GHz |q2M| (cm-1)ξ\n-1ξ = 100 nm10-41\n0-35\n×10-3ΔHTMS/H'2 (Oe-1) (×10-4)f\nFMR (GHz)α = 10-2(\nq || M)(q ^ M)(b) |q| (cm-1)q\n ^ Mωm (GHz)q\n || MD\negeneracyH = 1 kOe(a)\nFIG. 5. (a) Two-magnon scattering linewidth contribution\nfor values of Gilbert damping \u000b= 10\u00002;5\u000210\u00003;10\u00003;and\n10\u00004. The inset shows magnon dispersions for an applied\n\feld ofH= 1 kOe. (b) Contours of the degenerate mode\nwavenumber q2Min the \flm plane as a function of wavevector\nangle relative to the magnetization for fFMR = 16, 24, and\n32 GHz. The dashed circle indicates the wavenumber of a\ndefect with size \u0018= 100 nm.\nthe Gilbert damping from the TMS linewidth contribu-\ntions. We therefore describe the TMS mechanism in more\ndetail in the following section in order to analyze the IP\nlinewidths in Fig. 4 and extract the Gilbert damping.\nC. Two-magnon scattering model\nThe TMS mechanism leads to a characteristic nonlin-\near frequency dependence of \u0001 H[11, 12]. In Fig. 4,\nthe IP \u0001His not a linear function of frequency, but\npossesses the \\knee\" behavior characteristic of the fre-\nquency dependence of linewidths dominated by the TMS\nmechanism. We have \ft our data to the TMS model\ndescribed by McMichael and Krivosik [12], in which theTMS linewidth \u0001 HTMS is given by [26, 27]\n\u0001HTMS =\r2\u00182H02\ndf=dHjfFMRZ\n\u00000qCq(\u0018)\u000e\u000b(!\u0000!q)d2q;(3)\nwhere \u0000 0qis the defect-mediated interaction term be-\ntween magnons at wavevector 0 and q,Cq(\u0018) = (1 +\n(q\u0018)2)\u00003=2is the correlation function of the magnetic sys-\ntem with correlation length \u0018, andH0is the magnitude\nof the characteristic inhomogeneity (units of magnetic\n\feld). The \u000e\u000b-function in Eq. 3 selects only the magnon\nscattering channels that conserve energy. In the limit of\nzero intrinsic damping, it is identical to the Dirac delta\nfunction, but for \fnite \u000bit is replaced by a Lorentzian\nfunction of width \u000e!= (2\u000b!=\r )d!=dH . The magnon\ndispersion relation determining !qis the usual Damon-\nEshbach thin \flm result [26, 28] with the addition of mag-\nnetocrystalline anisotropy sti\u000bness \feld terms extracted\nfrom the dependence of the resonance \feld on the applied\nfrequency for the IP con\fguration. The \flm thickness\nda\u000bects the states available for two-magnon scattering\nthrough the dispersion relation, namely, the linear term\nwhich gives rise to negative group velocity for small q\n(/\u0000qd). The IP FMR linewidth data shown in Fig. 4\nwere \ft to Eq. 1 (with Eq. 3 used to evaluate \u0001 HTMS)\nwith\u0018,\u000b, andH0as \ftting parameters (shown in Table\nII). The correlation length \u0018remains approximately con-\nstant for di\u000berent in-plane directions, while the strength\nH0is larger for theh100idirections in the CMA and CFA\nsamples and the h110idirections in the CMS sample.\nSome degree of uncertainty results from this \ftting proce-\ndure, because for linewidth data collected over a limited\nfrequency range, \u0018and\u000bare not completely decoupled\nas \ftting parameters. In absolute terms, however, the\nlargest systematic errors come from the exchange sti\u000b-\nness, which is not well-known. The error bars given in\nTable II were calculated by varying the exchange sti\u000b-\nness over the range 400 meV \u0017A2to 800 meV \u0017A2, and\nrecording the change in the \ft parameters. This range of\nvalues was chosen based on previous Brillouin light scat-\ntering measurements of the exchange sti\u000bness in similar\nHeusler compounds [29, 30]. In addition, we note that\nin Eq. 1 \u0001 H0is taken to be isotropic, with the value\ngiven by the PP linewidth measurements shown in Fig.\n2. Although certain realizations of inhomogeneity may\nresult in an anisotropic \u0001 H0(see Ref. [14] for a good\ndiscussion), doing so here would only serve to create an\nadditional \ftting parameter.\nD. E\u000bect of low intrinsic damping\nThe e\u000bect of low intrinsic damping on the two-magnon\nlinewidth can be seen in Fig. 5(a). As \u000bdecreases, with\nall other parameters \fxed, \u0001 HTMS steadily increases\nand becomes increasingly nonlinear (and eventually non-\nmonotonic) with frequency. In particular, a \\knee\" in\nthe frequency dependence becomes more pronounced for6\nTABLE II. Summary of the \ftting parameters used to \ft the\nin-plane data of Fig. 4 (black squares and red circles) to Eqs.\n1 and 3. CFA refers to the unannealed Co 2FeAl sample.\nSample (Field Direction) \u000b(\u000210\u00003)\u0018(nm)H0(Oe)\nCMSh110i 1:6\u00060:2 40\u000625 55\u000630\nCMSh100i 1:5\u00060:1 40\u000625 30\u000615\nCMAh110i 3:1\u00060:2 70\u000620 30\u00065\nCMAh100i 4:7\u00060:4 55\u000610 90\u00065\nCFAh110i 2:0\u00060:3 20\u000610 175\u000660\nCFAh100i N/A N/A N/A\nlow damping (see e.g. Fig. 5(a) curve for \u000b= 10\u00004). The\nphysics giving rise to the knee behavior is illustrated in\nFig. 5(b). The TMS process scatters magnons from zero\nto non-zero wavevector at small q. There is assuemd\nto be su\u000ecient disorder to allow for the momentum q\nto be transferred to the magnon system. There will al-\nways be, however, a length scale \u0018below which the disor-\nder decreases, so that the \flm becomes e\u000bectively more\nuniform at large wavevectors. The corresponding FMR\nfrequencies are those for which the contours of constant\nfrequency (the \fgure eights in Fig. 5) in q-space have ex-\ntrema atq\u0018\u0018\u00001. The TMS rate is also determined by\nthe interplay of the magnon density of states, the e\u000bec-\ntive area in q-space occupied by the modes that conserve\nenergy, and the Gilbert damping. The knee behavior is\nmore pronounced for low \u000bdue to the increased weight\nof the van Hove singularity coming from the tips of the\n\fgure eights, in the integrand of Eq. 3. Although a larger\nwindow of energies, set by the width of \u000e\u000b, is available for\nlarger\u000b, this smears out the singularity in the magnon\ndensity of states, removing the sharp knee in the TMS\nlinewidth as a function of frequency. The PP measure-\nment con\frms that all of these epitaxial Heusler \flms lie\nwithin the range \u000b<2\u000210\u00003. Ferromagnetic \flms with\nultralow\u000bare therefore increasingly prone to large TMS\nlinewidths (particularly for metals with large Ms). The\nTMS linewidths will also constitute a larger fraction of\nthe total linewidth due to a smaller contribution from the\nGilbert damping. In practice, this is why experimental\nreports [7, 22, 23] of ultralow \u000bhave almost all utilized\nthe PP geometry.\nE. Discussion\nThe results of the IP linewidth \fts to Eqs. 1 and 3 are\nsummarized in Table II. In the case of CMS, the high-\nfrequency slopes in Fig. 4(a) approach the same value\nalong each direction, as would be expected when the fre-\nquency is large enough for the TMS wavevector to exceed\nthe inverse of any defect correlation length. In this limit,\n\u000bis isotropic (within error limits).\nNext, we discuss the CMA IP data shown in Fig. 4(b)\nand Table II. It is clear from this \fgure that a good \ftcan be obtained along both h100iandh110idirections.\nIn Table II it can be seen that the value of the defect cor-\nrelation length \u0018is approximately the same along both\ndirections. However, the values of \u000bwe obtain from \ft-\nting to Eqs. 1 and 3 do not agree well with the PP value\nof\u000b001= 1:8\u000210\u00003(Fig. 2). Anisotropic values of \u000b\nhave been both predicted [5, 10] and observed [31], and\nan anisotropic \u000bis possibly the explanation of our best-\n\ft results. The in-plane h100iand [001] directions are\nequivalent in the bulk, so the anisotropy would neces-\nsarily be due to an interface anisotropy energy [31] or\nperhaps a tetragonal distortion due to strain [32].\nFinally, we discuss the CFA linewidths shown in Fig.\n4(c) and Table II. This sample has by far the largest two-\nmagnon scattering contribution, which is likely related\nto the anomalously large inhomogeneous broadening and\nlow intrinsic damping [see Fig. 5(a)] observed in the PP\nmeasurement. A good \ft of the data was obtained when\nthe \feld was applied along the h110idirection. Notably,\nthe IPh110ibest \ft value of 2 :1\u000210\u00003is nearly a factor\nof 3 larger than the \u000b001upper bound on the same sample\n(Table I), strongly suggesting an anisotropic Gilbert \u000b. A\nstriking anisotropy in the IP linewidth was revealed upon\nrotating the magnetization to the h100iorientation. For\ntheh100icase, which yielded the largest TMS linewidths\nmeasured in this family of \flms, we were not able to \ft\nthe data to Eq. 3 using a set of physically reasonable in-\nput parameters. We believe that this is related to the\nconsideration that higher order terms in the inhomoge-\nneous magnetic energy (see Ref. [26]) need to be taken\ninto account. Another reason why this may be the case is\nthat the model of McMichael and Krivosik [12] assumes\nthe inhomogeneities to be grain-like, whereas the samples\nare epitaxial [see Fig. 1(a)]. Atomic force microscopy im-\nages of these samples [Fig. 1(d)] imply that grains, if they\nexist, are much larger than the defect correlation lengths\nlisted in Table II, which are of order 10's of nm. We also\nnote that there does not appear to be a correlation be-\ntween the strength of two-magnon scattering H0and the\ncubic anisotropy \feld 2 K1=Ms, which would be expected\nfor grain-induced two-magnon scattering.\nV. SUMMARY AND CONCLUSION\nWe conclude by discussing the successes and limita-\ntions of the McMichael and Krivosik [12] model in an-\nalyzing our epitaxial Heusler \flm FMR linewidth data.\nWe have shown that two-magnon scattering is the ex-\ntrinsic linewidth-broadening mechanism in our samples.\nAny model which takes this as its starting point will\npredict much of the qualitative behavior we observe,\nsuch as the knee in the frequency dependence and the\nlarge linewidths IP for low \u000b\flms. The TMS model\nused in this article (for the purpose of separating TMS\nand Gilbert linewidth contributions) is, however, only\nas accurate as its representation of the inhomogeneous\nmagnetic \feld and the underlying assumption for the7\nfunctional form of Cq(\u0018). Grain-like defects are as-\nsumed, which essentially give a random magnetocrys-\ntalline anisotropy \feld. We did not, however, explicitly\nobserve grains in our samples with AFM, at least below\nlengthscales of\u001810\u0016m [Fig. 1(d)]. Mis\ft dislocations, a\nmuch more likely candidate in our opinion, would cause\nan e\u000bective inhomogeneous magnetic \feld which could\nhave a more complicated spatial pro\fle and therefore\nlead to anisotropic two-magnon scattering (see Ref. [14]).\nThe perturbative nature of the model also brings its own\nlimitations, and we believe that the CFA h100idata, for\nwhich we cannot obtain a satisfactory \ft, are exemplary\nof a breakdown in the model for strong TMS. Future\nwork should go into methods of treating the two-magnon\nscattering di\u000berently based on the type of crystalline de-\nfects present, which will in turn allow for a more reli-\nable extraction of the Gilbert damping \u000band facilitate\nthe observation of anisotropic Gilbert damping, enabling\nquantitative comparison to \frst-principles calculations.\nRegardless of the limitations of the model, we empha-\nsize three critical observations drawn from the linewidth\nmeasurements presented in this article. First, in all cases\nwe observe large and anisotropic TMS linewidth contri-\nbutions, which imply inhomogeneity correlation length-\nscales of order tens-to-hundreds of nanometers. The mi-\ncroscopic origin of these inhomogeneities is the subjectof ongoing work, but are likely caused by arrays of mis\ft\ndislocations [14]. The relatively large lengthscale of these\ndefects may cause them to be easily overlooked in epi-\ntaxial \flm characterization techniques such as XRD and\ncross-sectional HAADF-STEM, but they still strongly in-\n\ruence magnetization dynamics. These defects and their\nin\ruence on the FMR linewidth through TMS complicate\ndirect observation of Kambersk\u0013 y's model for anisotropic\nand (in the case of Heusler compounds) ultralow intrinsic\ndamping in metallic ferromagnets. Second, we observed\nlow intrinsic damping through our PP measurement,\nwhich was<2\u000210\u00003for all of our samples. Finally, we\nhave presented the mechanism by which FMR linewidths\nin ultralow damping \flms are particularly likely to be en-\nhanced by TMS, the anisotropy of which may dominate\nany underlying anisotropic Kambersk\u0013 y damping.\nThis work was supported by NSF under DMR-1708287\nand by SMART, a center funded by nCORE, a Semi-\nconductor Research Corporation program sponsored by\nNIST. 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The terahertz magnetic field induces a large change (~40%) in the spontaneous magnetization. The frequency of the antife rromagnetic resonance \ndecreases in proportion to the square of the magnetization change. A modified \nLandau-Lifshitz-Gilbert equation with a phenomenological nonlinear damping term \nquantitatively reproduced the nonlinear dynamics. \nPACS: 75.78.Jp, 76.50.+g, 78.47.-p, 78.67.Pt \n    21. Introduction \nUltrafast control of magnetization dynamics by a femtosecond optical laser pulse has \nattracted considerable attention from the persp ective of fundamental physics and technological \napplications of magnetic recording and inform ation processing [1]. The first observation of \nsubpicosecond demagnetization of a fe rromagnetic nickel film demonstrated that a femtosecond \nlaser pulse is a powerful stimulus of ultrafast magnetization dynamics [2], and it has led to numerous theoretical and experimental inves tigations on metallic and semiconducting magnets \n[3-8]. The electronic state created by the laser pulse has a strongly nonequilibrium distribution \nof free electrons, which consequently leads to  demagnetization or even magnetic reversal \n[1,2,9-11]. However, the speed of the magnetizat ion change is limited by the slow thermal \nrelaxation and diffusion, and an alternative t echnique without the limits of such a thermal \ncontrol and without excessive thermal energy would be desirable.  \nIn dielectric magnetic media, carrier heating hardly occurs, since no free electrons are present \n[12]. Consequently, great effort has been devoted  to clarifying the spin dynamics in magnetic \ndielectrics by means of femtosecond laser pulses. A typical method for nonthermal optical \ncontrol of magnetism is the inverse Faraday effect, where circularly polarized intense laser \nirradiation induces an effective magnetic field in  the medium. Recently, new optical excitation \nmethods avoiding the thermal effect such as the ma gneto-acoustic effect is also reported [13,14]. \nIn particular, these techniques have been used in  many studies on antiferromagnetic dielectrics \nbecause compared with ferromagne ts, antiferromagnets have inhe rently higher spin precessional \nfrequencies that extend into the terahertz (THz) regime [12,15]. Additionally, ultrafast \nmanipulation of the antiferromagnetic order parame ter may be exploited in order to control the \nmagnetization of an adjacent ferromagnet through the exchange interaction [16]. The THz wave \ngeneration technique is possibly a new way of optical spin control through direct magnetic \nexcitation without undesirable thermal effects [17-19]. As yet however, no technique has been \nsuccessful in driving magnetic motion excited directly by a magnetic field into a nonlinear \ndynamics regime that would presumably be fo llowed by a magnetization reversal [20-22]. \n \nIn our previous work [23], we demonstrated that the THz magnetic field can be resonantly \nenhanced with a split ring resonator (SRR) and may become a tool for the efficient excitation of \na magnetic resonance mode of antiferromagnetic dielectric HoFeO\n3. We applied a Faraday \nrotation technique to detect the magnetization ch ange but the observed Faraday signal averaged  3the information about inhomogeneous magnetiza tion induced by localized THz magnetic field \nof the SRR over the sample thickness [23]. In th is Letter, we have developed a time-resolved \nmagneto-optical Kerr effect (MOKE) micr oscopy in order to access the extremely \nfield-enhanced region, sample surface near th e SRR structure. As a result, the magnetic \nresponse deviates from the linear response in the strong THz magnetic field regime, remarkably \nshowing a redshift of the antiferromagnetic r esonance frequency that is proportional to the \nsquare of the magnetization change. The observe d nonlinear dynamics could be reproduced with \na modified Landau-Lifshitz-Gilbert (LLG) e quation having an additional phenomenological \nnonlinear damping term.  \n2. Experimental setup \nFigure 1 shows the experimental setup of MOKE microscopy with a THz pump pulse \nexcitation. Intense single-cycle THz pulses were  generated by optical rectification of \nnear-infrared (NIR) pulses in a LiNbO\n3 crystal [24-26]; the maximum peak electric field was \n610 kV/cm at focus. The sample was a HoFeO 3 single crystal polished to a thickness of 145 µm, \nwith a c-cut surface in the Pbnm  setting [27]. (The x-, y-, and z-axes are parallel to the \ncrystallographic a-, b-, and c-axes, respectively. ) Before the THz pump excitation, we applied a \nDC magnetic field to the sample to saturate its magnetization along the crystallographic c-axis. We fabricated an array of SRRs on the crystal surface by using gold with a thickness of 250 nm. \nThe incident THz electric field, parallel to the metallic arm with the SRR gap (the x-axis), drove \na circulating current that resulted in a strong  magnetic near-field normal to the crystal surface \n[23,28,29]. The SRR is essentially subwavelength LC circuit, and the current induces magnetic \nfield B\nnr oscillating with the LC resonance frequency (the Q-factor is around 4). The right side \nof the inset in figure 1 shows the spatial distribut ion of the magnetic field of the SRR at the LC \nresonance frequency as calculated by the fin ite-difference time-domain (FDTD) method. \nAround the corner the current density in the metal is very high, inducing the extremely \nenhanced magnetic field in the HoFeO 3 [29]. \n \nAt room temperature, the two magnetizations mi (i=1,2) of the different iron sublattices in \nHoFeO 3 are almost antiferromagnetically aligned along the x-axis with a slight canting angle \n0(=0.63°) owing to the Dzyaloshinskii fiel d and form a spontaneous magnetization MS along \nthe z-axis [30]. In the THz region, ther e are two antiferromagnetic resonance modes \n(quasiantiferromagnetic (AF) and quasiferroma gnetic (F) mode [31]). The magnetic field Bnr  4generated along the z-axis in our setup causes AF-mode motion; as illustrated in figure 2(a), the \nZeeman torque pulls the spins along the y-ax is, thereby triggering precessional motions of mi \nabout the equilibrium directions. The precessional motions cause the macroscopic magnetization M=m\n1+m2 to oscillate in the z-direction [32,33]. The resultant magnetization \nchange Mz(t) modulates the anti-symmetric off-diagonal element of the dielectric tensor \nεxyaሺൌ െεyxaሻ and induces a MOKE signal (Kerr ellipticity change \u001f [34,35] (see Appendix A \nfor the detection scheme of the MOKE measuremen t). The F-mode oscillation is also excited by \nTHz magnetic field along the x or y-axis. Howe ver, the magnetization deviations associated \nwith the F-mode, Mx and My, do not contribute to the MOKE in our experimental geometry, \nwhere the probe light was incident no rmal to the c-cut surface of HoFeO 3 (the xy-plane) [34,35]. \nIn addition, the amplitude of the F-mode is much smaller than AF-mode because the F-mode \nresonance frequency ( F~0.37 THz) differs from the LC resonance frequency ( LC~0.56 THz). \n-10010x position (µm)\n-10 0 10\ny position (µm)\nTHz pump HoFeO3 \nz y \nx \nSRR  \nObjective lens \nNonpolarized \nbeam splitter Quarter \nwave plateWollaston  \nprism \nLens  Balanced photodiodes \nVisible probe Bin Bnr \nEin \n-10 0 10\ny position (µm)\n120\n80\n40\n0\nFigure 1. Schematic setup of THz pump-visible MOKE measurement. The left side\nof the inset shows the photograph of SRR fabricated on the c-cut surface of the\nHoFeO 3 crystal and the white solid line indicat es the edge of the SRR. The red soli d\nand blue dashed circles indicate the probe spots for the MOKE measurement. The\nright side of the inset shows the spatia l distribution of the enhancement facto r\ncalculated by the FDTD method, i.e., the ratio between the Fourier amplitude at LC\nof the z-component of Bnr (at z=0) and the incident THz pulse Bin.  5To detect the magnetization change induced onl y by the enhanced magnetic field, the MOKE \nsignal just around the corner of the SRR (indicated by the red circle in figure 1’s inset), where \nthe magnetic field is enhanced 50-fold at the LC  resonance frequency, was measured with a 400 \nnm probe pulse focused by an objective lens (spot diameter of ~1.5 µm). Furthermore, although \nthe magnetic field reaches a maximum at the surface and decreases along the z-axis with a \ndecay length of lTHz~5 µm, the MOKE measurement in refl ection geometry, in contrast to the \nFaraday measurement in transmission [23], can evaluate the magnetization change induced only \nby the enhanced magnetic field around the sample  surface since the penetration depth of 400 nm \nprobe light for typical orthoferrites is on the orde r of tens of nm [35]. (The optical refractive \nindices of rare-earth orthoferrites in th e near ultraviolet region including HoFeO 3 are similar to \neach other, regardless of the rare-earth ion speci es, because it is mostly determined by the strong \noptical absorption due to charge transfer and orbital promotion transitions inside the FeO 6 \ntetragonal cluster [35].) All experiments in this  study were performed at room temperature. \n \n3. Results and discussions   \nFigure 2(a) (upper panel) shows the calculated  temporal magnetic waveform together with \nthe incident magnetic field. The maximum peak am plitude is four times that of the incident THz \npulse in the time domain and reaches 0.91 T. Th e magnetic field continues to ring until around \n25 ps after the incident pulse has decayed away. The spectrum of the pulse shown in figure 2(c) \nhas a peak at the LC resonance frequency ( LC=0.56 THz) of the SRR, which is designed to \ncoincide with the resonance frequency of the AF-mode ( νAF0=0.575 THz). Figure 2(a) (lower \npanel) shows the time development of the MOKE signal  for the highest THz excitation \nintensity (pump fluence I of 292 µJ/cm2 and maximum peak magnetic field Bmax of 0.91 T). The \ntemporal evolution of  is similar to that of the Faraday rotation measured in the previous \nstudy and the magnetization oscillates harmonically with a period of ~2 ps [23], implying that \nthe THz magnetic field coherently drives the AF-mode motion.  \n As shown in figure 2(b), as th e incident pump pulse intensity increases, the oscillation period \nbecomes longer. The Fourier transform spectra  of the MOKE signals for different pump \nintensities are plotted in figure 2(c). As the ex citation intensity increases, the spectrum becomes \nasymmetrically broadened on the lower freque ncy side and its peak frequency becomes \nredshifted. Figure 2(d) plots the center-of-mass fre quency (open circles) and the integral (closed \ncircles) of the power spectrum P(\n) as a function of incident pulse fluence. The center  6frequency monotonically redshifts and P() begins to saturate. As shown in figure 2(c), the \nMOKE spectra obtained at the center of the SRR (indicated in the inset of figure 1) does not \nshow a redshift even for the highest intensity excitation, suggesting that the observed redshift \noriginates from the nonlinearity of the precessional spin motion rather than that of the SRR \nresponse. We took the analytic signal approach  (ASA) to obtain the time development of the \ninstantaneous frequency (t) (figure 3(c)) and the envelope amplitude 0(t) (figure 3(d)) from \nthe measured magnetization change (t)=Mz(t)/|MS| (figure 3(b)) (see Appendix B for the \ndetails of the analytic signal approach). As is  described in the Appendix C, the MOKE signal \n6\n4\n2\n0\nIntegral of P( ) \n(arb. units)\n300 200 100 0\nFluence (µJ/cm2)0.575\n0.570\n0.565Frequency (THz)1.0\n0.8\n0.6\n0.4\n0.2\n0.0Intensity P( ) (arb. units)\n0.60 0.58 0.56 0.54\nFrequency (THz)  50%\n   100%\n 100% (x 3.7)\n (center)\n \n  10%\n |Bnr|2\n \nz \nm2 m1 \nM \nx Bnr \ny (a) (c) \n(d) (b) -0.020.000.02∆(degrees)\n40 30 20 10 0\nTime (ps)1.0\n0.5\n0.0\n-0.5B (T)  Bnr\n  Bin (x 3)\n0.08\n0.06\n0.04\n0.02\n0.00∆(degrees)\n24 20 16 12 8\nTime (ps)10%100%\nFigure 2. (a) Upper panel: Incident magnetic field of the THz pump pulse Bin estimated by\nelectro-optic sampling (dashed line) and the THz magnetic near-field Bnr calculated by the\nFDTD method (solid line). The illustration s hows the magnetization motion for the AF-mode.\nLower panel: The MOKE signal for a pump fluence of 292 µJ/cm2 (100%). (b) Comparison o f\ntwo MOKE signals for different pump fluences, vertically offset for clarity. (c) The FFT powe r\nspectrum of the magnetic near-field Bnr (black solid line). The spectra P() of the MOKE\nsignals for a series of pump fluences obtained at th e corner (solid lines) and at the center (blue\ndashed circle in the inset of figure 1) for a pump fluence of 100% (dashed line). Each spectru m\nof the MOKE signal is normalized by the peak amplitude at the corner for a pump fluence o f\n100%. (d) Intensity dependence of the center-of-m ass frequency (open circles) and the integral\n(closed circles) of the P().  7(t) is calibrated to the magnetization change (t) by using a linear relation, i.e., (t)=g(t), \nwhere g (=17.8 degrees−1) is a conversion coefficient. The tim e resolved experiment enables us \nto separate the contributions of the applied magnetic field and magnetization change to the \nfrequency shift in the time domain. A comparison of the temporal profiles between the driving \nmagnetic field (figure 3(a)) and the frequency e volution (figure 3(c)) shows that for the low \npump fluence (10%, closed blue circles), the frequency is redshifted only when the magnetic \nfield persists ( t < 25 ps), and after that, it recovers  to the constant AF mode frequency \n(νAF0=0.575 THz). This result indicates that the signals below t = 25 ps are affected by the \npersisting driving field and the redshift may orig inate from the forced oscillation. As long as the \n0.575\n0.570\n0.565\n0.560\n0.555Frequency (THz)\n50 40 30 20 10 0\nTime (ps)-0.4-0.20.00.20.4Magnetization \nchange 0.5\n0.0\n-0.5Bnr (T)\n0.4\n0.3\n0.2\n0.1\n0.0Amplitude 0  Experiment\n100%\n  10%\n  Experiment\n100%\nSimulation\n100% (1=0)\n100% \n  10% Simulation(a) \n(b) \n(c) \n(d) \nFigure 3. (a) FDTD calculated magnetic field Bnr for pump fluence of 100%. (b) Temporal\nevolution of the magnetization change obtained from the experimental data (gray circles) an d\nthe LLG model (red line). (c) Instantaneous frequencies and (d) envelope amplitudes fo r\npump fluences of 100% and 10% obtained by the analytic signals calculated from the\nexperimental data (circles) and the LLG simulation with nonlinear damping paramete r\n(1=1×10−3, solid lines) and without one ( 1=0, dashed line).  8magnetic response is under the linear regime, the instantaneous frequency is independent on the \npump fluence. However, for the high pump fluenc e (100%) a redshift (a maximum redshift of \n~15 GHz relative to the constant frequency νAF0) appears in the delay time ( t < 25 ps) and even \nafter the driving field decays away ( t > 25 ps) the frequency continues to be redshifted as long \nas the amplitude of the magnetization change is large. These results suggest that the frequency \nredshift in the high intensity case depends on the magnitude of the magnetization change, \nimplying that its origin is a nonlinear precessional spin motion with a large amplitude. \n \nThe temperature increase due to  the THz absorption (for HoFeO 3 T=1.7×10−3 K, for gold \nSRR T=1 K) is very small (see Appendix D). In ad dition, the thermal relaxation of the spin \nsystem, which takes more than a nanosecond [36], is much longer than the frequency \nmodulation decay (~50 ps) in figure 3(c). Therefore, laser heating can be ignored as the origin of the redshift.  \n \nFigure 4 shows a parametric plot of the instantaneous frequency \n(t) and envelope amplitude \n0(t) for the high pump fluence (100%). The instantaneous frequency shift for t > 25 ps has a \nsquare dependence on the amplitude, i.e., νAF=νAF0(1െCζ02). To quantify the relationship \nbetween the redshift and magnetization change, it  would be helpful to have an analytical \nexpression of the AF mode frequency AF as a function of the magnetization change, which is \nderived from the LLG equation based on the two- lattice model [32,33]. The dynamics of the \nsublattice magnetizations mi (i=1,2), as shown in the inset of figure 2(a), are described by \n \n dRi\ndt=െγ\n(1+α2)ቀRi×[B(t)+Beff,i]െαRi×൫Ri×[B(t)+Beff,i]൯ቁ,  (1) \n \nwhere Ri=mi/m0 (m0=|mi|) is the unit directional vector of the sublattice magnetizations, \n=1.76×1011 s−1T−1 is the gyromagnetic ratio, V(Ri) is the free energy of the iron spin system \nnormalized with m0, and Beff,i is the effective magnetic field given by െ∂V/∂Ri (i=1,2) (see \nAppendix E). The second term represents the ma gnetization damping with the Gilbert damping \nconstant \u001f \n \nSince Beff,i depends on the sublattice magnetizations mi and the product of these quantities \nappears on the right side of Eq. (1), the LLG e quation is intrinsically nonlinear. If the angle of  9the sublattice magnetization precession is sufficien tly small, Eq. (1) can be linearized and the \ntwo fixed AF- and F-modes for the weak excitation can be derived. However, as shown in figure \n3(b), the deduced maximum magnetization change  reaches ~0.4, corresponding to precession \nangles of 0.25° in the xz-plane and 15° in th e xy-plane. Thus, the magnetization change might \nbe too large to use the linear approximation. For such a large magnetization motion, assuming \nthe amplitude of the F-mode is zero and =0 in Eq. (1), the AF mode frequency AF in the \nnonlinear regime can be deduced as  \n \n νAF =νAF0ට1ିζ02tan2β0\nK(D),       ( 2 )  \n D =ඨ\t\t\t\t\t\tζ02(rAF2ି1) tan2β0\n1ିζ02tan2β0,      ( 3 )  \n \n0.575\n0.570\n0.565Frequency (THz)\n0.4 0.3 0.2 0.1 0.0\nAmplitude 0Experiment\n t > 25 ps\n t < 25 ps\n \n Analytic Solution\n 2nd order expansion\n \nFigure 4.  Relation between instantaneous frequency  and envelope amplitude 0 obtaine d\nfrom the magnetization change; for t < 25 ps (open circles) and for t > 25 ps (closed circles),\nthe analytic solution (blue line) and second orde r expansion of the analytic solution (gree n\ndashed line). Errors are estim ated from the spatial inhomogeneity of the driving magnetic\nfield (see Appendix H).  10where K(D) is the complete elliptic integral of the first kind, rAF(≈60) is the ellipticity of the \nsublattice magnetization precession trajectory of the AF-mode (see Appendix F), and 0 is the \namplitude of the (t). As shown in figure 4, the analytic solution can be approximated by the \nsecond order expansion νAF≈νAF0(1െtan2β0(rAF2െ1)ζ024⁄) and matches the observed redshift \nfor t > 25 ps, showing that the frequency appr oximately decreases with the square of (t). The \ndiscrepancy of the experimental data from the theoretical curve ( t < 25 ps) may be due to the \nforced oscillation of the AF-mode caused by the driving field.  \n \nTo elaborate the nonlinear damping effects, we compared the measured (t) with that \ncalculated from the LLG equation with the damping term. As shown in figures 3(c) and 3(d), the \nexperiment for the high intensity excitation devi ates from the simulation with a constant Gilbert \ndamping (dashed lines) even in the t > 25 ps time region, suggesting nonlinear damping \nbecomes significant in the large amplitude region. To describe the nonlinear damping \nphenomenologically, we modified the LLG equa tion so as to make the Gilbert damping \nparameter depend on the displacement of th e sublattice magnetization from its equilibrium \nposition, (Ri)=0+1Ri. As shown in figures 3(b)-(d), the magnetization change (t) derived \nwith Eq. (1) (solid line) with the damping parameters ( 0=2.27×10−4 and 1=1×10−3) nicely \nreproduces the experiments for both the high (100%) and low (10%) excitations.1 These results \nsuggest that the nonlinear damping plays a signifi cant role in the large amplitude magnetization \ndynamics. Most plausible mechanism for the nonlinear damping is four magnon scattering \nprocess, which has been introdu ced to quantitatively evaluate the magnon mode instability of \nferromagnet in the nonlinear response regime [37]. \n \n4. Conclusions  \nIn conclusion, we studied the nonlinear magnetization dynamics of a HoFeO 3 crystal excited \nby a THz magnetic field and measured by MOKE microscopy. The intense THz field can induce \nthe large magnetization change (~40%), and the ma gnetization change can be kept large enough \n                                            \n1 The damping parameter 0 (=2.27×10−4) and conversion coefficient  g (=17.8 degrees−1) are \ndetermined from the least-squares fit of the calculated result without the nonlinear damping \nparameter 1 to the experimental MOKE signal for the low pump fluence of 29.2 µJ/cm2. The \nnonlinear damping parameter 1 (=1×10−3) is obtained by fitting the experimental result for the \nhigh intensity case ( I=292 µJ/cm2) with the values of 0 and g obtained for the low excitation \nexperiment. The estimated value of g is consistent with the stat ic MOKE measurement; the Kerr \nellipticity induced by the spontaneous magnetization MS is ~0.05 degrees ( g~20 degrees−1). See \nAppendix G for details on the static Kerr measurement.  11to induce the redshift even after the field has gone , enabling us to separate the contributions of \nthe applied magnetic field and ma gnetization change to the frequency shift in the time domain. \nThe resonance frequency decreases in proportion to  the square of the magnetization change. A \nmodified LLG equation with a phenomenologi cal nonlinear damping term quantitatively \nreproduced the nonlinear dynamics. This suggest s that a nonlinear spin relaxation process \nshould take place in a strongly driven regime. Th is study opens the way to the study of the \npractical limits of the speed and efficiency of magnetization reversal, which is of vital \nimportance for magnetic recording and information processing technologies. \n   12Acknowledgments \nWe are grateful to Shintaro Takayoshi, Masah iro Sato, and Takashi Oka for their discussions \nwith us. This study was supported by a J SPS grants (KAKENHI 26286061 and 26247052) and \nIndustry-Academia Collaborative R&D grant fro m the Japan Science and Technology Agency \n(JST). \n   13Appendix A. Detection sche me of MOKE measurement \nWe show the details of the detection scheme  of the MOKE measurement. A probe pulse for \nthe MOKE measurement propagates along the z direction. By using the Jones vector [38], an electric field E\n0 of the probe pulse polarized linearly along the x-axis is described as  \n \n E0 =ቀ1\n0ቁ.       ( A . 1 )  \n \nThe probe pulse E1 reflected from the HoFeO 3 surface becomes elliptically polarized with a \npolarization rotation angle and a ellipticity angle . It can be written as \n \n E1 =R(െ߶)MR(θ)E0ൌ൬cos θ cos ߶െ\t݅ sin θ sin ߶\ncos θ sin ߶\t݅ sin θ cos ߶൰,  (A.2) \n  \nwhere  M is the Jones matrix describing \u001f\u001f phase retardation of the y component with \nrespect to the x component \n \nM=ቀ10\n0െiቁ,       ( A . 3 )  \n \nand R(ψ) is the rotation matrix \n \nR(ψ)=൬cosψ sinψ\nെsinψcosψ൰.      (A.4)  \n \nThe reflected light passes through the quarter wave plate, which is arranged such that its fast \naxis is tilted by an angle of 45° to the x-axis. The Jones matrix of the wave plate is given by \n \nRቀെπ\n4ቁMRቀπ\n4ቁ.        ( A . 5 )  \n Thus, the probe light E\n2 after the quarter wave plate is described as follows, \n  14E2 = ൬E2,x\nE2,y൰=Rቀെπ\n4ቁMRቀπ\n4ቁE1  \n=1\n2൬cosሺθ߶ሻെsin (θെ߶+)i(െcosሺθെ߶ሻsin (θ߶))\ncosሺθെ߶ሻsin (θϕ)+i(c o sሺθ߶ሻsin (θെ߶))൰.  (A.6)  \n \nThe Wollaston prism after the quarter wave plat e splits the x and y-polarization components of \nthe probe light E2. The spatially separated two pulses are incident to the balanced detector and \nthe detected probe pulse intensity ratio of the di fferential signal to the total corresponds to the \nKerr ellipticity angle as follows,    \n〈หாమ,ೣหమ〉ି〈หாమ,หమ〉\n〈หாమ,ೣหమ〉ା〈หாమ,หమ〉ൌെsin2θ.      ( A . 7 )  \n \nIn the main text, we show the Kerr ellipticity change =w−wo, where the ellipticity angles \n(w and wo) are respectively obtained with and without the THz pump excitation. \n \nAppendix B. Analytic signal approach and short time Fourier transform \nThe Analytic signal approach (ASA) allows the extraction of the time evolution of the \nfrequency and amplitude by a simple procedure and assumes that the signal contains a single \noscillator component. In our study, we measure only the MOKE signal originating from the \nAF-mode and it can be expected that the single oscillator assumption is valid. In the ASA, the \ntime profile of the magnetization change (t) is converted into an analytic signal (t), which is a \ncomplex function defined by using the Hilbert transform [39]; \n \nψ(t)=ζ0(t)exp( i߶(t))=ζ(t)+i ζ෨(t),     (B.1) \nζ෨(t)ൌ1\nπ pζ(t)\ntିτ∞\n-∞ dτ.       ( B . 2 )  \n \nwhere the p is the Cauthy principal value. The real part of (t) corresponds to (t). The real \nfunction 0(t) and (t) represent the envelope amplitude  and instantaneous phase of the \nmagnetization change. The instantaneous frequency (t)(=2(t)) is given by (t)=d(t)/dt. In \nthe analysis, we averaged 0(t) and (t) over a ten picosecond time range. \n \nTo confirm whether the ASA gives appropriate results, as shown in figure B.1 we compare  15them with those obtained by the short time Fourie r transform (STFT). As shown in figure B.1(a), \nthe time-frequency plot shows only one oscillato ry component of the AF-mode. As shown in \nfigures B.1(b) and (c), the instantaneous freque ncies and amplitudes obtained by the ASA and \nthe STFT are very similar. Because the ASA provides us the instantaneous amplitude with a \nsimple procedure, we showed the time evolu tions of frequency and amplitude derived by the \nASA in the main text. \n \nAppendix C. Determination of conversion coefficient g and linear damping parameter 0 \nThe conversion coefficient  g and the linear damping parameter 0(=) in Eq. (1) are \ndetermined by fitting the experimental MOKE signal (t) for the low pump fluence of 29.2 \nµJ/cm2 with the LLG calculation of the magnetization change (t). Figure C.1 shows the MOKE \nsignal (t) (circle) and the calculated magnetization change (t) (solid line). From the \nleast-squares fit of the calculated result to th e experiment by using a linear relation, i.e., \n(t)=g(t), we obtained the parameters g(=17.8 degrees−1) and 0(=2.27×10−4). 0.575\n0.570\n0.565\n0.560\n0.555\n0.550Frequency (THz)\n50 40 30 20 100\nTime (ps)ASA\n 100%\n   10%\nSTFT\n 100%\n   10%\n 1.0\n0.8\n0.6\n0.4\n0.2\n0.0\nFourier am plitude (arb. units)\n50 40 30 20 100\nTime (ps)0.4\n0.3\n0.2\n0.1\n0.0Amplitude 0ASA\n   100%\n     10% \nSTFT\n   100%\n     10%\n (a) (b) (c) \n1.2\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0Frequency (THz)\n5040302010\nTime (ps)(arb. units)\n1.0 0.0\nFigure B.1.  (a) Time-dependence of the power spectrum of the magnetization \noscillation for the highest THz excitation ( I=292 µJ/cm2) obtained by the STFT. \nComparison of (b) instantaneous frequencies and (c) amplitudes obtained by the ASA \nand STFT with a time window with FWHM of 10 ps.   16 \n \nAppendix D. Laser heating effect \nThe details of the calculation of the temperature change are as follows: \n \nFor HoFeO 3:  \nThe absorption coefficient abs of HoFeO 3 at 0.5 THz is ~4.4 cm−1 [40]; the fluence IHFO \nabsorbed by HoFeO 3 can be calculated as IHFO=I(1−exp(−absd)), where d (=145 µm) is the \nsample thickness and I is the THz pump fluence. For the highest pump fluence, I=292 µJ/cm2, \nIHFO is 18.1 µJ/cm2. Since the sample thickness is much smaller than the penetration depth, \nd≪abs−1, we assume that the heating of the sample due to the THz absorption is homogeneous. \nBy using the heat capacity Cp of 100 J mol−1 K−1 [27], and the molar volume v of ~1.4×102 \ncm3/mol [27], the temperature change T can be estimated as\u001f T=IHFOv/Cpd ~1.7×10−3 K. \n \nFor gold resonator (SRR): \nThe split ring resonator has an absorption band (center frequency ~0.56 THz, band width ~50 \nGHz) originated from the LC resonance (figure 2( c)). Assuming the SRR absorbs all incident \nTHz light in this frequency band, the absorbed energy accounts for 3 % of the total pulse energy. \nHence, for the highest THz pump fluence, I=292 µJ/cm2, the fluence absorbed by the SRR is \nIgold=8.76 µJ/cm2. By using the heat capacity Cp of 0.13 J g−1 K−1 [41], the number of the SRRs \nper unit area N of 4×104 cm−2, and the mass of the SRR m of 1.6×10−9 g, the temperature change -10x10-3-50510degrees)\n50 40 30 20 10 0\nTime (ps)-0.10.00.1\nMagnetization \nchange  Experiment\n Simulation\nFigure C.1.  Experimentally observed MOKE signal \u001f(circle) and LLG simulatio n\nresult of the magnetization change \u001f(solid line) for the pump fluence of 29.2\nµJ/cm2.  17T can be estimated as\u001f T=Igold/CpNm ~ 1 K \n \nAppendix E. Free energy of HoFeO 3 \nThe free energy F of the iron spin (Fe3+) system based on the two-lattice model is a function \nof two different iron sublattice magnetizations mi, and composed of the exchange energy and \none-site anisotropy energy [32,33]. The free en ergy normalized by the sublattice magnetization \nmagnitude, V=F/m0 (m0=|mi|), can be expanded as a power series in the unit directional vector of \nthe sublattice magnetizations, Ri=mi/m0=(Xi,Yi,Zi). In the magnetic phase 4 (T > 58K), the \nnormalized free energy is given as follows [32,33]: \n \nV=ER1·R2+D(X1Z2െX2Z1)െAxx(X12+X22)െAzz(Z12+Z22),  (E.1) \n \nwhere E(=6.4×102 T) and D(=1.5×10 T) for HoFeO 3 are respectively the symmetric and \nantisymmetric exchange field [42]. Axx and Azz are the anisotropy constants. As mentioned in \nAppendix F, the temperature dependent values of the anisotropy constants can be determined \nfrom the antiferromagnetic resonance frequencies. The canting angle of Ri to the x-axis β0 \nunder no magnetic field is given by \n \ntan 2β0=D\nE+AxxିAzz.        ( E . 2 )  \n \nAppendix F. Linearized resonance modes and anisotropy constants ( Axx and Azz) \nThe nonlinear LLG equation of Eq. (1) can be  linearized and the two derived eigenmodes \ncorrespond to the AF and F-mode. The sublatti ce magnetization motion for each mode is given \nby the harmonic oscillation of mode coordinates; for the AF-mode ( QAF, \nPAF)=((X1−X2)s i nβ0+(Z1+Z2)c o sβ0, Y1−Y2), and for the F-mode ( QF, \nPF)=((X1+X2)sinβ0−(Z1−Z2)cosβ0, Y1+Y2), \n \nQAF=AAFcosωAFt,       ( F . 1 )  \nPAF=AAFrAFsinωAFt,      ( F . 2 )  \n \nQF=AFcosωFt,       ( F . 3 )  \nPF=AFrAFsinωFt,       ( F . 4 )   18 \nwhere AAF,F represents the amplitude of each mode. AF,F, and rAF,F are the resonance frequencies \nand ellipticities, which are given by    \n \nωAF=γට(b+a)(d-c),        ( F . 5 )  \nωF=γට(b-a)(d+c),        ( F . 6 )  \n rAF=γටሺௗିሻ\n(b+a),        ( F . 7 )  \n rF=γටሺௗାሻ\n(b-a),        ( F . 8 )  \n \nwhere =1.76×1011 s−1T−1 is the gyromagnetic ratio, and \n \n a=െ2Axxcos2β0െ2Azzsin2β0െEcos 2β0െDsin 2β0,  (F.9) \n b=E,         ( F . 1 0 )  \n c=2Axxcos2β0��2Azzcos2β0+Ecos 2β0+Dsin 2β0,    (F.11) \n d=െEcos 2β0െDsin 2β0.       ( F . 1 2 )  \n \nSubstituting the literature values of the exchange fields ( E=6.4×102 T and D=1.5×10 T [42]) and \nthe resonance frequencies at room temperature ( AF/2=0.575 THz and F/2=0.37 THz) to \nEqs. (F.5) and (F.6), Axx and Azz can be determined to 8.8×10−2 T and 1.9×10−2 T. \n \nAppendix G. MOKE measurement for the spontaneous magnetization \nFigure G.1 shows time-development of the MOKE signals for the different initial condition \nwith oppositely directed magnetization. We applied the static magnetic field (~0.3 T) to saturate \nthe magnetization along the z-axis before the TH z excitation. The spontaneous magnetization of \nsingle crystal HoFeO a can be reversed by the much smaller magnetic field (~0.01 T) because of \nthe domain wall motion [27]. Then, we separately measured the static Kerr ellipticity angle \n\u001f\u001f\u001f\u001f\u001f\u001f  and THz induced ellipticity change  for different initial magnetization Mz=±Ms \nwithout the static magnetic field \u001f In figure G.1 we plot the summation of the time resolved \nMOKE signal \u001fand the static Kerr ellipticity \u001f\u001f\u001f\u001f\u001f\u001f  The sings of the ellipticity offset angle  19\u001f\u001f\u001f\u001f\u001f\u001f  for the different spontaneous magnetization (±M S) are different and their magnitudes \nare ~0.05 degrees. The conversion coefficient g(=1/~\u001f/0.05 degrees) is estimated to be ~20 \ndegrees−1, which is similar to the value dete rmined by the LLG fitting (~17.8 degrees−1). In the \ncase of the AF-mode excitation, the phases of the magnetization oscillations are in-phase \nregardless of the direction of the spontaneous magnetization M=±Ms, whereas they are \nout-of-phase in the case of the F-mode excitation. We can explain this claim as follows: In the \ncase of AF-mode excitation, the external THz magne tic field is directed along the z-direction as \nshown in the inset of figure 2(a), the signs of the torques acting on the sublattice magnetization \nmi (i=1,2) depends on the direction of mi, however, the resultant oscillation of the macroscopic \nmagnetization M= m1+m2 along the z-direction has same phase for the different initial condition \nM=±Ms. In the case of the F-mode excitation with the external THz magnetic field along the x \nor y-direction, the direction of the torques acting on the magnetization M depends on the initial \ndirection and the phase of the F-mode osc illation changes depending on the sign of the \nspontaneous magnetization ±Ms. \n \nAppendix H. Influence of the spatial distri bution of magnetic field on magnetization \nchange \nAs shown in the inset of figure 1, the pump magnetic field  strongly localizes near the metallic \narm of the SRR and the magnetic field strength significantly depends on the spatial position r \nwithin the probe pulse spot area. The intensity distribution of the probe pulse Iprobe(r) has an 0.05\n0.00\n-0.05Kerr ellipticity  (degrees)\n25 20 15 10 5\nTime (ps) +MS\n -MS\n \nFigure G.1. The MOKE signals, the temporal change of the Kerr ellipticity , measured \nfor different initial conditions with oppositely directed magnetizations.       20elongated Gaussian distribution with spatial widt hs of 1.1 µm along the x-axis and 1.4 µm along \nthe y-axis [full width at half maximum (FWHM)  intensity]. The maximum magnetic field is 1.2 \ntimes larger than the minimum one in the spot  diameter, causing the different magnetization \nchange dynamics at different positions. To take into account this spatial inhomogeneity to the \nsimulation, the spatially weighted average of magnetization change ζ̅(t) has to be calculated as \nfollows:  \n \n ζ̅(t)=ζ(r,t)Iprobe(r)dr\nIprobe(r)ௗr ,       ( H . 1 )  \n \nwhere (r,t) is a magnetization change at a position r and time t.  \n \nFigure H.1(a) shows the simulation result of the spatially averaged magnetization change ζ̅(t) \nand the non-averaged (r0,t) without the nonlinear damping term ( 1=0), where r0 denotes the \npeak position of Iprobe(r). For the low excitation intensity (10%), ζ̅(t) is almost the same as \n(r0,t) as shown in figure H.1(a). On the other hand, for the high excitation intensity, the spatial \ninhomogeneity of magnetization change dyna mics induces a discrepancy between the ζ̅(t) and (a) (b) (c) \n-0.100.000.10Magnetization change\n50 40 30 20 100\nTime (ps)-0.6-0.4-0.20.00.20.4 Averaged\n Non-averaged\n Averaged\n Non-averaged100%  10%\n0.575\n0.570\n0.565\n0.560\n0.55550 40 30 20 100\nTime (ps)0.575\n0.570\n0.565\n0.560\n0.555Frequency (THz) Averaged\n Non-averaged\n Experiment\n Averaged\n Non-averaged\nExperiment\n100%  10%\n0.4\n0.3\n0.2\n0.1\n0.0\n50 40 3020 100\nTime (ps)0.12\n0.08\n0.04\n0.00Amplitude   Averaged\n Non-averaged\n Experiment\n Averaged\n Non-averaged\n Experiment100%  10%\nFigure H.1.  Comparison of the spatially averag ed and non-averaged magnetization \nchange for the different pump fluences of 10% and 100%. (a) Temporal evolutions of \nthe magnetization change, (b) instantaneous  frequencies and (c) normalized envelope \namplitudes. Open circles show the experimental results.      21(r0,t). This discrepancy is caused by the quasi-i nterference effect between the magnetization \ndynamics with different frequencies and amplit udes at different positions. Figures H.1(b) and \n(c) show the instantaneous freque ncy and envelope amplitude obt ained from the data shown in \nfigure H.1(a) by using analytic signal approach with the experimental result. For the averaged \nmagnetization change, the frequency redshift is more emphasized (figure H.1(b)) and the decay \ntime becomes shorter (figure H.1(c)). Nonetheless,  neither spatially averaged nor non-averaged \nsimulation reproduces the experimental result of  the instantaneous frequency (figure H.1(b)) \nwithout nonlinear damping term.  \n \n   22References \n[1] Kirilyujk A, Kimel A V and Rasing T 2010 Rev. Mod. Phys.  82 2731 \n[2] Beaurepaire E, Merle J-C, Daunois A and Bigot J-Y 1996 Phys. Rev. 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Phys. Soc. Jpn.  57 4418 "    },    {        "title": "1401.6467v2.Wavenumber_dependent_Gilbert_damping_in_metallic_ferromagnets.pdf",        "content": "arXiv:1401.6467v2  [cond-mat.mtrl-sci]  24 Jan 2016Wavenumber-dependent Gilbert damping in metallic ferroma gnets\nY. Li and W. E. Bailey\nDept. of Applied Physics & Applied Mathematics,\nColumbia University, New York NY 10027, USA\n(Dated: November 21, 2021)\nA wavenumber-dependentdissipative term to magnetization dynamics, mirroring the conservative\nterm associated with exchange, has been proposed recently f or ferromagnetic metals. We present\nmeasurements ofwavenumber-( k-)dependentGilbert dampinginthree metallic ferromagnet s, NiFe,\nCo, and CoFeB, using perpendicular spin wave resonance up to 26 GHz. In the thinnest films\naccessible, where classical eddy-current damping is negli gible, size effects of Gilbert damping for the\nlowest and first excited modes support the existence of a k2term. The new term is clearly separable\nfrom interfacial damping typically attributed to spin pump ing. Higher-order modes in thicker films\ndo not show evidence of enhanced damping, attributed to a com plicating role of conductivity and\ninhomogeneous broadening. Our extracted magnitude of the k2term, ∆α∗\nkE= ∆α∗\n0+A∗\nkk2where\nA∗\nk=0.08-0.1 nm2in the three materials, is an order of magnitude lower than th at identified in prior\nexperiments on patterned elements.\nThe dynamical behavior of magnetization for ferro-\nmagnets (FMs) can be described by the Landau-Lifshitz-\nGilbert (LLG) equation[1]:\n˙ m=−µ0|γ|m×Heff+αm×˙ m (1)\nwhereµ0is the vacuum permeability, m=M/Msis the\nreduced magnetization unit vector, Heffis the effective\nmagnetic field, γis the gyromagnetic ratio, and αis the\nGilbert damping parameter. The LLG equation can be\nequivalently formulated, for small-angle motion, in terms\nof a single complex effective field along the equilibrium\ndirection, as ˜Heff=Heff-iαω/|γ|; damping torque is in-\ncluded in the imaginary part of ˜Heff.\nFor all novel spin-transport related terms to the LLG\nidentified so far[2–7], each real (conservative) effective\nfield term is mirrored by an imaginary (dissipative)\ncounterpart. In spin-transfer torque, there exist both\nconventional[2, 3] and field-like[8] terms in the dynamics.\nIn spin-orbit torques (spin Hall[4] and Rashba[6] effect)\ndampinglike and fieldlike components have been theoret-\nically predicted[9] and most terms have been experimen-\ntally identified[5, 6]. For pumped spin current[7], theory\npredictsrealandimaginaryspinmixingconductances[10]\ng↑↓\nrandg↑↓\niwhich introduce imaginary and real effective\nfields, respectively.\nIt is well known that the exchange interaction, respon-\nsible for ferromagnetism, contributes a real effective field\n(fieldlike torque) quadratic in wavenumber kfor spin\nwaves[11]. It isthennaturaltoaskwhetheracorrespond-\ning imaginary effective field might exist, contributing a\ndampinglike torque to spin waves. Theoretically such an\ninteraction has been predicted due to the intralayer spin-\ncurrent transport in a spin wave[12–15], reflected as an\nadditional term in Eq. (1):\n˙ m=···−(|γ|σ⊥/Ms)m×∇2˙ m (2)\nwhereσ⊥is the transverse spin conductivity. This term\nrepresents a continuum analog of the well-established in-terlayer spin pumping effect[7, 16, 17]. For spin wave\nresonance (SWR) with well-defined wavenumber k, Eq.\n(2) generates an additional Gilbert damping ∆ α(k) =\n(|γ|σ⊥/Ms)k2. In this context, Gilbert damping refers\nto an intrinsic relaxation mechanism in which the field-\nswept resonance linewidth is proportional to frequency.\nRemarkably, the possible existence of such a term has\nnot been addressed in prior SWR measurements. Previ-\nous studies of ferromagneticresonance (FMR) linewidths\nof spin waves[18–21] were typically operated at fixed fre-\nquency, not allowing separation of intrinsic (Gilbert) and\nextrinsic linewidths. Experiments have been carried out\non thick FM films, susceptible to a large eddy current\ndamping contribution[22]. Any wavenumber-dependent\nlinewidth broadening in these systems has been at-\ntributed to eddy currents or inhomogeneous broadening,\nnot intrinsic torques which appear in the LLG equation.\nIn this Manuscript, wepresent a study of wavenumber-\ndependent Gilbert damping in the commonly applied\nferromagnetic films Ni 79Fe21(Py), Co, and CoFeB. A\nbroad range of film thicknesses (25-200 nm) has been\nstudied in order to exclude eddy-current effects. We\nobserve a thickness-dependent difference in the Gilbert\ndamping for uniform and first excited spin wave modes\nwhich is explained well by the intralayer spin pump-\ning model[14]. Corrections for interfacial damping, or\nconventional spin pumping, have been applied and are\nfound to be small. The measurements show that the\nwavenumber-dependent damping, as identified in contin-\nuousfilms, isinreasonableagreementwith thetransverse\nspin relaxation lengths measured in Ref. [23], but an or-\nder of magnitude smaller than identified in experiments\non sub-micron patterned Py elements[24].\nTwo different types of thin-film heterostructures were\ninvestigated in this study. Films were deposited by\nUHV sputtering with conditions given in Ref. [23, 25].\nMultilayers with the structure Si/SiO 2(substrate)/Ta(5\nnm)/Cu(5 nm)/ FM(tFM)/Cu(5 nm)/Ta(5 nm), where2\nFM= Py, Co and CoFeB and tFM= 25-200 nm, were\ndesigned to separate the effects of eddy-current damp-\ning and the intralayer damping mechanism proposed in\nEq. (2). The minimum thickness investigated here is\nour detection threshold for the first SWR mode, 25 nm.\nA second type of heterostructure focused on much thin-\nner Py films, with the structure Si/SiO 2(substrate)/Ta(5\nnm)/Cu(5 nm)/Py( tPy)/Cu(5 nm)/ X(5 nm),tPy= 3-30\nnm. Here the cap layer X= Ta or SiO 2was changed,\nfor two series of this type, in order to isolate the effect\nof interfacial damping (spin pumping) from Cu/Ta inter-\nfaces.\nTo study the Gilbert damping behavior of finite-\nwavenumber spin waves in the samples, we have\nexcited perpendicular standing spin wave resonance\n(PSSWR)[26] using a coplanar waveguide from 3 to 26\nGHz. The spin-wave mode dispersion is given by the\nKittel equation ω(k)/|γ|=µ0(Hres−Ms+Hex(k)); the\neffective field from exchange, µ0Hex(k) = (2Aex/Ms)k2\nwithAexas the exchange stiffness, gives a precise mea-\nsurement of the wavenumber excited ((Fig. 1 inset)).\nPSSWR modes are indexed by the number of nodes p,\nwithk=pπ/tFMin the limit of unpinned surface spins.\nThe full-width half-maximum linewidth, ∆ H1/2, is fit-\nted using µ0∆H1/2(ω) =µ0∆H0+ 2αω/|γ|to extract\nthe Gilbert damping α. Forp= 1 modes we fix µ0∆H0\nas the values extracted from the corresponding p= 0\nmodes for ( tFM≤40 nm), because frequency ranges are\nreduced due to large exchange fields. In unconstrained\nfits for films of this thickness, the inhomogeneous broad-\neningµ0∆H0of thep= 1 modes does not exhibit a\ndiscernible trend with 1 /t2\nFM(ork2)[19–21], justifying\nthis approximation[27].\nTo fit our data, we have solved Maxwell’s equations\nand the LLG equation (Eq. 1), including novel torques\nsuch as those given in Eq. (2), according to the method\nof Rado[28]. The model (designated ’EM+LLG’) is de-\nscribed in the Supplemental Information. Values calcu-\nlated using the EM+LLG model are shown with curves\nin Fig. 1 and dashed lines in Fig. 4. Comparison with\nsuch a model has been necessary since in our first type of\nsample series, tFM= 25-200 nm, eddy-current damping\nis negligible for thinner films (25 nm), the Akk2contri-\nbution is negligible for thicker films (200 nm), but the\ntwo effects coexist for the intermediate region.\nIn Fig. 1(a-c)we comparethe measuredGilbert damp-\ning for the uniform ( p= 0,αu) and first excited ( p= 1,\nαs)spinwavemodes. Thedominantthickness-dependent\ncontribution to Gilbert damping of the uniform modes of\nPy, Co, and CoFeB is clearly due to eddy currents which\narequadraticinthickness. Notethateddy-currentdamp-\ning is negligible for the thinnest films investigated (25\nnm), but quite significant for the thickest films (200 nm).\nThis term sums with the bulk Gilbert damping α0[29].\nThe simulation of αu, shown by black curves in Fig. 1,\nmatches closely with the analytical expression for bulkand eddy-current damping only[30] of αu=αu0+αE0,\nwhereαE0=µ2\n0γMst2\nFM/12ρcdenotes the eddy-current\ndamping for uniform modes. Fittings of αuyield resis-\ntivitiesρc= 16.7, 26.4 and 36.4 µΩ·cm for Py, Co and\nCoFeB, respectively.\nUnlike the uniform-mode damping, the 1st SWR\n(×10 -3 )\nuu\ns\nu\ns(a) (b) \n(c) Co \nCoFeB Py μ0HB (T). . .p=0 p=1\nPy 75nm  μ 0Hex \nx50\ns\n(×10 -3 )\nkus-=\nFIG. 1. Thickness dependence of αuandαsfor (a) Py, (b)\nCo and (c) CoFeB thin films. Curves are calculated from\na combined solution of Maxwell’s equations and the LLG\n(EM+LLG). For αuthe values of µ0Ms,α(Table I), effective\nspin mixing conductance (Supplemental Information Sectio n\nC) g-factor (2.12 for Py and CoFeB and 2.15 for Co) and ρc\n(from analytical fitting) are used. For αsthe values of ∆ α∗\nkE\nand ∆α∗\nk0(Table I) are also included in the simulation. Inset:\n10 GHz FMR spectra of p= 0 and p= 1 modes in Py 75 nm\nfilm.\nmode damping αsis found to exhibit a minimum as a\nfunction of thickness. For decreasing thicknesses below\n75 nm,αsis increased. This behavior indicates an addi-\ntional source of Gilbert damping for the 1st SWR modes.\nIn CoFeB the increased αsis less visible in Fig. 1(c) due\nto fluctuations in damping for samples of different thick-\nness, but is evident in the difference, αs−αu, plotted in\nFig. 2.\nIn order to isolate this new damping mechanism, we\nplot in Fig. 2 the increased damping for the 1st SWR\nmode, ∆ αk=αs−αu, side-by-side with exchange field\nµ0Hexas a function of ( π/tFM)2taken as the wavenum-\nberk2. When π/tFMis large, a linear k2dependence\nof ∆αkin all three ferromagnets mirrors the linear de-\npendence of µ0Hexonk2. This parallel behavior reflects\nthe wavenumber-dependent imaginary and real effective\nfields acting on magnetization, respectively. To quantify\nthe quadratic wavenumber term in ∆ ��k, we also show\nthe eddy-current-corrected values ∆ αkE=∆αk−∆αEin\nFig. 2(a). Here ∆ αE=αE1−αE0denotes the differ-\nence in eddy current damping between p= 1 and p= 0\nmodes according to the theory of Ref. [30], for weak sur-\nface pinning, where αE1≈0.23αE0(See Supplemental\nInformation for more details). We then fit this eddy-3(×10 -3 )\nPy \nCo \nCoFeB \nPy 150 nm (a)\n(b)\n(π/t FM )2 (×10 16  m -2 )Py \nCo \nCoFeB kk\nFIG. 2. Imaginary (damping, a) and real (exchange, b) ef-\nfective fields as a function of k2for Py, Co and CoFeB. (a)\nAdditional SWR damping ∆ αk(circle) and eddy-current cor-\nrected value ∆ αkE(cross) as a function of ( π/tFM)2. Solid\nlines are guides to eye and dashed lines are fits to Eq. (3). (b)\nExchange field µ0Hexas a function of ( π/tFM)2((pπ/tFM)2,\np=0-6, for Py 150 nm). Lines are fits to µ0Hex= (2A/Ms)k2.\ncurrent-corrected value to a linearization of Eq. (2), as:\n∆αkE= ∆αk0+Akk2(3)\nwithAk=|γ|σ⊥/Msand ∆αk0a constant offset. The\nvalues of Akestimated this way are 0 .128±0.022 nm2,\n0.100±0.011 nm2and 0.100±0.018 nm2for Py, Co and\nCoFeB.\nRecently, Kapelrud et al.[31] have predicted that\ninterface-localized (e.g. spin-pumping) damping terms\nwill also be increased in SWR, with interfacial terms\nforp≥1 modes a factor of two greater than those for\nthep= 0 mode. Using the second series of thinner\nPy films, we have applied corrections for the interfacial\nterm to our data, and find that these effects introduce\nonly a minor ( ∼20%) correction to the estimate of\nAk. Thep= 0 mode damping associated with the\nCu/Ta interface has been measured from the increase\nin damping upon replacement of SiO 2with Ta at the\ntop surface (Fig. 3, inset). Here Cu/SiO 2is taken as\na reference with zero interfacial damping; insulating\nlayers have been shown to have no spin pumping\ncontribution[32]. We find the damping enhancement to\nbe inversely proportional to tFM, indicating an interfa-\ncial damping term quantified as spin pumping into Ta[7]\nwith ∆αsp=γ¯h(g↑↓/S)/4πMstFM. Using the values\nin Table I yields the effective spin mixing conductance\nasg↑↓\nPy/Cu/Ta/S=2.5 nm−2, roughly a factor of three\nsmaller than that contributed by Cu/Pt interfaces[17].\nUsing the fitted g↑↓\nFM/Cu/Ta/S, we calculate andcorrect for the additional spin pumping contribution to\ndamping of the p= 1 mode, 2∆ αsp(from top and bot-\ntom interfaces). The corrected values for the 1st SWR\ndamping enhancement, ∆ α∗\nkE= ∆αkE−2∆αsp, are\nplotted for Py(25-200nm)in Fig. 3. These correctionsdo\nnot change the result significantly. We fit the k2depen-\ndence of ∆ α∗\nkEto Eq. (3) to extract the corrected values\nA∗\nkand ∆α∗\nk0. The fitted value, A∗\nk= 0.105±0.021 nm2\nfor Py, is slightly smaller than the uncorrected value\nAk. Other extracted interfacial-corrected values A∗\nkare\nlisted in Table I. Note that the correction of wavenumber\nby finite surface anisotropy will only introduce a small\ncorrection of AkandA∗\nkwithin errorbars. We also\nshow the EM+LLG numerical simulation results for\nthe uniform modes and the first SWR modes in Fig. 1\n(solid curves). Those curves coincide with the analytical\nexpressions of eddy-current damping plus k2damping\n(not shown) and fit the experimental data points nicely.\nThe negative offsets ∆ α∗\nk0between uniform modes\n(π/t FM )2 (×10 16  m -2 )(×10 -3 )*(= )\n**u0 u0 \ntFM  (nm)\nFIG. 3. Interfacial damping correction for Py. Main panel:\n∆αkEand ∆α∗\nkEas a function of ( π/tFM)2. Dashed lines\nare fits to k2-dependent equation as Eq. (3); ∆ α∗\nk0are ex-\ntracted from ∆ α∗\nkEfits.Inset:size effect of uniform-modes\nGilbert damping in Py/Cu/Ta and Py/Cu/SiO 2samples (cir-\ncles). The dashed curve is the theoretical reproduction of\nPy/Cu/SiO 2usingαu0+ ∆αsp(tFM). The shadow is the\nsame reproduction using αu0+ ∆αsp(tFM) +A∗\nkk2where\nthe error of shadow is from A∗\nk. Here kis determined by\nAexk2= 2Ks/tFM.\nand spin wave modes for Py and CoFeB are attributed\nto resistivitylike intrinsic damping[33]: because ˙ mis\naveraged through the whole film for uniform modes\nand maximized at the interfaces for unpinned boundary\ncondition, the SWR mode experiences a lower resistivity\nnear low-resistivity Cu and thus a reduced value of\ndamping. For Co a transition state between resistivity-\nlike and conductivitylike mechanisms[34] corresponds to\nnegligible ∆ α∗\nk0as observed in this work.\nIn addition to the thickness-dependent comparison of\np= 0 and p= 1 modes, we have also measured Gilbert4\ndamping for a series of higher-order modes in a thick\nPy (150 nm) film. Eddy-current damping ( αE∼0.003)\nis the dominant mode-dependent contribution in this\nfilm. The wavenumber kfor the mode p= 6 is roughly\nequal to that for the first SWR, p= 1, in the 25 nm\nfilm. Resonance positions are plotted with the dashed\nlines in Fig. 2(b), as a function of k, and are in good\nagreement with those found from the p= 1 data. In Fig.\n4 we plot the mode-related Gilbert damping αpup to\np= 6, which gradually decreases as pincreases. We have\nagain conducted full numerical simulations using the\nEM+LLG method with ( A∗\nk= 0.105 nm2) or without\n(A∗\nk= 0) the intralayer spin pumping term, shown in\nred and black crosses, respectively. Neither scenario fits\nthe data closely; an increase at p= 3 is closer to the\nmodel including the k2mechanism, but experimental α\natp= 6 falls well below either calculation.\nWe believe there are two possibilities why the α∝p2\ndamping term is not evident in this configuration. First,\nthe effective exchange field increases with p, resulting\nin a weaker (perpendicular) resonance field at the same\nfrequency. When the perpendicular biasing field at\nresonance is close to the saturation field, the spins\nnear the boundary are not fully saturated, which might\nproduce an inhomogeneous linewidth broadening at\nlower frequencies and mask small Gilbert contributions\nfrom wavenumber effect. From the data in Fig. 4 inset\nthe high- pSWR modes is more affected by this inhomo-\ngeneous broadening and complicate the extraction of k2\ndamping. Second, high- pmodes in thick films are close\nto the anomalous conductivity regime, kλM∼1, where\nλMis the electronic mean free path. The Rado-type\nmodel such as that applied in Fig. 4 is no longer valid\nin this limit[35], beyond which Gilbert damping has\nbeen shown to decrease significantly in Ni and Co[36].\nBased on published ρλMproducts for Py[37] and our\nexperimental value of ρc= 16.7µΩ·cm, we find λM∼8\nnm andkλM∼1 for the p= 6 mode in Py 150 nm. For\nthe 1st SWR mode in Py 25 nm, on the other hand,\neddy currents are negligible and the anomalous behavior\nis likely suppressed due to surface scattering, which\nreducesλM.\nAn important conclusion of our work is that the\nintralayer spin pumping, as measured classically through\nPSSWR, is indeed present but more than 10 times\nsmaller than estimated in single nanoscale ellipses[24].\nThe advantages of the PSSWR measurements presented\nin this manuscript are that the one-dimensional mode\nprofile is well-defined, two-magnon effects are reduced,\nif not absent[39], and there are no lithographic edges to\ncomplicate the analysis. The lower estimates of A∗\nkfrom\nPSSWR aresensible, basedonphysicalparametersofPy,\nCo, and CoFeB. The polarization of continuum-pumped\nspins in a nearly uniformly magnetized film, like that\nof pumped spin current in a parallel-magnetized F/N/F\nstructure, is transverse to the magnetization[14]. Fromthe measured transverse spin conductance σ⊥we extract\nthat the relaxation lengths of pumping intralayer spin\ncurrent are 0.8-1.9 nm for the three ferromagnets[27], in\ngood agreement with the small transverse spin coherence\nlengths found in these same ferromagneticmetals[23, 40].\nFinally, we show that the magnitude of the intralayer\nmT p=1, tFM =25-200 nm \np=0-6, t FM =150 nm \n(pπ/t FM )2 (×1016  m -2 )\np*\n*\nFIG.4. Mode-dependentdamping αpfor Py(150nm), 0 ≤p≤\n6. Crosses are EM+LLG calculated values with and without\nthe wavenumber-dependent damping term. Inset: Inhomoge-\nneous broadening ∆ H0vs 0≤p≤6, 150nm film. Larger,\nk-dependent values are evident, compared with those in the\nthickness series ( tFM=25-200 nm).\nspin pumping identified here is consistent with the\ndamping size effect notattributable to interlayer spin\npumping, in layers without obvious spin sinks. For the\np= 0 mode, a small but finite wavenumber is set by the\nsurface anisotropy through[30, 41] Aexk2= 2Ks/tFM.\nThe damping enhancement due to intralayer spin\npumping will, like the interlayer spin pumping, be\ninverse in thickness, leading to an ’interfacial’ term as\nα= 2Ks(A∗\nk/Aex)t−1\nFM. This contribution is indicated\nby the grey shadow in Fig. 3 insetand provides a\ngood account of the additional size effect in the SiO 2-\ncapped film. Here we use Ks=0.11 mJ/m2extracted\nby fitting the thickness-dependent magnetization to\nµ0Meff=µ0Ms−4Ks/MstFM. While alternate\ncontributions to the observed damping size effect for the\nSiO2-capped film cannot be ruled out, the data in Fig.\n3insetplace an upper bound on A∗\nk.\nIn summary, we have identified a wavenumber-\ndependent, Gilbert-type damping contribution to spin\nwaves in nearly uniformly magnetized, continuous\nfilms of the metallic ferromagnets Py, Co and CoFeB\nusing classical spin wave resonance. The term varies\nquadratically with wavenumber, ∆ α∼A∗\nkk2, with the\nmagnitude, A∗\nk∼0.08-0.10 nm2, amounting to ∼20% of\nthe bulk damping in the first excited mode of a 25 nm\nfilm of Py or Co, roughly an order of magnitude smaller\nthan previously identified in patterned elements. The\nmeasurements quantify this texture-related contribution5\nto magnetization dynamics in the limit of nearly homo-\ngeneous magnetization.\nµ0Ms(T)α0Aex(J/m) A∗\nk(nm2)∆α∗\n0\nPy 1.00 0.0073 1.2×10−110.11±0.02 -0.0008\nCo 1.47 0.0070 3.1×10−110.08±0.01 -0.0002\nCoFeB 1.53 0.0051 1.8×10−110.09±0.02 -0.0011\nTABLE I. Fit parameters extracted from resonance fields and\nlinewidths of uniform and 1st SWR modes. Values of A∗\nk\nand ∆α∗\n0for Co and CoFeB are calculated using the spin\nmixing conductances measured in FM/Cu/Pt[17]. See the\nSupplemental Material for details.\n[1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[2] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1\n(1996).\n[3] L. 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Understanding the magnetic \ndamping in these ma terials is crucial, but normal Ferromagnetic Resonance (FMR) measurements \nface some limitations. The desire to quantify the damping in materials with PMA has resulted in \nthe adoption of Time -Resolved Magneto -optical Kerr Effect (TR -MOKE) measurements. In t his \npaper, we discuss the angle and field dependent signals in TR -MOKE, and utilize a numerical \nalgorithm based on the Landau -Lifshitz -Gilbert (LLG) equation to provide information on the \noptimal conditions to run TR -MOKE measu rements . \n \nI. INTRODUCTION  \nSpintronics utilizing perpendicular magneti c anisotropy (PMA) are very promising for the \nadvancement of computer memory, logic, and storage. Due to the time scale of magnetic switching \nin these devices (~ 1 ns), it is crucial to understand the ultrafast dy namic magnetization, which \nbehave according to the Landau -Lifshitz -Gilbe rt (LLG) equation. The application of this equation to understand magnetization dynamics requires knowledge of the magnetic anisotropy and the \nGilbert damping (α). While anisotropy can  be determined through magnetostatic measurements, \nextracting α requires measurements that can capture the dynamic magnetization at time scales \nfaster than magnetic switching. To date, the most common method to do this is through frequency \ndomain measureme nts of ferr omagnetic resonance ( FMR ). By measuring the resonance frequency \nand linewidth as a function of field, FMR can probe both the magnetic anisotropy a nd Gilbert \ndamping . As spintronic  applications  begin to use materials  with large PMA, the use  of another \ntechnique, time -resolved magneto -optical Kerr effect (TR -MOKE), has increased. This technique \n(which is essentially a time -domain FMR measurement technique) is able to measure at higher \nresonance frequencies and external fields, which allows ex tremely hard mag netic materials to be \nmeasured . \nThere are many papers discussing TR -MOKE measurements for measuring the Gilbert \nDamping. Most of these papers utilize similar polar MOKE measurement techniques, but there is \noften a large variation in both  the Hext range for measurements and in the angle of external field. \nWhile some papers utilize in -plane external field because of its well -understood frequency \ndependence, others choose to apply the field at a chosen angle away from the surface normal . It \nhas been theorized and shown in measurements that the process of applying the field at some angle \nbetween 0 and 90°  is beneficial to increase the TR -MOKE signal amplitude, but the explanations \nas to why this occurs are lacking . In this paper, we aim to discuss why the signal depends on the \nangle of external field and calculate the optimal angle for conducting TR -MOKE measurements \nof damping on magnetic materials with PMA.  \n \n II. FINITE DIFFERENCE METHOD LANDAU -LIFSHITZ -GILBERT EQUATIONS  \nSimulations in this  work utilize a finite difference approach to solve the LLG equation \n(Eq. 1) with an explicit solution for the magnetization vector ( M) as a function of time following \nthe forward Euler method.  \neff\nsdd\ndt M dt    MMM H M\n          (1) \nwhere M is the magneti zation vector with a magnitude of Ms (the saturation magnetization), γ is \nthe gyromagnetic ratio, Heff is the effective magnetic field, and α is the Gilbert damping parameter. \nThe vector Heff is determined by taking the gradient of the magnetic free energy  density ( F) with \nrespect to the magnetization direction (\neff FM H ). The scalar quantity F is the summation of \ncontributions from Zeeman energy (from the external magnetic field, Hext), perpendicular uniaxial \nmagnetic anisotropy ( Ku), and  the demagnetizing field (assuming the sample is a magnetic thin \nfilm).  \nWhile Eqn. 1 is often used to describe magneto -dynamics due to the use of α, it is not \nconducive to numerical solutions of this ordinary differential equation. To simplify the \ndevelopment of computational algorithms, it is preferential to utilize the Landau -Lifshitz equation \n(Eq. 2).  \n  eff eff 2\ns'd\ndt M    MM H M M H\n.             (2) \nThe coefficients in Eq. 2 can be related to the previously defined constants in Eqs. 3 and 4 [1]. \n2'\n1\n\n\n           (3) \ns'M \n           (4) In equilibrium, M is parallel to Heff, and so the magnetization does not precess . If the \nmagnetization is removed from the equilibrium direction, it will begin precessing around the \nequilibrium direction, finally damping towards equilibrium at a rate determined by the magnitude \nof α (shown in Fig. 1) .  \n \nFigure 1 . A three -dimensional representation of the magnetization vector ( M) precessing around the equilibrium \ndirection ( θ) displayed on the surface of a sphere of radius Ms. The equilibrium direction  is controlled by the magnitude \nand direction ( θH) of the external magnetic field vec tor (Hext). The change in the z-component of magnetization (Δ Mz) \nis proportional to the TR -MOKE signal.  \n \nTo initiate precession, a thermal  demagnetization process  is applied , emulating  TR-MOKE \nmeasurements. For TR -MOKE measurements, a “pump” laser pulse increases the temperature at \nan ultrafast time scale, causing a thermal demagnetization (a decrease in Ms caused by temperature) \n[2, 3] . This thermal demagnetization temporarily moves the equilibrium direction causing the \nmagnetization to begin precession, which is continued even when Ms has recovered to its original \nstate.  Here, the demagnetization process is treated as a step decrease in Ms that lasts for 2.5 ps \nbefore an instant recovery to the initial value.  All signal analysis discussed in this work  is following \nthe recovery of Ms. \nFor polar MOKE measurements, the projected magnetization in the z -direction ( Mz, \nthrough -plane magnetization ) is proportional to the Kerr rotation  [4]. The projection of Mz in time \nduring precession will appear is a decaying sinusoid  (\n sin exp /zM t t t      ), which is \nalso captured by TR -MOKE measurements. The amplitude of the precession will greatly depend \non the applied field magnitude and angle, which is also carried into TR -MOKE signal. By \nanalyzing the precession as a function of field and angle, the precession amplitude (delta Mz) can \nbe extracted. Figure 2 shows the process of extracting the amplitude as a function of angle for two \ndifferent regions of magnetic field. Tracking this signal amplitude as a function of θH, reveals  that \nthe precession (and thus the signal) will be maximized for a certain θH as shown in Fig. 2(b). \nMaximizing the oscillation implies that it will be beneficial to maximize the “magnetic torque” \nterm (M × Heff, which prefers a large angle between M and Heff), but it also important to factor in \nthat TR -MOKE measures the projection of the magnetization along the z-direction (which prefers \nθ = 90°). Because of this, the value of θH,MAX  requires weighing inputs from both the magnetic \ntorque and the z-direction projection of  magnetization.  \n \nFigure 2. For specific conditions, the LLG simulation will produce a time -dependent magnetization vector. The \ndifference between the maximum and mini mum of the z-component of magnetization in time (Δ Mz) provides \ninformation about the strength of the TR -MOKE signal. These simulations are conducted for a range of θH resulting \nin the curves in (b). The trend of signal with increasing θH also depends on th e magnitude of the external field relative \nto Hk,eff, as shown by the black ( Hk,eff < Hext) and red ( Hk,eff > Hext) lines.  \nDepending on whether the field ratio ( Hext/Hk,eff) the angular dependence on magnitude \nwill drastically change. For Hext<Hk,eff, the magnetization will be in equilibrium between the \nperpendicular direction and the in -plane direction (0  ≤ θ ≤ 90°). Maximizing the magnetic torque \nand projection in the z -direction in these cases will cause Hext applied in -plane (θH = 90°) to be the \noptimal setup [shown by the red line in Fig. 4(b)]. Once Hext exceeds Hkeff, the Stoner -Wolfarth \nminimum energy model predicts that the magnetization will approach the direction of external \nfield, but never align (except along θH = 0 or 90 °). Because these two directions will have no \nmagnetic torque, there should be no magnetic precession, and thus there will be amplitude minima \nat these extremes. Between these two angles, the two effects for optimizing signal will complete, \nleading to a n amplit ude maximum at an angle that depends on the size of the ratio.  \nFigure 3 shows a contour plot of the dependence on  signal amplitude as a function of both \nthe magnitude of Hext and θH. The highest amplitude of precession will occur near Hk,eff when the \nfield is applied in the film plane. If the external field is greater than Hk,eff, it is beneficial to conduct \nthe measurement at an angle that is out of the film plane. To reveal this trend, the dotted red line \nin Fig. 3 indicates the angle of maximum signal ( θH,MAX ) as a function at specified field ratios. \nNote that the curve does not follow the gradient of signal vs. θH and Hext. This is due to the \ndefinition of θH,MAX  as the value of θH that maximizes signal for a given Hext , instead of a \nmaximization of signal with both parameters.  Based on these results, measurement conditions can \nbe tuned to maximize the signal based on the field ratio. For example, if the maximum strength of \nthe magnetic field is only 2 Hk,eff, then it would be beneficial to set θH > 60°. Furthermore, \nmeasurements conducted at a constant field and a varied magnetic field angle, should not \nnecessarily conduct the measurement at the highest possible Hext if the goal is to maximize SNR.   \nFigure 3. A contour plot of the relative signal size  as a function of field ratio ( Hext/Hk,eff) and θH where a value of “1” \nindicates the maximum possible signal. The dotted line shows the θH where the signal is maximized at a specific field \nratio.  \n \nFor field -swept measurements, (where the angle is held constant and the field is swept) \nFig. 4 should provide a simple guide for maximizing signals  (a summary of θH,MAX  in Fig. 3). To \nfurther assist in the design of TR -MOKE signals to maximize SNR, we suggest a simplified \nestimation for the determination of th e amplitude of TR -MOKE signal. Equation 5 predicts the \nprecession amplitude based on the equilibrium direction ( θ, from Fig. 1) and the external field \nangle. The magnitude of Hext is integrated into Eq. 5 through the θ through Eq. 6 whic h provides \nthe mini mum energy condition.  \nH\nssin sinzM\nM  \n      (5) \n  ext H k,eff2 sin sin 2HH  \n         (6) \nThis simplified expression is based on the product of the two components for signal \nmaximization previously discussed: the projection of the magnetization in the z -direction , \n sin , \nand the magnetic torque, \nH sin . While the simplified expression presented in Eq. 2 cannot \ncapture all the details of a more complex LLG simulation, it is more than accurate enough for an \ninitial estimate of θH,MAX , as shown by the comparison in Fig. 4.  \n \nFigure 4. The trend of θH,MAX  at a given field ratio. The open circles indicate results from the LLG simulation discussed \nin Section I, while the red curve is the simplified model from Eq. 5.  \n \nIII. COMPARING SIMULATION RESULTS TO TR -MOKE MEASUREMENTS  \nTo verify the precited results for the m aximum TR -MOKE signal amplitude, a series of \nmeasurements were conducted on a 300 °C post -annealed W/CoFeB/MgO film (see our previous \npublication for more information ). After conducting measurements, the thermal background was \nsubtracted leaving purely the decaying sinusoidal term. The oscillation amplitude from \nmeasurement was calculated as shown in Fig. 2a. Results from four values of Hext and six value s \nof θH are summarized in Fig. 5.  \n \nFigure 5. Normalized TR -MOKE oscillation amplitudes directly for a W/CoFeB/MgO  when Hext is 4, 6, 8, and \n10 kOe. The open red circles show the measurement data (a line between points is provided to guide the eye) while \nthe black curves indicate the results from the LLG simulations for a material with Hk,eff ≈ 6 kOe.  \n  \nComparisons between the trends predicted simula tions and measurement results show \nremarkable agreement. As expected, the signal amplitude decreases with increasing angle for \nHext < Hk,eff (Hk,eff ≈ 6 kOe ) and decreases with increasing angle for Hext > Hk,eff. These \nmeasurements can even capture the predicted peak of amplitude at nearly the same θH for fields \nnear Hk,eff. For the 6 kOe measurements, there is a slight deviation in the amount of decay in signal \nstrength for decreasing θH (simulations predict a s lower decrease). This is most likely due to an \ninhomogeneous broadening effect (i.e. the Hk,eff in the sample has a distribution of values) leading \nto a deviation from theory near Hk,eff. While the θH in the setup used in this experiment was limited, \nthese  results verify that the excellent agreement between simulation and measurement.  \n \nIV.  CONCLUSION  \nIn conclusion, we utilized a numerical approach to calculate the dynamic  response of \nmagnetization to a demagnetization process. We find that the size of the magnetic precession, and \nthus the size of the TR -MOKE signal depends on the angle and amplitude of the external field \n(relative to Hk,eff). To verify the results of these  simulations, we conducted measurements on a \nW/CoFeB/MgO sample with perpendicular magnetic anisotropy. The results of the measurements \nshow that the magnitude of the TR -MOKE signal shows good agreement with our prediction. \nThese results should assist to m aximize the SNR in TR-MOKE measurements.  \n \nACKNOWLEDGEMENTS  \nThis work is supported by C -SPIN  (award #: 2013 -MA-2381) , one of six centers of STARnet, a \nSemiconductor Research Corporation progra m, sponsored by MARCO and DARPA.  \n \nREFERENCES  \n[1] Iida, S., 1963, \"The difference between gilbert's and landau -lifshitz's equations,\" Journal of \nPhysics and Chemistry of Solids, 24(5), pp. 625 -630. \n[2] van Kampen, M., Jozsa, C., Ko hlhepp, J. T., LeClair, P., Lagae, L., de Jonge, W. J. M., and \nKoopmans, B., 2002, \"All -Optical Probe of Coherent Spin Waves,\" Physical Review Letters, \n88(22), p. 227201.  \n[3] Zhu, J., Wu, X., Lattery, D. M., Zheng, W., and Wang, X., 2017, \"The Ultrafast Laser Pump -\nProbe Technique for Thermal Characterization of Materials With Micro/Nanostructures,\" \nNanoscale and Microscale Thermophysical Engineering, 21(3), pp. 177 -198. \n[4] You, C. -Y., and Shin, S. -C., 1998, \"Generalized analytic formulae for magneto -optical Kerr \neffects,\" Journal of Applied Physics, 84(1), pp. 541 -546. \n "    },    {        "title": "1401.1672v1.Dynamic_exchange_via_spin_currents_in_acoustic_and_optical_modes_of_ferromagnetic_resonance_in_spin_valve_structures.pdf",        "content": "1 \n Dynamic exchange via spin currents in acoustic and optical modes of \nferromagnetic resonance in spin -valve structures  \n \nA.A. Timopheev1, Yu.G. Pogorelov2, S. Cardoso3, P.P. Freitas3, G.N. Kakazei2,4, N.A . Sobolev1 \n1Departamento de Física and  I3N, Universidade de Aveiro, 3810 -193 Aveiro, Portugal  \n2IFIMUP and IN -Institute of Nanoscience and Nanotechnology, Departamento de Física e Astronomia, \nUniversidade do Porto, 4169 -007 Porto, Portugal  \n3INESC -MN and IN -Institute of Nanoscience and Nanotechn ology, 1000 -029 Lisbon, Portugal  \n4Institute of Magnetism, NAS of Ukraine, 03142 Kiev, Ukraine  \n     \ne-mail:  andreyt@ua.pt  \nTwo ferromagnetic layer s magnetically decoupled by a thick normal metal spacer layer can be, \nnevertheless, dynamically coupled via spin currents emitted by the spin -pump and absorbed through the \nspin-torque effects at the neighboring interfaces. A decrease of damping in both layers due to a partial \ncompensation of the angular momentum leakage in e ach layer was previously observed at the coincidence \nof the two ferromagnetic resonances. In case of non -zero magnetic coupling, such a dynamic exchange \nwill depend on the mutual precession of the magnetic moments in the layers. A difference in the linewid th \nof the resonance peaks is expe cted for the acoustic and optical regimes of precession. However, the \ninterlayer coupling hybridizes the resonance responses of the layers and therefore can also change their \nlinewidths. The interplay between the two mechan isms has never been considered before. In the present \nwork, the joint influence of the hybridization and non -local damping on the linewidth has been studied in \nweakly coupled NiFe/CoFe/Cu/CoFe/MnIr spin -valve multilayers. It has been found that the dynamic  \nexchange by spin currents is different in the optical and acoustic modes, and this difference is dependent \non the interlayer coupling strength. In contrast to the acoustic precession mode, the dynamic exchange in \nthe optical mode works as an additional da mping source. A simulation in the framework of the Landau -\nLifshitz -Gilbert formalism for two ferromagnetic layers coupled magnetically and by spin currents has \nbeen done to separate the effects of the non -local damping from the resonance modes hybridizatio n. In \nour samples both mechanisms bring about linewidth changes of the same order of magnitude, but lead to \na distinctly different angular behavior. The obtained results are relevant  for a broad class of coupled \nmagnetic multilayers  with ballistic regime o f the spin transport . \n    \n1. Introduction  \nSpin current, a flow of angular momentum , is a basic concept in spintronics  and spin caloritronics [1, \n2]. Spin current generation is experimentally accessible via spin pumping [3-5], spin Seebek effect [6], spin \nHall effect [7, 8] and acoustic wave propagation in the case of magnetic insulators [9]. The spin -orbit \ninteraction plays a fundamental role in these effects. The presence of a spin current in a normal metal 2 \n (NM) or semiconductor can be detected by the inverse spin Hall effect [10-12] or as a change of the \neffective damping in an adjacent ferromagnetic (FM) layer [3-5]. The latter effect allows one to alter the \nswitching field of the FM layer and even sustain  a stable precession in it [13-15]. It is hard to overestimate \nthe fundamental and practical importance of the issues emerging from the investigation  of the spin \ncurrents.  \nA precessing magnetic moment in a FM layer acts as a spin battery [16] injecting a pu re spin current in \na neighboring NM layer through the FM/NM interface. This spin current can then  return to  the NM/FM \ninterface bringing the carried angular momentum  back to the precessing spins of the FM layer. Depending \non the spin -orbit interaction stre ngth and the layer thickness, the normal metal will absorb a certain part \nof the angular momentum flow via the spin -flip relaxation processes. Thus, the backflow through the \nNM/FM interface will be always weaker than the direct flow, which results in an enhanced  precession  \ndamping  [3-5]. The spin diffusion length of the normal metal and the spin mixing  interface conductance \ncan be evaluated in this way [3-5, 17]. \nAn interesting result has been obtained for a FM/NM/FM trilayer [18] having non -identical FM layers. \nThe asymmetry provided different angular dependences of the ferromagnetic resonance (FMR) fields of \nthe FM layers. When the external magnetic field was directed at an angle for which the FMR peak \npositions coincide, a narrowing of both resonances w as observed. The explanation of this effect is that, \nfor the case of separately precessing FM layers, the spin current generated in  a precessing FM layer is \nabsorbed in the other, non -resonating FM layer, which causes , in a full analogy to the written  abov e, a \ndamping enhancement,  while for the case of a mutual resonant precession this spin current leakage is \npartially compensated by the spin current from the other FM layer. In this experiment , the NM spacer was \nthin enough for the spin current to be consid erable at the second NM/FM interface, but thick enough to \nexclude any possible magnetic coupling between the FM layers.  \nIndeed, the magnetic coupling between two FM layers complicates the analysis of the spin -current -\ninduced non -local damping. If the coupling is strong enough, the resonance response of the system is \nrepresented by the collective acoustic and optical modes w hich are the in -phase and out -of-phase mutual \nprecession modes in the  FM layers. There is no separate precession in such a regime – the precession in \none layer drags the magnetic moment in the other one. Moreover, the linewidths of the resonance peaks \nare dependent o n the field separation betwe en them, and usually these parameters are angular dependent. \nAnd finally, the interaction fundamentally forbids the peaks to have a crossing point, i.e. the anticrossing \nis a characteristic feature here. The stronger the interlayer coupling, the larger is the anticrossing \nseparation between  the modes. From this point of view, the difference of damping  for the acoustic and \noptical modes in  a FM/NM/FM trilayer as a result of a dynamic spin currents exchange, theoreticall y \npredicted by Kim and Chappert [19], seems to be experimentally unachievable. Nevertheless , in several \nrecent papers [20-22] experimental observations of this effect  have been already claimed.  There  is, 3 \n however, a full ignorance of the fact that the FMR p eaks hybridization will also influence the linewidth \neven if a separate measurement of the precession in each layer can be done.  \nMotivated by this, we have performed  a comprehensive  study of weakly coupled spin -valve (SV) \nmultilayers, where the hybridizati on is weak and the layers behave almost independently, conserving at \nthe same time the main features of the acoustic and optical modes of the collective magnetic response. \nOne important objective is to separate the hybridization -induced change of the FMR linewidth from the \nspin-current -induced one and to check in this way the difference between the spin -current -induced \ndamping in the optical and acoustic regime s of precession. W e present an  experimental study of the FMR \nin NiFe/CoFe/Cu/CoFe/MnIr SV  multilayers  conducted using a standard X -band EPR spectrometer. Our \nstudy is acc ompanied by a simulation of the microwave absorption in such a magnetically coupled system \nin the presence of dynamical exchange by spin currents  in the framework of the Landau -Lifshitz -Gilbert \nformalism.  \n \n2. Experimental details  \nFMR was measured  at room temp erature using a Bruker ESP 300E E SR spectrometer at a microwave \nfrequency  of 9.67 GHz. The f irst derivative of the microwave absorp tion by the magnetic field  was \nregistered. For each sample, a series of in-plane FMR spectra were collected for different ang les of  the \nmagnetic field in the film plane with respect to the internal exchange bias  field. Each FMR spectrum , \nexperimentally measured or simulated,  was fitted by Lorentzian functions to obtain  angular dependences \nof the resonance field and linewidth.  The least -squares method was employed.  \nThe SV multilayers were grown by the ion -beam deposition in a Nordiko  3000 system.  The cobalt -\niron fixed layer is exchange coupled to the MnIr  antiferromagnet (AF), the free layer is a bilayer \ncomposed of a permalloy  and a cobalt -iron sublayers, and the copper spacer separates the free and fixed \nlayers. Two series of samples were used in the study:  \n1) Glass / Ta(30 Å) / Ni 80Fe20(30 Å) / Co 80Fe20(25 Å) / Cu ( dCu) / Co 80Fe20(25 Å) / Mn 82Ir18(80 Å) / \nTa(30 Å) – the average thickness of the copper spacer , dCu, varies from 17 to 28 Å in 1  Å steps.  \n2) Glass / Ta(30 Å) / Ni80Fe20(56 ‒ dF) / Co 80Fe20(dF) / Cu(22 Å) / Co 80Fe20(25 Å) / Mn 82Ir18(80 Å) / \nTa(50 Å)  – the relative thicknesses of the permalloy and cobal -iron sublayers var y within the \n56 Å thick free layer by setting the parameter dF to 8, 16, 24, 32 and 40  Å. \nAdditionally, separate free layers ( Glass / Ta(30 Å) / Ni80Fe20(56 ‒ dF) / Co 80Fe20(dF) / Cu(22 Å) / \nTa(50 Å)) of the first and second series were grown to serve as reference samples.  \nThe first series was already studied in Refs. [23, 24]. It has been shown that the samples with tCu > 16 Å \nare in the weak coupling regime, and the main interlayer coupling mechanism here is  Néel’s “orange -\npeel” magnetostatic interaction  [25]. When the copper spacer thickness grows from 17 to 28 Å, the 4 \n interlayer coupling energy is reduced from 1.1×102 erg/cm2 to 4×103 erg/cm2, which corresponds to a \nvariation of the effective interaction field on the free layer from  17 to 6 Oe.   \nThe second series has a fixed metallic spacer thickness, tCu = 22 Å, while the free layer effective \nmagnetization, 4π Meff, determined by the Kittel formula, gradually varies from 15 kG to 8.5 kG.  In this \nway the angular dependence of the free layer resonance field can be  vertically  shifted  with respect to that \nof the fixed layer . \n \n3. Simulation of the microwave absorption spectrum  \nA SV is considered as a system of two cou pled FM layers  consisting of a free and a fixed layer with \nthe thicknesses d1, d2, volume saturation magnetization s Ms1, Ms2, and in -plane uniaxial magnetic \nanisotropy constants K1, K2, respectively. The e xchange coupling of the fixe d layer to the AF  layer with \nthe interface coupling energy Eex is defined by a unidirectional anisotropy with the effective field \nex 2 s2E d M\n. The e asy axes of all three anisotropies lay in  the sample plane and have the same direction \nalong the magnetic field applied at annealing. The m agnetizations in  both layers are assumed  to be  \nuniform, thus the bilayer magnetic state is completel y described by the unit vectors \nˆˆ,12mm  of their \ninstantaneous directions. The layers are coupled by the Heisenberg exchange interaction , Eic.  \nThen the magnetic energy density per unit area of the considered system can be written as:  \n \n  \n\n22\ntot 1 s1 1 ext s1 mw s1\n22\ns2 2 ext s2\n2 icex\nmw s2\n2 s2ˆˆ ˆ ˆ ˆ ˆ ˆ 2\nˆ ˆ ˆ ˆ ˆ 2\nˆˆ .ˆˆˆ  U d M K H M h M\nM K H M\ndEEhMdM\n    \n   \n1 1 1 0 1 1\n2 2 2 0\n12\n2 2 2m ·n m ·û m ·h m ·h\nm ·n m ·û m ·h\nm ·m\nm ·h m ·û  (1) \nThere are also included four unit vectors determining the spatial orientation of the effective fields:  the \neasy axis \nˆûn  of the uniaxial and unidirectional anisotropies  (here \nˆn  is the normal to the multilayer \nplane) , the direction \nˆ\n0h  of the external magnetic field Hext, and the direction \n1ˆh  of the microwave \nmagnetic  field hmw.  \nThe spin -pump / spin -sink mechanism  in our SVs is considered as follows. The CoFe/Cu and \nCu/CoFe interfaces are assumed to be identical and t o give rise to an effective spin mixing conductance in \nthe FM1/NM/FM2 structure characterized  by the parameter  AFNF [26] which is in  a gener ic case \ndependent on the relative magnetization orienta tions in the layers, \nˆˆ,12mm . Since the copper spacer is \nmuch thinner than the  the spin -diffusion length ( λsd ~ 0.4 µm at T = 300  K), the transfer of the angular \nmomentum from one FM layer to the other occurs in a purely ballistic regime, i.e. the spin current emitted 5 \n at the first CoFe /Cu interface is fully absorbed at the second Cu/CoFe interface . The spin current \nbackflow is not considered separately: it just renormalizes the parameter  AFNF. The spin -pump  / spin-\ntorque induced damping \nsp  for each layer is influenced by its thickness, saturation magnetization and g-\nfactor.  The dynamics of such a structure can be described by a system of coupled Landau -Lifshitz -Gilbert \nequations with additional spin -pump / spin -torque induced Gilbert -like damping  terms [5]: \n \n1 eff sp\ntot\neff\nFNF\nsp B\nsˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ,\n1,ˆ\n,4\n, 1,2,\n.i\niis\ni\nit t t t\nU\ndM\nAgdM\nij\nij  \n                   \n\n\n\nm m m mm H m m m\nHmii\ni\nij i i i\ni i i i j\ni  (2) \nThe microwave field, \nmwˆjthe1h , is linear ly polarized and directed along the multilayer \nnormal, \n1ˆˆ||hn , while the external static magnetic field lies in the film plane,  \n0ˆˆhn , making an angle h \nwith the system’s easy axis \nû . A linear response of the system relates to small angle deviations from the \nequilibrium, \n2ˆ ˆ ˆ ˆ ,    1 1 2 1 1 2 2mδm , m δm δm m , δm m . The  complex vectors \njte12δm , δm  \ncan be found from a linear 4 ×4 system by Eqs. (2) linearized near the equilibrium . This system is too \ncomplicated for an analytical treatment but easily solved numerically using  a standard  desktop computer. \nA certain simplification can be achieved using spherical coordinates. The microwave absorption is \nproportional to the imaginary part of the microwave susceptibility in the direction of the  microwave field : \n \n 1 s1 2 s2 212\n1 2 Cu mw 1 2ˆˆ\nIm()d M d M dd\nd d d h d d        1 1 1δm h δm h . (3) \nTo treat the volume microwa ve susceptibility of a SV, the metallic  spacer width, dCu, was added in Eq. \n(3). Then a full cycle of calculations in each simulation consists of : i) finding the equilibrium orientation \nof the magnetic moments by the minimization of Eq. (1) ; ii) numerical solution of Eq. (2) linearized near \nthe equilibrium; iii) combining the obtained precession amplitudes in the volume susceptibility by Eq. (3) . \nThe separate susceptibility of each layer can be obtained if the thickness of the other layer is set to zero at \nthe last step of calculations. This can be useful in the analysis of experimental data obtained by the \nelement -specific X -ray magnetic circular dichroism, time -resolved Kerr microscopy and other techniques \nallowing to separately measure the microwave responses  of the layers [22, 27, 28]. 6 \n The m agnetic parameters in our simulations were set in accordance  to the experiment. In the studied \nsamples , the in -plane effective fields of the free and fixed layers are several times lower than the \nresonance field of the free layer ( Hres > 600 Oe), whose FMR linewidth will be the main discussion  issue  \nin the present paper. This implies that at the free layer’s resonance conditions the magnetic fi eld almost \naligns both magnetic moments. Thus, the dynamic exchange via spin currents will be considered in the \ncollinear regime , and the parameter AFNF is assumed to be independent of the in -plane magnetic field \norientation.  \n \n4. General features of the FM R in both SV series  \nThe d ynamic s of two coupled FM layers can be described in terms of acoustic and optical modes, a \nhybridized response of the system to the exciting microwave field. These modes are the in-phase and out -\nof-phase mutual precession of the magnetic moments in the FM layers.  The acoustic mode bears averaged \nmagnetic parameters of the system, while the optical one gives information about the system’s \nasymmetry. The interlayer coupling shifts the optical mode away from the acoustic one, therefor e, the \ncoupling strength can be determined if the other effective fields in the system are known. However, this is \na strong coupling regime which has few similarities with the FMR of standard SV multilayers, including \nthe samples used in this study, where the effective inte rlayer coupling does not exceed  several tens of \nOersted.   \n \n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0fixed layerfree layerspin valve\n  '', arb. units\nHext, kOeh= 0o,\nEic=10-2erg/cm2,\nhmw= 1 Oe,\n/2 = 9.67 GHz.\n \nFig. 1.  FMR spectrum calculated for a SV in the weak coupling regime (top curve). The m iddle and \nbottom curves show separated responses from the free and fixed layers in the SV. The layer  parameters \ncorrespond to the  first series of SVs: d1 = 5.5nm, α1 = 0.012, Ms1 = 1155 emu/cm3, K1 = 5.7103 \nerg/cm3; d2 = 2.5 nm, α2 = 0.055, Ms2 = 1175 emu/cm3, K2 = 1.7104 erg/cm3, Eex = 0.094 erg/cm2 and \nEic = 0.01 erg/cm2. \n 7 \n The samples  under study are in a weak coupling regime provided by Néel’s “orange -peel” \nmagnetostatic interaction [25]. The determined effective interlayer coupling field, acting from one layer to \nanother, is in the 10 to 30 Oe range for both layers [24] in all sample s of the two series. The main \ninteraction effect is a constant decrease of the resonance field in each laye r. This and other  related effects \nare thoroughly discussed in Ref.  [24].  \nTo support the ideology of the weak coupling regime, a simulation of the microwave response has \nbeen done using a parameter  set for the first series and the interlayer coupling stren gth Eic = 0.01 erg/cm2. \nThe spin -pump  / spin-sink mechanism was switched off: αsp1 = αsp2 = 0. Fig. 1 shows a typical microwave \nabsorption spectrum of a SV multilayer and respective separated responses  of each layer in it. The \nmagnetic moments are precessing almost independently , and therefore each peak can be associated with \nthe precession of the magnetization in a specific layer . The a symmetry of the thicknesses, damping  \nparameter s and magnetizations is clearly manifested  in these spectra. A fixed layer  with half the thickness \nof the free one  is much easier dragged by the precessing free layer. However , the inverse effect, a drag of \nthe free layer by the reson ance precession in the fixe d layer, is not so pronounced: only a small \nasymm etry on the wings of the free layer peak is observed. A four times strong er damping, mainly that \ndue to the contact with an antiferromagnet [24], produces a much lower precession amplitude of the fixed \nlayer. The situation gets even worse because the free layer is twice  as thick as the fixed one, t hus, the \neffective interlayer coupling field, acting on the free layer from the preces sing fixed layer, is about two \ntimes  lower. Leaping ahead, it is evid ent that the spin -pump  / spin-torque effect will be more pronounced \nin the free layer . \nA very important feature is that, despite the almost independent precession of the layers, an optical -\nlike and acoustic -like behavior is still present in the dynamics. A precessing layer drags the magnetization \nof the other layer either in the “in -phase” or in the “out -of-phase” regime . For the case of \nferromagnetically coupled layers, the optical mode (an out -of-phase mutual precession) has , in a given \nmagnetic field,  a higher precession frequen cy than the acoustic mode (an in -phase mutual precession). \nTherefore , the optical mode will be observe d, at a given microwave frequency,  in lower r esonance fields. \nA specifics of the first sample series is that, for the parallel and antiparallel orientatio ns of Hext (φh = 0º \nand 180 º), the resonance field of the fixed layer is respectively lower (~  300 Oe) or higher (~  1000 Oe) \nthan th at of the free layer (~  700 Oe in both cases). As seen from Fig. 1 , this brings  about  an interesting \nbehavior: the precession of the free layer in the parallel Hext (Hext > 0, φh = 0º) drags the fixed layer “in -\nphase”, while in the antiparallel orientation ( Hext < 0, φh = 0º) it drags the fixed layer “out -of-phase”, i.e. \nin the optical mode.   \nIt is evident that the switching between the acoustic and optical “drag” regimes would disappear with \nthe fixed layer resonance peak being  below that of the free layer. This justifies our choice of the sample \nseries: a variation of the interlayer coupling in the first series sho uld influence the intensity of the dragged 8 \n precession, while varying the effective magnetization of the free layer in the second series will tune the \nresonance field of the free layer with respect to that of the fixed one.  \nFig. 2  shows the evolution of the angular dependences of the resonance field in both series . The \ngeneral properties of the samples are as follow s. The effective field of unidirectional anisotropy for the \nfixed layer is about 300 Oe , and it is the main in -plane anisotropic contribution he re. The free layer has a \nweak in -plane unidirectional anisotropy of 5 to 20 Oe, var ying with the NiFe/CoFe composition. The \nmagnetic parameters of the free layer are less fluctuating than those of the fixed one since the former  is \nthicker and always deposi ted on the same surface. The i ncreased roughness of the fixed layer also \nstrongly influe nces the AF /FM interface, giving rise to fluctuations not only of the fixed layer ’s effective \nmagnetization but also of the exchange bias coupling. It is hard  as well  to prepare reference samples for \nthe fixed layer . Our previous investigation ha s shown that a separately deposited fixed layer has \nconsiderably different magnetic parameters [24]. The s trong angular variation of the resonance field and \nthe direct contact wi th the AF has also a strong  influence  on the angular dependence of the linewidth even \nin a separately deposited fixed layer. Moreover, as the linewidth is extracted using the least -squares \nmethod, the accuracy of the fitti ng for the low -intensity peak stemming from the fixed layer will be m uch \nlower than for the free layer . Due to  these reasons and the asymmetry discussed above, the following  \ndiscussion of the experimental results is mostly focused on the linewidth, ∆Hfr, of the free -layer -related  \npeak and on its angular dependence, ∆Hfr(φh). \n \n0 60 120 1800,40,60,81,0\nfixed layer's peaks\n  Hres, kOe\nh, deg. dCu= 28 A\n dCu= 24 A\n dCu= 21 A\n dCu= 17 Afree layer's peaks\n   \n0 60 120 1800,20,40,60,81,0\nfree layer's \npeaks\nfixed layer's\n peaks\n  Hres, kOe\nh, deg.Ni80Fe20/Co80Fe20:\n 48 A /   8A\n 40 A / 16A\n 32 A / 24A\n 16 A / 40A  \nFig. 2.  Angular dependences of the FMR peaks for the free and fixed layer: the first series where the \ninterlayer coupling strength is varied by gradual ly changing the metal  spacer thickness dCu (left panel); \nthe second series where the mean FMR  field of the free layer  is varied by gradual ly changing the free \nlayer effective magnetization, Ms1 (right panel).  \n \n5. Analysis of angular dependences  9 \n Additional reference samples  which completely duplicate the free layer and the next nearest \nnonmagnetic layers in each SV sample  have been grown and used as a reference in the analysis of the \nangular dependences of the free layer FMR linewidth, ∆Hfr(φh). It has been found that < ∆Hfr> (averaged \nover the whole φh range) of each reference sample is at least 20% lower than < ∆Hfr> in the corresponding \nSV sample. However, the increased damping in the presence of a second FM layer (i.e. fixed layer) \ncannot be uniquely associ ated with the spin -pump  / spin -sink mechanism [5, 26], because a non -zero \ninterlayer coupling causes a hybridization of the resonance modes. Though the layers are weakly  coupled, \neach layer’s resonance mode bears a small portion of the magnetic behavior of the layer coupled to it. As \nthe free layer’s damping parameter is several times lower than the fixed -layer -related  one, the observed \nFMR line broadening in the SV can have both origins , and it demands a quantitative analysis. At the s ame \ntime, the shape of the ∆Hfr(φh) dependence in the SV samples deserves  additional attention .  \n \n0 60 120 18064728088\n  \nHfr, Oe\nh, deg. dCu=28A,\n dCu=24A,\n dCu=21A,\n dCu=17A.  \nreference sample\n   \n16 18 20 22 24 26 28481216Relative step height ,  %\ndCu, Å0.011 0.0088 0.0076 0.0064 0.0052 0.01 Eic, erg/cm2\n0.004\n72747678808284\n<Hfr >, Oe  \nFig. 3.  Linewidth of the free layer in the first sample series.  Left panel: The angular dependence for \ndifferent copper spacer thicknesses. The reference sample  curve  does not show step s. Right panel: The \nrelative step height  and mean linewidth versus the interlayer coupling strength . \n \nFig. 3  shows experimental results obtained on the first series of s amples, where the interlayer \ncoupling has been gradually tuned by changing the copper spacer thickness. The reference layer does not \nshow any noticeable ∆Hfr(φh) dependence . In contrast, a step -like shape of  the ∆Hfr(φh) dependence has \nbeen observed in all  SVs. A noticeable growth of ∆Hfr is observed for the antiparal lel orientation of the \nmagnetic field (90 º < φh < 270 º). The transition from the weaker  damped to the stronger  damped regime is \nquite smooth and occurs within the angular range where the fixed layer peak crosses the free layer’s one \n(see Fig. 2 ). The relative step height in the ∆Hfr(φh) dependence has been found to decrease w ith \nincreasing  copper spacer  thickness , dCu. In other words, with decreasing interlayer coupling, assumed to \nbe the only parameter influenc ing the free layer in this series, the observed step height  also decreases. As \nseen from Fig. 3 , the relative step height monotonously decreases from 12% to  4% with decreasing \ninterlayer coupling. It should be noted that, among the other extracted SV parameters analyzed as a 10 \n function of dCu, this one has the smoothest  dependence. As an example, we show the thickness \ndependence of < ∆Hfr> averaged over the whole [0,  360º] range of angles ( Fig. 3 ). Though the scattering \nof experimental points is several times higher, this parameter also shows a tendency to decrease, whose \nnature is hard to identify at present. A degree of resonance modes hybridization, weaken ing with \ndecreasing interlayer coupling, seems to be the most probable source of this effect.  The free layer \nresonance precession drags the magnetic moment of the fixed layer , and this could be itself an additional \nsource of increased linewidth. A more det ailed discussion o f a simultaneous influence of hybridization \nand spin -pump  / spin-sink effects on the linewidth will be given in the next Section.  \n0 60 120 180556065707580859095\nNi80Fe20/Co80Fe20:  48 A /   8A,  40 A / 16A,\n 32 A / 24A,  24 A / 32A,  16 A / 40A.\n  \nH, Oe\nh, Deg.reference samples\n \nFig. 4.  Angular dependences of the linewidth for the free layer in the second sa mple series and in the \nrespective reference samples.  \nIn the second SV series , an increase of the < ∆Hfr> parameter in comparison with the reference layers \nis also clearly seen (see Fig. 4 ). At the same time, the observed step -like ∆Hfr(φh) dependence has \nrevealed  additional features. The step from the weaker  damped to strong er damped regime is shifted to \nhigher angles as the mean resonance field of the free layer get s higher. The observed shift completely \nmatches that of the crossing angle, i.e., the angle  where the resonances of the free and fixed layers \ncoincide (see Fig. 2 ). The most important feature is the absence of step -like behavior in  the ∆Hfr(φh) \ndependence for the sample with  the Ni80Fe20(48 Å) / Co 80Fe20(8 Å) free layer. Fig. 2 shows that the \nresonances are not crossing there at all: the free layer’s resonance field is always higher than the  fixed \nlayer’s one.  \nAs compared to the first series, there are also additional peculiarities in the ∆Hfr(φh) dependences, \ndistorting the step -like shape. Th ey, however, are linked to the intrinsic  angular dependence of ∆Hfr of a \nconcrete free layer. An analysis of the reference samples shows that the increase of the Co 80Fe20 / Ni80Fe20 \nthickness ratio causes a noticeable increase in the angular variation of ∆Hfr. Also a considerable variation \nof the damping parameter is observed in the reference samples , however , of a nonsystematic character . \nThese intrinsic features, as seen from Fig. 4 , are conserved also in the SV samples.  \n 11 \n  \nThus, the observed experimental results can be res umed as follows. When the fixed layer resonance \nfield is higher than the free layer’s one, the linewidth of the free layer  peak, ∆H fr, get s larger. The \nrespective angular dependence,  ∆H fr(φh), shows a step -like shape with the threshold an gular position \ncorresponding to the crossing region of the free and fixed layer resonances. The step height decreas es \nwith decreas ing interlayer coupling strength. This effect is absent in the reference samples containing \nonly the free layer , as well as it  disappears in the SVs where the resonances of the free and fixed layers do \nnot cross.  \n \n6. Hybridization versus non -local damping  \nTo clarify the interpretation of the experiment, a series of in -plane FMR spectra w ere simulated as a \nfunction of the in -plane magnetic field direction φh employing the formalism described in Sec. 3 . The \nsimulated spectr a display the resonance peaks by the free and fixed layer (as shown, e.g., in Fig. 1 ). By \nfitting a set of overlapping Lor entzians to the simulated spectrum, the resonance peaks’ parameters were \ndeduced. Then the angular dependence of the linewidth of the free layer, ∆Hfr(φh), was analyzed. For the \nfirst sample series, the layer  parameters and coupling were determined in our previous work  [24] on \nexactly the same samples. For the second series, these parameters were chosen to reproduce the \nexperiment as close as possible , and the interlayer coupling was fixed to Eic = 0.01 erg/cm2 in all SV s. \nFluctuating parameters of the fixe d layer and a slight variation of the internal damping of the free layer \n0 60 120 18066697281848790\n00 - SP\nIC - SP\nIC - 00\n  Hfr, Oe\nh, deg.00 - 00 \nFig. 5.  Simulated angular dependences of the free layer’s FMR linewidth. Four different regimes are \nshown: “00 -00”: Eic=  0 and \nFNFA  = 0; “IC -00”: Eic = 0.01 erg/cm2 and \nFNFA  = 10; “00 -SP”: Eic = 0 \nand \nFNFA  = 1.11015 cm‒2; “IC -SP”: Eic = 0.01 erg/cm2 and \nFNFA  = 1.11015 cm‒2. The layer  \nparameters refer to the first series of SVs, as they are already listed in the caption to Fig. 1 . 12 \n noted in the experiment were ignored in the simulation. In both series, the effective spin -mixing \nconductance for the whole FM/NM/FM structure is assumed  to be \nFNFA  = 1.11015 cm2 (which is \nslightly lower than in case of a single Co/Cu interface ~ 1.41015 cm2 [26]), in units of e2/h. \nRelative contributions of the hybridization and spin -pump  / spin-sink effects to the linewidth of a \nweakly coupled SV system are the central  object of th e present  investigation. Referring to a SV from the \nfirst series, we have done four different simulations (see Fig. 5 ) of the ∆Hfr(φh) dependences. First , both \nthe interlayer coupling (IC) and the spin mixing conductivity (SP) were set to zero (the “00 -00” curve). \nThis has demonstrated that the fitting procedure  correctly extracts the linewidth , and the free layer’s ∆Hfr \ndoes not depend on the peaks separa tion between the free and fixed layers (when fully uncoupled). It has \nbeen found that a small increase of ∆Hfr is observed in the crossing region. This increase, however, is \nlower than 0.3%, thus being at least one order of magnitude lower than the other f actors relevant for the \n∆Hfr(φh) dependence , both in the experiment and simulation. Therefore , this factor was ignored in the \nabove experimental data and will be omitted  in the further  consideration s.  \nThe next simulation has been made with Eic = 0.01 erg/ cm2 and \nFNFA  = 0 (the “IC -00” curve). In this \ncase, a noticeable increase (~  7%) in ∆Hfr is observed in the crossing region. This effect can be only \nattributed to an enhanced hybridization of the resonance peaks in th is region. When increasing the \nlinewidth of the free layer peak, the hybridizat ion also makes the fixed layer  peak narrower. The \ndependence of the hybridization degree on the distance between  the resonance peaks is also responsible \nfor the fact that  the ∆Hfr value for the antiparallel orientation ( φh = 180 º) is slightly higher ( by ~ 1.3%) \nthan that for the parallel orientation ( φh = 0º). As seen from Fig. 2 , the resonance peaks are indeed closer \nto each other in the antiparallel orientation . It is worth noti ng that the shape of the ∆Hfr(φh) dependence is \nquite different from the exp erimentally observed step -like profil e.  \nA pure spin -pump / spin -sink regime has been set in the next simulation, i.e. with Eic = 0 and \nFNFA  = \n1.11015 cm‒2. The corresponding ∆Hfr(φh) dependence is labeled “00 -SP”. In comparison with the \npreviously discussed regime, ∆Hfr is depressed (by ~  2%) in the crossing region. This effect was observed \nexperimentally in a FM/NM/FM system and has been interpreted as a p artial compensation of the spin \ncurrent leakage which occurs when  both FM layers are in resonance precession [5] and thus emit the spin \ncurrents. Without discussing t his in details, we note only two points: i) due to the considerably  thicker \nFM layers in our SVs , the observed effect is much weaker than in the above mentioned paper  [5]. Since \nthe spin torque effect is of interfacial origin, its influence scales with the  inverse layer thickness; ii) the \nspin-pump  / spin-sink and hybridi zation effect s work in the opposite  senses  in the crossing region.  \n \n 13 \n \n0.000 0.013 0.0266080100120\n0FNFA\n    h=00\n   h=1800\n  \nHfr, Oe\nEic, erg/cm2\n15 21.1 10 cmFNFA  \nFig. 6.  Linewidth of the free layer in the parallel and antiparallel orientation versus the interlayer \ncoupling strength simulated through spin conductiv ity (and without it). The layer  parameters are set \nfor the first series of SVs, as they are already listed in the caption to Fig. 1 . \nThe last simulation, labeled “IC -SP”, shows a simultaneous action of the interlayer coupling and sp in-\npump / spin -sink effect,  i.e. Eic = 0.01 erg/cm2 and \nFNFA  = 1.11015 cm‒2. As seen from Fig. 5 , there is a \ngood agreement with the experiment. The step size in the ∆Hfr(φh) dependence is ~  8%, also very close to \nthe experimental values. In the parallel orientation ( φh = 0º), the ∆Hfr value  is almost the same as in the \ncrossing region for the case of the pure spin -pump  / spin-sink effect. This means that a partial \ncompensation of the spin current leakage takes place in the whole range of angles for the acoustic regime \nof precession ( ‒90º < φh < 90º). On the contrary , in t he optical regime ( ‒110º > φh > 110 º) the free layer \nsuffers additional damping, absent in the previously discussed “00 -SP” simulation. The explanation is as \nfollow s. The p recession can be  geometrically separated in a transversal and a longitudinal component of \nmagnetization with respect to its equilibrium orientation. The c onservation of angular momentum allows \nthe same separation for the generated spin current. For a small -angle precession, the transversal \ncomponent of magnetization (  sin(θprec)) is larger than the longitudinal one ( sin2(θprec/2)). The \ntransversal part varies in time, while the longitudinal does not (at least in the linear response \napproximation , neglecting, e. g., a possible nutation). The i mportance of the time -dependent  transversal \npart of the spin current has been recently sho wn in Ref.  [29]. Both components are transferred by the spin \ncurrent from one FM layer to the other. In the acoustic precession mode (a s well as in the crossing point \nfor the “00 -SP” case), the transversal component of the spin current from the second layer is in -phase \nwith the transversal part of that from the first layer. Therefore , the spin current absorbed at the interface \nshould act in an “anti -damping” manne r. On the contrary, in the optical pr ecession regi me the transversal \ncomponent of the absorbed spin current is out -of-phase with the magnetic moment precession , and \ntherefore an extra damping occurs. An increase of the non-local damping in the optical precession regime \nin a magnetically  coupled FM/NM/FM trilayer has been predicted by Kim in Ref.  [19]. Probably this \neffect was observed in several  papers [20-22]. However , its interpretation in these papers fully ignores the \nhybridization of resonance modes , and therefore it is hard to draw  some clear  conclusions.  14 \n The weak interlayer coupling and an almost symmetri cal position of the free layer  peak with respect \nto the fixed one in the first SV series play an important role  in the non -local damping effect. Fig. 6  shows \nthe calculated ∆Hfr parameter versus the interlayer coupling strength for φh = 0º and φh = 180 º, with and \nwithout spin -pump  / spin-sink effect. It is seen that, for Eic < 0.013 erg/cm2, the increase of ∆Hfr occurs  \nmerely due to the non-local damping effect, while for a stronger coupling  the hybridization takes a \ncomparable role , and the se two contributions are hardly separable in a real experiment . From this \nsimulation  it is also seen that the dynamic exchang e via spin currents is quite different in the optical and \nacoustic precession modes . The i ncrease of ∆Hfr due to increasing hybridization is suppressed in the \nacoustic mode ( φh = 0º) by “anti -damping”, i.e., in-phase interaction between the transversal components \nof magnetization and the absorbed spin current. On the  contrary, in the  optical  precession  mode  (φh = \n180º) the effect of non -local damping is considerably enhanced, as the transversal components of the \nprecessing magnetization and of the absorbed spin current are out -of-phase.  \n \n0 60 120 1807580859095\n2\n15 2\nFNF0.01erg/cm ,\n1.1 10 cm .icE\nA \n\nMs1 = 1600 emu/cm3Ms1 = 1200 emu/cm3Ms1 = 1000 emu/cm3Ms1 = 800 emu/cm3 \n \nHfr, Oe\nh, deg.Ms1 = 700 emu/cm3\n   \n0 60 120 180666870727476\n  \nHfr, Oe\nh, deg. Ms1 = 700 emu/cm3\n Ms1 = 800 emu/cm3\n Ms1 = 1000 emu/cm3\n Ms1 = 1200 emu/cm3\n Ms1 = 1600 emu/cm3\n2\nFNF0.01erg/cm ,\n0.icE\nA\n  \nFig. 7.  Angular behavior of the linewidth in the second series of SVs, with a gradual variation of the \neffective magnetization of the free layer , simulated considering the  spin conductivity and without it. For \nthe red and black curves, the fixed layer resonance does  not cross that of the free layer  anymore . The \nparameters set is the same as for the first se ries and with Ms2 = 1525 emu/cm3 and Eex = 0.12 erg/cm2. \n \nA simulation of the ∆Hfr(φh) dependence in the second series, where the effective magnetization of \nthe free layer,  Ms1, is gradually changed, complete s the discussion. A comparison of the simulation ( Fig. \n7) with the experiment ( Fig. 2 , right panel) allows one to conclude that the effects of non -local  damping \nare also clearly seen here. First, when the free layer ’s saturation magnetization is such low that the fixed \nlayer pe ak does not cros s the free layer resonance, and therefor e, the precessing free layer drags the fixed \nlayer always in -phase (ac oustic mode), a characteristic step -like feature in the ∆Hfr(φh) dependence \ndisappears. I n these regime, the calculated ∆Hfr(φh) dependences are fundament ally different , \nirrespectively of whether the spi n conductivity exist s in the system or not. For the case of \nFNFA  = 0, the 15 \n fixed layer  peak approaching the free layer  one at φh = 180 º induces an enhanced hybridization , and ∆Hfr \ngrow s, while for \nFNFA  = 1.11015 cm‒2 the enhanced hybridization is fully suppressed by the described \nabove “anti -damping” feature of the acoustical mode of precession in the presence of spin conductivity. A \ndecrease of ∆Hfr is observed when the fixed layer peak is approaching. T he closer i s the fixed layer  \nresonance  to the free  layer  one, the higher is the precession amplitude in the fixed layer , and thus the \nhigher is the generated spin current. Therefore , a decrease of ∆Hfr is observed. Another distinct feature of \nthe non -local damping is  a continuous growth of the low -angle part of the ∆Hfr(φh) dependence (which \ncorresponds to the acoustical  precession mode) with decreas ing Ms1. As Ms1 decreases, all effective fields \narising from the interface , as well as the spin torque emerging from the absorbed spin current , will \nincrease. For the case of zero spin conductivity, the low -angle part of the ∆Hfr(φh) dependence remains \nalways the same. Both these features are clearly seen in  the experiment ( Fig. 2 , right panel ). \n \n7. Conclusions  \nIn-plane angular dependences of the free layer’s FMR linewidth have been studied in two series of \nspin-valve  multilayers , where the free and fixed layers are weakly coupled by N éel’s “orange peel” \nmagneto static interaction. In the first series, the interlayer coupling strength was  varied by changing the \nmetal  spacer thickness, while in the second series the in -plane resonance field of the free layer was tuned \nby changing the Ni 80Fe20/Co 80Fe20 thickness rat io.  \nThe main experimental results are as follow s. The a ngular dependence of the linewidth of the free \nlayer displays a characteristic step -like feature. When the resonance field of the fixed layer is higher than \nthat of the free layer , the damping increase s. The transition from the weakly  damped to strongly  damped \nregime occurs in the angular region of the peaks crossing. The reference  samples, containing only a free \nlayer and an adjacent nonmagnetic layer, do not show such a behavior. Similarly, no step is observed in \nthe samples from the second series , where the fixed layer  peak does not cross that of the free layer at all. \nThe step size decreases with decreasing interlayer coupling strength .  \nA comparison with simulation s has shown th at the observed effect is due to the non -local damping \neffect. In the weakly coupled regime, the hybridization of the resonance peaks is low , and each peak can \nbe attributed to the resonance precession of a particular layer. At the same time, due to a non -zero \nmagnetic coupling, the resonant precession in one layer induces a small correlated precession (“drag”) in \nthe other one. Depending on the relative fi eld position of the free layer  resonance peak with respect to the \nfixed one, the fixed layer  magnetic moment is “dragged” either in the acoustic -like (“in -phase” precession \nin both layers) or optical -like (“out -of-phase” mutual precession) regime. Therefore, varying the in -plane \nangle between the external magnetic field and the exchange bias field  and chan ging in this way the \nrelative peaks field position , one can switch between these two regimes. In case of ballistic regime of spin \ntransport, a dditionally to the time -independent longitudinal component, the spin current generated by the 16 \n dragged fixed layer has a time -varying transversal component, being “in-phase” or “out -of-phase” with \nthe time -varying transversal component of the free layer’s precessing magnetization. The resulting spin -\ntorque effect on the free layer will be either of “anti -damping” or “e xtra-damping” type, experimentally \nobservable as an additional increase/decrease of the linewidth in the antiparallel/parallel orientation. It is \nworth noting  that the acoustic regime is in a full analogy to the case of a magnetically  uncoupled \nFM1/NM/FM2 system [5] when the resonances coincide.  Another important point is  that diffusive regime \nof the spin transport will suppress the above described effects  due to averaging of transversal  components \nof the spin currents . \nOur study has also shown that the hybridization effect on the linewidth is of the same magnitude as \nthe non -local damping effect in the case of weak interlayer coupling , and that the hybridization fully \ndominates in the case of strongly coupled magnetic layer s. A separation of these two contributions, \nhowever, is possible due to  their different angula r behavior. In general case, contribution of the \nhybridization to the linewidth parameter will be dependent on degree of asymmetry of layers.  Thus, one \ncan expec t that, if the free and fixed layers would have the same damping , the influence of the \nhybridization would be considerably suppressed.  \n \nAcknowledgements  \nThis work was partially supported by the FCT of Portugal through the projects PEst/CTM/LA0025/2011, \nRECI/FIS -NAN/0183/2012, PTDC /CTM -NAN/112672/2009,  PTDC/FIS/120055/2010 , and grants \nSFRH/BPD/74086/2010 (A.A.T.)  and IF/00981/2013 (G.N.K.) as well as by the Euro pean FP7 project \n“Mold -Nanonet” .17 \n References  \n1 S. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura, Spin Current  (OUP Oxford, 2012).  \n2 E. Y. Tsymbal and I. Zutic, Handbook of Spin Transport and Magnetism  (Taylor & Francis, \n2011).  \n3 S. Mizukami, Y. Ando, and T. Miyazaki, Japanese Journal of Applied P hysics 40, 580 (2001).  \n4 R. Urban, G. Woltersdorf, and B. Heinrich, Physical Review Letters 87, 217204 (2001).  \n5 B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. Bauer, Physical \nReview Letters 90, 187601 (2003).  \n6 K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, \nNature 455, 778 (2008).  \n7 Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004).  \n8 J. Wunderlich, B. Kaestner, J. Sinova, and T. Ju ngwirth, Physical Review Letters 94, 047204 \n(2005).  \n9 K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrands, S. Maekawa, and E. Saitoh, \nNature materials 10, 737 (2011).  \n10 A. A. Bakun, B. P. Zakharchenya, A. A. Rogachev, M. N. Tkachuk, and V. G. Fle ǐsher, JETP \nLetters 40, 464 (1984).  \n11 E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Applied Physics Letters 88, 182509 (2006).  \n12 S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006).  \n13 L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. 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Abrudan, F. Brüssing, K. Gr oss, C. Luo, K. Westerholt, H. Zabel, F. Radu, and I. \nA. Garifullin, Physical Review B 86, 144422 (2012).  \n23 A. A. Timopheev, N. A. Sobolev, Y. G. Pogorelov, S. A. Bunyaev, J. M. Teixeira, S. Cardoso, P. \nP. Freitas, and G. N. Kakazei, J Appl Phys 113, 17D7 13 (2013).  \n24 A. A. Timopheev, N. A. Sobolev, Y. G. Pogorelov, A. V. Talalaevskij, J. M. Teixeira, S. \nCardoso, P. P. Freitas, and G. N. Kakazei, J Appl Phys 114, 023906 (2013).  \n25 L. Nèel, Compt. Rend. 255, 1676 (1962).  \n26 M. Zwierzycki, Y. Tserkovnyak, P. Kelly, A. Brataas, and G. Bauer, Physical Review B 71, \n064420 (2005).  \n27 M. K. Marcham, et al., Physical Review B 87, 180403 (2013).  \n28 O. Mosendz, G. Woltersdorf, B. Kardasz, B. Heinrich, and C. Back, Physical Review B 79, \n224412 (2009).  \n29 H. Jiao and G. E. W. Bauer, Physical Review Letters 110, 217602 (2013).  \n \n \n "    },    {        "title": "1709.02295v1.Tunable_spin_pumping_in_exchange_coupled_magnetic_trilayers.pdf",        "content": "arXiv:1709.02295v1  [cond-mat.mes-hall]  7 Sep 2017Tunable spin pumping in exchange coupled magnetic trilayer s\nM. Fazlali,1M. Ahlberg,1M. Dvornik,1and J.˚Akerman1,2\n1Department of Physics, University of Gothenburg, 412 96, Go thenburg, Sweden\n2Department of Materials and Nano Physics, School of Informa tion and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, 164 40 Kist a, Sweden\n(Dated: September 8, 2017)\nMagnetic thin films at ferromagnetic resonance (FMR) leak an gular momentum, which may be\nabsorbedbyadjacent layers. This phenomenon, knownas spin pumping, is manifested byanincrease\nin the resonance linewidth (∆ H), and the closely related Gilbert damping. Another effect of this\ntransfer of spin currents is a dynamical and long-range coup ling that can drive two magnetic layers\ninto a collective precession when their FMR frequencies coi ncide. A collective behavior is also\nfound in magnetic trilayers with interlayer exchange coupl ing (IEC). In this study we investigate\nthe interplay between IEC and spin pumping, using Co/Cu/Py p seudo-spin values. We employ\nbroadband FMR spectroscopy to explore both the frequency an d coupling-strength dependence\nof ∆H. Our observations show that there exists a cut-off frequency , set by the IEC strength,\nbelow which the precession is truly collective and the spin p umping is suppressed. These results\ndemonstrate that it is possible to control the spin pumping e fficiency by varying the frequency or\nthe interlayer exchange coupling.\nPseudo-spin valves are the building blocks of many\nspintronicdevices, suchasnanocontactspin-torquenano-\noscillators [1–6]. These devices not only show great\npromise for microwave [7, 8] and magnonic [9] applica-\ntions, but have also allowed for the exploration of spin\ntransfer torque driven propagating spin waves [10, 11]\nand magnetodynamical solitons [11–15]. Two corner-\nstones for further development towards applications, and\nto open new routes to answer fundamental questions are\nto tailor the magnetic damping and to control the flow of\nspin currents. The concept of spin pumping describes\nhow the leakage of angular momentum (spin current)\nfrom a precessing magnetic film may be absorbed at the\ninterface to another magnetic/non-magnetic layer, which\nprovides an additional damping term [16–18]. The di-\nmensionless damping coefficient is then given by α=\nα(0)+αsp, whereα(0)is the intrinsic damping of the pre-\ncessing layer and αspis the spin-pumping-induced term.\nWhile this effect has been studied extensively [19], the\nmajority of those investigation has focused on the regime\nwherethe staticcouplingbetweenthe layersisveryweak.\nThe staticinterlayerexchangecoupling(IEC, J′), is an\noscillatorycouplingpresentwhentwoferromagneticfilms\n(FMs) are separated by a sufficiently thin nonmagnetic\nlayer (NM) [20, 21]. The coupling will either promote\na parallel or antiparallel configuration of the magnetiza-\ntion in the films, depending on the spacerlayerthickness.\nIt can be described within the Ruderman-Kittel-Kasuya-\nYosida framework [22] and is therefore often referred to\nas RKKY-interaction. For sufficiently strong J′, the IEC\ncan separate the ferromagnetic resonance (FMR) of a\nFM/NM/FM trilayer into two modes, called the acoustic\n(in-phase) and optical (out-of-phase) mode [23]. While\nthe static IEC is oscillating and short-ranged in nature,\nthere also exists a dynamic and long-ranged coupling be-\ntween magnetic layers. Two magnetic films that precessat the same frequency can synchronize and display a col-\nlective behavior in the presence of spin pumping [18, 21].\nThe exploration of the interplay between the static and\ndynamic interlayer exchange coupling hence has great\nprospects of finding new exciting physics.\nWhile the literature includes studies on spin pumping\nin coupled layers [24–28], most focus on one or a few\nfrequencies. Here we investigate spin pumping in a wide\nfrequency range of 3-37 GHz using a broadband FMR\nsetup and examine four regimes: strong, intermediate,\nweak,andzeroIEC.ThesamplesarebasedonCo/Cu/Py\ntrilayers, where the thickness of the Cu spacer sets the\nstrength of the interlayer interaction.\nWe show that the mode hybridization between the lay-\ners, leading to acoustical and optical modes, is not only\ndependent ontheIECbut alsoonthe field andfrequency.\nThe collective nature of the precession is clear at low\nfields, as reflected by the relative amplitude of the sig-\nnals and the field dependence of the resonance frequency\n(fr). At higher applied fields the layers instead behave\nas single films subjected to an effective field, which scales\nwith the interlayercoupling. This transition, from collec-\ntive to single layer precession, is accompanied by changes\nin the slope of ∆ Hvs.fr, i.e. the damping, and we at-\ntribute those changes to the spin pumping between the\nlayers. The results demonstrate that it is possible to\nengineer a cut-off frequency , below which the spin pump-\ning is effectively turned off. At higher frequencies the\nspin pumping and the concomitant damping gradually\nincreases until it reach a constant value. This effect can\nbe used to tailorthe behaviorof spintronic devices by the\nstrength of the IEC.\nThe samples were prepared on oxidized Si-substrates\nusing magnetronsputtering and havethe following struc-\nture: substrate/seed/Co(80 ˚A)/Cu(dCu)/Py(45 ˚A)/cap,\nwheredCu= 0−40˚A. Single Py (Ni 80Fe20) and Co films2\nwere also prepared, using the same seed and cap layers.\nThe seed layer consisted of Pd(80 ˚A)/Cu(150 ˚A) and the\ncap layer was Cu(30 ˚A)/Pd(30 ˚A). These layers were in-\ncluded in the sample structure for three reasons: i) to\nfacilitate the growth of the Co film, ii) to avoid oxida-\ntion, and iii) to stay close to a material stack commonly\nused in the fabrication of spin-torque nano-oscillators.\nUsing Py and Co with well separated resonance fre-\nquencies also allows us to observe both the acoustic and\noptical modes in the coupled regimes – identical layers\nonly display one resonance [23]. Moreover, by drawing\nthe analogy with classical harmonic oscillators, we ex-\npect the collective modes to gradually split from the free\nrunningresonancesofthe individual layers(see, e.g., Sec-\ntion 2.2 in [29]). Thus, for weak coupling, the acoustical\nand optical modes will have their intensities mostly con-\ncentrated in the Py and Co layer, respectively. As the\ncoupling increases, the collective modes span more over\nboth layers, which leads to an enhancement of the acous-\ntical mode net response and a suppression of the opti-\ncal mode intensity. The measurements were done using\na NanOsc Instruments PhaseFMR-40 broadband FMR\nspectrometer. The external in-plane field was swept at\nfixed frequencies, varied step-wise from 3 to 37 GHz, and\nthe acquired instrument signal represents the derivative\nofthe absorptionpeaks. The derivativeofanasymmetric\nLorentzian[30,31]wasfit tothe signal,providingthe res-\nonance field ( µ0Hr), the full width half maximum (∆ H)\nand the amplitude ( A) of the absorption peak. The Kit-\ntel equation [32] was subsequently fit to the resonance\nfrequencies of the single layer samples:\nfr=γµ0\n2π/radicalbig\n(Hr+Hadd)(Hr+Hadd+Meff) (1)\nwhereµ0is the permabillity of free space, γis the gyro-\nmagnetic ratio, Meffis the effective magnetization, Hr\nis the (applied) resonance field, and Haddis the sum\nof additional in-plane fields, mainly represented by the\nanisotropy.\nTheresultsofthefitsgavethefollowingmagneticprop-\nerties of Py: γ/2π=29.0GHz/T and µ0Meff=0.89T, and\nof Co:γ/2π=30.8 GHz/T and µ0Meff=1.56 T. The ad-\nditional fields are in the order of 2 mT for both samples.\nThere are examples in the literature where the Kittel\nequation has also been used to fit the FMR of exchange\ncoupled layers, and it has been claimed that the magni-\ntude of the IEC can be extracted from the fitted value\nofHadd[27, 33]. However, we observed that this method\nnot only gave quite poor fits, but the J′values deter-\nmined independently from HPy\naddandHCo\naddalso differed\nsignificantly.\nInstead, we used an approach where the relation be-\ntweenfrandHris derived from the free energy of the\nsystem, giving the following expression [34–36]:\naω4+cω2+eω= 0 (2)\nFigure 1. FMR frequency vs. field for a trilayer with a\n7.5˚A Cu spacer. The data (filled symbols) is compared to\nthree different models. The behavior at all fields is well de-\nscribed by a fit of Eq. (2) to the data, using J′= 0.4 mJ/m2\n(solid lines). The transition from collective to single-la yer-like\nprecession is illustrated by two fits using the Kittel equati on,\nEq. (1), where MSandγare fixed either to the values of sin-\ngle Py and Co (dashed lines), or to the volume mean of these\nlayers (dotted lines). The additional field ( Hadd) was used as\na free parameter in those fits.\nwhereω= 2πfr, and the coefficients a,c, andecontain\nthe interlayer coupling, the magnetic properties, as well\nas the thickness of the magnetic layers. We have followed\nthe equations presented in Ref. [35] and [36], and the\nresults show a clear correspondence with the data.\nNevertheless, the Kittel equation still sheds some light\non the nature of the oscillations. In Fig. 1 the fits of\nEq. (2) are therefore supplemented with calculations us-\ning the Kittel equation. At high fields, frfollows the\npredictions of Eq. (1) with magnetizations and gyromag-\nneticratiosequaltothoseofthesinglefilms; atlowfields,\nfrinstead matches the behavior of an effective medium\nwith an Meffandγgiven by the volume average of the\ntwo materials. These limiting cases hence imply a transi-\ntion from a high frregion, where the inherent properties\nof each layer dominates, to a low frregion governed by\ncollective motion.\nTheextractedvaluesoftheIECarepresentedinFig.2.\nThe layer thicknesses and magnetizations were fixed dur-\ningthefitstoEq.(2), while J′,γPy, andγCowereallowed\nto vary. The resulting values of the IEC were essentially\nunchangedifthegyromagneticratiosweretreatedascon-\nstants, but the goodness-of-fit was worse. The coupling\nis ferromagnetic for all tCuin contrast to the familiar\nbehavior where J′is expected to oscillate between posi-\ntive and negative values. We can therefore conclude that\nthe interactions between the layers are not only given by\nRKKYcontributions, but alsoincludes e.g., N´ eel (orange3\nFigure 2. IEC strength ( J′) vs. spacer layer thickness, as\ndetermined from fits of the field dependence of frto Eq. (2).\nInset) FMR spectra of the Cu-7.5 ˚A sample in Fig.1 (thick\nblack lines) together with fits of two asymmetric Lorentzian s\n(cyan lines), and the spectra of the single layer Co (dashed\nlines) and Py (dotted lines). The left and right plots show\nthe absorption at 20 and 35 GHz, respectively.\npeel) coupling [21]. This interpretation is strengthened\nby the fact that we observe a minimum at the thickness\n(10˚A) where a negative maximum should occur [37, 38].\nThe linewidth of the Py and Co resonances for rep-\nresentative samples are shown in Fig. 3(a) and (b), re-\nspectively. The Cu-0 ˚A and the Cu-0.5 ˚A samples have\nstrongIEC and display only the acoustic mode, while the\nnet response of their optical mode is suppressed. The\nfrequency dependence of the linewidth is linear at all\nfrequencies. For samples in the intermediate and weak\ncoupling regimes (Cu-7.5–8.8 ˚A; Cu-12.5–19 ˚A) two reso-\nnances are observedand ∆ Hvs.frhave a rather concave\nshape, which is more pronounced for Co. The linewidth\nof the single Co layer is also non-Gilbert-like, as it sat-\nurates at a constant value at low fr. This behavior is\ninherited in the trilayer samples. However, ∆ HCoof the\ncoupled layers not only flattens out, but also increases\nat low frequencies. This implies that the mode becomes\nmore optical-like, since the optical mode is expected to\nhave a much larger linewidth compared to a single layer\ndue to mutual, out-of-phase, spin pumping [25, 39, 40].\nThere is also some additional variation of αCovs. Cu\nthickness, probably due to strain induced effects beyond\nour control. We have consequently chosen to not dwell\non the linewidth and damping associated with Co, but\ndo note that a large ∆ His consistent with an optical\ncharacter of the mode.\nIf we follow ∆ HPyof the samples with intermedi-\nate/weak IEC from high to low frequencies, we see that\ntheinitialconstantslopeisreducedatacertainfrequency\n(finfl) marked by a vertical line in Fig. 3(a). It is note-\nworthy that finflalso corresponds to the inflection point\nof ∆HCo(Fig. 3(b)). The relative resonance intensities\nFigure 3. ∆ Hvs.frfor the (a) Py and b) Co resonances.\nThe vertical lines (in both (a) and (b)) mark the inflection\npoint of the Co linewidth, which is marks the transition from\ncollective to single-layer-like behavior. The arrows show the\nCo frequency at zero applied field. Below this frequency ther e\nis only one resonance in the system. The insets are schematic\nillustrations of the magnetodynamics in the Py (gray) and\nCo (red) layers at different frequencies, for intermediate I EC\n(Cu-7.5˚A).\n(see Fig. 4 and inset of Fig. 2) also change drastically\naroundfinfl. Both these effects are clear signs of a tran-\nsition from a high-frequency region where the modes are\nassociated with the individual Py and Co layers, to a\nlow-frequency region where the precession is truly col-\nlective. ∆ HPydecreases since it predominantly repre-\nsents acoustic, in-phase, oscillations and the spin cur-\nrents hence cancel, while they instead add up in the opti-\ncal out-of-phase mode, resulting in an increasing ∆ HCo.\nWhen the Co/optical mode disappears, marked by ar-\nrows in Fig. 3(a), the slope of ∆ HPyagain becomes con-\nstantand followsthe single film behavior. The absenceof\nspin-pumping enhanced damping implies that both lay-\nersprecessin-phase andthat the sum ofthe spin currents\nis virtually zero [18].\nThe Cu-40 ˚A represents the samples without IEC and\nin those systems ∆ His linearin f, as expected for a pure\nspin pumping effect. The IEC not only influences the\nlinewidth, but also the signal amplitude. The measured4\nspectra (3-37GHz) of the Cu-40 ˚A and Cu-8.8 ˚A samples\nare presented in Fig. 4(a) and 4(b), respectively. The\nintensity of the signal is dependent on both the probed\nmagnetic moment and the frequency. We have there-\nfore normalized both resonances at each frequency to the\nhighest amplitude. The Co mode of the Cu-40 ˚A sample\nis stronger than the Py mode at all frequencies, as ex-\npected considering the higher magnetization and greater\nthickness of the Co layer. In contrast, the presence of\nIEC in the Cu-8.8 ˚A sample gives rise to a different pic-\nture, asitsComodequicklydecreasesinamplitudebelow\n≈15 GHz. This reveals the transition to a region where\nthe Py and Co show distinct acoustic and optical mode\ncharacteristics, in accordance with the interpretation of\nthe frequency dependence of the linewidth and the shape\nof thefrvs.Hrcurves.\nThe damping parameter αis determined by the rela-\ntion ∆H(fr) = ∆H0+ 4παfr/γ, where ∆ H0is a zero-\nfrequency offset [41]. The top panel of Fig. 5 shows αPy\nextracted from ∆ Hin the linear region at high frequen-\ncies. The damping increaseswith IEC strength, asshown\nby the peaks around tCu= 16 and 8 ˚A, since the precess-\ning Py layer is dragged by the exchange field from the\nstatic Co layer. For even thinner tCuonly the acoustic\nmode is present and αis low. Nonetheless, all samples\nshowing two resonances has a higher damping compared\ntothesinglelayer,and αisconstantforthesampleswith-\nout coupling. Hence, the main source of the increased\ndamping must be spin pumping.\nThe results are summarized in the bottom panel of\nFig. 5, which illustrates how the evolution of the reso-\nnances from Co and Py-like to truly acoustic and optical\nmodes is associated with a reduction, and eventually a\ncancellation, of the spin pumping effects. Different fre-\nquency regimes are therefore characterized by dissimilar\ndamping parameters and different levels of spin currents\nexchanged by the magnetic layers. This effect can be\nused to tailor the behavior of spintronic devices by the\nstrength of the IEC. It is worth noting that the IEC is\nnot only set within the growth process, but can also be\ntunedex situ, for example by loading the sample with\nhydrogen [42]. It could therefore be possible to switch on\nand off the spin pumping in a device under operation.\nIn summary, we have investigated spin pumping in ex-\nchange coupled magnetic layers using broadband FMR\nspectroscopy. We observe a frequency dependence of the\nnature of the resonance modes. They display a single-\nlayer like behavior at high frequencies and transform to\ncharacteristiccollectivemodesas frdecreases. Thistran-\nsition is accompanied by a reduction of the effective spin\npumping and damping. The results demonstrate that\nit is possible to engineer a cut-off frequency, using the\nstrength of the IEC, below which the spin pumping is\nminimized.\nWe acknowledge financial support from ERC Start-\ning Grant 307144 “Mustang”, the Swedish Foundation\nFigure 4. Color map of the normalized FMR response for (a)\na sample with zero, and (b) weak, interlayer coupling. The\ndata broadening along the x-axis is an artifact arising from\nthe conversion of line scans to a field/frequency/amplitude\nmatrix. (a) For decoupled layers, the Py amplitude is lower\nthan that of Co at all frequencies. (b) With IEC the relative\nintensities change significantly below 15 GHz. The modifica-\ntion of the intensities mirrors the transition from single- layer\nlike behavior at high frequencies/fields, to a region where t he\nprecession is truly collective with acoustic (high amplitu de)\nand optical (low amplitude) modes.\nFigure 5. Top) The damping parameter ( α) of the Py lay-\ners. Only data at frequencies well above finflwas used to ex-\ntractα. Bottom) The inflection point of the Co/optical-mode\nlinewidth ( finfl, cyan squares), which marks the transition\nfrom ordinary to reduced spin pumping, and the Co/optical-\nmode frequency at zero field ( f0, blue circles), below which\nthe spin pumping between the magnetic layers is absent.\nfor Strategic Research (SSF) Successful Research Lead-\ners program, the Swedish Research Council (VR), the\nG¨ oran Gustafsson foundation, and the Knut and Alice\nWallenberg Foundation.5\n[1] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang,\nM. Seck, V. Tsoi, and P. 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